“The partial least squares algorithm: a truncated Cayley-Hamilton series approximation used to solve the regression problem”

Authors: David Di Ruscio,
Affiliation: Telemark University College
Reference: 1998, Vol 19, No 3, pp. 117-140.

Keywords: Least squares, Cayley-Hamilton, PLS

Abstract: In this paper it is shown that the PLS algorithm for univariate data is equivalent to using a truncated Cayley-Hamilton polynomial expression of degree 1 less than a less than r for the matrix inverse inv(X´X) in R^(rxr) used to compute the LS solution. Furthermore, the a coefficients in this polynomial are computed as the LS optimal solution (minimizing parameters) to the prediction error. The resulting solution is non-iterative. The solution can be expressed in terms of one matrix inverse and is given by BPLS = Ka * inv(Ka´*X´*X*Ka)*Ka´*X´*Y where Ka in R^(rxr) is the controllability (Krylov) matrix for the pair (X´X,X´Y).

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  title={{The partial least squares algorithm: a truncated Cayley-Hamilton series approximation used to solve the regression problem}},
  author={Di Ruscio, David},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}