Penalty Matrix for Higher Order Differences

Description

Computes the block-diagonal penalty matrix penalizing higher-order differences.

Usage

Penalty.matrix(m, order = 2)

Arguments

Details

For each block of size m[j], and default order = 2, computes:

v^\top P_j v = \sum_{i=3}^{m[j]} (v_i - 2v_{i-1} + v_{i-2})^2.

The final penalty matrix is block-diagonal composed of blocks P_j.

Value

Penalty matrix (numeric matrix) of dimension sum(m) x sum(m).

References

Kraemer, N., Boulesteix, A.-L., & Tutz, G. (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94, 60-69. https://doi.org/10.1016/j.chemolab.2008.06.009

Examples

P <- Penalty.matrix(c(6, 4), order = 2)

Nonlinear Transformation via B-Splines

Description

Transforms each column of a numeric matrix (or vector) into a new basis defined by B-spline functions.

Usage

X2s(X, Xtest = NULL, deg = 3, nknot = NULL, reduce.knots = FALSE)

Arguments

Value

A list containing:

Z

Design matrix for training data (B-spline coefficients).

Ztest

Design matrix for test data.

sizeZ

Vector of number of basis functions for each column.

References

Kraemer, N., Boulesteix, A.-L., & Tutz, G. (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94, 60-69. https://doi.org/10.1016/j.chemolab.2008.06.009

Examples

X <- matrix(rnorm(100), ncol = 5)
Xtest <- matrix(rnorm(300), ncol = 5)
result <- X2s(X, Xtest)

Extract Regression Coefficients from a mypls Object

Description

Returns the regression coefficients (without intercept) from an object of class mypls, typically produced by the function jack.ppls.

Usage

## S3 method for class 'mypls'
coef(object, ...)

Arguments

Details

This method returns the vector of regression coefficients associated with the penalized PLS fit stored in the mypls object. These coefficients can be used together with the variance-covariance matrix returned by vcov.mypls to construct confidence intervals or hypothesis tests.

Value

A numeric vector containing the regression coefficients corresponding to the penalized PLS model.

References

N. Kraemer, A.-L. Boulesteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94(1), 60–69. doi:10.1016/j.chemolab.2008.06.009

See Also

vcov.mypls, jack.ppls

Examples

n <- 50  # number of observations
p <- 5   # number of variables
X <- matrix(rnorm(n * p), ncol = p)
y <- rnorm(n)

pls.object <- penalized.pls.cv(X, y)
my.jack <- jack.ppls(pls.object)
my.coef <- coef(my.jack)
print(my.coef)

Near-Infrared (NIR) Spectroscopy of Biscuit Doughs

Description

This dataset contains measurements from a quantitative NIR spectroscopy experiment designed to evaluate the feasibility of using NIR spectra to estimate the chemical composition of unbaked biscuit doughs.

Usage

data(cookie)

Format

A list of 2 data-frames of 72 observations:

NIR

NIR reflectance spectrum values from 1100 to 2498 nm on 700 columns.

constituents

Percentage of fat, sucrose, dry flour and water in the 72 samples.

Details

Two sets of samples were prepared with variations in a standard biscuit recipe to produce a broad range for each of the four ingredients of interest: fat, sucrose, dry flour, and water.

The first 40 samples correspond to a calibration (training) set, and the remaining 32 samples form a validation (prediction) set. Sample 23 (training) and sample 21 (test) are known outliers.

Each sample is represented by an NIR reflectance spectrum composed of 700 values measured between 1100 and 2498 nanometers, at 2 nm intervals. The last 4 columns represent the percentage of each constituent.

References

• P.J. Brown, T. Fearn, and M. Vannucci (2001). Bayesian Wavelet Regression on Curves with Applications to a Spectroscopic Calibration Problem. Journal of the American Statistical Association, 96, pp. 398–408.

• B.G. Osborne, T. Fearn, A.R. Miller, and S. Douglas (1984). Application of Near-Infrared Reflectance Spectroscopy to Compositional Analysis of Biscuits and Biscuit Dough. Journal of the Science of Food and Agriculture, 35, pp. 99–105.

