13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aliasing (factorial experiments) | 4/7 | https://en.wikipedia.org/wiki/Aliasing_(factorial_experiments) | reference | science, encyclopedia | 2026-05-05T09:48:49.670316+00:00 | kb-cron |
where the equation is modulo
s
{\displaystyle s}
if
s
{\displaystyle s}
is prime, and is in the finite field
G
F
(
s
)
{\displaystyle GF(s)}
if
s
{\displaystyle s}
is a power of a prime. Such equations are called defining equations of the fraction. When the defining equation or equations are homogeneous, the fraction is said to be principal. One defining equation yields a fraction of size
s
k
−
1
{\displaystyle s^{k-1}}
, two independent equations a fraction of size
s
k
−
2
,
{\displaystyle s^{k-2},}
and so on. Such fractions are generally denoted as
s
k
−
r
{\displaystyle s^{k-r}}
designs. The half-fractions described above are
2
3
−
1
{\displaystyle 2^{3-1}}
designs. The notation often includes the resolution as a subscript, in Roman numerals; the above fractions are thus
2
I
I
I
3
−
1
{\displaystyle 2_{III}^{3-1}}
designs. Associated to each expression
a
1
t
1
+
⋯
+
a
k
t
k
{\displaystyle a_{1}t_{1}+\cdots +a_{k}t_{k}}
is another, namely
A
1
a
1
⋯
A
k
a
k
{\displaystyle A_{1}^{a_{1}}\cdots A_{k}^{a_{k}}}
, which rewrites the coefficients as exponents. Such expressions are called "words", a term borrowed from group theory. (In a particular example where
k
{\displaystyle k}
is a specific number, the letters
A
,
B
,
C
…
{\displaystyle A,B,C\ldots }
are used, rather than
A
1
,
A
2
,
A
3
…
{\displaystyle A_{1},A_{2},A_{3}\ldots }
.) These words can be multiplied and raised to powers, where the word
I
=
A
1
0
⋯
A
k
0
{\displaystyle I=A_{1}^{0}\cdots A_{k}^{0}}
acts as a multiplicative identity, and they thus form an abelian group
G
{\displaystyle \mathbb {G} }
, known as the effects group. When
s
{\displaystyle s}
is prime, one has
W
s
=
I
{\displaystyle W^{s}=I}
for every element (word)
W
∈
G
{\displaystyle W\in \mathbb {G} }
; something similar holds in the prime-power case. In
2
k
{\displaystyle 2^{k}}
factorial experiments, each element of
G
{\displaystyle \mathbb {G} }
represents a main effect or interaction. In
s
k
{\displaystyle s^{k}}
experiments with
s
>
2
{\displaystyle s>2}
, each one-letter word represents the main effect of that factor, while longer words represent components of interaction. An example below illustrates this with
s
=
3
{\displaystyle s=3}
. To each defining expression (the left-hand side of a defining equation) corresponds a defining word. The defining words generate a subgroup
H
{\displaystyle \mathbb {H} }
of
G
{\displaystyle \mathbb {G} }
that is variously called the alias subgroup, the defining contrast subgroup, or simply the defining subgroup of the fraction. Each element of
H
{\displaystyle \mathbb {H} }
is a defining word since it corresponds to a defining equation, as one can show. The effects represented by the defining words are completely lost in the fraction while all other effects are preserved. If
H
=
{
I
,
W
1
,
…
,
W
ℓ
}
{\displaystyle \mathbb {H} =\{I,W_{1},\ldots ,W_{\ell }\}}
, say, then the equation
I
=
W
1
=
⋯
=
W
ℓ
{\displaystyle I=W_{1}=\cdots =W_{\ell }}
is called the defining relation of the fraction. This relation is used to determine the aliasing structure of the fraction: If a given effect is represented by the word
W
{\displaystyle W}
, then its aliases are computed by multiplying the defining relation by
W
{\displaystyle W}
, viz.,
W
=
W
W
1
=
⋯
=
W
W
ℓ
,
{\displaystyle W=WW_{1}=\cdots =WW_{\ell },}
where the products
W
W
i
{\displaystyle WW_{i}}
are then simplified. This relation indicates complete (not partial) aliasing, and W is unaliased with all other effects listed in
G
{\displaystyle \mathbb {G} }
.
=== Example 1 === In either of the
2
3
−
1
{\displaystyle 2^{3-1}}
fractions described above, the defining word is
A
B
C
{\displaystyle ABC}
, since the exponents on these letters are the coefficients of
t
1
+
t
2
+
t
3
{\displaystyle t_{1}+t_{2}+t_{3}}
. The
A
B
C
{\displaystyle ABC}
effect is completely lost in the fraction, and the defining subgroup
H
{\displaystyle \mathbb {H} }
is simply
{
I
,
A
B
C
}
{\displaystyle \{I,ABC\}}
, since squaring does not generate new elements
(
(
A
B
C
)
2
=
A
2
B
2
C
2
=
I
)
{\displaystyle ((ABC)^{2}=A^{2}B^{2}C^{2}=I)}
. The defining relation is thus
I
=
A
B
C
{\displaystyle I=ABC}
, and multiplying both sides by
A
{\displaystyle A}
gives
A
=
A
2
B
C
{\displaystyle A=A^{2}BC}
; which simplifies to
A
=
B
C
,
{\displaystyle A=BC,}
the alias relation seen earlier. Similarly,
B
=
A
C
{\displaystyle B=AC}
and
C
=
A
B
{\displaystyle C=AB}
. Note that multiplying both sides of the defining relation by
A
B
,
A
C
{\displaystyle AB,AC}
and
B
C
{\displaystyle BC}
does not give any new alias relations. For comparison, the
2
3
−
1
{\displaystyle 2^{3-1}}
fraction with defining equation
t
1
+
t
2
=
0
(
mod
2
)
{\displaystyle t_{1}+t_{2}=0{\pmod {2}}}
has the defining word
A
B
{\displaystyle AB}
(i.e.,
A
1
B
1
C
0
{\displaystyle A^{1}B^{1}C^{0}}
). The effect
A
B
{\displaystyle AB}
is completely lost, and the defining relation is
I
=
A
B
{\displaystyle I=AB}
. Multiplying this by
A
{\displaystyle A}
, by
C
{\displaystyle C}
, and by
A
C
{\displaystyle AC}
gives the alias relations
A
=
B
{\displaystyle A=B}
,
C
=
A
B
C
{\displaystyle C=ABC}
, and
A
C
=
B
C
{\displaystyle AC=BC}
among the six remaining effects. This fraction only has resolution 2 since all effects (except
A
B
{\displaystyle AB}
) are preserved but two main effects are aliased. Finally, solving the defining equation
t
1
+
t
2
=
0
(
mod
2
)
{\displaystyle t_{1}+t_{2}=0{\pmod {2}}}
yields the fraction {000, 001, 110, 111}. One may verify all of this by sorting the table above on column
A
B
{\displaystyle AB}
.