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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)

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