for this section, i need – i want – to match davis’ signs and sizes. this is just temporary.
instead of my …
i need to change the sign of the 2nd column and drop the 3rd:
instead of my …
i need to change the sign of the 1st column and drop the last two:
having those, it’s quickest to recompute and from the definitions. recall that
(yes, those are davis’ signs.) then we have
(yes, those too are davis’ signs.)
he has two plots. one is the so-called Q-mode loadings, the two nonzero columns of davis’ :
it would be informative for me to draw the so-called R-mode scores at this time, the two nonzero columns of davis’ :
i have arranged for Mathematica® to use the same imagesize and aspect ratio. gee, this is the same picture, except for the scales. (in fact, the book drew the Q-mode loadings but labeled it R-mode scores. we see that it’s an understandable error.)
the other plot he drew was the R-mode loadings, the two nonzero columns of his :
and i know you know we need to plot the Q-mode scores right now, the two nonzero columns of :
again, the scales have changed, but this is the same as the previous plot.
writing a little carelessly (exactly what do i mean by ?), the algebraic statements of what i have just shown you are :
hang on a second. suppose we compute and ; these two equations say that we can compute from and from . but computing directly gave us an eigenvector matrix U; computing directly gave us an eigenvector matrix V. these two equations say that we could have gotten without ever computing V.
if we have so many observations that V is huge, this is a useful trick.
in addition, however, it leaves one wondering what’s to be gained from Q-mode analysis. we have not established that there’s nothing to be gained; but we can get whatever it is, if anything, without ever forming . (and yes, there is something to be gained. believe me!)
to confirm those relationships among the cut-down matrices, we need a 2×2 matrix instead of a 3×3 or 4×4 namely:
let’s confirm . the RHS is:
and the LHS is:
which is what we wanted to see.
similarly, for , the RHS is
and the LHS is
as it should be.
so: if we cut down the matrices and we use davis’ signs, we see that the Q-mode FA could have been computed from the R-mode..
next, we will look at that using the SVD.