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+title: "Formal ontology"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Formal_ontology"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:47.552517+00:00"
+instance: "kb-cron"
+---
+
+In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased (domain- and application-independent) view on reality, which can help the modeler of domain- or application-specific ontologies to avoid possibly erroneous ontological assumptions encountered in modeling large-scale ontologies.
+By maintaining an independent view on reality, a formal (upper level) ontology gains the following properties:
+
+indefinite expandability:
+the ontology remains consistent with increasing content.
+content and context independence:
+any kind of 'concept' can find its place.
+accommodate different levels of granularity.
+
+== Historical background ==
+Theories on how to conceptualize reality date back as far as Plato and Aristotle. The term 'formal ontology' itself was coined by Edmund Husserl in the second edition of his Logical Investigations (1900–01), where it refers to an ontological counterpart of formal logic. Formal ontology for Husserl embraces an axiomatized mereology and a theory of dependence relations, for example between the qualities of an object and the object itself. 'Formal' signifies not the use of a formal-logical language, but rather: non-material, or in other words domain-independent (of universal application). Husserl's ideas on formal ontology were developed especially by his Polish student Roman Ingarden in his Controversy over the Existence of the World. The relations between the Husserlian tradition of formal ontology and the Polish tradition of mereology are set forth in Parts and Moments. Studies in Logic and Formal Ontology, edited by Barry Smith. 
+
+== Existing formal ontologies (foundational ontologies) ==
+
+BFO – Basic Formal Ontology
+GFO – General Formal Ontology
+BORO – Business Objects Reference Ontology
+CIDOC Conceptual Reference Model
+Cyc (Cyc is not just an upper ontology, it also contains many mid-level and specialized ontologies as well)
+UMBEL – Upper Mapping and Binding Exchange Layer, a subset of OpenCyc
+DOLCE – Descriptive Ontology for Linguistic and Cognitive Engineering
+SUMO – Suggested Upper Merged Ontology
+
+== Common terms in formal (upper-level) ontologies ==
+The differences in terminology used between separate formal upper-level ontologies can be quite substantial, but most formal upper-level ontologies apply one foremost dichotomy: that between endurants and perdurants.
+
+=== Endurant ===
+Also known as continuants, or in some cases as "substance", endurants are those entities that can be observed-perceived as a complete concept, at no matter which given snapshot of time.
+Were we to freeze time we would still be able to perceive/conceive the entire endurant.
+Examples include material objects (such as an apple or a human), and abstract "fiat" objects (such as an organization, or the border of a country).
+
+=== Perdurant ===
+Also known as occurrents, accidents or happenings, perdurants are those entities for which only a part exists if we look at them at any given snapshot in time.
+When we freeze time we can only see a part of the perdurant. Perdurants are often what we know as processes, for example: "running". If we freeze time then we only see a part of the running, without any previous knowledge one might not even be able to determine the actual process as being a process of running. Other examples include an activation, a kiss, or a procedure.
+
+=== Qualities ===
+In a broad sense, qualities can also be known as properties or tropes.
+Qualities do not exist on their own, but they need another entity (in many formal ontologies this entity is restricted to be an endurant) which they occupy. Examples of qualities and the values they assume include colors (red color), or temperatures (warm).
+Most formal upper-level ontologies recognize qualities, attributes, tropes, or something related, although the exact classification may differ. Some see qualities and the values they can assume (sometimes called quale) as a separate hierarchy besides endurants and perdurants (example: DOLCE). Others classify qualities as a subsection of endurants, e.g. the dependent endurants (example: BFO). Others consider property-instances or tropes that are single characteristics of individuals as the atoms of the ontology, the simpler entities of which all other entities are composed, so that all the entities are sums or bundles of tropes.
+
+== Formal versus nonformal ==
+In information science an ontology is formal if it is specified in a formal language, otherwise it is informal.
+In philosophy, a separate distinction between formal and nonformal ontologies exists, which does not relate to the use of a formal language.
+
+=== Example ===
+An ontology might contain a concept representing 'mobility of the arm'. In a nonformal ontology, a concept like this can often be classified as for example a 'finding of the arm', right next to other concepts such as 'bruising of the arm'. This method of modeling might create problems with increasing amounts of information, as there is no foolproof way to keep hierarchies like this, or their descendant hierarchies (one is a process, the other is a quality) from entangling or knotting.
+In a formal ontology, there is an optimal way to properly classify this concept, it is a kind of 'mobility', which is a kind of quality/property (see above). As a quality, it is said to inhere in independent endurant entities (see above), as such, it cannot exist without a bearer (in the case the arm).
+
+== Applications for formal (upper-level) ontologies ==
+
+=== Formal ontology as a template to create novel specific domain ontologies ===
+Having a formal ontology at your disposal, especially when it consists of a Formal upper layer enriched with concrete domain-independent 'middle layer' concepts, can really aid the creation of a domain specific ontology.
+It allows the modeller to focus on the content of the domain specific ontology without having to worry on the exact higher structure or abstract philosophical framework that gives his ontology a rigid backbone. Disjoint axioms at the higher level will prevent many of the commonly made ontological mistakes made when creating the detailed layer of the ontology.

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+title: "Formal ontology"
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+---
+
+=== Formal ontology as a crossmapping hub: crossmapping taxonomies, databases and nonformal ontologies ===
+Aligning terminologies and ontologies is not an easy task. The divergence of the underlying meaning of word descriptions and terms within different information sources is a well known obstacle for direct approaches to data integration and mapping. One single description may have a completely different meaning in one data source when compared with another. This is because different databases/terminologies often have a different viewpoint on similar items. They are usually built with a specific application-perspective in mind and their hierarchical structure represents this.
+A formal ontology, on the other hand, represents entities without a particular application scope. Its hierarchy reflects ontological principles and a basic class-subclass relation between its concepts. A consistent framework like this is ideal for crossmapping data sources.
+However, one cannot just integrate these external data sources in the formal ontology. A direct incorporation would lead to corruption of the framework and principles of the formal ontology.
+A formal ontology is a great crossmapping hub only if a complete distinction between the content and structure of the external information sources and the formal ontology itself is maintained. This is possible by specifying a mapping relation between concepts from a chaotic external information source and a concept in the formal ontology that corresponds with the meaning of the former concept.
+Where two or more external information sources map to one and the same formal ontology concept a crossmapping/translation is achieved, as you know that those concepts—no matter what their phrasing is—mean the same thing.
+
+=== Formal ontology to empower natural language processing ===
+In ontologies designed to serve natural language processing (NLP) and natural language understanding (NLU) systems, ontology concepts are usually connected and symbolized by terms. This kind of connection represents a linguistic realization. 
+Terms are words or a combination of words (multi-word units), in different languages, used to describe in natural language an element from reality, and hence connected to that formal ontology concept that frames this element in reality.
+The lexicon, the collection of terms and their inflections assigned to the concepts and relationships in an ontology, forms the 'ontology interface to natural language', the channel through which the ontology can be accessed from a natural language input.
+
+=== Formal ontology to normalize database/instance data ===
+The great thing about a formal ontology, in contrast to rigid taxonomies or classifications, is that it allows for indefinite expansion. Given proper modeling, just about any kind of conceptual information, no matter the content, can find its place.
+To disambiguate a concept's place in the ontology, often a context model is useful to improve the classification power. The model typically applies rules to surrounding elements of the context to select the most valid classification.
+
+== See also ==
+Mereology
+Ontology (information science)
+Upper ontology
+
+== References ==

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+title: "Mathematics"
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+category: "reference"
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+date_saved: "2026-05-05T03:56:46.299650+00:00"
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+---
+
+Mathematics is a field of study that discovers and organizes methods, theories, and theorems that are developed and proved either in response to the needs of empirical sciences or the needs of mathematics itself. There are many areas of mathematics, including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form),  geometry (the study of shapes and spaces that contain them), analysis (the study of approximating continuous changes), and set theory (presently used as a foundation for all mathematics).
+Mathematics involves the description and manipulation of abstract objects that are either abstractions from nature or purely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to prove the properties of objects through proofs, which consist of a succession of applications of deductive rules to already established results. These results, called theorems, include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.
+Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application but often find practical applications later.
+Mathematical written records first appeared in Ancient Egypt and Mesopotamia, but the concept of proof and its associated mathematical rigor began in Ancient Greek mathematics, exemplified in Euclid's Elements. Mathematics was primarily divided into geometry and arithmetic until the 16th and 17th centuries, when algebra and infinitesimal calculus evolved into new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematic use of the axiomatic method, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics.
+
+== Areas of mathematics ==
+
+Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the study and manipulation of numbers, and geometry, regarding the study of shapes. Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.
+Beginning with the Renaissance, two more areas became predominant. New mathematical notation led to modern algebra which, roughly speaking, begins with the study and manipulation of algebraic expressions. Calculus, consisting of the two subfields differential calculus and integral calculus, originated with geometry but evolved into the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areas—arithmetic, geometry, algebra, and calculus—endured until the end of the 19th century. Other areas that were previously studied by mathematicians, such as celestial mechanics and solid mechanics, are now considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the 17th century.
+At the end of the 19th century, the foundational crisis in mathematics and the systematic use of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations.
+
+=== Number theory ===
+
+Number theory evolved from the manipulation of numbers, that is, natural numbers 
+  
+    
+      
+        (
+        
+          N
+        
+        )
+        ,
+      
+    
+    {\displaystyle (\mathbb {N} ),}
+  
+ and later expanded to integers 
+  
+    
+      
+        (
+        
+          Z
+        
+        )
+      
+    
+    {\displaystyle (\mathbb {Z} )}
+  
+ and rational numbers 
+  
+    
+      
+        (
+        
+          Q
+        
+        )
+        .
+      
+    
+    {\displaystyle (\mathbb {Q} ).}
+  
+ Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations. The study of numbers arguably dates back to ancient Babylon and probably China, but developed into a distinct discipline in Ancient Greece. Two prominent early number theorists were Euclid and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.
+Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra. Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven despite considerable effort.
+Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), Diophantine analysis, and transcendence theory (problem oriented).
+
+=== Geometry ===

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+date_saved: "2026-05-05T03:56:46.299650+00:00"
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+---
+
+Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.
+A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or are part of the definition of the subject of study (axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.
+The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the three-dimensional Euclidean space.
+Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This constituted a major change of paradigm: Instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.
+Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.
+In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space.
+Today's subareas of geometry include:
+
+Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
+Affine geometry, the study of properties relative to parallelism and independent from the concept of length.
+Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions.
+Manifold theory, the study of shapes that are not necessarily embedded in a larger space.
+Riemannian geometry, the study of distance properties in curved spaces.
+Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials.
+Topology, the study of properties that are kept under continuous deformations.
+Algebraic topology, the use in topology of algebraic methods, mainly homological algebra.
+Discrete geometry, the study of finite configurations in geometry.
+Convex geometry, the study of convex sets, which takes its importance from its applications in optimization.
+Complex geometry, the geometry obtained by replacing real numbers with complex numbers.
+
+=== Algebra ===
+
+  
+Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise.
+Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.
+Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether,
+and popularized by Van der Waerden's book Moderne Algebra.
+Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:

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+instance: "kb-cron"
+---
+
+group theory
+field theory
+vector spaces, whose study is essentially the same as linear algebra
+ring theory
+commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry
+homological algebra
+Lie algebra and Lie group theory
+Boolean algebra, which is widely used for the study of the logical structure of computers
+The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory. The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.
+
+=== Calculus and analysis ===
+
+Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. It is fundamentally the study of the relationship between variables that depend continuously on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.
+Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:
+
+Multivariable calculus
+Functional analysis, where variables represent varying functions
+Integration, measure theory and potential theory, all strongly related with probability theory on a continuum
+Ordinary differential equations
+Partial differential equations
+Numerical analysis, mainly devoted to the computation of solutions of ordinary and partial differential equations that arise in many applications
+
+=== Discrete mathematics ===
+
+Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms—especially their implementation and computational complexity—play a major role in discrete mathematics.
+The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.
+Discrete mathematics includes:
+
+Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes.
+Graph theory and hypergraphs
+Coding theory, including error correcting codes and a part of cryptography
+Matroid theory
+Discrete geometry
+Discrete probability distributions
+Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
+Discrete optimization, including combinatorial optimization, integer programming, constraint programming
+
+=== Mathematical logic and set theory ===
+
+The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.
+Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory. In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigor.
+This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.
+The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic (which explicitly lacks the law of excluded middle).
+These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, formal verification, program analysis, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.
+
+=== Computational mathematics ===

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+instance: "kb-cron"
+---
+
+Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Part of computational mathematics involves numerical analysis, which is the study of methods for problems in analysis using functional analysis and approximation theory. Numerical analysis broadly includes the study of approximation and discretization, with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing, also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.
+
+== History ==
+
+=== Etymology ===
+The word mathematics comes from the Ancient Greek word máthēma (μάθημα), meaning 'something learned, knowledge, mathematics', and the derived expression mathēmatikḗ tékhnē (μαθηματικὴ τέχνη), meaning 'mathematical science'. It entered the English language during the Late Middle English period through French and Latin. 
+Traditionally, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established.
+In Latin and English, until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine's warning that Christians should beware of mathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians.
+The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.
+
+=== Ancient ===
+In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years.Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa. Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.The Ishango bone, according to scholar Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed. Megalithic structures located in Nabta Playa, Upper Egypt featured astronomy, calendar arrangements in alignment with the heliacal rising of Sirius and supported calibration of the yearly calendar for the annual Nile flood. Ancient Nubians established a system of geometric rules which served as the basis for initial sunclocks. Nubians also exercised a trigonometric methodology comparable to their Egyptian counterparts.Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.
+By the 5th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements, covers geometry and number theory and is widely considered the most successful and influential textbook of all time. Another notable mathematician of antiquity is Archimedes of Syracuse (c. 287 – c. 212 BC). He developed methods for calculating the surface area and volume of solids of revolution, including using the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner reminiscent of modern calculus. Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hipparchus of Nicaea, 2nd century BC), and the beginnings of pre-modern algebra (Diophantus, 3rd century AD).
+
+The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.
+
+=== Medieval and later ===

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+tags: "science, encyclopedia"
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+instance: "kb-cron"
+---
+
+During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.
+During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation, the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems.
+
+Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.
+Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."
+
+== Symbolic notation and terminology ==
+
+Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts. Operation and relations are generally represented by specific symbols or glyphs, such as + (plus), × (multiplication), 
+  
+    
+      
+        ∫
+      
+    
+    {\textstyle \int }
+  
+ (integral), = (equal), and < (less than). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrases and formulas play the role of clauses.
+Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corollary.
+Numerous technical terms used in mathematics are neologisms, such as polynomial and homeomorphism. Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, "or" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive or"). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module is flat" and "a field is always a ring".

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+title: "Mathematics"
+chunk: 6/10
+source: "https://en.wikipedia.org/wiki/Mathematics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:46.299650+00:00"
+instance: "kb-cron"
+---
+
+== Relationship with sciences ==
+Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model.
+There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation. In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation). However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence.
+
+=== Pure and applied mathematics ===
+
+Until the 19th century, there was no clear distinction between pure and applied mathematics as understood today. The distinction between developing mathematics for its own sake or for its applications was rather fluid: natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture, and astronomy, but both subjects quickly stood on their own. Later, Isaac Newton used infinitesimal calculus in part to help explain the movement of the planets and his law of gravitation. Moreover, since antiquity, most mathematicians were also scientists, and many scientists were also mathematicians. Nonetheless, the Western tradition of pure mathematics traces its roots back to Ancient Greece. The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks.
+In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure mathematics. This led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.
+The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere. Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory".
+An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.
+In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas. The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics". However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge.

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+title: "Mathematics"
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+source: "https://en.wikipedia.org/wiki/Mathematics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:46.299650+00:00"
+instance: "kb-cron"
+---
+
+=== Unreasonable effectiveness ===
+The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner. It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
+A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem. A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It was almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.
+In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four.
+A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon 
+  
+    
+      
+        
+          Ω
+          
+            −
+          
+        
+        .
+      
+    
+    {\displaystyle \Omega ^{-}.}
+  
+ In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.
+
+=== Specific sciences ===
+
+==== Physics ====
+
+Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly, and is also considered to be the motivation of major mathematical developments.
+
+==== Computing ====
+
+Computing is closely related to mathematics in several ways. Theoretical computer science is considered to be mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in cryptography and coding theory. Discrete mathematics is useful in many areas of computer science, such as complexity theory, information theory, and graph theory. In 1998, the Kepler conjecture on sphere packing seemed to also be partially proven by computer.
+
+==== Statistics and other decision sciences ====
+
+The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods such as, and especially, probability theory. Statisticians generate data with random sampling or randomized experiments.
+Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.
+
+==== Biology and chemistry ====
+
+Biology uses probability extensively in fields such as ecology or neurobiology.  Most discussion of probability centers on the concept of evolutionary fitness. Ecology heavily uses modeling to simulate population dynamics, study ecosystems such as the predator-prey model, measure pollution diffusion, or to assess climate change. The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations.
+Statistical hypothesis testing is run on data from clinical trials to determine whether a new treatment works. Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.
+
+==== Earth sciences ====
+
+Structural geology and climatology use probabilistic models to predict the risk of natural catastrophes. Similarly, meteorology, oceanography, and planetology also use mathematics due to their heavy use of models.
+
+==== Social sciences ====
+
+Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, economics, sociology, and psychology.

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+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:46.299650+00:00"
+instance: "kb-cron"
+---
+
+Often the fundamental postulate of mathematical economics is that of the rational individual actor – Homo economicus (lit. 'economic man'). In this model, the individual seeks to maximize their self-interest, and always makes optimal choices using perfect information. This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual calculations are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms. Some reject or criticize the concept of Homo economicus. Economists note that real people have limited information, make poor choices, and care about fairness and altruism, not just personal gain.
+Without mathematical modeling, it is hard to go beyond statistical observations or untestable speculation. Mathematical modeling allows economists to create structured frameworks to test hypotheses and analyze complex interactions. Models provide clarity and precision, enabling the translation of theoretical concepts into quantifiable predictions that can be tested against real-world data.
+At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, Nikolai Kondratiev discerned the ~50-year-long Kondratiev cycle, which explains phases of economic growth or crisis. Towards the end of the 19th century, mathematicians extended their analysis into geopolitics. Peter Turchin developed cliodynamics in the 1990s.
+Mathematization of the social sciences is not without risk. In the controversial book Fashionable Nonsense (1997), Sokal and Bricmont denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences. The study of complex systems (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy.
+
+== Philosophy ==
+
+=== Reality ===
+The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.
+Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views.
+
+ Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...
+Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics (as Platonism assumes mathematics exists independently, but does not explain why it matches reality).
+
+=== Proposed definitions ===
+
+There is no general consensus about the definition of mathematics or its epistemological status—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do". A common approach is to define mathematics by its object of study.
+Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given. With the large number of new areas of mathematics that have appeared since the beginning of the 20th century, defining mathematics by its object of study has become increasingly difficult. For example, in lieu of a definition, Saunders Mac Lane in Mathematics, form and function summarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes:
+
+the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems.  Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas.
+Another approach for defining mathematics is to use its methods. For example, an area of study is often qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely logical deduction.
+
+=== Rigor ===

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+instance: "kb-cron"
+---
+
+Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of inference rules, without any use of empirical evidence and intuition. Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' concision, rigorous proofs can require hundreds of pages to express, such as the 255-page Feit–Thompson theorem. The emergence of computer-assisted proofs has allowed proof lengths to further expand.  The result of this trend is a philosophy of the quasi-empiricist proof that can not be considered infallible, but has a probability attached to it.
+The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs.
+At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks. It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.
+Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.
+
+== Training and practice ==
+
+=== Education ===
+
+Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines. Prominent careers for professional mathematicians include mathematics teacher or professor, statistician, actuary, financial analyst, economist, accountant, commodity trader, or computer consultant.
+Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia. Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE. The oldest known mathematics textbook is the Rhind papyrus, dated from c. 1650 BCE in Egypt. Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized oral tradition since the Vedic period (c. 1500 – c. 500 BCE). In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy.
+Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium. Formal instruction in pedagogy began with Jesuit schools in the 16th and 17th century. Most mathematical curricula remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was L'Enseignement Mathématique, which began publication in 1899. The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component—initially for its military applications. While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.
+During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics. Some students studying mathematics may develop an apprehension or fear about their performance in the subject. This is known as mathematical anxiety, and is considered the most prominent of the disorders impacting academic performance. Mathematical anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.

