correlation coefficient - dot and cross products

Question

I refer to 'Thirteen Ways to Look at the Correlation Coefficient' by Rodgers & Nicewander (1988), specifically to 2 of the 13 ways of looking at the correlation coefficient, namely: No. 6 "CORRELATION AS THE MEAN CROSSPRODUCT OF STANDARDIZED VARIABLES" and No. 8 "CORRELATION AS A FUNCTION OF THE ANGLE BETWEEN THE TWO VARIABLE VECTORS".

My issue is that No. 6 does NOT appear to be to be a crossproduct (i.e. a vector product between vectors resulting in a pseudovector). It appears to me to be a sum of products of Z-scores. I don't dispute that it is a computationally correct version of the coefficient, just that it is not a crossproduct.

Similarly, No. 8 DOES appear to be a product between vectors, namely a normalised dot (or inner) product resulting in the cosine of the angle between vectors. However, it is not referred to in that way even though it would appear to be correct to do so.

I am mystified as to where vector terminology is used in the first case, apparently incorrectly, whilst it is not used in the second case when it apparently would have been correct to do so.

Could somebody please explain or am I just missing something?

Answer 1

It may be a conflict-of-terminology issue.

Cross-product is not an unambiguous term (e.g. see here which includes a number of different meanings - yet doesn't seem to include the intended sense).

They don't mean the vector-cross-product $Z_1 × Z_2$.

In the sense they meant it, $Z_{1,i}Z_{2,i}$ is a cross product. This is common in stats

e.g. see here or here or here

You also often see $\sum_i Z_{1,i}Z_{2,i}=Z_1^\top Z_2=Z_1\cdot Z_2$ called a cross-product*.

Consequently $\frac{1}{N}\sum_i Z_{1,i}Z_{2,i}=Z_1\cdot Z_2/N$ might be called either a "mean cross product" (in the first sense) or say a "normalized cross product" (in the second sense).

* Compare with say the R function crossprod which means $Z_1\cdot Z_2$.