Examples

data(cookie) # load data
X <- cookie$NIR       # NIR spectra
Y <- cookie$constituents     # constituent values
Xtrain <- X[1:40, ]; Ytrain <- Y[1:40, ]   # calibration set
Xtest <- X[41:72, ]; Ytest <- Y[41:72, ]   # validation set

Plot Penalized PLS Components for Spline-Transformed Data

Description

This function applies a nonlinear regression model using penalized Partial Least Squares (PLS) on B-spline transformed variables, then visualizes each additive component.

Usage

graphic.ppls.splines(
  X,
  y,
  lambda = NULL,
  add.data = FALSE,
  select = FALSE,
  ncomp = 1,
  deg = 3,
  order = 2,
  nknot = NULL,
  reduce.knots = FALSE,
  kernel = TRUE,
  window.size = c(3, 3)
)

Arguments

Details

This function first transforms the input data X and a test grid Xtest using B-spline basis functions, then fits a penalized PLS model using these transformed variables. Each additive component (i.e., variable effect) is then plotted individually.

If add.data = TRUE, the actual observations are plotted on top of the corresponding fitted component functions. While this can help visualize the fit, note that only the sum of all fitted components approximates y, and not each component individually.

The function is intended for exploratory visualization and should be used after appropriate model selection using, e.g., ppls.splines.cv.

Value

A numeric vector of regression coefficients for the final penalized PLS model.

References

N. Kraemer, A.-L. Boulesteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94(1), 60–69. doi:10.1016/j.chemolab.2008.06.009

See Also

ppls.splines.cv, X2s, penalized.pls, Penalty.matrix

Examples

# Load Boston housing data
library(MASS)
data(Boston)
y <- Boston[, 14]
X <- Boston[, -14]
X <- X[, -4]  # remove categorical variable
X <- as.matrix(X)

# Plot with variable selection and original data
graphic.ppls.splines(
  X, y, lambda = 100, ncomp = 5,
  add.data = TRUE, select = TRUE, window.size = c(3, 4)
)

# Plot without variable selection and without data
graphic.ppls.splines(
  X, y, lambda = 100, ncomp = 5,
  add.data = FALSE, select = FALSE, window.size = c(3, 4)
)

Jackknife Estimation for Penalized PLS Coefficients

Description

This function computes jackknife estimates (mean and covariance) of the regression coefficients obtained from a cross-validated Penalized Partial Least Squares (PPLS) model.

Usage

jack.ppls(
  ppls.object,
  ncomp = ppls.object$ncomp.opt,
  index.lambda = ppls.object$index.lambda
)

Arguments

Details

The jackknife estimates are computed using the array of regression coefficients obtained in each cross-validation fold. The function returns both the mean coefficients and the associated variance-covariance matrix.

If the requested number of components ncomp or the lambda index index.lambda exceeds the available dimensions of the coefficients.jackknife array, they are adjusted to their maximum allowable values, with a message.

Note: This jackknife procedure is not discussed in Kraemer et al. (2008), but it is useful for statistical inference, such as confidence intervals or hypothesis tests.

Value

An object of class "mypls", which is a list containing:

coefficients

The mean regression coefficients across cross-validation splits.

covariance

The estimated covariance matrix of the coefficients.

k

Number of cross-validation folds used.

ncomp

Number of components used in estimation.

index.lambda

Index of the lambda value used.

References

N. Kraemer, A.-L. Boulesteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94(1), 60–69. doi:10.1016/j.chemolab.2008.06.009

See Also

penalized.pls.cv, coef.mypls, vcov.mypls, ttest.ppls

Examples

data(cookie)  # load example data
X <- as.matrix(cookie$NIR)  # NIR spectra
y <- cookie$constituents$fat    # extract one constituent

pls.object <- penalized.pls.cv(X, y, ncomp = 10, kernel = TRUE)
my.jack <- jack.ppls(pls.object)
coef(my.jack)
vcov(my.jack)

Normalize a Numeric Vector to Unit Length

Description

Returns the input vector normalized to have unit Euclidean norm (i.e., length equal to 1).

Usage

normalize.vector(v)

Arguments

Details

This function performs:

v_\text{normalized} = \frac{v}{\sqrt{\sum v_i^2}}

It is primarily used to normalize weight vectors or component directions in Partial Least Squares algorithms.