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+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:46.299650+00:00"
+instance: "kb-cron"
+---
+
+=== Psychology (aesthetic, creativity and intuition) ===
+The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process. An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.
+Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzles. This aspect of mathematical activity is emphasized in recreational mathematics.
+Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to elegance, which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in A Mathematician's Apology expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetics. Paul Erdős expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis.
+Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are created (as in art) or discovered (as in science). The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
+
+== Cultural impact ==
+
+=== Artistic expression ===
+
+Notes that sound well together to a Western ear are sounds whose fundamental frequencies of vibration are in simple ratios. For example, an octave doubles the frequency and a perfect fifth multiplies it by 
+  
+    
+      
+        
+          
+            3
+            2
+          
+        
+      
+    
+    {\textstyle {\frac {3}{2}}}
+  
+.
+
+Humans, as well as some other animals, find symmetric patterns to be more beautiful. Mathematically, the symmetries of an object form a group known as the symmetry group. For example, the group underlying mirror symmetry is the cyclic group of two elements, 
+  
+    
+      
+        
+          Z
+        
+        
+          /
+        
+        2
+        
+          Z
+        
+      
+    
+    {\displaystyle \mathbb {Z} /2\mathbb {Z} }
+  
+. A Rorschach test is a figure invariant by this symmetry, as are butterfly and animal bodies more generally (at least on the surface). Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea. Fractals possess self-similarity.
+
+=== Popularization ===
+
+Popular mathematics is the act of presenting mathematics without technical terms. Presenting mathematics may be hard since the general public suffers from mathematical anxiety and mathematical objects are highly abstract. However, popular mathematics writing can overcome this by using applications or cultural links. Despite this, mathematics is rarely the topic of popularization in printed or televised media.
+
+=== Awards and prize problems ===
+
+The most prestigious award in mathematics is the Fields Medal, established by Canadian John Charles Fields in 1936 and awarded every four years (except around World War II) to up to four individuals. It is considered the mathematical equivalent of the Nobel Prize.
+Other prestigious mathematics awards include:
+
+The Abel Prize, instituted in 2002 and first awarded in 2003
+The Chern Medal for lifetime achievement, introduced in 2009 and first awarded in 2010
+The AMS Leroy P. Steele Prize, awarded since 1970
+The Wolf Prize in Mathematics, also for lifetime achievement, instituted in 1978
+A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list has achieved great celebrity among mathematicians, and at least thirteen of the problems (depending how some are interpreted) have been solved.
+A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. To date, only one of these problems, the Poincaré conjecture, has been solved, by the Russian mathematician Grigori Perelman.
+
+== See also ==
+
+== Notes ==
+
+== References ==
+
+=== Citations ===
+
+=== Other sources ===
+
+== Further reading ==

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+title: "Organoid intelligence"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Organoid_intelligence"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:48.926200+00:00"
+instance: "kb-cron"
+---
+
+Organoid intelligence (OI) is an emerging field of study in computer science and biology that develops and studies biological wetware computing using 3D cultures of human brain cells (or brain organoids) and brain-machine interface technologies. Such technologies may be referred to as OIs or the nervous filesystem.
+Organoid intelligent computer systems can be an example of biohybrid systems.
+
+
+== Differences with non-organic computing ==
+As opposed to traditional non-organic silicon-based approaches, OI seeks to use lab-grown cerebral organoids to serve as "biological hardware". While these structures are still far from being able to think like a regular human brain and do not yet possess strong computing capabilities, OI research currently offers the potential to improve the understanding of brain development, learning and memory, potentially finding treatments for neurological disorders such as dementia.
+Thomas Hartung, a professor from Johns Hopkins University, argued in 2023 that "while silicon-based computers are certainly better with numbers, brains are better at learning." He noted that transistor density in computer chip may be approaching its limits, whereas brains, being wired differently, are more energy-efficient and can store large amounts of information.
+Some researchers claim that even though human brains are slower than machines at processing simple information, they are far better at processing complex information as brains can deal with fewer and more uncertain data, perform both sequential and parallel processing, being highly heterogenous, use incomplete datasets, and is said to outperform non-organic machines in decision-making.
+Training OIs involve the process of biological learning (BL) as opposed to machine learning (ML) for AIs.
+
+
+== Bioinformatics in OI ==
+OI generates complex biological data, necessitating sophisticated methods for processing and analysis. Bioinformatics provides the tools and techniques to decipher raw data, uncovering the patterns and insights. Researchers have developed a platform named Neuroplatform for experimenting remotely with brain organoids via an API.
+
+
+== Intended functions ==
+Brain-inspired computing hardware aims to emulate the structure and working principles of the brain and could be used to address current limitations in AI technologies. However, brain-inspired silicon chips are still limited in their ability to fully mimic brain function, as most examples are built on digital electronic principles. One study performed OI computation (which they termed Brainoware) by sending and receiving information from the brain organoid using a high-density multielectrode array. By applying spatiotemporal electrical stimulation, nonlinear dynamics, and fading memory properties, as well as unsupervised learning from training data by reshaping the organoid functional connectivity, the study showed the potential of this technology by using it for speech recognition and nonlinear equation prediction in a reservoir computing framework.
+
+
+== Ethical concerns ==
+While researchers are hoping to use OI and biological computing to complement traditional silicon-based computing, there are also questions about the ethics of such an approach. Concerns include the possibility that an organoid could develop sentience or consciousness, and the question of the relationship between a stem cell donor (for growing the organoid) and the respective OI system.
+
+
+== See also ==
+AI effect
+Biohybrid system
+Cerebral organoid
+Artificial intelligence
+Hybrot
+Organ-on-a-chip
+Biological intelligence
+
+
+== References ==

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+title: "Oxford model"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Oxford_model"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:50.239737+00:00"
+instance: "kb-cron"
+---
+
+The Oxford Model or the Oxford macro econometric Model was a Macroeconomic model created by Lawrence Klein and Sir James Ball. It included a Phillips-type relation and led to an "explosion" of macroeconometric forecasting.
+
+
+== References ==

data/en.wikipedia.org/wiki/Pattern_recognition-0.md

@@ -0,0 +1,25 @@
+---
+title: "Pattern recognition"
+chunk: 1/4
+source: "https://en.wikipedia.org/wiki/Pattern_recognition"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:51.565383+00:00"
+instance: "kb-cron"
+---
+
+Pattern recognition is the task of assigning a class to an observation based on patterns extracted from data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess PR capabilities but their primary function is to distinguish and create emergent  patterns.   PR has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. Pattern recognition has its origins in statistics and engineering; some modern approaches to pattern recognition include the use of machine learning, due to the increased availability of big data and a new abundance of processing power.
+Pattern recognition systems are commonly trained from labeled "training" data. When no labeled data are available, other algorithms can be used to discover previously unknown patterns. KDD and data mining have a larger focus on unsupervised methods and stronger connection to business use. Pattern recognition focuses more on the signal and also takes acquisition and signal processing into consideration. It originated in engineering, and the term is popular in the context of computer vision: a leading computer vision conference is named Conference on Computer Vision and Pattern Recognition.
+In machine learning, pattern recognition is the assignment of a label to a given input value. In statistics, discriminant analysis was introduced for this same purpose in 1936. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes (for example, determine whether a given email is "spam"). Pattern recognition is a more general problem that encompasses other types of output as well. Other examples are regression, which assigns a real-valued output to each input; sequence labeling, which assigns a class to each member of a sequence of values (for example, part of speech tagging, which assigns a part of speech to each word in an input sentence); and parsing, which assigns a parse tree to an input sentence, describing the syntactic structure of the sentence.
+Pattern recognition algorithms generally aim to provide a reasonable answer for all possible inputs and to perform "most likely" matching of the inputs, taking into account their statistical variation. This is opposed to pattern matching algorithms, which look for exact matches in the input with pre-existing patterns. A common example of a pattern-matching algorithm is regular expression matching, which looks for patterns of a given sort in textual data and is included in the search capabilities of many text editors and word processors.
+
+== Overview ==
+
+A modern definition of pattern recognition is:
+
+The field of pattern recognition is concerned with the automatic discovery of regularities in data through the use of computer algorithms and with the use of these regularities to take actions such as classifying the data into different categories.
+Pattern recognition is generally categorized according to the type of learning procedure used to generate the output value. Supervised learning assumes that a set of training data (the training set) has been provided, consisting of a set of instances that have been properly labeled by hand with the correct output. A learning procedure then generates a model that attempts to meet two sometimes conflicting objectives: Perform as well as possible on the training data, and generalize as well as possible to new data (usually, this means being as simple as possible, for some technical definition of "simple", in accordance with Occam's Razor, discussed below). Unsupervised learning, on the other hand, assumes training data that has not been hand-labeled, and attempts to find inherent patterns in the data that can then be used to determine the correct output value for new data instances. A combination of the two that has been explored is semi-supervised learning, which uses a combination of labeled and unlabeled data (typically a small set of labeled data combined with a large amount of unlabeled data). In cases of unsupervised learning, there may be no training data at all.
+Sometimes different terms are used to describe the corresponding supervised and unsupervised learning procedures for the same type of output. The unsupervised equivalent of classification is normally known as clustering, based on the common perception of the task as involving no training data to speak of, and of grouping the input data into clusters based on some inherent similarity measure (e.g. the distance between instances, considered as vectors in a multi-dimensional vector space), rather than assigning each input instance into one of a set of pre-defined classes. In some fields, the terminology is different. In community ecology, the term classification is used to refer to what is commonly known as "clustering".
+The piece of input data for which an output value is generated is formally termed an instance. The instance is formally described by a vector of features, which together constitute a description of all known characteristics of the instance. These feature vectors can be seen as defining points in an appropriate multidimensional space, and methods for manipulating vectors in vector spaces can be correspondingly applied to them, such as computing the dot product or the angle between two vectors. Features typically are either categorical (also known as nominal, i.e., consisting of one of a set of unordered items, such as a gender of "male" or "female", or a blood type of "A", "B", "AB" or "O"), ordinal (consisting of one of a set of ordered items, e.g., "large", "medium" or "small"), integer-valued (e.g., a count of the number of occurrences of a particular word in an email) or real-valued (e.g., a measurement of blood pressure). Often, categorical and ordinal data are grouped together, and this is also the case for integer-valued and real-valued data. Many algorithms work only in terms of categorical data and require that real-valued or integer-valued data be discretized into groups (e.g., less than 5, between 5 and 10, or greater than 10).
+
+=== Probabilistic classifiers ===

data/en.wikipedia.org/wiki/Pattern_recognition-1.md

@@ -0,0 +1,433 @@
+---
+title: "Pattern recognition"
+chunk: 2/4
+source: "https://en.wikipedia.org/wiki/Pattern_recognition"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:51.565383+00:00"
+instance: "kb-cron"
+---
+
+Many common pattern recognition algorithms are probabilistic in nature, in that they use statistical inference to find the best label for a given instance. Unlike other algorithms, which simply output a "best" label, often probabilistic algorithms also output a probability of the instance being described by the given label. In addition, many probabilistic algorithms output a list of the N-best labels with associated probabilities, for some value of N, instead of simply a single best label. When the number of possible labels is fairly small (e.g., in the case of classification), N may be set so that the probability of all possible labels is output. Probabilistic algorithms have many advantages over non-probabilistic algorithms:
+
+They output a confidence value associated with their choice. (Note that some other algorithms may also output confidence values, but in general, only for probabilistic algorithms is this value mathematically grounded in probability theory. Non-probabilistic confidence values can in general not be given any specific meaning, and only used to compare against other confidence values output by the same algorithm.)
+Correspondingly, they can abstain when the confidence of choosing any particular output is too low.
+Because of the probabilities output, probabilistic pattern-recognition algorithms can be more effectively incorporated into larger machine-learning tasks, in a way that partially or completely avoids the problem of error propagation.
+
+=== Number of important feature variables ===
+Feature selection algorithms attempt to directly prune out redundant or irrelevant features. A general introduction to feature selection which summarizes approaches and challenges, has been given. The complexity of feature-selection is, because of its non-monotonous character, an optimization problem where given a total of 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+ features the powerset consisting of all 
+  
+    
+      
+        
+          2
+          
+            n
+          
+        
+        −
+        1
+      
+    
+    {\displaystyle 2^{n}-1}
+  
+ subsets of features need to be explored. The Branch-and-Bound algorithm does reduce this complexity but is intractable for medium to large values of the number of available features 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+
+Techniques to transform the raw feature vectors (feature extraction) are sometimes used prior to application of the pattern-matching algorithm. Feature extraction algorithms attempt to reduce a large-dimensionality feature vector into a smaller-dimensionality vector that is easier to work with and encodes less redundancy, using mathematical techniques such as principal components analysis (PCA). The distinction between feature selection and feature extraction is that the resulting features after feature extraction has taken place are of a different sort than the original features and may not easily be interpretable, while the features left after feature selection are simply a subset of the original features.
+
+== Problem statement ==
+The problem of pattern recognition can be stated as follows: Given an unknown function 
+  
+    
+      
+        g
+        :
+        
+          
+            X
+          
+        
+        →
+        
+          
+            Y
+          
+        
+      
+    
+    {\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}}
+  
+ (the ground truth) that maps input instances 
+  
+    
+      
+        
+          x
+        
+        ∈
+        
+          
+            X
+          
+        
+      
+    
+    {\displaystyle {\boldsymbol {x}}\in {\mathcal {X}}}
+  
+ to output labels 
+  
+    
+      
+        y
+        ∈
+        
+          
+            Y
+          
+        
+      
+    
+    {\displaystyle y\in {\mathcal {Y}}}
+  
+, along with training data 
+  
+    
+      
+        
+          D
+        
+        =
+        {
+        (
+        
+          
+            x
+          
+          
+            1
+          
+        
+        ,
+        
+          y
+          
+            1
+          
+        
+        )
+        ,
+        …
+        ,
+        (
+        
+          
+            x
+          
+          
+            n
+          
+        
+        ,
+        
+          y
+          
+            n
+          
+        
+        )
+        }
+      
+    
+    {\displaystyle \mathbf {D} =\{({\boldsymbol {x}}_{1},y_{1}),\dots ,({\boldsymbol {x}}_{n},y_{n})\}}
+  
+ assumed to represent accurate examples of the mapping, produce a function 
+  
+    
+      
+        h
+        :
+        
+          
+            X
+          
+        
+        →
+        
+          
+            Y
+          
+        
+      
+    
+    {\displaystyle h:{\mathcal {X}}\rightarrow {\mathcal {Y}}}
+  
+ that approximates as closely as possible the correct mapping 
+  
+    
+      
+        g
+      
+    
+    {\displaystyle g}
+  
+. (For example, if the problem is filtering spam, then 
+  
+    
+      
+        
+          
+            x
+          
+          
+            i
+          
+        
+      
+    
+    {\displaystyle {\boldsymbol {x}}_{i}}
+  
+ is some representation of an email and 
+  
+    
+      
+        y
+      
+    
+    {\displaystyle y}
+  
+ is either "spam" or "non-spam"). In order for this to be a well-defined problem, "approximates as closely as possible" needs to be defined rigorously. In decision theory, this is defined by specifying a loss function or cost function that assigns a specific value to "loss" resulting from producing an incorrect label. The goal then is to minimize the expected loss, with the expectation taken over the probability distribution of 
+  
+    
+      
+        
+          
+            X
+          
+        
+      
+    
+    {\displaystyle {\mathcal {X}}}
+  
+. In practice, neither the distribution of 
+  
+    
+      
+        
+          
+            X
+          
+        
+      
+    
+    {\displaystyle {\mathcal {X}}}
+  
+ nor the ground truth function 
+  
+    
+      
+        g
+        :
+        
+          
+            X
+          
+        
+        →
+        
+          
+            Y
+          
+        
+      
+    
+    {\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}}
+  
+ are known exactly, but can be computed only empirically by collecting a large number of samples of 
+  
+    
+      
+        
+          
+            X
+          
+        
+      
+    
+    {\displaystyle {\mathcal {X}}}
+  
+ and hand-labeling them using the correct value of 
+  
+    
+      
+        
+          
+            Y
+          
+        
+      
+    
+    {\displaystyle {\mathcal {Y}}}
+  
+ (a time-consuming process, which is typically the limiting factor in the amount of data of this sort that can be collected). The particular loss function depends on the type of label being predicted. For example, in the case of classification, the simple zero-one loss function is often sufficient. This corresponds simply to assigning a loss of 1 to any incorrect labeling and implies that the optimal classifier minimizes the error rate on independent test data (i.e. counting up the fraction of instances that the learned function 
+  
+    
+      
+        h
+        :
+        
+          
+            X
+          
+        
+        →
+        
+          
+            Y
+          
+        
+      
+    
+    {\displaystyle h:{\mathcal {X}}\rightarrow {\mathcal {Y}}}
+  
+ labels wrongly, which is equivalent to maximizing the number of correctly classified instances). The goal of the learning procedure is then to minimize the error rate (maximize the correctness) on a "typical" test set.
+For a probabilistic pattern recognizer, the problem is instead to estimate the probability of each possible output label given a particular input instance, i.e., to estimate a function of the form
+
+  
+    
+      
+        p
+        (
+        
+          
+            l
+            a
+            b
+            e
+            l
+          
+        
+        
+          |
+        
+        
+          x
+        
+        ,
+        
+          θ
+        
+        )
+        =
+        f
+        
+          (
+          
+            
+              x
+            
+            ;
+            
+              θ
+            
+          
+          )
+        
+      
+    
+    {\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})=f\left({\boldsymbol {x}};{\boldsymbol {\theta }}\right)}
+  
+
+where the feature vector input is 
+  
+    
+      
+        
+          x
+        
+      
+    
+    {\displaystyle {\boldsymbol {x}}}
+  
+, and the function f is typically parameterized by some parameters 
+  
+    
+      
+        
+          θ
+        
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}}
+  
+. In a discriminative approach to the problem, f is estimated directly. In a generative approach, however, the inverse probability 
+  
+    
+      
+        p
+        (
+        
+          
+            x
+          
+          
+            |
+          
+          
+            
+              l
+              a
+              b
+              e
+              l
+            
+          
+        
+        )
+      
+    
+    {\displaystyle p({{\boldsymbol {x}}|{\rm {label}}})}
+  
+ is instead estimated and combined with the prior probability 
+  
+    
+      
+        p
+        (
+        
+          
+            l
+            a
+            b
+            e
+            l
+          
+        
+        
+          |
+        
+        
+          θ
+        
+        )
+      
+    
+    {\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}
+  
+ using Bayes' rule, as follows:

