Partial Least Squares Regression via NIPALS (Internal)

This function is called internally by pls.regression and is not intended to be used directly. Use pls.regression(..., calc.method = "NIPALS") instead.

Performs Partial Least Squares (PLS) regression using the NIPALS (Nonlinear Iterative Partial Least Squares) algorithm. This method estimates the latent components (scores, loadings, weights) by iteratively updating the X and Y score directions until convergence. It is suitable for cases where the number of predictors is large or predictors are highly collinear.

Usage

NIPALS.pls(x, y, n.components = NULL)

Arguments

x

A numeric matrix or data frame of predictors (X). Should have dimensions n × p.

y

A numeric matrix or data frame of response variables (Y). Should have dimensions n × q.

n.components

Integer specifying the number of PLS components to extract. If NULL, it defaults to qr(x)$rank.

Value

A list with the following elements:

model.type

Character string indicating the model type ("PLS Regression").

T

Matrix of X scores (n × H).

U

Matrix of Y scores (n × H).

W

Matrix of X weights (p × H).

C

Matrix of normalized Y weights (q × H).

P_loadings

Matrix of X loadings (p × H).

Q_loadings

Matrix of Y loadings (q × H).

B_vector

Vector of regression scalars (length H), one for each component.

coefficients

Matrix of regression coefficients in original data scale (p × q).

intercept

Vector of intercepts (length q). Always zero here due to centering.

X_explained

Percent of total X variance explained by each component.

Y_explained

Percent of total Y variance explained by each component.

X_cum_explained

Cumulative X variance explained.

Y_cum_explained

Cumulative Y variance explained.

Details

The algorithm standardizes both x and y using z-score normalization. It then performs the following for each of the n.components latent variables:

1. Initializes a random response score vector \(u\).

2. Iteratively:

   • Updates the X weight vector \(w = E^\top u\), normalized.

   • Computes the X score \(t = E w\), normalized.

   • Updates the Y loading \(q = F^\top t\), normalized.

   • Updates the response score \(u = F q\).

   • Repeats until \(t\) converges below a tolerance threshold.

3. Computes scalar regression coefficient \(b = t^\top u\).

4. Deflates residual matrices \(E\) and \(F\) to remove current component contribution.

After component extraction, the final regression coefficient matrix \(B_{original}\) is computed and rescaled to the original data units. Explained variance is also computed component-wise and cumulatively.

References

Wold, H., & Lyttkens, E. (1969). Nonlinear iterative partial least squares (NIPALS) estimation procedures. Bulletin of the International Statistical Institute, 43, 29–51.

Examples

if (FALSE) { # \dontrun{
X <- matrix(rnorm(100 * 10), 100, 10)
Y <- matrix(rnorm(100 * 2), 100, 2)
model <- pls.regression(X, Y, n.components = 3, calc.method = "NIPALS")
model$coefficients
} # }