Note: If the input vector has zero norm, the function returns NaN due to division by zero.

Value

A numeric vector of the same length as v, with unit norm.

See Also

penalized.pls, penalized.pls.default, penalized.pls.kernel

Examples

v <- c(3, 4)
normalize.vector(v)  # returns c(0.6, 0.8)

v2 <- rnorm(10)
sqrt(sum(normalize.vector(v2)^2))  # should be 1

Predict New Data Using a Penalized PLS Model

Description

Given a fitted penalized PLS model and new test data, this function predicts the response for all components. If true response values are provided, it also returns the mean squared error (MSE) for each component.

Computes the regression coefficients for a Penalized Partial Least Squares (PPLS) model, using either a classical NIPALS algorithm or a kernel-based version. Optionally allows block-wise variable selection.

Performs k-fold cross-validation to evaluate and select the optimal penalization parameter lambda and the number of components ncomp in a PPLS model.

Computes the regression coefficients using the standard (NIPALS-based) version of Penalized PLS. This function is typically called internally by penalized.pls.

Computes the regression coefficients using the kernel-based version of Penalized PLS, especially useful when the number of predictors exceeds the number of observations (p >> n).

Computes the regression coefficients of a Penalized Partial Least Squares (PPLS) model using block-wise selection, where each component is restricted to use variables from only one block.

Usage

new.penalized.pls(ppls, Xtest, ytest = NULL)

penalized.pls(
  X,
  y,
  P = NULL,
  ncomp = NULL,
  kernel = FALSE,
  scale = FALSE,
  blocks = 1:ncol(X),
  select = FALSE
)

penalized.pls.cv(
  X,
  y,
  P = NULL,
  lambda = 1,
  ncomp = NULL,
  k = 5,
  kernel = FALSE,
  scale = FALSE
)

penalized.pls.default(X, y, M = NULL, ncomp)

penalized.pls.kernel(X, y, M = NULL, ncomp)

penalized.pls.select(X, y, M = NULL, ncomp, blocks)

Arguments

Details

The fitted model ppls contains intercepts and regression coefficients for each number of components (from 1 to ncomp). The function computes:

• the matrix of predicted values for each component (as columns),

• and, if ytest is provided, a vector of mean squared errors for each component.

The prediction is performed as:

\hat{y}^{(i)} = X_\text{test} \cdot \beta^{(i)} + \text{intercept}^{(i)},

for each number of components i = 1, \ldots, ncomp.

This function centers X and y, and optionally scales X, then computes PPLS components using one of:

• the classical NIPALS algorithm (kernel = FALSE), or

• the kernel representation (kernel = TRUE), often faster when p > n (high-dimensional case).

When a penalty matrix P is supplied, a transformation M = (I + P)^{-1} is computed internally. The algorithm then maximizes the penalized covariance between Xw and y:

\text{argmax}_w \; \text{Cov}(Xw, y)^2 - \lambda \cdot w^\top P w

The block-wise selection strategy (when select = TRUE) restricts the weight vector w at each iteration to be non-zero in a single block, selected greedily.

The function splits the data into k cross-validation folds, and for each value of lambda and number of components up to ncomp, computes the mean squared prediction error.

The optimal parameters are selected as those minimizing the prediction error across all folds. Internally, for each fold and lambda value, the function calls penalized.pls to fit the model and new.penalized.pls to evaluate predictions.

The returned object can be further used for statistical inference (e.g., via jackknife) or prediction.

The method is based on iteratively computing latent directions that maximize the covariance with the response y. At each step:

• A weight vector w is computed as w = M X^\top y (if penalization is used).

• The latent component t = X w is extracted and normalized.

• The matrix X is deflated orthogonally with respect to t.

The final regression coefficients are computed via a triangular system using the bidiagonal matrix R = T^\top X W, and backsolving:

\beta = W L (T^\top y),

where L = R^{-1}.

The kernel PPLS algorithm is based on representing the model in terms of the Gram matrix K = X M X^\top (or simply K = X X^\top if M = NULL). The algorithm iteratively computes orthogonal latent components t_i in sample space.

