Let Y be a set of observations. In the following notes Y is always finite. A ~#{factor} on Y (or a #~{factorization} of Y) is a mapping of Y to a finite set _ F #: Y &rightarrow. $F.
We generally refer to this as "the factor F". For the time being we use $F to designate the co-domain of the mapping. It is important to realize that specifying the co-domain does not specify the factor, i.e. it is the mapping itself which specifies the factor. When the concepts become more familiar, we will drop the over-line and also refer to the co-domain as F.
The elements ~f &in. $F are called the ~{levels} of the factor. The factor induces a partition of the set Y into the subsets
F^{-1} ( ~f ) = \{ ~y &in. Y | F ( ~y ) = ~f \}.
These subsets are sometimes refered to as levels of F.
Two factors found with any finite set of observations are:
If ~f is a level of the factor F, let ~{n_f } _ denote the number of elements of Y mapped to ~f:
~{n_f } _ = _ &hash.\{ ~y | F (~y) = ~f \} _ = _ &hash. F^{-1} ( ~f )
The number of non-empty levels of the factor F, is denoted |F|:
|F| _ = _ &hash.\{ ~f &in. $F | ~{n_f } &neq. 0 \}
obviously
|F| _ = _ &hash.$F _ _ _ _ <=> _ _ _ _ F is surjective
A factor, F, is said to be ~#{balanced} if it has the same number of elements mapped to each level. i.e. F is balanced if:
~{n_f } _ = _ ~{n_g } _ _ _ _ _ _ _ _ any _ ~f, ~g &in. F
In which case _ ~{n_f } _ = _ &hash.Y #/ &hash.$F , _ &forall. ~f &in. $F.
If F and G are two factors on the same set, Y, then F is said to be ~#{finer} than G if &exist. a map &phi. #: $F &rightarrow. $G, such that G = &phi. _{&circle.} F .
[Equivalently we say that G is ~#{coarser} than F.] _ I.e. the levels of F are completely contained or ~{nested} in the levels of G.
We write _ F >= G . _ [ F finer than G ].
Let _ @A_F #:= \{ F^{-1}(M) | M &subseteq. $F \} _ _ i.e. the subsets of Y which are reverse images of ~{subsets} of $F under F.
For any element of @A_F , B &in. @A_F , say, we have:
B _ = _ F^{-1}(M) _ = _ union{F^{-1}( ~f ), _ ~f &in. M,} _ _ _ _ (some M &subseteq. $F)
I.e. B is the union of levels of F. _ If _ B, C &in. @A_F _ then _ B &union. C &in. @A_F , _ B &intersect. C &in. @A_F , _ and _ &empty. &in. @A_F. _ So @A_F is an algebra of subsets of Y generated by the levels of F.
If F >= G where G = &phi. _{&circle.} F _ then F^{-1}( ~f ) &subseteq. G^{-1}( ~g ) _ &forall. ~f such that _ &phi.( ~f ) = ~g . _ So
G^{-1}( ~g ) _ = _ union{ F^{-1}( ~f ) , \{ ~f | &phi.( ~f ) = ~g \},}
If _ B &in. @A_G _ then, for some subset M &subseteq. G,
B _ _ = _ _ union{G^{-1}( ~g ), _ ~g &in. M,} _ _ = _ _ union{, _ ~g &in. M,} rndb{union{ F^{-1}( ~f ) , \{ ~f | &phi.( ~f ) = ~g \}, _ }}
So _ B &in. @A_F _ _ I.e. _ F >= G _ &iff. _ @A_F &supseteq. @A_G.
Note that _ _ I >= F >= O _ _ for any factor F.
Note that there is divergence amongst authors about the symbols used to define the partial order between factors. I use "F > G" to denote "F (strictly) finer than G". Other authors will use "F < G" or "F &subset. G". I prefer the " > " to "&subset." as this makes clear that it is a partial ordering. I've chosen " > " rather than " < " because intuitively I think of I > O, and not the other way round. Also we will soon define the ~{minimum} of two factors F and G, where F and G are finer than their minimum. I prefer to write F >= _ min \{F,G\}.
Perhaps it will help intuitively to think: _ _ F finer than G, _ _ F > G _ _ then F has more levels than G.
Two factors, F and G, on the same set Y, are ~{equivalent} _ if _ F >= G _ and _ G >= F, _ in which case @A_F &eq. @A_G. _ I.e. they induce the same partition on Y. However, equivalence does not imply identity, as F and G could differ on the number of empty levels.
If F and G are two factors on the same set, Y, then the ~{cross factor} or ~{product factor} of F and G is defined by the set $F # $G, and the mapping {F # G} : Y &rightarrow. $F # $G given by:
(F # G)(~y) _ = _ (F (~y ), G(~y ) )
It is fairly obvious that _ F # G >= F _ and _ F # G >= G. _ [ F = &theta. _{&circle.} (F # G) where &theta.(~x, ~y) = ~x, _ etc.]
Suppose _ H >= F _ and _ H >= G _ then _ H >= F # G .
Let
F _ = _ &phi._1 _{&circle.} H _ _ _ _ and _ _ _ _ G _ = _ &phi._2 _{&circle.} H
then
{F # G}(~y) _ = _ (F (~y ), G(~y ) )
_ _ _ _ _ _ = _ (&phi._1 _{&circle.} H (~y ) , &phi._2 _{&circle.} H(~y ) ) _ = _ &phi. _{&circle.} H (~y )
where
&phi.(~x) _ = _ ( &phi._1(~x) , &phi._2(~x) )
So _ H >= F # G .
As shown above, if _ H >= F _ and _ H >= G _ then _ H >= F # G . I.e. F # G is the coarsest factor which is finer than both F and G. For this reason, F # G is known as the ~#{maximum} of the factors F and G.
F # G >= F _ => _ @A_{F # G} &supseteq. @A_F , _ F # G >= G _ => _ @A_{F # G} &supseteq. @A_G. _ So _ @A_{F # G} &supseteq. @A_F &union. @A_G, _ in fact, as F # G is the coarsest such factor
@A_{F # G} _ = _ @A_F &union. @A_G
Let F and G be factors (on Y) with corresponding algebras @A_F and @A_G _ [@A_F = \{ F^{-1}(M) | M &subseteq. F \} _ etc.].
Then the factor corresponding to the algebra @A_F &intersect. @A_G is called the ~{minimum} of the factors F and G and is denoted F &min. G.
We have _ F >= F &min. G, _ G >= F &min. G, _ and if _ H =< F , _ H =< G, _ then _ H =< F &min. G .
Unlike the maximum, which is just the cross-product of two factors, the minimum is difficult to envisage, especially as most cases the minimum of two factors will be the null factor.
Here is an example of a simple case of a non-trivial minimum.
A group of students are noted for their hair colour and eye colour:
These observations are summarized on the right.
Then the minimum of hair colour and eye colour has two levels, one containing all the blonds (both blue and green eyed) and the other containing all the brown eyed (both dark and red haired).