data/en.wikipedia.org/wiki/Pattern_recognition-2.md

@@ -0,0 +1,613 @@
+---
+title: "Pattern recognition"
+chunk: 3/4
+source: "https://en.wikipedia.org/wiki/Pattern_recognition"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:51.565383+00:00"
+instance: "kb-cron"
+---
+
+  
+    
+      
+        p
+        (
+        
+          
+            l
+            a
+            b
+            e
+            l
+          
+        
+        
+          |
+        
+        
+          x
+        
+        ,
+        
+          θ
+        
+        )
+        =
+        
+          
+            
+              p
+              (
+              
+                
+                  x
+                
+                
+                  |
+                
+                
+                  
+                    l
+                    a
+                    b
+                    e
+                    l
+                    ,
+                    
+                      θ
+                    
+                  
+                
+              
+              )
+              p
+              (
+              
+                
+                  l
+                  a
+                  b
+                  e
+                  l
+                  
+                    |
+                  
+                  
+                    θ
+                  
+                
+              
+              )
+            
+            
+              
+                ∑
+                
+                  L
+                  ∈
+                  
+                    all labels
+                  
+                
+              
+              p
+              (
+              
+                x
+              
+              
+                |
+              
+              L
+              )
+              p
+              (
+              L
+              
+                |
+              
+              
+                θ
+              
+              )
+            
+          
+        
+        .
+      
+    
+    {\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})={\frac {p({{\boldsymbol {x}}|{\rm {label,{\boldsymbol {\theta }}}}})p({\rm {label|{\boldsymbol {\theta }}}})}{\sum _{L\in {\text{all labels}}}p({\boldsymbol {x}}|L)p(L|{\boldsymbol {\theta }})}}.}
+  
+
+When the labels are continuously distributed (e.g., in regression analysis), the denominator involves integration rather than summation:
+
+  
+    
+      
+        p
+        (
+        
+          
+            l
+            a
+            b
+            e
+            l
+          
+        
+        
+          |
+        
+        
+          x
+        
+        ,
+        
+          θ
+        
+        )
+        =
+        
+          
+            
+              p
+              (
+              
+                
+                  x
+                
+                
+                  |
+                
+                
+                  
+                    l
+                    a
+                    b
+                    e
+                    l
+                    ,
+                    
+                      θ
+                    
+                  
+                
+              
+              )
+              p
+              (
+              
+                
+                  l
+                  a
+                  b
+                  e
+                  l
+                  
+                    |
+                  
+                  
+                    θ
+                  
+                
+              
+              )
+            
+            
+              
+                ∫
+                
+                  L
+                  ∈
+                  
+                    all labels
+                  
+                
+              
+              p
+              (
+              
+                x
+              
+              
+                |
+              
+              L
+              )
+              p
+              (
+              L
+              
+                |
+              
+              
+                θ
+              
+              )
+              d
+              ⁡
+              L
+            
+          
+        
+        .
+      
+    
+    {\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})={\frac {p({{\boldsymbol {x}}|{\rm {label,{\boldsymbol {\theta }}}}})p({\rm {label|{\boldsymbol {\theta }}}})}{\int _{L\in {\text{all labels}}}p({\boldsymbol {x}}|L)p(L|{\boldsymbol {\theta }})\operatorname {d} L}}.}
+  
+
+The value of 
+  
+    
+      
+        
+          θ
+        
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}}
+  
+ is typically learned using maximum a posteriori (MAP) estimation. This finds the best value that simultaneously meets two conflicting objects: To perform as well as possible on the training data (smallest error-rate) and to find the simplest possible model. Essentially, this combines maximum likelihood estimation with a regularization procedure that favors simpler models over more complex models. In a Bayesian context, the regularization procedure can be viewed as placing a prior probability 
+  
+    
+      
+        p
+        (
+        
+          θ
+        
+        )
+      
+    
+    {\displaystyle p({\boldsymbol {\theta }})}
+  
+ on different values of 
+  
+    
+      
+        
+          θ
+        
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}}
+  
+. Mathematically:
+
+  
+    
+      
+        
+          
+            θ
+          
+          
+            ∗
+          
+        
+        =
+        arg
+        ⁡
+        
+          max
+          
+            θ
+          
+        
+        p
+        (
+        
+          θ
+        
+        
+          |
+        
+        
+          D
+        
+        )
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}^{*}=\arg \max _{\boldsymbol {\theta }}p({\boldsymbol {\theta }}|\mathbf {D} )}
+  
+
+where 
+  
+    
+      
+        
+          
+            θ
+          
+          
+            ∗
+          
+        
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}^{*}}
+  
+ is the value used for 
+  
+    
+      
+        
+          θ
+        
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}}
+  
+ in the subsequent evaluation procedure, and 
+  
+    
+      
+        p
+        (
+        
+          θ
+        
+        
+          |
+        
+        
+          D
+        
+        )
+      
+    
+    {\displaystyle p({\boldsymbol {\theta }}|\mathbf {D} )}
+  
+, the posterior probability of 
+  
+    
+      
+        
+          θ
+        
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}}
+  
+, is given by
+
+  
+    
+      
+        p
+        (
+        
+          θ
+        
+        
+          |
+        
+        
+          D
+        
+        )
+        =
+        
+          [
+          
+            
+              ∏
+              
+                i
+                =
+                1
+              
+              
+                n
+              
+            
+            p
+            (
+            
+              y
+              
+                i
+              
+            
+            
+              |
+            
+            
+              
+                x
+              
+              
+                i
+              
+            
+            ,
+            
+              θ
+            
+            )
+          
+          ]
+        
+        p
+        (
+        
+          θ
+        
+        )
+        .
+      
+    
+    {\displaystyle p({\boldsymbol {\theta }}|\mathbf {D} )=\left[\prod _{i=1}^{n}p(y_{i}|{\boldsymbol {x}}_{i},{\boldsymbol {\theta }})\right]p({\boldsymbol {\theta }}).}
+  
+
+In the Bayesian approach to this problem, instead of choosing a single parameter vector 
+  
+    
+      
+        
+          
+            θ
+          
+          
+            ∗
+          
+        
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}^{*}}
+  
+, the probability of a given label for a new instance 
+  
+    
+      
+        
+          x
+        
+      
+    
+    {\displaystyle {\boldsymbol {x}}}
+  
+ is computed by integrating over all possible values of 
+  
+    
+      
+        
+          θ
+        
+      
+    
+    {\displaystyle {\boldsymbol {\theta }}}
+  
+, weighted according to the posterior probability:
+
+  
+    
+      
+        p
+        (
+        
+          
+            l
+            a
+            b
+            e
+            l
+          
+        
+        
+          |
+        
+        
+          x
+        
+        )
+        =
+        ∫
+        p
+        (
+        
+          
+            l
+            a
+            b
+            e
+            l
+          
+        
+        
+          |
+        
+        
+          x
+        
+        ,
+        
+          θ
+        
+        )
+        p
+        (
+        
+          θ
+        
+        
+          |
+        
+        
+          D
+        
+        )
+        d
+        ⁡
+        
+          θ
+        
+        .
+      
+    
+    {\displaystyle p({\rm {label}}|{\boldsymbol {x}})=\int p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})p({\boldsymbol {\theta }}|\mathbf {D} )\operatorname {d} {\boldsymbol {\theta }}.}
+  
+
+=== Frequentist or Bayesian approach to pattern recognition ===
+The first pattern classifier – the linear discriminant presented by Fisher – was developed in the frequentist tradition. The frequentist approach entails that the model parameters are considered unknown, but objective. The parameters are then computed (estimated) from the collected data. For the linear discriminant, these parameters are precisely the mean vectors and the covariance matrix. Also the probability of each class 
+  
+    
+      
+        p
+        (
+        
+          
+            l
+            a
+            b
+            e
+            l
+          
+        
+        
+          |
+        
+        
+          θ
+        
+        )
+      
+    
+    {\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}
+  
+ is estimated from the collected dataset. Note that the usage of 'Bayes' rule' in a pattern classifier does not make the classification approach Bayesian.
+Bayesian statistics has its origin in Greek philosophy where a distinction was already made between the 'a priori' and the 'a posteriori' knowledge. Later Kant defined his distinction between what is a priori known – before observation – and the empirical knowledge gained from observations. In a Bayesian pattern classifier, the class probabilities 
+  
+    
+      
+        p
+        (
+        
+          
+            l
+            a
+            b
+            e
+            l
+          
+        
+        
+          |
+        
+        
+          θ
+        
+        )
+      
+    
+    {\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}
+  
+ can be chosen by the user, which are then a priori. Moreover, experience quantified as a priori parameter values can be weighted with empirical observations – using e.g., the Beta- (conjugate prior) and Dirichlet-distributions. The Bayesian approach facilitates a seamless intermixing between expert knowledge in the form of subjective probabilities, and objective observations.
+Probabilistic pattern classifiers can be used according to a frequentist or a Bayesian approach.
+
+== Uses ==
+
+Within medical science, pattern recognition is the basis for computer-aided diagnosis (CAD) systems. CAD describes a procedure that supports the doctor's interpretations and findings. Other typical applications of pattern recognition techniques are automatic speech recognition, speaker identification, classification of text into several categories (e.g., spam or non-spam email messages), the automatic recognition of handwriting on postal envelopes, automatic recognition of images of human faces, or handwriting image extraction from medical forms. The last two examples form the subtopic image analysis of pattern recognition that deals with digital images as input to pattern recognition systems.
+Optical character recognition is an example of the application of a pattern classifier. The method of signing one's name was captured with stylus and overlay starting in 1990. The strokes, speed, relative min, relative max, acceleration and pressure is used to uniquely identify and confirm identity. Banks were first offered this technology, but were content to collect from the FDIC for any bank fraud and did not want to inconvenience customers.
+Pattern recognition has many real-world applications in image processing. Some examples include:

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+title: "Pattern recognition"
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+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:51.565383+00:00"
+instance: "kb-cron"
+---
+
+identification and authentication: e.g., license plate recognition, fingerprint analysis, face detection/verification, and voice-based authentication.
+medical diagnosis: e.g., screening for cervical cancer (Papnet), breast tumors or heart sounds;
+defense: various navigation and guidance systems, target recognition systems, shape recognition technology etc.
+mobility: advanced driver assistance systems, autonomous vehicle technology, etc.
+In psychology, pattern recognition is used to make sense of and identify objects, and is closely related to perception. This explains how the sensory inputs humans receive are made meaningful. Pattern recognition can be thought of in two different ways. The first concerns template matching and the second concerns feature detection. A template is a pattern used to produce items of the same proportions. The template-matching hypothesis suggests that incoming stimuli are compared with templates in the long-term memory. If there is a match, the stimulus is identified. Feature detection models, such as the Pandemonium system for classifying letters (Selfridge, 1959), suggest that the stimuli are broken down into their component parts for identification. One observation is a capital E having three horizontal lines and one vertical line.
+
+== Algorithms ==
+Algorithms for pattern recognition depend on the type of label output, on whether learning is supervised or unsupervised, and on whether the algorithm is statistical or non-statistical in nature. Statistical algorithms can further be categorized as generative or discriminative.
+
+=== Classification methods (methods predicting categorical labels) ===
+
+Parametric:
+
+Linear discriminant analysis
+Quadratic discriminant analysis
+Maximum entropy classifier (aka logistic regression, multinomial logistic regression): Note that logistic regression is an algorithm for classification, despite its name. (The name comes from the fact that logistic regression uses an extension of a linear regression model to model the probability of an input being in a particular class.)
+Nonparametric:
+
+Decision trees, decision lists
+Kernel estimation and K-nearest-neighbor algorithms
+Naive Bayes classifier
+Neural networks (multi-layer perceptrons)
+Perceptrons
+Support vector machines
+Gene expression programming
+
+=== Clustering methods (methods for classifying and predicting categorical labels) ===
+
+Categorical mixture models
+Hierarchical clustering (agglomerative or divisive)
+K-means clustering
+Correlation clustering
+Kernel principal component analysis (Kernel PCA)
+
+=== Ensemble learning algorithms (supervised meta-algorithms for combining multiple learning algorithms together) ===
+
+Boosting (meta-algorithm)
+Bootstrap aggregating ("bagging")
+Ensemble averaging
+Mixture of experts, hierarchical mixture of experts
+
+=== General methods for predicting arbitrarily-structured (sets of) labels ===
+Bayesian networks
+Markov random fields
+
+=== Multilinear subspace learning algorithms (predicting labels of multidimensional data using tensor representations) ===
+Unsupervised:
+
+Multilinear principal component analysis (MPCA)
+
+=== Real-valued sequence labeling methods (predicting sequences of real-valued labels) ===
+
+Kalman filters
+Particle filters
+
+=== Regression methods (predicting real-valued labels) ===
+
+Gaussian process regression (kriging)
+Linear regression and extensions
+Independent component analysis (ICA)
+Principal components analysis (PCA)
+
+=== Sequence labeling methods (predicting sequences of categorical labels) ===
+Conditional random fields (CRFs)
+Hidden Markov models (HMMs)
+Maximum entropy Markov models (MEMMs)
+Recurrent neural networks (RNNs)
+Dynamic time warping (DTW)
+
+== See also ==
+Adaptive resonance theory – Theory in neuropsychology
+Black box – System where only the inputs and outputs can be viewed, and not its implementation
+Cache language model
+Compound-term processing
+Computer-aided diagnosis – Type of diagnosis assisted by computers
+Contextual image classification
+Data mining – Process of analyzing large data sets
+Deep learning – Branch of machine learning
+Grey box model – Mathematical data production model with limited structure
+Information theory – Scientific study of digital information
+List of datasets for machine learning research
+List of numerical-analysis software
+List of numerical libraries
+Neocognitron – Type of artificial neural network
+Perception – Interpretation of sensory information
+Perceptual learning – Process of learning better perception skills
+Predictive analytics – Statistical techniques analyzing facts to make predictions about unknown events
+Prior knowledge for pattern recognition
+Sequence mining – Data mining techniquePages displaying short descriptions of redirect targets
+Template matching – Technique in digital image processing
+
+== References ==
+
+== Further reading ==
+Fukunaga, Keinosuke (1990). Introduction to Statistical Pattern Recognition (2nd ed.). Boston: Academic Press. ISBN 978-0-12-269851-4.
+Hornegger, Joachim; Paulus, Dietrich W. R. (1999). Applied Pattern Recognition: A Practical Introduction to Image and Speech Processing in C++ (2nd ed.). San Francisco: Morgan Kaufmann Publishers. ISBN 978-3-528-15558-2.
+Schuermann, Juergen (1996). Pattern Classification: A Unified View of Statistical and Neural Approaches. New York: Wiley. ISBN 978-0-471-13534-0.
+Godfried T. Toussaint, ed. (1988). Computational Morphology. Amsterdam: North-Holland Publishing Company. ISBN 978-1-4832-9672-2.
+Kulikowski, Casimir A.; Weiss, Sholom M. (1991). Computer Systems That Learn: Classification and Prediction Methods from Statistics, Neural Nets, Machine Learning, and Expert Systems. San Francisco: Morgan Kaufmann Publishers. ISBN 978-1-55860-065-2.
+Duda, Richard O.; Hart, Peter E.; Stork, David G. (2000). Pattern Classification (2nd ed.). Wiley-Interscience. ISBN 978-0-471-05669-0.
+Jain, Anil.K.; Duin, Robert.P.W.; Mao, Jianchang (2000). "Statistical pattern recognition: a review". IEEE Transactions on Pattern Analysis and Machine Intelligence. 22 (1): 4–37. Bibcode:2000ITPAM..22....4J. CiteSeerX 10.1.1.123.8151. doi:10.1109/34.824819. S2CID 192934.
+An introductory tutorial to classifiers (introducing the basic terms, with numeric example)
+Kovalevsky, V. A. (1980). Image Pattern Recognition. New York, NY: Springer New York. ISBN 978-1-4612-6033-2. OCLC 852790446.
+
+== External links ==
+The International Association for Pattern Recognition
+List of Pattern Recognition web sites
+Journal of Pattern Recognition Research Archived 2008-09-08 at the Wayback Machine
+Pattern Recognition Info
+Pattern Recognition (Journal of the Pattern Recognition Society)
+International Journal of Pattern Recognition and Artificial Intelligence Archived 2004-12-11 at the Wayback Machine
+International Journal of Applied Pattern Recognition
+Open Pattern Recognition Project, intended to be an open source platform for sharing algorithms of pattern recognition
+Improved Fast Pattern Matching Improved Fast Pattern Matching

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+title: "Perceptual control theory"
+chunk: 1/7
+source: "https://en.wikipedia.org/wiki/Perceptual_control_theory"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:52.953091+00:00"
+instance: "kb-cron"
+---
+
+Perceptual control theory (PCT) is a model of behavior based on the properties of negative feedback control loops. A control loop maintains a sensed variable at or near a reference value by means of the effects of its outputs upon that variable, as mediated by physical properties of the environment. In engineering control theory, reference values are set by a user outside the system. An example is a thermostat. In a living organism, reference values for controlled perceptual variables are endogenously maintained. Biological homeostasis and reflexes are simple, low-level examples. The discovery of mathematical principles of control introduced a way to model a negative feedback loop closed through the environment (circular causation), which spawned perceptual control theory. It differs fundamentally from some models in behavioral and cognitive psychology that model stimuli as causes of behavior (linear causation). PCT research is published in experimental psychology, neuroscience, ethology, anthropology, linguistics, sociology, robotics, developmental psychology, organizational psychology and management, and a number of other fields. PCT has been applied to design and administration of educational systems, and has led to a psychotherapy called the method of levels.
+
+== Principles and differences from other theories ==
+The perceptual control theory is deeply rooted in biological cybernetics, systems biology and control theory and the related concept of feedback loops. Unlike some models in behavioral and cognitive psychology it sets out from the concept of circular causality. It shares, therefore, its theoretical foundation with the concept of plant control, but it is distinct from it by emphasizing the control of the internal representation of the physical world.
+The plant control theory focuses on neuro-computational processes of movement generation, once a decision for generating the movement has been taken. PCT spotlights the embeddedness of agents in their environment. Therefore, from the perspective of perceptual control, the central problem of motor control consists in finding a sensory input to the system that matches a desired perception.
+
+== History ==
+PCT has roots in the 19th-century physiological insights of Claude Bernard and Walter Bradford Cannon, and in the fields of control systems engineering and cybernetics. Classical negative feedback control was worked out by engineers in the 1930s and 1940s, and further developed by Wiener, Ashby, and others in the early development of the field of cybernetics. Beginning in the 1950s, William T. Powers applied the concepts and methods of engineered control systems to biological control systems, and developed the experimental methodology of PCT.
+A key insight of PCT is that the controlled variable is not the output of the system (the behavioral actions), but its input, that is, a sensed and transformed function of some state of the environment that the control system's output can affect. Because these sensed and transformed inputs may appear as consciously perceived aspects of the environment, Powers labelled the controlled variable "perception". The theory came to be known as "Perceptual Control Theory" or PCT rather than "Control Theory Applied to Psychology" because control theorists often assert or assume that it is the system's output that is controlled. In PCT it is the internal representation of the state of some variable in the environment—a "perception" in everyday language—that is controlled. The basic principles of PCT were first published by Powers, Clark, and MacFarland as a "general feedback theory of behavior" in 1960, with credits to cybernetic authors Wiener and Ashby. It has been systematically developed since then in the research community that has gathered around it. Initially, it was overshadowed by the cognitive revolution (later supplanted by cognitive science), but has now become better known.
+Powers and other researchers in the field point to problems of purpose, causation, and teleology at the foundations of psychology which control theory resolves. From Aristotle through William James and John Dewey it has been recognized that behavior is purposeful and not merely reactive, but how to account for this has been problematic because the only evidence for intentions was subjective. As Powers pointed out, behaviorists following Wundt, Thorndike, Watson, and others rejected introspective reports as data for an objective science of psychology. Only observable behavior could be admitted as data. Such behaviorists modeled environmental events (stimuli) as causing behavioral actions (responses). This causal assumption persists in some models in cognitive psychology that interpose cognitive maps and other postulated information processing between stimulus and response but otherwise retain the assumption of linear causation from environment to behavior, which Richard Marken called an "open-loop causal model of behavioral organization" in contrast to PCT's closed-loop model.
+Another, more specific reason that Powers observed for psychologists' rejecting notions of purpose or intention was that they could not see how a goal (a state that did not yet exist) could cause the behavior that led to it. PCT resolves these philosophical arguments about teleology because it provides a model of the functioning of organisms in which purpose has objective status without recourse to introspection, and in which causation is circular around feedback loops.

