Functions that deal with the singular value decomposition of an output Y, for use with Gaussian process lists
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the output to decompose, where each column of
optionally, the number of principle component weights to keep
Utilizes the singular value decomposition (SVD) of
Y, Y = UDVprime. Columns of
Y should correspond to a single k-dimensional observation (e.g., functional output of a computer model, evaluated at a particular input).
For a k x m matrix
Y, and r = min(k,m), in the complete SVD,
U is k x r,
D is r x r, containing the singular values along the diagonal, and
Vprime is r x m. The output
Y is approximated by keeping l < r singular values, keeping a UD matrix of dimension k x l, and the
Vprime matrix of dimension l x m. Each column of
Vprime now contains l principle component weights, which can be used to reconstruct the functional output.
pcweights returns a list with components:
the UD matrix corresponding to the number of principle components kept
The Vprime matrix corresponding to the number of principle components kept
Note: the number of principle component weights kept is equal to dim(UD)
getSingularValues returns a matrix containing the singular values of
numSingularValues returns the minimum number of singular values accounting for
cutoff percent of the variation in
singularValueImportance returns a matrix where element i corresponds to the percentage of total variation in
Y accounted for by the first i singular values
these functions are utilized by
mlegp to fit Gaussian processes to principle component weights
Garrett M. Dancik email@example.com
Heitmann, K., Higdon, D., Nakhleh, C., Habib, S., 2006. Cosmic Calibration. The Astrophysical Journal, 646, 2, L1-L4.
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