PCA / FA malinowski: example 5. target testing

Recall that we computed the SVD X = u\ w\ v^T\ of this matrix:

X = \left(\begin{array}{lll} 2 & 3 & 4 \\ 1 & 0 & -1 \\ 4 & 5 & 6 \\ 3 & 2 & 1 \\ 6 & 7 & 8\end{array}\right)

and we found that the w matrix was

w = \left(\begin{array}{lll} 16.2781 & 0. & 0. \\ 0. & 2.45421 & 0. \\ 0. & 0. & 0. \\ 0. & 0. & 0. \\ 0. & 0. & 0.\end{array}\right)

Because w has only two nonzero entries, we know that X is of rank 2. Its three columns only span a 2D space.

Given a column of data x (a variable, in this example, of length 5), Malinowski wants to know if it is in that 2D space. As he puts it, “if the suspected test vector [x] is a real factor, then the regeneration \hat{x} = R\ t will be successful.” He gives us a formula for computing t; by a successful regeneration, he means that \hat{x}\ is close to x.
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