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+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:52.953091+00:00"
+instance: "kb-cron"
+---
+
+== Example ==
+A simple negative feedback control system is a cruise control system for a car. A cruise control system has a sensor which "perceives" speed as the rate of spin of the drive shaft directly connected to the wheels. It also has a driver-adjustable 'goal' specifying a particular speed. The sensed speed is continuously compared against the specified speed by a device (called a "comparator") which subtracts the currently sensed input value from the stored goal value. The difference (the error signal) determines the throttle setting (the accelerator depression), so that the engine output is continuously varied to prevent the speed of the car from increasing or decreasing from that desired speed as environmental conditions change.
+If the speed of the car starts to drop below the goal-speed, for example when climbing a hill, the small increase in the error signal, amplified, causes engine output to increase, which keeps the error very nearly at zero. If the speed begins to exceed the goal, e.g. when going down a hill, the engine is throttled back so as to act as a brake, so again the speed is kept from departing more than a barely detectable amount from the goal speed (brakes being needed only if the hill is too steep). The result is that the cruise control system maintains a speed close to the goal as the car goes up and down hills, and as other disturbances such as wind affect the car's speed. This is all done without any planning of specific actions, and without any blind reactions to stimuli. Indeed, the cruise control system does not sense disturbances such as wind pressure at all, it only senses the controlled variable, speed. Nor does it control the power generated by the engine, it uses the 'behavior' of engine power as its means to control the sensed speed.
+The same principles of negative feedback control (including the ability to nullify effects of unpredictable external or internal disturbances) apply to living control systems. Implications of these principle are e.g. intensively studied by biological and medical cybernetics and systems biology.
+The thesis of PCT is that animals and people do not control their behavior; rather, they vary their behavior as their means for controlling their perceptions, with or without external disturbances. This is harmoniously consistent with the historical and still widespread assumption that behavior is the final result of stimulus inputs and cognitive plans.
+
+== The methodology of modeling, and PCT as model ==
+
+The principal datum in PCT methodology is the controlled variable. The fundamental step of PCT research, the test for controlled variables, begins with the slow and gentle application of disturbing influences to the state of a variable in the environment which the researcher surmises is already under control by the observed organism. It is essential not to overwhelm the organism's ability to control, since that is what is being investigated. If the organism changes its actions just so as to prevent the disturbing influence from having the expected effect on that variable, that is strong evidence that the experimental action disturbed a controlled variable. It is crucially important to distinguish the perceptions and point of view of the observer from those of the observed organism. It may take a number of variations of the test to isolate just which aspect of the environmental situation is under control, as perceived by the observed organism.
+PCT employs a black box methodology. The controlled variable as measured by the observer corresponds quantitatively to a reference value for a perception that the organism is controlling. The controlled variable is thus an objective index of the purpose or intention of those particular behavioral actions by the organism—the goal which those actions consistently work to attain despite disturbances. With few exceptions, in the current state of neuroscience this internally maintained reference value is seldom directly observed as such (e.g. as a rate of firing in a neuron), since few researchers trace the relevant electrical and chemical variables by their specific pathways while a living organism is engaging in what we externally observe as behavior. However, when a working negative feedback system simulated on a digital computer performs essentially identically to observed organisms, then the well understood negative feedback structure of the simulation or model (the white box) is understood to demonstrate the unseen negative feedback structure within the organism (the black box).
+Data for individuals are not aggregated for statistical analysis; instead, a generative model is built which replicates the data observed for individuals with very high fidelity (0.95 or better). To build such a model of a given behavioral situation requires careful measurements of three observed variables:
+
+A fourth value, the internally maintained reference r (a variable ′setpoint′), is deduced from the value at which the organism is observed to maintain qi, as determined by the test for controlled variables (described at the beginning of this section).
+With two variables specified, the controlled input qi and the reference r, a properly designed control system, simulated on a digital computer, produces outputs qo that almost precisely oppose unpredictable disturbances d to the controlled input. Further, the variance from perfect control accords well with that observed for living organisms. Perfect control would result in zero effect of the disturbance, but living organisms are not perfect controllers, and the aim of PCT is to model living organisms. When a computer simulation performs with >95% conformity to experimentally measured values, opposing the effect of unpredictable changes in d by generating (nearly) equal and opposite values of qo, it is understood to model the behavior and the internal control-loop structure of the organism.
+By extension, the elaboration of the theory constitutes a general model of cognitive process and behavior. With every specific model or simulation of behavior that is constructed and tested against observed data, the general model that is presented in the theory is exposed to potential challenge that could call for revision or could lead to refutation.

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+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:52.953091+00:00"
+instance: "kb-cron"
+---
+
+== Mathematics ==
+To illustrate the mathematical calculations employed in a PCT simulation, consider a pursuit tracking task in which the participant keeps a mouse cursor aligned with a moving target on a computer monitor.
+The model assumes that a perceptual signal within the participant represents the magnitude of the input quantity qi. (This has been demonstrated to be a rate of firing in a neuron, at least at the lowest levels.) In the tracking task, the input quantity is the vertical distance between the target position T and the cursor position C, and the random variation of the target position acts as the disturbance d of that input quantity. This suggests that the perceptual signal p quantitatively represents the cursor position C minus the target position T, as expressed in the equation p=C–T.
+Between the perception of target and cursor and the construction of the signal representing the distance between them there is a delay of τ milliseconds, so that the working perceptual signal at time t represents the target-to-cursor distance at a prior time, t – τ. Consequently, the equation used in the model is
+1. p(t) = C(t–τ) – T(t–τ)
+The negative feedback control system receives a reference signal r which specifies the magnitude of the given perceptual signal which is currently intended or desired. (For the origin of r within the organism, see under "A hierarchy of control", below.) Both r and p are input to a simple neural structure with r excitatory and p inhibitory. This structure is called a "comparator". The effect is to subtract p from r, yielding an error signal e that indicates the magnitude and sign of the difference between the desired magnitude r and the currently input magnitude p of the given perception. The equation representing this in the model is:
+2. e = r–p
+The error signal e must be transformed to the output quantity qo (representing the participant's muscular efforts affecting the mouse position). Experiments have shown that in the best model for the output function, the mouse velocity Vcursor is proportional to the error signal e by a gain factor G (that is, Vcursor = G*e). Thus, when the perceptual signal p is smaller than the reference signal r, the error signal e has a positive sign, and from it the model computes an upward velocity of the cursor that is proportional to the error.
+The next position of the cursor Cnew is the current position Cold plus the velocity Vcursor times the duration dt of one iteration of the program. By simple algebra, we substitute G*e (as given above) for Vcursor, yielding a third equation:
+3. Cnew = Cold + G*e*dt
+These three simple equations or program steps constitute the simplest form of the model for the tracking task. When these three simultaneous equations are evaluated over and over with similarly distributed random disturbances d of the target position that the human participant experienced, the output positions and velocities of the cursor duplicate the participant's actions in the tracking task above within 4.0% of their peak-to-peak range, in great detail.
+This simple model can be refined with a damping factor d which reduces the discrepancy between the model and the human participant to 3.6% when the disturbance d is set to maximum difficulty.
+3'. Cnew = Cold + [(G*e)–(d*Cold)]*dt
+Detailed discussion of this model in (Powers 2008) includes both source and executable code, with which the reader can verify how well this simple program simulates real behavior. No consideration is needed of possible nonlinearities such as the Weber-Fechner law, potential noise in the system, continuously varying angles at the joints, and many other factors that could afflict performance if this were a simple linear model. No inverse kinematics or predictive calculations are required. The model simply reduces the discrepancy between input p and reference r continuously as it arises in real time, and that is all that is required—as predicted by the theory.

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+title: "Perceptual control theory"
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+category: "reference"
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+date_saved: "2026-05-05T03:56:52.953091+00:00"
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+---
+
+== Distinctions from engineering control theory ==
+In the artificial systems that are specified by engineering control theory, the reference signal is considered to be an external input to the 'plant'. In engineering control theory, the reference signal or set point is public; in PCT, it is not, but rather must be deduced from the results of the test for controlled variables, as described above in the methodology section. This is because in living systems a reference signal is not an externally accessible input, but instead originates within the system. In the hierarchical model, error output of higher-level control loops, as described in the next section below, evokes the reference signal r from synapse-local memory, and the strength of r is proportional to the (weighted) strength of the error signal or signals from one or more higher-level systems.
+In engineering control systems, in the case where there are several such reference inputs, a 'Controller' is designed to manipulate those inputs so as to obtain the effect on the output of the system that is desired by the system's designer, and the task of a control theory (so conceived) is to calculate those manipulations so as to avoid instability and oscillation. The designer of a PCT model or simulation specifies no particular desired effect on the output of the system, except that it must be whatever is required to bring the input from the environment (the perceptual signal) into conformity with the reference. In Perceptual Control Theory, the input function for the reference signal is a weighted sum of internally generated signals (in the canonical case, higher-level error signals), and loop stability is determined locally for each loop in the manner sketched in the preceding section on the mathematics of PCT (and elaborated more fully in the referenced literature). The weighted sum is understood to result from reorganization.
+Engineering control theory is computationally demanding, but as the preceding section shows, PCT is not. For example, contrast the implementation of a model of an inverted pendulum in engineering control theory with the PCT implementation as a hierarchy of five simple control systems.
+
+== A hierarchy of control ==
+
+Perceptions, in PCT, are constructed and controlled in a hierarchy of levels. For example, visual perception of an object is constructed from differences in light intensity or differences in sensations such as color at its edges. Controlling the shape or location of the object requires altering the perceptions of sensations or intensities (which are controlled by lower-level systems). This organizing principle is applied at all levels, up to the most abstract philosophical and theoretical constructs.
+The Russian physiologist Nicolas Bernstein independently came to the same conclusion that behavior has to be multiordinal—organized hierarchically, in layers. A simple problem led to this conclusion at about the same time both in PCT and in Bernstein's work. The spinal reflexes act to stabilize limbs against disturbances. Why do they not prevent centers higher in the brain from using those limbs to carry out behavior? Since the brain obviously does use the spinal systems in producing behavior, there must be a principle that allows the higher systems to operate by incorporating the reflexes, not just by overcoming them or turning them off. The answer is that the reference value (setpoint) for a spinal reflex is not static; rather, it is varied by higher-level systems as their means of moving the limbs (servomechanism). This principle applies to higher feedback loops, as each loop presents the same problem to subsystems above it.
+Whereas an engineered control system has a reference value or setpoint adjusted by some external agency, the reference value for a biological control system cannot be set in this way. The setpoint must come from some internal process. If there is a way for behavior to affect it, any perception may be brought to the state momentarily specified by higher levels and then be maintained in that state against unpredictable disturbances. In a hierarchy of control systems, higher levels adjust the goals of lower levels as their means of approaching their own goals set by still-higher systems. This has important consequences for any proposed external control of an autonomous living control system (organism). At the highest level, reference values (goals) are set by heredity or adaptive processes.
+
+== Reorganization in evolution, development, and learning ==
+If an organism controls inappropriate perceptions, or if it controls some perceptions to inappropriate values, then it is less likely to bring progeny to maturity, and may die. Consequently, by natural selection successive generations of organisms evolve so that they control those perceptions that, when controlled with appropriate setpoints, tend to maintain critical internal variables at optimal levels, or at least within non-lethal limits. Powers called these critical internal variables "intrinsic variables" (Ashby's "essential variables").
+The mechanism that influences the development of structures of perceptions to be controlled is termed "reorganization", a process within the individual organism that is subject to natural selection just as is the evolved structure of individuals within a species.
+This "reorganization system" is proposed to be part of the inherited structure of the organism. It changes the underlying parameters and connectivity of the control hierarchy in a random-walk manner. There is a basic continuous rate of change in intrinsic variables which proceeds at a speed set by the total error (and stops at zero error), punctuated by random changes in direction in a hyperspace with as many dimensions as there are critical variables. This is a more or less direct adaptation of Ashby's "homeostat", first adopted into PCT in the 1960 paper and then changed to use E. coli's method of navigating up gradients of nutrients, as described by Koshland (1980).
+Reorganization may occur at any level when loss of control at that level causes intrinsic (essential) variables to deviate from genetically determined set points. This is the basic mechanism that is involved in trial-and-error learning, which leads to the acquisition of more systematic kinds of learning processes.
+
+== Psychotherapy: the method of levels (MOL) ==

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+title: "Perceptual control theory"
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+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:52.953091+00:00"
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+---
+
+The reorganization concept has led to a method of psychotherapy called the method of levels (MOL). Using MOL, the therapist aims to help the patient shift his or her awareness to higher levels of perception in order to resolve conflicts and allow reorganization to take place.
+
+== Neuroscience ==
+
+=== Learning ===
+Currently, no one theory has been agreed upon to explain the synaptic, neuronal or systemic basis of learning. Prominent since 1973, however, is the idea that long-term potentiation (LTP) of populations of synapses induces learning through both pre- and postsynaptic mechanisms. LTP is a form of Hebbian learning, which proposed that high-frequency, tonic activation of a circuit of neurones increases the efficacy with which they are activated and the size of their response to a given stimulus as compared to the standard neurone (Hebb, 1949). These mechanisms are the principles behind Hebb's famously simple explanation: "Those that fire together, wire together".
+LTP has received much support since it was first observed by Terje Lømo in 1966 and is still the subject of many modern studies and clinical research. However, there are possible alternative mechanisms underlying LTP, as presented by Enoki, Hu, Hamilton and Fine in 2009, published in the journal Neuron. They concede that LTP is the basis of learning. However, they firstly propose that LTP occurs in individual synapses, and this plasticity is graded (as opposed to in a binary mode) and bidirectional. Secondly, the group suggest that the synaptic changes are expressed solely presynaptically, via changes in the probability of transmitter release. Finally, the team predict that the occurrence of LTP could be age-dependent, as the plasticity of a neonatal brain would be higher than that of a mature one. Therefore, the theories differ, as one proposes an on/off occurrence of LTP by pre- and postsynaptic mechanisms and the other proposes only presynaptic changes, graded ability, and age-dependence.
+These theories do agree on one element of LTP, namely, that it must occur through physical changes to the synaptic membrane/s, i.e. synaptic plasticity. Perceptual control theory encompasses both of these views. It proposes the mechanism of 'reorganisation' as the basis of learning. Reorganisation occurs within the inherent control system of a human or animal by restructuring the inter- and intraconnections of its hierarchical organisation, akin to the neuroscientific phenomenon of neural plasticity. This reorganisation initially allows the trial-and-error form of learning, which is seen in babies, and then progresses to more structured learning through association, apparent in infants, and finally to systematic learning, covering the adult ability to learn from both internally and externally  generated stimuli and events. In this way, PCT provides a valid model for learning that combines the biological mechanisms of LTP with an explanation of the progression and change of mechanisms associated with developmental ability.
+Powers in 2008 produced a simulation of arm co-ordination. He suggested that in order to move your arm, fourteen control systems that control fourteen joint angles are involved, and they reorganise simultaneously and independently. It was found that for optimum performance, the output functions must be organised in a way so as each control system's output only affects the one environmental variable it is perceiving. In this simulation, the reorganising process is working as it should, and just as Powers suggests that it works in humans, reducing outputs that cause error and increasing those that reduce error. Initially, the disturbances have large effects on the angles of the joints, but over time the joint angles match the reference signals more closely due to the system being reorganised. Powers suggests that in order to achieve coordination of joint angles to produce desired movements, instead of calculating how multiple joint angles must change to produce this movement the brain uses negative feedback systems to generate the joint angles that are required. A single reference signal that is varied in a higher-order system can generate a movement that requires several joint angles to change at the same time.

