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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Facet theory | 5/8 | https://en.wikipedia.org/wiki/Facet_theory | reference | science, encyclopedia | 2026-05-05T09:54:10.375224+00:00 | kb-cron |
Variables processed by Facet Theory measurement operations to be described below, evenly cover the attribute content universe. To ensure such coverage, Facet Theory measurement operations are often performed not on the sample of the observed items themselves, but rather on composite variables that represent facet elements that had been validated by Faceted SSA. The sample of individuals is rich enough to allow existing score-profiles of the processed variables to be observed. In the resulting measurement, order relations among individuals should preserve sufficiently well order relations (including comparability and incomparability; see below) between individuals' profiles of the processed variables. The result of the measurement operation yields the smallest number of scales; The resultant scales represent fundamental variables whose interpretation derives from the contents of the observed items, but does not depend on the particular sample of items observed. Partial order analysis of observed data. Let observed items v1,...,vn with a common-meaning range (CMR) represent an investigated content universe; let A1,...,An be their ranges with each Aj ordered from high to low with respect to the common meaning; and let A = A1×A2 × ... × An be the cartesian product of all the range facets, Aj (j = 1,...,n). A system of observations is a mapping P → A from the observed subjects P to A, that is, each subject pi gets a score from each Aj (j = 1,...,n), or pi → [ai1,ai2, ..., ain]
≡
{\displaystyle \equiv }
a(pi). The point a(pi) in A is also called the profile of pi, and the subset A′ of A (
A
′
⊆
A
{\displaystyle A'\subseteq A}
) of observed profiles is called a scalogram. Facet Theory defines relations between profiles as follows: Two different profiles ai = [ai1,ai2,...,ain] and aj = [aj1,aj2,...,ajn], are comparable, denoted by aiSaj, with ai greater than aj, ai > aj, if and only if aik ≥ ajk for k = 1, ..., n, and aik′ > ajk′ for some k. Two different profiles are incomparable, denoted by ai $ aj, if neither ai > aj nor aj > ai. A, and therefore its subset A′, form a partially ordered set. Facet Theoretical measurement consists in mapping points a(pi) of A' into a coordinate space X of the lowest dimensionality while preserving observed order relations, including incomparability: Definition. The p.o. dimensionality of scalogram A' is the smallest m (m ≤ n) for which there exist m facets X1 ... Xm (each Xi is ordered) and there exists a 1-1 mapping Q:X′ → A′ from X′ (
X
′
⊆
X
=
X
1
×
⋯
×
X
m
{\textstyle X'\subseteq X=X_{1}\times \cdots \times X_{m}}
) to A′ such that a > a′ if and only if x > x′ whenever Q maps points x, x′ in X′ to points a, a′ ∈ A. The coordinate scales, Xi (i = 1, ..., m) represent underlying fundamental variables whose meanings must be inferred in any specific application. The well known Guttman scale [24] (example: 1111, 1121, 1131, 2131, 2231, 2232) is simply a 1-d scalogram, i.e. one all of whose profiles are comparable. The procedure of identifying and interpreting the coordinate scales X1...Xm is called multiple scaling. multiple scaling is facilitated by partial order scalogram analysis by base coordinates (POSAC) for which algorithms and computer programs have been devised. In practice, a particular dimensionality is attempted and a solution that best accommodates the order-preserving condition is sought. The POSAC/LSA program finds an optimal solution in 2-d coordinate space, then goes on to analyze by Lattice Space Analysis (LSA) the role played by each of the variables in structuring the POSAC 2-space, thereby facilitating interpretation of the derived coordinate scales, X1, X2. Recent developments include the algorithms for computerized partitioning of the POSAC space by the range facet of each variable, which induces meaningful intervals on the coordinate scales, X, Y.
=== Example 3. TV watching patterns: analysis of simplified survey data === Source: Members of a particular population were asked four questions: whether they watched TV the night before for an hour at 7 PM (hour 1), at 8 PM (hour 2), at 9 PM (hour 3) and at 10 PM (hour 4). A positive answer to a question was recorded as 1, and a negative answer, as 0. Thus, for example, the profile 1010 represents a person who watched TV at 7 PM and at 9 PM but not at 8 PM and at 10 PM. Suppose that out of the 16 combinatorially possible profiles, only the following eleven profiles were observed empirically: 0000, 1000, 0100, 0010, 0001, 1100, 0110, 0011, 1110, 0111, 1111. Figure 3 is an order-preserving mapping of these profiles into a 2-dimensional coordinate space.