First let me explain what I mean by "transitive" Suppose that the price of product A and the price of product B has a correlation of .5 Suppose also that the price of product B and product C has a correlation of .5
One might thing that if correlation(A,B) = .5 and correlation(B,C) = .5 then correlation(A,C) > 0, but with an intuitive example we can see why this is not the case:
Suppose that A is a fruit salad made of papaya and banana, suppose also that B is a fruit salad made of banana and strawberry and, finally, suppose that C is strawberries and cream. Clearly, when the price of banana goes up, both the price of A and B will go up; when the price of strawberry goes up the price of B and C will go up; but since A and C have no ingredients in common, their price isn't correlated.
My question is which statistical concept captures this intuitive idea. Is it the concept of dimensions?
What do I mean by transitive? Let me define the binary operator X corr? Y as 1 if correlation(X,Y) != 0 and 0 otherwise, so:
- A corr? B = 1,
- B corr? C = 1, but
- A corr? C could be = 0
If corr? was transitive, then A corr? C = 1