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+date_saved: "2026-05-05T03:56:52.953091+00:00"
+instance: "kb-cron"
+---
+
+=== Hierarchical organisation ===
+Botvinick in 2008 proposed that one of the founding insights of the cognitive revolution was the recognition of hierarchical structure in human behavior. Despite decades of research, however, the computational mechanisms underlying hierarchically organized behavior are still not fully understood. Bedre, Hoffman, Cooney & D'Esposito in 2009 proposed that the fundamental goal in cognitive neuroscience is to characterize the functional organization of the frontal cortex that supports the control of action.
+Recent neuroimaging data has supported the hypothesis that the frontal lobes are organized hierarchically, such that control is supported in progressively caudal regions as control moves to more concrete specification of action. However, it is still not clear whether lower-order control processors are differentially affected by impairments in higher-order control when between-level interactions are required to complete a task, or whether there are feedback influences of lower-level on higher-level control.
+Botvinik in 2008 found that all existing models of hierarchically structured behavior share at least one general assumption – that the hierarchical, part–whole organization of human action is mirrored in the internal or neural representations underlying it. Specifically, the assumption is that there exist representations not only of low-level motor behaviors, but also separable representations of higher-level behavioral units. The latest crop of models provides new insights, but also poses new or refined questions for empirical research, including how abstract action representations emerge through learning, how they interact with different modes of action control, and how they sort out within the prefrontal cortex (PFC).
+Perceptual control theory (PCT) can provide an explanatory model of neural organisation that deals with the current issues. PCT describes the hierarchical character of behavior as being determined by control of hierarchically organized perception. Control systems in the body and in the internal environment of billions of interconnected neurons within the brain are responsible for keeping perceptual signals within survivable limits in the unpredictably variable environment from which those perceptions are derived. PCT does not propose that there is an internal model within which the brain simulates behavior before issuing commands to execute that behavior. Instead, one of its characteristic features is the principled lack of cerebral organisation of behavior. Rather, behavior is the organism's variable means to reduce the discrepancy between perceptions and reference values which are based on various external and internal inputs. Behavior must constantly adapt and change for an organism to maintain its perceptual goals. In this way, PCT can provide an explanation of abstract learning through spontaneous reorganisation of the hierarchy. PCT proposes that conflict occurs between disparate reference values for a given perception rather than between different responses, and that learning is implemented as trial-and-error changes of the properties of control systems, rather than any specific response being reinforced. In this way, behavior remains adaptive to the environment as it unfolds, rather than relying on learned action patterns that may not fit.
+Hierarchies of perceptual control have been simulated in computer models and have been shown to provide a close match to behavioral data. For example, Marken conducted an experiment comparing the behavior of a perceptual control hierarchy computer model with that of six healthy volunteers in three experiments. The participants were required to keep the distance between a left line and a centre line equal to that of the centre line and a right line. They were also instructed to keep both distances equal to 2 cm. They had 2 paddles in their hands, one controlling the left line and one controlling the middle line. To do this, they had to resist random disturbances applied to the positions of the lines. As the participants achieved control, they managed to nullify the expected effect of the disturbances by moving their paddles. The correlation between the behavior of subjects and the model in all the experiments approached 0.99. It is proposed that the organization of models of hierarchical control systems such as this informs us about the organization of the human subjects whose behavior it so closely reproduces.
+
+== Robotics ==
+PCT has significant implications for Robotics and Artificial Intelligence. W.T. Powers introduced the application of PCT to robotics in 1978, early in the availability of home computers.
+
+ The comparatively simple architecture, a hierarchy of perceptual controllers, has no need for complex models of the external world, inverse kinematics, or computation from input-output mappings. Traditional approaches to robotics generally depend upon the computation of actions in a constrained environment. Robots designed this way are inflexible and clumsy, unable to cope with the dynamic nature of the real world. PCT robots inherently resist and counter the chaotic, unpredictable disturbances to their controlled inputs which occur in an unconstrained environment. The PCT robotics architecture has recently been applied to a number of real-world robotic systems including robotic rovers, balancing robot and robot arms. Some commercially available robots which demonstrate good control in a naturalistic environment use a control-theoretic architecture which requires much more intensive computation. For example, Boston Dynamics has said that its robots use historically leveraged model predictive control.

data/en.wikipedia.org/wiki/Perceptual_control_theory-6.md

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+---
+title: "Perceptual control theory"
+chunk: 7/7
+source: "https://en.wikipedia.org/wiki/Perceptual_control_theory"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:52.953091+00:00"
+instance: "kb-cron"
+---
+
+== Current situation and prospects ==
+The preceding explanation of PCT principles provides justification of how this theory can provide a valid explanation of neural organisation and how it can explain some of the current issues of conceptual models.
+Perceptual control theory currently proposes a hierarchy of 11 levels of perceptions controlled by systems in the human mind and neural architecture. These are: intensity, sensation, configuration, transition, event, relationship, category, sequence, program, principle, and system concept. Diverse perceptual signals at a lower level (e.g. visual perceptions of intensities) are combined in an input function to construct a single perception at the higher level (e.g. visual perception of a color sensation). The perceptions that are constructed and controlled at the lower levels are passed along as the perceptual inputs at the higher levels. The higher levels in turn control by adjusting the reference levels (goals) of the lower levels, in effect telling the lower levels what to perceive.
+While many computer demonstrations of principles have been developed, the proposed higher levels are difficult to model because too little is known about how the brain works at these levels. Isolated higher-level control processes can be investigated, but models of an extensive hierarchy of control are still only conceptual, or at best rudimentary.
+Perceptual control theory has not been widely accepted in mainstream psychology, but has been effectively used in a considerable range of domains in human factors, clinical psychology, and psychotherapy (the "Method of Levels"), it is the basis for a considerable body of research in sociology, and it has formed the conceptual foundation for the reference model used by a succession of NATO research study groups.
+Recent approaches use principles of perceptual control theory to provide new algorithmic foundations for artificial intelligence and machine learning.
+
+== Selected bibliography ==
+Cziko, Gary (1995). Without miracles: Universal selection theory and the second Darwinian revolution. Cambridge, MA: MIT Press (A Bradford Book). ISBN 0-262-53147-X
+Cziko, Gary (2000). The things we do: Using the lessons of Bernard and Darwin to understand the what, how, and why of our behavior. Cambridge, MA: MIT Press (A Bradford Book). ISBN 0-262-03277-5
+Forssell, Dag (Ed.), 2016. Perceptual Control Theory, An Overview of the Third Grand Theory in Psychology: Introductions, Readings, and Resources. Hayward, CA: Living Control Systems Publishing. ISBN 978-1938090134.
+Mansell, Warren (Ed.), (2020). The Interdisciplinary Handbook of Perceptual Control Theory: Living Control Systems IV. Cambridge: Academic Press. ISBN 978-0128189481.
+Marken, Richard S. (1992) Mind readings: Experimental studies of purpose. Benchmark Publications: New Canaan, CT.
+Marken, Richard S. (2002) More mind readings: Methods and models in the study of purpose. Chapel Hill, NC: New View. ISBN 0-944337-43-0
+Pfau, Richard H. (2017). Your Behavior: Understanding and Changing the Things You Do. St. Paul, MN: Paragon House. ISBN 9781557789273
+Plooij, F. X. (1984). The behavioral development of free-living chimpanzee babies and infants. Norwood, N.J.: Ablex.
+Plooij, F. X. (2003). "The trilogy of mind". In M. Heimann (Ed.), Regression periods in human infancy (pp. 185–205). Mahwah, NJ: Erlbaum.
+Powers, William T. (1973). Behavior: The control of perception. Chicago: Aldine de Gruyter. ISBN 0-202-25113-6. [2nd exp. ed. = Powers (2005)].
+Powers, William T. (1989). Living control systems. [Selected papers 1960–1988.] New Canaan, CT: Benchmark Publications. ISBN 0-9647121-3-X.
+Powers, William T. (1992). Living control systems II. [Selected papers 1959–1990.] New Canaan, CT: Benchmark Publications.
+Powers, William T. (1998). Making sense of behavior: The meaning of control.  New Canaan, CT: Benchmark Publications. ISBN 0-9647121-5-6.
+Powers, William T. (2005). Behavior: The control of perception. New Canaan: Benchmark Publications. ISBN 0-9647121-7-2. [2nd exp. ed. of Powers (1973). Chinese tr. (2004) Guongdong Higher Learning Education Press, Guangzhou, China. ISBN 7-5361-2996-3.]
+Powers, William T. (2008). Living Control Systems III: The fact of control. [Mathematical appendix by Dr. Richard Kennaway. Includes computer programs for the reader to demonstrate and experimentally test the theory.] New Canaan, CT: Benchmark Publications. ISBN 978-0-9647121-8-8.
+Powers, William. T., Clark, R. K., and McFarland, R. L. (1960). "A general feedback theory of human behavior [Part 1; Part 2]. Perceptual and Motor Skills 11, 71–88; 309–323.
+Powers, William T. and Runkel, Philip J. 2011. Dialogue concerning the two chief approaches to a science of life: Word pictures and correlations versus working models. Hayward, CA: Living Control Systems Publishing ISBN 0-9740155-1-2.
+Robertson, R. J. & Powers, W.T. (1990). Introduction to modern psychology: the control-theory view. Gravel Switch, KY: Control Systems Group.
+Robertson, R. J., Goldstein, D.M., Mermel, M., & Musgrave, M. (1999). Testing the self as a control system: Theoretical and methodological issues. Int. J. Human-Computer Studies, 50, 571–580.
+Runkel, Philip J[ulian]. 1990. Casting Nets and Testing Specimens: Two Grand Methods of Psychology. New York: Praeger. ISBN 0-275-93533-7. [Repr. 2007, Hayward, CA: Living Control Systems Publishing ISBN 0-9740155-7-1.]
+Runkel, Philip J[ulian]. (2003). People as living things. Hayward, CA: Living Control Systems Publishing ISBN 0-9740155-0-4
+Taylor, Martin M. (1999). "Editorial: Perceptual Control Theory and its Application," International Journal of Human-Computer Studies, Vol 50, No. 6, June 1999, pp. 433–444.
+
+=== Sociology ===
+McClelland, Kent (1994). "Perceptual Control and Social Power". Sociological Perspectives. 37 (4): 461–496. doi:10.2307/1389276. JSTOR 1389276. S2CID 144872350.
+McClelland, Kent (2004). "The Collective Control of Perceptions: Constructing Order from Conflict". International Journal of Human-Computer Studies. 60: 65–99. doi:10.1016/j.ijhcs.2003.08.003.
+McClelland, Kent and Thomas J. Fararo, eds. (2006). Purpose, Meaning, and Action: Control Systems Theories in Sociology. New York: Palgrave Macmillan.
+McPhail, Clark. 1991. The Myth of the Madding Crowd. New York: Aldine de Gruyter.
+
+== References ==
+
+== External links ==
+
+=== Articles ===
+PCT for the Beginner by William T. Powers (2007)
+The Dispute Over Control theory by William T. Powers (1993) – requires access approval 
+Demonstrations of perceptual control by Gary Cziko (2006)
+
+=== Audio ===
+Interview with William T. Powers on origin and history of PCT (Part One – 20060722 (58.7M) Archived 2016-03-04 at the Wayback Machine
+Interview with William T. Powers on origin and history of PCT (Part Two – 20070728  (57.7M) Archived 2016-03-04 at the Wayback Machine
+
+=== Videos ===
+Demonstration of a robot arm with visual servoing and pressure control based on principles of PCT
+
+=== Websites ===
+The International Association for Perceptual Control Systems – The IAPCT website.
+PCTWeb Archived 2020-01-30 at the Wayback Machine – Warren Mansell's comprehensive website on PCT.
+Living Control Systems Publishing – resources and books about PCT.
+Mind Readings – Rick Marken's website on PCT, with many interactive demonstrations.
+Method of Levels – Timothy Carey's website on the Method of Levels.
+Perceptual Robots – The PCT methodology and architecture applied to robotics.
+ResearchGate Project – Recent research products.

data/en.wikipedia.org/wiki/Prescriptive_analytics-0.md

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+---
+title: "Prescriptive analytics"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Prescriptive_analytics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:54.207294+00:00"
+instance: "kb-cron"
+---
+
+Prescriptive analytics is a form of business analytics which suggests decision options for how to take advantage of a future opportunity or mitigate a future risk and shows the implication of each decision option. It enables an enterprise to consider "the best course of action to take" in the light of information derived from descriptive and predictive analytics.
+
+== Overview ==
+Prescriptive analytics is the third and final phase of business analytics, which also includes descriptive and predictive analytics. Referred to as the "final frontier of analytic capabilities", prescriptive analytics entails the application of mathematical and computational sciences and suggests decision options for how to take advantage of the results of descriptive and predictive phases.
+The first stage of business analytics is descriptive analytics, which still accounts for the majority of all business analytics today. Descriptive analytics looks at past performance and understands that performance by mining historical data to look for the reasons behind past success or failure. Most management reporting – such as sales, marketing, operations, and finance – uses this type of post-mortem analysis.
+
+The next phase is predictive analytics. Predictive analytics answers the question of what is likely to happen. This is where historical data is combined with rules, algorithms, and occasionally external data to determine the probable future outcome of an event or the likelihood of a situation occurring.
+The final phase is prescriptive analytics, which goes beyond predicting future outcomes but also suggesting actions to benefit from the predictions and showing the implications of each decision option.
+Prescriptive analytics uses algorithms and machine learning models to simulate various scenarios and predict the likely outcomes of different decisions. It then suggests the best course of action based on the desired outcome and the constraints of the situation. Prescriptive analytics not only anticipates what will happen and when it will happen, but also why it will happen. Further, prescriptive analytics suggests decision options on how to take advantage of a future opportunity or mitigate a future risk and shows the implication of each decision option. Prescriptive analytics incorporates both structured and unstructured data, and uses a combination of advanced analytic techniques and disciplines to predict, prescribe, and adapt. It can continually take in new data to re-predict and re-prescribe, thus automatically improving prediction accuracy and prescribing better decision options. Effective prescriptive analytics utilises hybrid data, a combination of structured (numbers, categories) and unstructured data (videos, images, sounds, texts), and business rules to predict what lies ahead and to prescribe how to take advantage of this predicted future without compromising other priorities. Basu suggests that without hybrid data input, the benefits of prescriptive analytics are limited.
+In addition to this variety of data types and growing data volume, incoming data can also evolve with respect to velocity, that is, more data being generated at a faster or a variable pace. Business rules define the business process and include objectives constraints, preferences, policies, best practices, and boundaries. Mathematical models and computational models are techniques derived from mathematical sciences, computer science and related disciplines such as applied statistics, machine learning, operations research, natural language processing, computer vision, pattern recognition, image processing, speech recognition, and signal processing. The correct application of all these methods and the verification of their results implies the need for resources on a massive scale including human, computational and temporal for every Prescriptive Analytic project. In order to spare the expense of dozens of people, high performance machines and weeks of work one must consider the reduction of resources and therefore a reduction in the accuracy or reliability of the outcome. The preferable route is a reduction that produces a probabilistic result within acceptable limits.
+All three phases of analytics can be performed through professional services or technology or a combination. In order to scale, prescriptive analytics technologies need to be adaptive to take into account the growing volume, velocity, and variety of data that most mission critical processes and their environments may produce.
+
+One criticism of prescriptive analytics is that its distinction from predictive analytics is ill-defined and therefore ill-conceived. 
+
+== History ==
+While the term prescriptive analytics was first coined by IBM, and was later trademarked by Texas-based company Ayata, the underlying concepts have been around for hundreds of years. The technology behind prescriptive analytics synergistically combines hybrid data, business rules with mathematical models and computational models. The data inputs to prescriptive analytics may come from multiple sources: internal, such as inside a corporation; and external, also known as environmental data. The data may be structured, which includes numbers and categories, as well as unstructured data, such as texts, images, sounds, and videos. Unstructured data differs from structured data in that its format varies widely and cannot be stored in traditional relational databases without significant effort at data transformation. More than 80% of the world's data today is unstructured, according to IBM.
+Ayata's trade mark was cancelled in 2018.
+
+== Applications in Oil and Gas ==
+Energy is the largest industry in the world ($6 trillion in size). The processes and decisions related to oil and natural gas exploration, development and production generate large amounts of data. Many types of captured data are used to create models and images of the Earth’s structure and layers 5,000 - 35,000 feet below the surface and to describe activities around the wells themselves, such as depositional characteristics, machinery performance, oil flow rates, reservoir temperatures and pressures. Prescriptive analytics software can help with both locating and producing hydrocarbons by taking in seismic data, well log data, production data, and other related data sets to prescribe specific recipes for how and where to drill, complete, and produce wells in order to optimize recovery, minimize cost, and reduce environmental footprint.

data/en.wikipedia.org/wiki/Prescriptive_analytics-1.md

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+---
+title: "Prescriptive analytics"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/Prescriptive_analytics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:54.207294+00:00"
+instance: "kb-cron"
+---
+
+=== Unconventional Resource Development ===
+With the value of the end product determined by global commodity economics, the basis of competition for operators in upstream E&P is the ability to effectively deploy capital to locate and extract resources more efficiently, effectively, predictably, and safely than their peers. In unconventional resource plays, operational efficiency and effectiveness is diminished by reservoir inconsistencies, and decision-making impaired by high degrees of uncertainty. These challenges manifest themselves in the form of low recovery factors and wide performance variations.
+Prescriptive Analytics software can accurately predict production and prescribe optimal configurations of controllable drilling, completion, and production variables by modeling numerous internal and external variables simultaneously, regardless of source, structure, size, or format. Prescriptive analytics software can also provide decision options and show the impact of each decision option so the operations managers can proactively take appropriate actions, on time, to guarantee future exploration and production performance, and maximize the economic value of assets at every point over the course of their serviceable lifetimes.
+
+=== Oilfield Equipment Maintenance ===
+In the realm of oilfield equipment maintenance, Prescriptive Analytics can optimize configuration, anticipate and prevent unplanned downtime, optimize field scheduling, and improve maintenance planning. According to General Electric, there are more than 130,000 electric submersible pumps (ESP's) installed globally, accounting for 60% of the world's oil production.  Prescriptive Analytics has been deployed to predict when and why an ESP will fail, and recommend the necessary actions to prevent the failure.
+In the area of health, safety and environment, prescriptive analytics can predict and preempt incidents that can lead to reputational and financial loss for oil and gas companies.
+
+=== Pricing ===
+Pricing is another area of focus. Natural gas prices fluctuate dramatically depending upon supply, demand, econometrics, geopolitics, and weather conditions. Gas producers, pipeline transmission companies and utility companies have a keen interest in more accurately predicting gas prices so that they can lock in favorable terms while hedging downside risk. Prescriptive analytics software can accurately predict prices by modeling internal and external variables simultaneously and also provide decision options and show the impact of each decision option.
+
+== Applications in maritime industry ==
+Common Structural Rules for  Bulk Carriers and Oil Tankers ( managed by IACS organisation )  intensively utilizes the term "prescriptive requirements"  as one of two main classes of checkable calculations by dedicated numerical tools and algorithms  for verifying safety of ship hull construction.
+
+== Applications in healthcare ==
+Multiple factors are driving healthcare providers to dramatically improve business processes and operations as the United States healthcare industry embarks on the necessary migration from a largely fee-for service, volume-based system to a fee-for-performance, value-based system. Prescriptive analytics is playing a key role to help improve the performance in a number of areas involving various stakeholders: payers, providers and pharmaceutical companies.
+Prescriptive analytics can help providers improve effectiveness of their clinical care delivery to the population they manage and in the process achieve better patient satisfaction and retention. Providers can do better population health management by identifying appropriate intervention models for risk stratified population combining data from the in-facility care episodes and home based telehealth.
+Prescriptive analytics can also benefit healthcare providers in their capacity planning by using analytics to leverage operational and usage data combined with data of external factors such as economic data, population demographic trends and population health trends, to more accurately plan for future capital investments such as new facilities and equipment utilization as well as understand the trade-offs between adding additional beds and expanding an existing facility versus building a new one.
+Prescriptive analytics can help pharmaceutical companies to expedite their drug development by identifying patient cohorts that are most suitable for the clinical trials worldwide - patients who are expected to be compliant and will not drop out of the trial due to complications. Analytics can tell companies how much time and money they can save if they choose one patient cohort in a specific country vs. another.
+In provider-payer negotiations, providers can improve their negotiating position with health insurers by developing a robust understanding of future service utilization. By accurately predicting utilization, providers can also better allocate personnel.
+
+== See also ==
+
+== Notes ==
+
+== References ==
+
+== Further reading ==
+
+== External links ==
+INFORMS' bi-monthly, digital magazine on the analytics profession
+Menon, Jai "Why Data Matters: Moving Beyond Prediction" IBM
+Global Openlabs for Performance-Enhancement Analytics and Knowledge System (GoPeaks)

