Partial Least Squares Regression via SVD (Internal)

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

Performs Partial Least Squares (PLS) regression using the Singular Value Decomposition (SVD) of the cross-covariance matrix. This method estimates the latent components by identifying directions in the predictor and response spaces that maximize their covariance, using the leading singular vectors of the matrix \(R = X^\top Y\).

Usage

SVD.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, defaults to qr(x)$rank.

Value

A list containing:

model.type

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

T

Matrix of predictor scores (n × H).

U

Matrix of response scores (n × H).

W

Matrix of predictor weights (p × H).

C

Matrix of normalized response weights (q × H).

P_loadings

Matrix of predictor loadings (p × H).

Q_loadings

Matrix of response loadings (q × H).

B_vector

Vector of scalar regression weights (length H).

coefficients

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

intercept

Vector of intercepts (length q). All zeros 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 begins by z-scoring both x and y (centering and scaling to unit variance). The initial residual matrices are set to the scaled values: E = X_scaled, F = Y_scaled.

For each component h = 1, ..., H:

1. Compute the cross-covariance matrix \(R = E^\top F\).

2. Perform SVD on \(R = U D V^\top\).

3. Extract the first singular vectors: \(w = U[,1]\), \(q = V[,1]\).

4. Compute scores: \(t = E w\) (normalized), \(u = F q\).

5. Compute loadings: \(p = E^\top t\), regression scalar \(b = t^\top u\).

6. Deflate residuals: \(E \gets E - t p^\top\), \(F \gets F - b t q^\top\).

After all components are extracted, a post-processing step removes components with zero regression weight. The scaled regression coefficients are computed using the Moore–Penrose pseudoinverse of the loading matrix \(P\), and then rescaled to the original variable units.

References

Abdi, H., & Williams, L. J. (2013). Partial least squares methods: Partial least squares correlation and partial least square regression. Methods in Molecular Biology (Clifton, N.J.), 930, 549–579. doi:10.1007/978-1-62703-059-5_23

de Jong, S. (1993). SIMPLS: An alternative approach to partial least squares regression. Chemometrics and Intelligent Laboratory Systems, 18(3), 251–263. doi:10.1016/0169-7439(93)85002-X

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 = "SVD")
model$coefficients
} # }