Exercises Unit E - Conceptual
Bachelor’s Degree Programme in Philosophy, International and Economic Studies, Ca’ Foscari University of Venice.
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Chapter 12, Exercise 6
We saw in Section 12.2.2 that the principal component loading and score vectors provide an approximation to a matrix, in the sense of (12.5). Specifically, the principal component score and loading vectors solve the optimization problem given in (12.6).
Now, suppose that the M principal component score vectors z_{im}, m = 1, \ldots, M, are known. Using (12.6), explain that each of the first M principal component loading vectors \phi_{jm}, m = 1, \dots, M, can be obtained by performing p separate least squares linear regressions.
In each regression, the principal component score vectors are the predictors, and one of the features of the data matrix is the response.