data/en.wikipedia.org/wiki/Queueing_theory-0.md

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+---
+title: "Queueing theory"
+chunk: 1/4
+source: "https://en.wikipedia.org/wiki/Queueing_theory"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:55.531499+00:00"
+instance: "kb-cron"
+---
+
+Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.
+Queueing theory has its origins in research by Agner Krarup Erlang, who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company. These ideas were seminal to the field of teletraffic engineering and have since seen applications in telecommunications, traffic engineering, computing, project management, and particularly industrial engineering, where they are applied in the design of factories, shops, offices, and hospitals.
+
+== Description ==
+Queueing theory is one of the major areas of study in the discipline of management science. Through management science, businesses are able to solve a variety of problems using different scientific and mathematical approaches. Queueing analysis is the probabilistic analysis of waiting lines, and thus the results, also referred to as the operating characteristics, are probabilistic rather than deterministic. The probability that n customers are in the queueing system, the average number of customers in the queueing system, the average number of customers in the waiting line, the average time spent by a customer in the total queuing system, the average time spent by a customer in the waiting line, and finally the probability that the server is busy or idle are all of the different operating characteristics that these queueing models compute. The overall goal of queueing analysis is to compute these characteristics for the current system and then test several alternatives that could lead to improvement. Computing the operating characteristics for the current system and comparing the values to the characteristics of the alternative systems allows managers to see the pros and cons of each potential option. These systems help in the final decision making process by showing ways to increase savings, reduce waiting time, improve efficiency, etc. The main queueing models that can be used are the single-server waiting line system and the multiple-server waiting line system, which are discussed further below. These models can be further differentiated depending on whether service times are constant or undefined, the queue length is finite, the calling population is finite, etc.
+
+== Single queueing nodes ==
+A queue or queueing node can be thought of as nearly a black box. Jobs (also called customers or requests, depending on the field) arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue.
+
+However, the queueing node is not quite a pure black box since some information is needed about the inside of the queueing node. The queue has one or more servers which can each be paired with an arriving job. When the job is completed and departs, that server will again be free to be paired with another arriving job.
+
+An analogy often used is that of the cashier at a supermarket. Customers arrive, are processed by the cashier, and depart. Each cashier processes one customer at a time, and hence this is a queueing node with only one server. A setting where a customer will leave immediately if the cashier is busy when the customer arrives, is referred to as a queue with no buffer (or no waiting area). A setting with a waiting zone for up to n customers is called a queue with a buffer of size n.
+
+=== Birth-death process ===
+
+The behaviour of a single queue (also called a queueing node) can be described by a birth–death process, which describes the arrivals and departures from the queue, along with the number of jobs currently in the system. If k denotes the number of jobs in the system (either being serviced or waiting if the queue has a buffer of waiting jobs), then an arrival increases k by 1 and a departure decreases k by 1.
+The system transitions between values of k by births and deaths, which occur at the arrival rates 
+  
+    
+      
+        
+          λ
+          
+            i
+          
+        
+      
+    
+    {\displaystyle \lambda _{i}}
+  
+ and the departure rates 
+  
+    
+      
+        
+          μ
+          
+            i
+          
+        
+      
+    
+    {\displaystyle \mu _{i}}
+  
+ for each job 
+  
+    
+      
+        i
+      
+    
+    {\displaystyle i}
+  
+. For a queue, these rates are generally considered not to vary with the number of jobs in the queue, so a single average rate of arrivals/departures per unit time is assumed. Under this assumption, this process has an arrival rate of 
+  
+    
+      
+        λ
+        =
+        
+          avg
+        
+        (
+        
+          λ
+          
+            1
+          
+        
+        ,
+        
+          λ
+          
+            2
+          
+        
+        ,
+        …
+        ,
+        
+          λ
+          
+            k
+          
+        
+        )
+      
+    
+    {\displaystyle \lambda ={\text{avg}}(\lambda _{1},\lambda _{2},\dots ,\lambda _{k})}
+  
+ and a departure rate of 
+  
+    
+      
+        μ
+        =
+        
+          avg
+        
+        (
+        
+          μ
+          
+            1
+          
+        
+        ,
+        
+          μ
+          
+            2
+          
+        
+        ,
+        …
+        ,
+        
+          μ
+          
+            k
+          
+        
+        )
+      
+    
+    {\displaystyle \mu ={\text{avg}}(\mu _{1},\mu _{2},\dots ,\mu _{k})}
+  
+.
+
+==== Balance equations ====
+The steady state equations for the birth-and-death process, known as the balance equations, are as follows. Here 
+  
+    
+      
+        
+          P
+          
+            n
+          
+        
+      
+    
+    {\displaystyle P_{n}}
+  
+ denotes the steady state probability to be in state n.
+
+  
+    
+      
+        
+          μ
+          
+            1
+          
+        
+        
+          P
+          
+            1
+          
+        
+        =
+        
+          λ
+          
+            0
+          
+        
+        
+          P
+          
+            0
+          
+        
+      
+    
+    {\displaystyle \mu _{1}P_{1}=\lambda _{0}P_{0}}
+  
+
+  
+    
+      
+        
+          λ
+          
+            0
+          
+        
+        
+          P
+          
+            0
+          
+        
+        +
+        
+          μ
+          
+            2
+          
+        
+        
+          P
+          
+            2
+          
+        
+        =
+        (
+        
+          λ
+          
+            1
+          
+        
+        +
+        
+          μ
+          
+            1
+          
+        
+        )
+        
+          P
+          
+            1
+          
+        
+      
+    
+    {\displaystyle \lambda _{0}P_{0}+\mu _{2}P_{2}=(\lambda _{1}+\mu _{1})P_{1}}
+  
+
+  
+    
+      
+        
+          λ
+          
+            n
+            −
+            1
+          
+        
+        
+          P
+          
+            n
+            −
+            1
+          
+        
+        +
+        
+          μ
+          
+            n
+            +
+            1
+          
+        
+        
+          P
+          
+            n
+            +
+            1
+          
+        
+        =
+        (
+        
+          λ
+          
+            n
+          
+        
+        +
+        
+          μ
+          
+            n
+          
+        
+        )
+        
+          P
+          
+            n
+          
+        
+      
+    
+    {\displaystyle \lambda _{n-1}P_{n-1}+\mu _{n+1}P_{n+1}=(\lambda _{n}+\mu _{n})P_{n}}
+  
+
+The first two equations imply
+
+  
+    
+      
+        
+          P
+          
+            1
+          
+        
+        =
+        
+          
+            
+              λ
+              
+                0
+              
+            
+            
+              μ
+              
+                1
+              
+            
+          
+        
+        
+          P
+          
+            0
+          
+        
+      
+    
+    {\displaystyle P_{1}={\frac {\lambda _{0}}{\mu _{1}}}P_{0}}
+  
+
+and
+
+  
+    
+      
+        
+          P
+          
+            2
+          
+        
+        =
+        
+          
+            
+              λ
+              
+                1
+              
+            
+            
+              μ
+              
+                2
+              
+            
+          
+        
+        
+          P
+          
+            1
+          
+        
+        +
+        
+          
+            1
+            
+              μ
+              
+                2
+              
+            
+          
+        
+        (
+        
+          μ
+          
+            1
+          
+        
+        
+          P
+          
+            1
+          
+        
+        −
+        
+          λ
+          
+            0
+          
+        
+        
+          P
+          
+            0
+          
+        
+        )
+        =
+        
+          
+            
+              λ
+              
+                1
+              
+            
+            
+              μ
+              
+                2
+              
+            
+          
+        
+        
+          P
+          
+            1
+          
+        
+        =
+        
+          
+            
+              
+                λ
+                
+                  1
+                
+              
+              
+                λ
+                
+                  0
+                
+              
+            
+            
+              
+                μ
+                
+                  2
+                
+              
+              
+                μ
+                
+                  1
+                
+              
+            
+          
+        
+        
+          P
+          
+            0
+          
+        
+      
+    
+    {\displaystyle P_{2}={\frac {\lambda _{1}}{\mu _{2}}}P_{1}+{\frac {1}{\mu _{2}}}(\mu _{1}P_{1}-\lambda _{0}P_{0})={\frac {\lambda _{1}}{\mu _{2}}}P_{1}={\frac {\lambda _{1}\lambda _{0}}{\mu _{2}\mu _{1}}}P_{0}}
+  
+.
+By mathematical induction,

data/en.wikipedia.org/wiki/Queueing_theory-1.md

@@ -0,0 +1,751 @@
+---
+title: "Queueing theory"
+chunk: 2/4
+source: "https://en.wikipedia.org/wiki/Queueing_theory"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:55.531499+00:00"
+instance: "kb-cron"
+---
+
+  
+    
+      
+        
+          P
+          
+            n
+          
+        
+        =
+        
+          
+            
+              
+                λ
+                
+                  n
+                  −
+                  1
+                
+              
+              
+                λ
+                
+                  n
+                  −
+                  2
+                
+              
+              ⋯
+              
+                λ
+                
+                  0
+                
+              
+            
+            
+              
+                μ
+                
+                  n
+                
+              
+              
+                μ
+                
+                  n
+                  −
+                  1
+                
+              
+              ⋯
+              
+                μ
+                
+                  1
+                
+              
+            
+          
+        
+        
+          P
+          
+            0
+          
+        
+        =
+        
+          P
+          
+            0
+          
+        
+        
+          ∏
+          
+            i
+            =
+            0
+          
+          
+            n
+            −
+            1
+          
+        
+        
+          
+            
+              λ
+              
+                i
+              
+            
+            
+              μ
+              
+                i
+                +
+                1
+              
+            
+          
+        
+      
+    
+    {\displaystyle P_{n}={\frac {\lambda _{n-1}\lambda _{n-2}\cdots \lambda _{0}}{\mu _{n}\mu _{n-1}\cdots \mu _{1}}}P_{0}=P_{0}\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}}
+  
+.
+The condition 
+  
+    
+      
+        
+          ∑
+          
+            n
+            =
+            0
+          
+          
+            ∞
+          
+        
+        
+          P
+          
+            n
+          
+        
+        =
+        
+          P
+          
+            0
+          
+        
+        +
+        
+          P
+          
+            0
+          
+        
+        
+          ∑
+          
+            n
+            =
+            1
+          
+          
+            ∞
+          
+        
+        
+          ∏
+          
+            i
+            =
+            0
+          
+          
+            n
+            −
+            1
+          
+        
+        
+          
+            
+              λ
+              
+                i
+              
+            
+            
+              μ
+              
+                i
+                +
+                1
+              
+            
+          
+        
+        =
+        1
+      
+    
+    {\displaystyle \sum _{n=0}^{\infty }P_{n}=P_{0}+P_{0}\sum _{n=1}^{\infty }\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}=1}
+  
+ leads to
+
+  
+    
+      
+        
+          P
+          
+            0
+          
+        
+        =
+        
+          
+            1
+            
+              1
+              +
+              
+                ∑
+                
+                  n
+                  =
+                  1
+                
+                
+                  ∞
+                
+              
+              
+                ∏
+                
+                  i
+                  =
+                  0
+                
+                
+                  n
+                  −
+                  1
+                
+              
+              
+                
+                  
+                    λ
+                    
+                      i
+                    
+                  
+                  
+                    μ
+                    
+                      i
+                      +
+                      1
+                    
+                  
+                
+              
+            
+          
+        
+      
+    
+    {\displaystyle P_{0}={\frac {1}{1+\sum _{n=1}^{\infty }\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}}}}
+  
+
+which, together with the equation for 
+  
+    
+      
+        
+          P
+          
+            n
+          
+        
+      
+    
+    {\displaystyle P_{n}}
+  
+ 
+  
+    
+      
+        (
+        n
+        ≥
+        1
+        )
+      
+    
+    {\displaystyle (n\geq 1)}
+  
+, fully describes the required steady state probabilities.
+
+=== Kendall's notation ===
+
+Single queueing nodes are usually described using Kendall's notation in the form A/S/c where A describes the distribution of durations between each arrival to the queue, S the distribution of service times for jobs, and c the number of servers at the node. For an example of the notation, the M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process (where inter-arrival durations are exponentially distributed) and have exponentially distributed service times (the M denotes a Markov process). In an M/G/1 queue, the G stands for general and indicates an arbitrary probability distribution for service times.
+
+=== Example analysis of an M/M/1 queue ===
+Consider a queue with one server and the following characteristics:
+
+  
+    
+      
+        λ
+      
+    
+    {\displaystyle \lambda }
+  
+: the arrival rate (the reciprocal of the expected time between each customer arriving, e.g. 10 customers per second)
+
+  
+    
+      
+        μ
+      
+    
+    {\displaystyle \mu }
+  
+: the reciprocal of the mean service time (the expected number of consecutive service completions per the same unit time, e.g. per 30 seconds)
+n: the parameter characterizing the number of customers in the system
+
+  
+    
+      
+        
+          P
+          
+            n
+          
+        
+      
+    
+    {\displaystyle P_{n}}
+  
+: the probability of there being n customers in the system in steady state
+Further, let 
+  
+    
+      
+        
+          E
+          
+            n
+          
+        
+      
+    
+    {\displaystyle E_{n}}
+  
+ represent the number of times the system enters state n, and 
+  
+    
+      
+        
+          L
+          
+            n
+          
+        
+      
+    
+    {\displaystyle L_{n}}
+  
+ represent the number of times the system leaves state n. Then 
+  
+    
+      
+        
+          |
+          
+            
+              E
+              
+                n
+              
+            
+            −
+            
+              L
+              
+                n
+              
+            
+          
+          |
+        
+        ∈
+        {
+        0
+        ,
+        1
+        }
+      
+    
+    {\displaystyle \left\vert E_{n}-L_{n}\right\vert \in \{0,1\}}
+  
+ for all n. That is, the number of times the system leaves a state differs by at most 1 from the number of times it enters that state, since it will either return into that state at some time in the future (
+  
+    
+      
+        
+          E
+          
+            n
+          
+        
+        =
+        
+          L
+          
+            n
+          
+        
+      
+    
+    {\displaystyle E_{n}=L_{n}}
+  
+) or not (
+  
+    
+      
+        
+          |
+          
+            
+              E
+              
+                n
+              
+            
+            −
+            
+              L
+              
+                n
+              
+            
+          
+          |
+        
+        =
+        1
+      
+    
+    {\displaystyle \left\vert E_{n}-L_{n}\right\vert =1}
+  
+).
+When the system arrives at a steady state, the arrival rate should be equal to the departure rate.
+Thus the balance equations
+
+  
+    
+      
+        μ
+        
+          P
+          
+            1
+          
+        
+        =
+        λ
+        
+          P
+          
+            0
+          
+        
+      
+    
+    {\displaystyle \mu P_{1}=\lambda P_{0}}
+  
+
+  
+    
+      
+        λ
+        
+          P
+          
+            0
+          
+        
+        +
+        μ
+        
+          P
+          
+            2
+          
+        
+        =
+        (
+        λ
+        +
+        μ
+        )
+        
+          P
+          
+            1
+          
+        
+      
+    
+    {\displaystyle \lambda P_{0}+\mu P_{2}=(\lambda +\mu )P_{1}}
+  
+
+  
+    
+      
+        λ
+        
+          P
+          
+            n
+            −
+            1
+          
+        
+        +
+        μ
+        
+          P
+          
+            n
+            +
+            1
+          
+        
+        =
+        (
+        λ
+        +
+        μ
+        )
+        
+          P
+          
+            n
+          
+        
+      
+    
+    {\displaystyle \lambda P_{n-1}+\mu P_{n+1}=(\lambda +\mu )P_{n}}
+  
+
+imply
+
+  
+    
+      
+        
+          P
+          
+            n
+          
+        
+        =
+        
+          
+            λ
+            μ
+          
+        
+        
+          P
+          
+            n
+            −
+            1
+          
+        
+        ,
+         
+        n
+        =
+        1
+        ,
+        2
+        ,
+        …
+      
+    
+    {\displaystyle P_{n}={\frac {\lambda }{\mu }}P_{n-1},\ n=1,2,\ldots }
+  
+
+The fact that 
+  
+    
+      
+        
+          P
+          
+            0
+          
+        
+        +
+        
+          P
+          
+            1
+          
+        
+        +
+        ⋯
+        =
+        1
+      
+    
+    {\displaystyle P_{0}+P_{1}+\cdots =1}
+  
+ leads to the geometric distribution formula
+
+  
+    
+      
+        
+          P
+          
+            n
+          
+        
+        =
+        (
+        1
+        −
+        ρ
+        )
+        
+          ρ
+          
+            n
+          
+        
+      
+    
+    {\displaystyle P_{n}=(1-\rho )\rho ^{n}}
+  
+
+where 
+  
+    
+      
+        ρ
+        =
+        
+          
+            λ
+            μ
+          
+        
+        <
+        1
+      
+    
+    {\displaystyle \rho ={\frac {\lambda }{\mu }}<1}
+  
+.
+
+=== Simple two-equation queue ===
+A common basic queueing system is attributed to Erlang and is a modification of Little's Law. Given an arrival rate λ, a dropout rate σ, and a departure rate μ, length of the queue L is defined as:
+
+  
+    
+      
+        L
+        =
+        
+          
+            
+              λ
+              −
+              σ
+            
+            μ
+          
+        
+      
+    
+    {\displaystyle L={\frac {\lambda -\sigma }{\mu }}}
+  
+.
+Assuming an exponential distribution for the rates, the waiting time W can be defined as the proportion of arrivals that are served. This is equal to the exponential survival rate of those who do not drop out over the waiting period, giving:
+
+  
+    
+      
+        
+          
+            μ
+            λ
+          
+        
+        =
+        
+          e
+          
+            −
+            W
+            
+              μ
+            
+          
+        
+      
+    
+    {\displaystyle {\frac {\mu }{\lambda }}=e^{-W{\mu }}}
+  
+
+The second equation is commonly rewritten as:
+
+  
+    
+      
+        W
+        =
+        
+          
+            1
+            μ
+          
+        
+        
+          l
+          n
+        
+        
+          
+            λ
+            μ
+          
+        
+      
+    
+    {\displaystyle W={\frac {1}{\mu }}\mathrm {ln} {\frac {\lambda }{\mu }}}
+  
+
+The two-stage one-box model is common in epidemiology.
+
+== History ==
+
+In 1909, Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory. He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920. In Kendall's notation:

