Performs partial least squares (PLS) regression for one or more response variables and one or more predictor variables.

The type double function is imsls_d_pls_regression.

A pointer to the array of length ix × iy containing the final PLS regression coefficient estimates for the mean-centered variables, where ix ≤ p is the number of predictor variables in the model, and iy ≤ h is the number of response variables. To release this space, use imsls_free. If the estimates cannot be computed, NULL is returned.

Function imsls_f_pls_regression performs partial least squares regression for a response matrix Y(ny × h) and a set of p explanatory variables, X(nx × p). imsls_f_pls_regression finds linear combinations of the predictor variables that have highest covariance with Y. In so doing, imsls_f_pls_regression produces a predictive model for Y using components (linear combinations) of the individual predictors. Other names for these linear combinations are scores, factors, or latent variables. Partial least squares regression is an alternative method to ordinary least squares for problems with many, highly collinear predictor variables. For further discussion see, for example, Abdi (2010), and Frank and Friedman (1993).

In Partial Least Squares (PLS), a score, or component matrix, T, is selected to represent both X and Y as in,

and

The matrices P and Q are the least squares solutions of X and Y regressed on T.

That is,

and

The columns of T in the above relations are often called X-scores, while the columns of P are the X-loadings. The columns of the matrix U in Y = UQT + G are the corresponding Y scores, where G is a residual matrix and Q, as defined above, contains the Y-loadings.

Restricting T to be linear in X, the problem is to find a set of weight vectors (columns of W) such that T = XW predicts both X and Y reasonably well.

Formally, W = [w1, ..., wm-1, wm, ...wM] where each wj is a column vector of length p, M ≤ p is the number of components, and where the m-th partial least squares (PLS) component wm solves:

Algorithms to solve the above optimization problem include NIPALS (nonlinear iterative partial least squares) developed by Herman Wold (1966, 1985) and numerous variations, including the SIMPLS algorithm of de Jong (1993). imsls_f_pls_regression implements the SIMPLS method. SIMPLS is appealing because it finds a solution in terms of the original predictor variables, whereas NIPALS reduces the matrices at each step. For univariate Y it has been shown that SIMPLS and NIPALS are equivalent (the score, loading, and weights matrices will be proportional between the two methods).

By default, imsls_f_pls_regression searches for the best number of PLS components using K-fold cross-validation. That is, for each M = 1, 2,…, p, imsls_f_pls_regression estimates a PLS model with M components using all of the data except a hold-out set of size roughly equal to nobs/k. Using the resulting model estimates, imsls_f_pls_regression predicts the outcomes in the hold-out set and calculates the predicted residual sum of squares (PRESS). The procedure then selects the next hold-out sample and repeats for a total of K times (i.e., folds). For further details see Hastie, et. al., pages 241-245 (2001).

For each response variable, imsls_f_pls_regression returns results for the model with lowest PRESS. The best model (the number of components giving lowest PRESS), generally will be different for different response variables.

When requested via the optional argument IMSLS_STD_ERRORS, imsls_f_pls_regression calculates modified jackknife estimates of the standard errors as described in Martens and Martens (2000).

The following artificial data set is provided in de Jong (1993).

The first call to imsls_f_pls_regression fixes the number of components to 3 for both response variables, and the second call performs K-fold cross validation. Note that because the number of folds is equal to n, imsls_f_pls_regression performs leave-one-out (LOO) cross-validation.

The data, as appears in S. Wold et al. (2001), is a single response variable, the “free energy of the unfolding of a protein”, while the predictor variables are 7 different, highly correlated measurements taken on 19 amino acids.