data/en.wikipedia.org/wiki/Queueing_theory-2.md

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+---
+title: "Queueing theory"
+chunk: 3/4
+source: "https://en.wikipedia.org/wiki/Queueing_theory"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:55.531499+00:00"
+instance: "kb-cron"
+---
+
+M stands for Markov or memoryless, and means arrivals occur according to a Poisson process
+D stands for deterministic, and means jobs arriving at the queue require a fixed amount of service
+k describes the number of servers at the queueing node (k = 1, 2, 3, ...)
+If the node has more jobs than servers, then jobs will queue and wait for service.
+The M/G/1 queue was solved by Felix Pollaczek in 1930, a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as the Pollaczek–Khinchine formula.
+After the 1940s, queueing theory became an area of research interest to mathematicians. In 1953, David George Kendall solved the GI/M/k queue and introduced the modern notation for queues, now known as Kendall's notation. In 1957, Pollaczek studied the GI/G/1 using an integral equation. John Kingman gave a formula for the mean waiting time in a G/G/1 queue, now known as Kingman's formula.
+Leonard Kleinrock worked on the application of queueing theory to message switching in the early 1960s and packet switching in the early 1970s. His initial contribution to this field was his doctoral thesis at the Massachusetts Institute of Technology in 1962, published in book form in 1964. His theoretical work published in the early 1970s underpinned the use of packet switching in the ARPANET, a forerunner to the Internet.
+The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival and service time distributions to be considered.
+Systems with coupled orbits are an important part in queueing theory in the application to wireless networks and signal processing.
+Modern day application of queueing theory concerns among other things product development where (material) products have a spatiotemporal existence, in the sense that products have a certain volume and a certain duration.
+Problems such as performance metrics for the M/G/k queue remain an open problem.
+
+== Service disciplines ==
+
+Various scheduling policies can be used at queueing nodes:
+
+First in, first out
+ Also called first-come, first-served (FCFS), this principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
+Last in, first out
+This principle also serves customers one at a time, but the customer with the shortest waiting time will be served first. Also known as a stack.
+Processor sharing
+Service capacity is shared equally between customers.
+Priority
+Customers with high priority are served first. Priority queues can be of two types: non-preemptive (where a job in service cannot be interrupted) and preemptive (where a job in service can be interrupted by a higher-priority job). No work is lost in either model.
+Shortest job first
+The next job to be served is the one with the smallest size.
+Preemptive shortest job first
+The next job to be served is the one with the smallest original size.
+Shortest remaining processing time
+The next job to serve is the one with the smallest remaining processing requirement.
+Service facility
+Single server: customers line up and there is only one server
+Several parallel servers (single queue): customers line up and there are several servers
+Several parallel servers (several queues): there are many counters and customers can decide for which to queue
+Unreliable server
+Server failures occur according to a stochastic (random) process (usually Poisson) and are followed by setup periods during which the server is unavailable. The interrupted customer remains in the service area until server is fixed.
+
+Customer waiting behavior
+Balking: customers decide not to join the queue if it is too long
+Jockeying: customers switch between queues if they think they will get served faster by doing so
+Reneging: customers leave the queue if they have waited too long for service
+Arriving customers not served (either due to the queue having no buffer, or due to balking or reneging by the customer) are also known as dropouts. The average rate of dropouts is a significant parameter describing a queue.
+
+== Queueing networks ==
+Queue networks are systems in which multiple queues are connected by customer routing. When a customer is serviced at one node, it can join another node and queue for service, or leave the network.
+For networks of m nodes, the state of the system can be described by an m–dimensional vector (x1, x2, ..., xm) where xi represents the number of customers at each node.
+The simplest non-trivial networks of queues are called tandem queues. The first significant results in this area were Jackson networks, for which an efficient product-form stationary distribution exists and the mean value analysis (which allows average metrics such as throughput and sojourn times) can be computed. If the total number of customers in the network remains constant, the network is called a closed network and has been shown to also have a product–form stationary distribution by the Gordon–Newell theorem. This result was extended to the BCMP network, where a network with very general service time, regimes, and customer routing is shown to also exhibit a product–form stationary distribution. The normalizing constant can be calculated with the Buzen's algorithm, proposed in 1973.
+Networks of customers have also been investigated, such as Kelly networks, where customers of different classes experience different priority levels at different service nodes. Another type of network are G-networks, first proposed by Erol Gelenbe in 1993: these networks do not assume exponential time distributions like the classic Jackson network.
+
+=== Routing algorithms ===
+
+In discrete-time networks where there is a constraint on which service nodes can be active at any time, the max-weight scheduling algorithm chooses a service policy to give optimal throughput in the case that each job visits only a single-person service node. In the more general case where jobs can visit more than one node, backpressure routing gives optimal throughput. A network scheduler must choose a queueing algorithm, which affects the characteristics of the larger network.

data/en.wikipedia.org/wiki/Queueing_theory-3.md

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+---
+title: "Queueing theory"
+chunk: 4/4
+source: "https://en.wikipedia.org/wiki/Queueing_theory"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:55.531499+00:00"
+instance: "kb-cron"
+---
+
+=== Mean-field limits ===
+Mean-field models consider the limiting behaviour of the empirical measure (proportion of queues in different states) as the number of queues m approaches infinity. The impact of other queues on any given queue in the network is approximated by a differential equation. The deterministic model converges to the same stationary distribution as the original model.
+
+=== Heavy traffic/diffusion approximations ===
+
+In a system with high occupancy rates (utilisation near 1), a heavy traffic approximation can be used to approximate the queueing length process by a reflected Brownian motion, Ornstein–Uhlenbeck process, or more general diffusion process. The number of dimensions of the Brownian process is equal to the number of queueing nodes, with the diffusion restricted to the non-negative orthant.
+
+=== Fluid limits ===
+
+Fluid models are continuous deterministic analogs of queueing networks obtained by taking the limit when the process is scaled in time and space, allowing heterogeneous objects. This scaled trajectory converges to a deterministic equation which allows the stability of the system to be proven. It is known that a queueing network can be stable but have an unstable fluid limit.
+
+=== Queueing applications ===
+Queueing theory finds widespread application in computer science and information technology. In networking, for instance, queues are integral to routers and switches, where packets queue up for transmission. By applying queueing theory principles, designers can optimize these systems, ensuring responsive performance and efficient resource utilization.
+Beyond the technological realm, queueing theory is relevant to everyday experiences. Whether waiting in line at a supermarket or for public transportation, understanding the principles of queueing theory provides valuable insights into optimizing these systems for enhanced user satisfaction. At some point, everyone will be involved in an aspect of queuing. What some may view to be an inconvenience could possibly be the most effective method.
+Queueing theory, a discipline rooted in applied mathematics and computer science, is a field dedicated to the study and analysis of queues, or waiting lines, and their implications across a diverse range of applications. This theoretical framework has proven instrumental in understanding and optimizing the efficiency of systems characterized by the presence of queues. The study of queues is essential in contexts such as traffic systems, computer networks, telecommunications, and service operations.
+Queueing theory delves into various foundational concepts, with the arrival process and service process being central. The arrival process describes the manner in which entities join the queue over time, often modeled using stochastic processes like Poisson processes. The efficiency of queueing systems is gauged through key performance metrics. These include the average queue length, average wait time, and system throughput. These metrics provide insights into the system's functionality, guiding decisions aimed at enhancing performance and reducing wait times.
+
+== See also ==
+
+== References ==
+
+== Further reading ==
+Gross, Donald; Carl M. Harris (1998). Fundamentals of Queueing Theory. Wiley. ISBN 978-0-471-32812-4. Online
+Zukerman, Moshe (2013). Introduction to Queueing Theory and Stochastic Teletraffic Models (PDF). arXiv:1307.2968.
+Deitel, Harvey M. (1984) [1982]. An introduction to operating systems (revisited first ed.). Addison-Wesley. p. 673. ISBN 978-0-201-14502-1. chap.15, pp. 380–412
+Gelenbe, Erol; Isi Mitrani (2010). Analysis and Synthesis of Computer Systems. World Scientific 2nd Edition. ISBN 978-1-908978-42-4.
+Newell, Gordron F. (1 June 1971). Applications of Queueing Theory. Chapman and Hall.
+Leonard Kleinrock, Information Flow in Large Communication Nets, (MIT, Cambridge, May 31, 1961) Proposal for a Ph.D. Thesis
+Leonard Kleinrock. Information Flow in Large Communication Nets (RLE Quarterly Progress Report, July 1961)
+Leonard Kleinrock. Communication Nets: Stochastic Message Flow and Delay (McGraw-Hill, New York, 1964)
+Kleinrock, Leonard (2 January 1975). Queueing Systems: Volume I – Theory. New York: Wiley Interscience. pp. 417. ISBN 978-0-471-49110-1.
+Kleinrock, Leonard (22 April 1976). Queueing Systems: Volume II – Computer Applications. New York: Wiley Interscience. pp. 576. ISBN 978-0-471-49111-8.
+Lazowska, Edward D.; John Zahorjan; G. Scott Graham; Kenneth C. Sevcik (1984). Quantitative System Performance: Computer System Analysis Using Queueing Network Models. Prentice-Hall, Inc. ISBN 978-0-13-746975-8.
+Jon Kleinberg; Éva Tardos (30 June 2013). Algorithm Design. Pearson. ISBN 978-1-292-02394-6.
+
+== External links ==
+
+Teknomo's Queueing theory tutorial and calculators
+Virtamo's Queueing Theory Course
+Myron Hlynka's Queueing Theory Page
+LINE: a general-purpose engine to solve queueing models

data/en.wikipedia.org/wiki/Risk_analysis_(business)-0.md

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+---
+title: "Risk analysis (business)"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Risk_analysis_(business)"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:56.759751+00:00"
+instance: "kb-cron"
+---
+
+Risk analysis is the process of identifying and assessing risks that may jeopardize an organization's success. It typically fits into a larger risk management framework.
+Diligent risk analysis helps construct preventive measures to reduce the probability of incidents from occurring, as well as counter-measures to address incidents as they develop to minimize negative impacts on the organization.
+A popular method to perform risk analysis on IT systems is called facilitated risk analysis process (FRAP).
+
+
+== Facilitated risk analysis process ==
+FRAP analyzes one system, application or segment of business processes at a time.
+FRAP assumes that additional efforts to develop precisely quantified risks are not cost-effective because:
+
+such estimates are time-consuming
+risk documentation becomes too voluminous for practical use
+specific loss estimates are generally not needed to determine if controls are needed.
+without assumptions, there is little risk analysis
+After identifying and categorizing risks, a team identifies the controls that could mitigate the risk. The decision for what controls are needed lies with the business manager. The team's conclusions as to what risks exist and what controls needed are documented, along with a related action plan for control implementation.
+Three of the most important risks a software company faces are: unexpected changes in revenue, unexpected changes in costs from those budgeted and the amount of specialization of the software planned. Risks that affect revenues can be: unanticipated competition, privacy, intellectual property right problems, and unit sales that are less than forecast. Unexpected development costs also create the risk that can be in the form of more rework than anticipated, security holes, and privacy invasions.
+Narrow specialization of software with a large amount of research and development expenditures can lead to both business and technological risks since specialization does not necessarily lead to lower unit costs of software. Combined with the decrease in the potential customer base, specialization risk can be significant for a software firm. After probabilities of scenarios have been calculated with risk analysis, the process of risk management can be applied to help manage the risk.
+Methods like applied information economics add to and improve on risk analysis methods by introducing procedures to adjust subjective probabilities, compute the value of additional information and to use the results in part of a larger portfolio management problem.
+
+
+== See also ==
+
+Benefit risk
+Optimism bias
+Reference class forecasting
+Extreme risk
+Risk management
+Peren–Clement index
+
+
+== References ==
+
+
+=== Further reading ===
+Doug Hubbard (1998). "Hurdling Risk". CIO Magazine.
+Hiram, E. C., Peren–Clement Index, 2012.
+Roebuck, K.: Risk Management Standards, 2011.
+Wankel, C.: Encyclopedia of Business in Today's World, 2009.
+
+
+== External links ==
+NIST SP 800-30 - Risk Management Guide for Information Technology Systems

data/en.wikipedia.org/wiki/Statistics-0.md

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+---
+title: "Statistics"
+chunk: 1/8
+source: "https://en.wikipedia.org/wiki/Statistics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:59.364659+00:00"
+instance: "kb-cron"
+---
+
+Statistics (from German: Statistik, orig. "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. Statistics is deeply related to subjects like physics, chemistry, geography, geopolitics, and especially mathematics.
+When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation.
+Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences made using mathematical statistics employ the framework of probability theory, which deals with the analysis of random phenomena.
+A standard statistical procedure involves the collection of data leading to a test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is rejected when it is in fact true, giving a "false positive") and Type II errors (null hypothesis fails to be rejected when it is in fact false, giving a "false negative"). Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.
+Statistical measurement processes are also prone to error with regard to the data they generate. Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of missing data or censoring may result in biased estimates, and specific techniques have been developed to address these problems.
+
+== Introduction ==
+
+"Statistics is both the science of uncertainty and the technology of extracting information from data." - featured in the International Encyclopedia of Statistical Science.Statistics is the discipline that deals with data, facts and figures with which meaningful information is inferred. Data may represent a numerical value, in form of quantitative data, or a label, as with qualitative data. Data may be collected, presented and summarised, in one of two methods called descriptive statistics. Two elementary summaries of data, singularly called a statistic, are the mean and dispersion. Whereas inferential statistics interprets data from a population sample to induce statements and predictions about a population.
+Statistics is regarded as a body of science or a branch of mathematics. It is based on probability, a branch of mathematics that studies random events. Statistics is considered the science of uncertainty. This arises from the ways to cope with measurement and sampling error as well as dealing with uncertanties in modelling. Although probability and statistics were once paired together as a single subject, they are conceptually distinct from one another. The former is based on deducing answers to specific situations from a general theory of probability, meanwhile statistics induces statements about a population based on a data set. Statistics serves to bridge the gap between probability and applied mathematical fields.
+Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is generally concerned with the use of data in the context of uncertainty and decision-making in the face of uncertainty. Statistics is indexed at 62, a subclass of probability theory and stochastic processes, in the Mathematics Subject Classification. Mathematical statistics is covered in the range 276-280 of subclass QA (science > mathematics) in the Library of Congress Classification.
+The word statistics ultimately comes from the Latin word Status, meaning "situation" or "condition" in society, which in late Latin adopted the meaning "state". Derived from this, political scientist Gottfried Achenwall, coined the German word statistik (a summary of how things stand). In 1770, the term entered the English language through German and referred to the study of political arrangements. The term gained its modern meaning in the 1790s in John Sinclair's works. In modern German, the term statistik is synonymous with mathematical statistics. The term statistic, in singular form, is used to describe a function that returns its value of the same name.
+
+== Statistical data ==
+
+=== Data collection ===

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+---
+title: "Statistics"
+chunk: 2/8
+source: "https://en.wikipedia.org/wiki/Statistics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:59.364659+00:00"
+instance: "kb-cron"
+---
+
+==== Sampling ====
+When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples. Statistics itself also provides tools for prediction and forecasting through statistical models.
+To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representative sampling ensures that inferences and conclusions can safely be extended from the sample to the population as a whole. A major problem lies in determining the extent to which the chosen sample is actually representative. Statistics offers methods to estimate and correct for any bias in the sample and data collection procedures. There are also methods of experimental design that can lessen these issues at the outset of a study, strengthening its ability to discern truths about the population.
+Sampling theory is part of the mathematical discipline of probability theory. Probability is used in mathematical statistics to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures. The use of any statistical method is valid only when the system or population under consideration satisfies the assumptions of the method. The difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts with the given parameters of a total population to deduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction—inductively inferring from samples to the parameters of a larger or total population.
+
+==== Experimental and observational studies ====
+A common goal for a statistical research project is to investigate causality, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables. There are two major types of causal statistical studies: experimental studies and observational studies. In both types of studies, the effect of differences of an independent variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types lies in how the study is actually conducted. Each can be very effective. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements with different levels using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Instead, data are gathered and correlations between predictors and response are investigated. While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data—like natural experiments and observational studies—for which a statistician would use a modified, more structured estimation method (e.g., difference in differences estimation and instrumental variables, among many others) that produce consistent estimators.
+
+===== Experiments =====
+The basic steps of a statistical experiment are:
+
+Planning the research, including finding the number of replicates of the study, using the following information:  preliminary estimates regarding the size of treatment effects, alternative hypotheses, and the estimated experimental variability. Consideration of the selection of experimental subjects and the ethics of research is necessary. Statisticians recommend that experiments compare (at least) one new treatment with a standard treatment or control, to allow an unbiased estimate of the difference in treatment effects.
+Design of experiments, using blocking to reduce the influence of confounding variables, and randomized assignment of treatments to subjects to allow unbiased estimates of treatment effects and experimental error. At this stage, the experimenters and statisticians write the experimental protocol that will guide the performance of the experiment and which specifies the primary analysis of the experimental data.
+Performing the experiment following the experimental protocol and analyzing the data following the experimental protocol.
+Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
+Documenting and presenting the results of the study.
+Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under the experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and blindness. The Hawthorne effect refers to finding that an outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was changed but because they were being observed.
+
+===== Observational study =====
+An example of an observational study is one that explores the association between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a cohort study, and then look for the number of cases of lung cancer in each group. A case-control study is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to participate and their exposure histories are collected.
+
+=== Types of data ===

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+---
+title: "Statistics"
+chunk: 3/8
+source: "https://en.wikipedia.org/wiki/Statistics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:59.364659+00:00"
+instance: "kb-cron"
+---
+
+Various attempts have been made to produce a taxonomy of levels of measurement. The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or Fahrenheit), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation.
+Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their numerical nature. Such distinctions can often be loosely correlated with data type in computer science, in that dichotomous categorical variables may be represented with the Boolean data type, polytomous categorical variables with arbitrarily assigned integers in the integral data type, and continuous variables with the real data type involving floating-point arithmetic. But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented.
+Other categorizations have been proposed. For example, Mosteller and Tukey (1977) distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998), van den Berg (1991).)
+The issue of whether it is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the transformation of variables and the precise interpretation of research questions. "The relationship between the data and what they describe merely reflects the fact that certain kinds of statistical statements may have truth values that are not invariant under some transformations. Whether a transformation is sensible to contemplate depends on the question one is trying to answer."
+
+== Methods ==
+
+=== Descriptive statistics ===
+
+A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information, while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics), in that descriptive statistics aims to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent.
+
+=== Inferential statistics ===
+
+Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates.  It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.
+
+==== Terminology and theory of inferential statistics ====
+
+===== Statistics, estimators and pivotal quantities =====
+Consider independent identically distributed (IID) random variables with a given probability distribution: standard statistical inference and estimation theory defines a random sample as the random vector given by the column vector of these IID variables. The population being examined is described by a probability distribution that may have unknown parameters.
+A statistic is a random variable that is a function of the random sample, but not a function of unknown parameters. The probability distribution of the statistic, though, may have unknown parameters. Consider now a function of the unknown parameter: an estimator is a statistic used to estimate such function. Commonly used estimators include sample mean, unbiased sample variance and sample covariance.
+A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution does not depend on the unknown parameter is called a pivotal quantity or pivot. Widely used pivots include the z-score, the chi square statistic and Student's t-value.
+Between two estimators of a given parameter, the one with lower mean squared error is said to be more efficient. Furthermore, an estimator is said to be unbiased if its expected value is equal to the true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at the limit to the true value of such parameter.
+Other desirable properties for estimators include: UMVUE estimators that have the lowest variance for all possible values of the parameter to be estimated (this is usually an easier property to verify than efficiency) and consistent estimators which converges in probability to the true value of such parameter.
+This still leaves the question of how to obtain estimators in a given situation and carry the computation, several methods have been proposed: the method of moments, the maximum likelihood method, the least squares method and the more recent method of estimating equations.

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+---
+title: "Statistics"
+chunk: 4/8
+source: "https://en.wikipedia.org/wiki/Statistics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:59.364659+00:00"
+instance: "kb-cron"
+---
+
+===== Null hypothesis and alternative hypothesis =====
+Interpretation of statistical information can often involve the development of a null hypothesis which is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time. The alternative hypothesis is the name of the hypothesis that contradicts the null hypothesis.
+The best illustration for a novice is the predicament encountered by a criminal trial. The null hypothesis, H0, asserts that the defendant is innocent, whereas the alternative hypothesis, H1, asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H0 (the status quo) stands in opposition to H1 and is maintained unless H1 is supported by evidence "beyond a reasonable doubt". However, "failure to reject H0" in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not necessarily accept H0 but fails to reject H0. While one can not "prove" a null hypothesis, one can test how close it is to being true with a power test, which tests for type II errors. The null hypothesis cannot be proven true because it is already assumed to be true when the test is being conducted.
+
+===== Error =====
+Working from a null hypothesis, two broad categories of error are recognized:
+
+Type I errors where the null hypothesis is falsely rejected, giving a "false positive".
+Type II errors where the null hypothesis fails to be rejected and an actual difference between populations is missed, giving a "false negative".
+Standard deviation refers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean, while Standard error refers to an estimate of difference between sample mean and population mean.
+A statistical error is the amount by which an observation differs from its expected value. A residual is the amount an observation differs from the value the estimator of the expected value assumes on a given sample (also called prediction).
+Mean squared error is used for obtaining efficient estimators, a widely used class of estimators. Root mean square error is simply the square root of mean squared error.
+
+Many statistical methods seek to minimize the residual sum of squares, and these are called "methods of least squares" in contrast to Least absolute deviations. The latter gives equal weight to small and big errors, while the former gives more weight to large errors. Residual sum of squares is also differentiable, which provides a handy property for doing regression. Least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares. Also in a linear regression model the non deterministic part of the model is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed in polynomial least squares, which also describes the variance in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve.
+Measurement processes that generate statistical data are also subject to error.  Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems.
+
+===== Interval estimation =====
+
+Most studies only sample part of a population, so results do not fully represent the whole population. Any estimates obtained from the sample only approximate the population value. Confidence intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval for a value is a range where, if the sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the true (population) value in 95% of all possible cases. This does not imply that the probability that the true value is in the confidence interval is 95%. From the frequentist perspective, such a claim does not even make sense, as the true value is not a random variable.  Either the true value is or is not within the given interval. However, it is true that, before any data are sampled and given a plan for how to construct the confidence interval, the probability is 95% that the yet-to-be-calculated interval will cover the true value: at this point, the limits of the interval are yet-to-be-observed random variables. One approach that does yield an interval that can be interpreted as having a given probability of containing the true value is to use a credible interval from Bayesian statistics: this approach depends on a different way of interpreting what is meant by "probability", that is as a Bayesian probability.
+In principle, confidence intervals can be symmetrical or asymmetrical. An interval can be asymmetrical because it works as a lower or upper bound for a parameter (left-sided interval or right sided interval), but it can also be asymmetrical if a two-sided interval is built violating symmetry around the estimate. Sometimes the bounds of a confidence interval are reached asymptotically, and these are used to approximate the true bounds.
+
+===== Significance =====
+
+Statistics rarely give a simple Yes/No type answer to the question under analysis. Interpretation often comes down to the level of statistical significance applied to the numbers and often refers to the probability of a value accurately rejecting the null hypothesis (sometimes referred to as the p-value).

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+---
+title: "Statistics"
+chunk: 5/8
+source: "https://en.wikipedia.org/wiki/Statistics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:59.364659+00:00"
+instance: "kb-cron"
+---
+
+The standard approach is to test a null hypothesis against an alternative hypothesis. A critical region is the set of values of the estimator that leads to refuting the null hypothesis. The probability of type I error is therefore the probability that the estimator belongs to the critical region given that null hypothesis is true (statistical significance) and the probability of type II error is the probability that the estimator does not belong to the critical region given that the alternative hypothesis is true. The statistical power of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false.
+Referring to statistical significance does not necessarily mean that the overall result is significant in real-world terms. For example, in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect, such that it is unlikely to help the patient noticeably.
+Although in principle the acceptable level of statistical significance may be subject to debate, the significance level is the largest p-value that allows the test to reject the null hypothesis. This test is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic. Therefore, the smaller the significance level, the lower the probability of committing type I error.
+Some problems are usually associated with this framework (See criticism of hypothesis testing):
+
+A difference that is highly statistically significant can still be of no practical significance, but it is possible to properly formulate tests to account for this. One response involves going beyond reporting only the significance level to include the p-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate the size or importance of the observed effect and can also seem to exaggerate the importance of minor differences in large studies. A better and increasingly common approach is to report confidence intervals. Although these are produced from the same calculations as those of hypothesis tests or p-values, they describe both the size of the effect and the uncertainty surrounding it.
+Fallacy of the transposed conditional, aka prosecutor's fallacy: criticisms arise because the hypothesis testing approach forces one hypothesis (the null hypothesis) to be favored, since what is being evaluated is the probability of the observed result given the null hypothesis and not probability of the null hypothesis given the observed result. An alternative to this approach is offered by Bayesian inference, although it requires establishing a prior probability.
+Rejecting the null hypothesis does not automatically prove the alternative hypothesis.
+As everything in inferential statistics it relies on sample size, and therefore under fat tails p-values may be seriously mis-computed.
+
+===== Examples =====
+Some well-known statistical tests and procedures are:
+
+=== Bayesian statistics ===
+
+An alternative paradigm to the popular frequentist paradigm is to use Bayes' theorem to update the prior probability of the hypotheses in consideration based on the relative likelihood of the evidence gathered to obtain a posterior probability. Bayesian methods have been aided by the increase in available computing power to compute the posterior probability using numerical approximation techniques like Markov Chain Monte Carlo.
+For statistically modelling purposes, Bayesian models tend to be hierarchical, for example, one could model each YouTube channel as having video views distributed as a normal distribution with channel dependent mean and variance 
+  
+    
+      
+        
+          
+            N
+          
+        
+        (
+        
+          μ
+          
+            i
+          
+        
+        ,
+        
+          σ
+          
+            i
+          
+        
+        )
+      
+    
+    {\displaystyle {\mathcal {N}}(\mu _{i},\sigma _{i})}
+  
+, while modeling the channel means as themselves coming from a normal distribution representing the distribution of average video view counts per channel, and the variances as coming from another distribution.
+The concept of using likelihood ratio can also be prominently seen in medical diagnostic testing.
+
+=== Exploratory data analysis ===
+
+Exploratory data analysis (EDA) is an approach to analyzing data sets to summarize their main characteristics, often with visual methods. A statistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling or hypothesis testing task.
+
+=== Mathematical statistics ===
+
+Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. All statistical analyses make use of at least some mathematics, and mathematical statistics can therefore be regarded as a fundamental component of general statistics.
+
+== History ==
+
+Formal discussions on inference date back to the mathematicians and cryptographers of the Islamic Golden Age between the 8th and 13th centuries. Al-Khalil (717–786) wrote the Book of Cryptographic Messages, which contains one of the first uses of permutations and combinations, to list all possible Arabic words with and without vowels. Al-Kindi's Manuscript on Deciphering Cryptographic Messages gave a detailed description of how to use frequency analysis to decipher encrypted messages, providing an early example of statistical inference for decoding. Ibn Adlan (1187–1268) later made an important contribution on the use of sample size in frequency analysis.
+Although the term statistic was introduced by the Italian scholar Girolamo Ghilini in 1589 with reference to a collection of facts and information about a state, it was the German Gottfried Achenwall in 1749 who started using the term as a collection of quantitative information, in the modern use for this science. The earliest writing containing statistics in Europe dates back to 1663, with the publication of Natural and Political Observations upon the Bills of Mortality by John Graunt. 
+Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its stat- etymology. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and natural and social sciences.

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+---
+title: "Statistics"
+chunk: 6/8
+source: "https://en.wikipedia.org/wiki/Statistics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:59.364659+00:00"
+instance: "kb-cron"
+---
+
+The mathematical foundations of statistics developed from discussions concerning games of chance among mathematicians such as Gerolamo Cardano, Blaise Pascal, Pierre de Fermat, and Christiaan Huygens. Although the idea of probability was already examined in ancient and medieval law and philosophy (such as the work of Juan Caramuel), probability theory as a mathematical discipline only took shape at the very end of the 17th century, particularly in Jacob Bernoulli's posthumous work Ars Conjectandi. This was the first book where the realm of games of chance and the realm of the probable (which concerned opinion, evidence, and argument) were combined and submitted to mathematical analysis. The method of least squares was first described by Adrien-Marie Legendre in 1805, though Carl Friedrich Gauss presumably made use of it a decade earlier in 1795.
+
+In the 1830s-1850s, "statistical offices" and national "statistical societies" were founded in Europe and America, and in the mid-19th century, the idea arose of "organized contacts between the statisticians of different countries although informal contacts occurred earlier". In those days, the name "statistics" referred mainly to "matters of state", and British statisticians were often called "statists". 
+Belgian scientist Adolphe Quetelet (1796–1874) introduced the notion of the "average man" (l'homme moyen) as a means of understanding complex social phenomena such as crime rates, marriage rates, and suicide rates. In 1853 Quetelet organised in Brussels the First International Statistical Congress in order to unify measurement in statistical research.
+The modern field of statistics emerged in the late 19th and early 20th century in three stages. The first wave, at the turn of the century, was led by the work of Francis Galton and Karl Pearson, who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. Galton's contributions included introducing the concepts of standard deviation, correlation, regression analysis and the application of these methods to the study of the variety of human characteristics—height, weight and eyelash length among others. Pearson developed the Pearson product-moment correlation coefficient, defined as a product-moment, the method of moments for the fitting of distributions to samples and the Pearson distribution, among many other things. Galton and Pearson founded Biometrika as the first journal of mathematical statistics and biostatistics (then called biometry), and the latter founded the world's first university statistics department at University College London.
+The second wave of the 1910s and 20s was initiated by William Sealy Gosset, and reached its culmination in the insights of Ronald Fisher, who wrote the textbooks that were to define the academic discipline in universities around the world. Fisher's most important publications were his 1918 seminal paper The Correlation between Relatives on the Supposition of Mendelian Inheritance (which was the first to use the statistical term, variance), his classic 1925 work Statistical Methods for Research Workers and his 1935 The Design of Experiments, where he developed rigorous design of experiments models. He originated the concepts of sufficiency, ancillary statistics, Fisher's linear discriminator and Fisher information. He also coined the term null hypothesis during the Lady tasting tea experiment, which "is never proved or established, but is possibly disproved, in the course of experimentation". In his 1930 book The Genetical Theory of Natural Selection, he applied statistics to various biological concepts such as Fisher's principle (which A. W. F. Edwards called "probably the most celebrated argument in evolutionary biology") and Fisherian runaway, a concept in sexual selection about a positive feedback runaway effect found in evolution.
+The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between Egon Pearson and Jerzy Neyman in the 1930s. They introduced the concepts of "Type II" error, power of a test and confidence intervals. Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling.
+Among the early attempts to measure national economic activity were those of William Petty in the 17th century. In the 20th century the uniform System of National Accounts was developed.
+Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology. The use of modern computers has expedited large-scale statistical computations and has also made possible new methods that are impractical to perform manually. Statistics continues to be an area of active research, for example on the problem of how to analyze big data.
+
+== Applications ==
+
+=== Applied statistics, theoretical statistics and mathematical statistics ===
+Applied statistics, sometimes referred to as Statistical science, comprises descriptive statistics and the application of inferential statistics. Theoretical statistics concerns the logical arguments underlying justification of approaches to statistical inference, as well as encompassing mathematical statistics. Mathematical statistics includes not only the manipulation of probability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics and the design of experiments.
+Statistical consultants can help organizations and companies that do not have in-house expertise relevant to their particular questions.
+
+=== Machine learning and data mining ===
+Machine learning models are statistical and probabilistic models that capture patterns in the data through use of computational algorithms.

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+title: "Statistics"
+chunk: 7/8
+source: "https://en.wikipedia.org/wiki/Statistics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T03:56:59.364659+00:00"
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+---
+
+=== Statistics in academia ===
+Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences, government, and business. Business statistics applies statistical methods in econometrics, auditing and production and operations, including services improvement and marketing research. A study of two journals in tropical biology found that the 12 most frequent statistical tests are: analysis of variance (ANOVA), chi-squared test, Student's t-test, linear regression, Pearson's correlation coefficient, Mann-Whitney U test, Kruskal-Wallis test, Shannon's diversity index, Tukey's range test, cluster analysis, Spearman's rank correlation coefficient and principal component analysis.
+A typical statistics course covers descriptive statistics, probability, binomial and normal distributions, test of hypotheses and confidence intervals, linear regression, and correlation. Modern fundamental statistical courses for undergraduate students focus on correct test selection, results interpretation, and use of free statistics software.
+
+=== Statistical computing ===
+
+The rapid and sustained increases in computing power starting from the second half of the 20th century have had a substantial impact on the practice of statistical science. Early statistical models were almost always from the class of linear models, but powerful computers, coupled with suitable numerical algorithms, caused an increased interest in nonlinear models (such as neural networks) as well as the creation of new types, such as generalized linear models and multilevel models.
+Increased computing power has also led to the growing popularity of computationally intensive methods based on resampling, such as permutation tests and the bootstrap, while techniques such as Gibbs sampling have made use of Bayesian models more feasible. The computer revolution has implications for the future of statistics with a new emphasis on "experimental" and "empirical" statistics. A large number of both general and special purpose statistical software are now available. Examples of available software capable of complex statistical computation include programs such as Mathematica, SAS, SPSS, and R.
+
+=== Business statistics ===
+
+In business, "statistics" is a widely used management- and decision support tool. It is particularly applied in financial management, marketing management, and production, services and operations management. Statistics is also heavily used in management accounting and auditing. The discipline of Management Science formalizes the use of statistics, and other mathematics, in business. (Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.)
+A typical "Business Statistics" course is intended for business majors, and covers descriptive statistics (collection, description, analysis, and summary of data), probability (typically the binomial and normal distributions), test of hypotheses and confidence intervals, linear regression, and correlation; (follow-on) courses may include forecasting, time series, decision trees, multiple linear regression, and other topics from business analytics more generally. Professional certification programs, such as the CFA, often include topics in statistics.
+
+== Specialized disciplines ==
+
+Statistical techniques are used in a wide range of types of scientific and social research, including: biostatistics, computational biology, computational sociology, network biology, social science, sociology and social research. Some fields of inquiry use applied statistics so extensively that they have specialized terminology. These disciplines include:
+
+In addition, there are particular types of statistical analysis that have also developed their own specialised terminology and methodology:
+
+Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for summarizing data, and to make data-driven decisions.
+
+== Misuse ==
+
+Misuse of statistics can produce subtle but serious errors in description and interpretation—subtle in the sense that even experienced professionals make such errors, and serious in the sense that they can lead to devastating decision errors. For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics.
+Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. The statistical significance of a trend in the data—which measures the extent to which a trend could be caused by random variation in the sample—may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and skepticism) that people need to deal with information in their everyday lives properly is referred to as statistical literacy.
+There is a general perception that statistical knowledge is all-too-frequently intentionally misused by finding ways to interpret only the data that are favorable to the presenter. A mistrust and misunderstanding of statistics is associated with the quotation, "There are three kinds of lies: lies, damned lies, and statistics". Misuse of statistics can be both inadvertent and intentional, and the book How to Lie with Statistics, by Darrell Huff, outlines a range of considerations. In an attempt to shed light on the use and misuse of statistics, reviews of statistical techniques used in particular fields are conducted (e.g. Warne, Lazo, Ramos, and Ritter (2012)).
+Ways to avoid misuse of statistics include using proper diagrams and avoiding bias. Misuse can occur when conclusions are overgeneralized and claimed to be representative of more than they really are, often by either deliberately or unconsciously overlooking sampling bias. Bar graphs are arguably the easiest diagrams to use and understand, and they can be made either by hand or with simple computer programs. Most people do not look for bias or errors, so they are not noticed. Thus, people may often believe that something is true even if it is not well represented. To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole. According to Huff, "The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism."
+To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case:

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+chunk: 8/8
+source: "https://en.wikipedia.org/wiki/Statistics"
+category: "reference"
+tags: "science, encyclopedia"
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+instance: "kb-cron"
+---
+
+Who says so? (Do they have an axe to grind?)
+How do they know? (Do they have the resources to know the facts?)
+What's missing? (Do they give us a complete picture?)
+Did someone change the subject? (Do they offer us the right answer to the wrong problem?)
+Does it make sense? (Is their conclusion logical and consistent with what we already know?)
+
+=== Misinterpretation: correlation ===
+
+The concept of correlation is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a data set often reveals that two variables (properties) of the population under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death, might find that poor people tend to have shorter lives than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation phenomena could be caused by a third, previously unconsidered phenomenon, called a lurking variable or confounding variable. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables.
+
+== See also ==
+
+Foundations and major areas of statistics
+
+== References ==
+
+== Further reading ==
+Lydia Denworth, "A Significant Problem: Standard scientific methods are under fire. Will anything change?", Scientific American, vol. 321, no. 4 (October 2019), pp. 62–67. "The use of p values for nearly a century [since 1925] to determine statistical significance of experimental results has contributed to an illusion of certainty and [to] reproducibility crises in many scientific fields. There is growing determination to reform statistical analysis... Some [researchers] suggest changing statistical methods, whereas others would do away with a threshold for defining "significant" results". (p. 63.)
+Barbara Illowsky; Susan Dean (2014). Introductory Statistics. OpenStax CNX. ISBN 978-1938168208.
+Stockburger, David W. "Introductory Statistics: Concepts, Models, and Applications". Missouri State University (3rd Web ed.). Archived from the original on 28 May 2020.
+OpenIntro Statistics Archived 16 June 2019 at the Wayback Machine, 3rd edition by Diez, Barr, and Cetinkaya-Rundel
+Stephen Jones, 2010. Statistics in Psychology: Explanations without Equations. Palgrave Macmillan. ISBN 978-1137282392.
+Cohen, J (1990). "Things I have learned (so far)" (PDF). American Psychologist. 45 (12): 1304–1312. doi:10.1037/0003-066x.45.12.1304. S2CID 7180431. Archived from the original (PDF) on 18 October 2017.
+Gigerenzer, G (2004). "Mindless statistics". Journal of Socio-Economics. 33 (5): 587–606. doi:10.1016/j.socec.2004.09.033. hdl:11858/00-001M-0000-0025-87C0-8.
+Ioannidis, J.P.A. (2005). "Why most published research findings are false". PLOS Medicine. 2 (4): 696–701. doi:10.1371/journal.pmed.0040168. PMC 1855693. PMID 17456002.
+
+== External links ==
+
+(Electronic Version): TIBCO Software Inc. (2020). Data Science Textbook.
+Online Statistics Education: An Interactive Multimedia Course of Study. Developed by Rice University (Lead Developer), University of Houston Clear Lake, Tufts University, and National Science Foundation.
+UCLA Statistical Computing Resources (archived 17 July 2006)
+Philosophy of Statistics from the Stanford Encyclopedia of Philosophy