Steps:

1. Initialize residual vector u = y, then normalize t = Ku.

2. Orthogonalize t with respect to previous components (if needed).

3. Repeat for ncomp components.

The regression coefficients are recovered as:

\beta = X^\top A, \quad \text{where } A = UU L (T^\top y),

with UU and TT the matrices of latent vectors and components, and L = R^{-1} the back-solved triangular system.

This function implements a sparse selection strategy inspired by sparse or group PLS. At each component iteration, it computes the penalized covariance between X and y, and selects the block k for which the mean squared weight of its variables is maximal:

\text{score}_k = \frac{1}{|B_k|} \sum_{j \in B_k} w_j^2

Only the weights corresponding to the selected block are retained, and all others are set to zero. The rest of the algorithm follows the classical NIPALS-like PLS with orthogonal deflation.

This procedure enhances interpretability by selecting only one block per component, making it suitable for structured variable selection (e.g., grouped predictors).

Value

A list containing:

ypred

A numeric matrix of predicted responses. Each column corresponds to a different number of PLS components.

mse

A numeric vector of mean squared errors, if ytest is provided. Otherwise NULL.

A list with components:

intercept

A numeric vector of intercepts for 1 to ncomp components.

coefficients

A numeric matrix of size ncol(X) x ncomp, each column being the coefficient vector for the corresponding number of components.

An object of class "mypls", a list with the following components:

error.cv

A matrix of mean squared errors. Rows correspond to different lambda values; columns to different numbers of components.

lambda

The vector of candidate lambda values.

lambda.opt

The lambda value giving the minimum cross-validated error.

index.lambda

The index of lambda.opt in lambda.

ncomp.opt

The optimal number of PLS components.

min.ppls

The minimum cross-validated error.

intercept

Intercept of the optimal model (fitted on the full dataset).

coefficients

Coefficient vector for the optimal model.

coefficients.jackknife

An array of shape ncol(X) x ncomp x length(lambda) x k, containing the coefficients from each CV split and parameter setting.

A list with:

coefficients

A matrix of size ncol(X) x ncomp, each column containing the regression coefficients for the first i components.

A list with:

coefficients

A matrix of size ncol(X) x ncomp, containing the estimated regression coefficients for each number of components.

A list with:

coefficients

A matrix of size ncol(X) x ncomp, containing the regression coefficients after block-wise selection.

References

N. Kraemer, A.-L. Boulesteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94(1), 60–69. doi:10.1016/j.chemolab.2008.06.009

N. Kraemer, A.-L. Boulesteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94(1), 60–69. doi:10.1016/j.chemolab.2008.06.009

N. Kraemer, A.-L. Boulesteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94(1), 60–69. doi:10.1016/j.chemolab.2008.06.009

See Also

penalized.pls, penalized.pls.cv, ppls.splines.cv

penalized.pls.cv, new.penalized.pls, ppls.splines.cv, Penalty.matrix

penalized.pls, new.penalized.pls, jack.ppls, ppls.splines.cv

penalized.pls, penalized.pls.kernel, normalize.vector

penalized.pls, penalized.pls.default, normalize.vector

penalized.pls, penalized.pls.cv, normalize.vector

Examples

set.seed(123)
X <- matrix(rnorm(50 * 200), ncol = 50)
y <- rnorm(200)

Xtrain <- X[1:100, ]
ytrain <- y[1:100]
Xtest <- X[101:200, ]
ytest <- y[101:200]

pen.pls <- penalized.pls(Xtrain, ytrain, ncomp = 10)
pred <- new.penalized.pls(pen.pls, Xtest, ytest)
head(pred$ypred)
pred$mse

## Example from Kraemer et al. (2008)
data(BOD)
X <- BOD[, 1]
y <- BOD[, 2]

Xtest <- seq(min(X), max(X), length = 200)
dummy <- X2s(X, Xtest, deg = 3, nknot = 20)  # Spline transformation
Z <- dummy$Z
Ztest <- dummy$Ztest
size <- dummy$sizeZ
P <- Penalty.matrix(size, order = 2)
lambda <- 200
number.comp <- 3

ppls <- penalized.pls(Z, y, P = lambda * P, ncomp = number.comp)
new.ppls <- new.penalized.pls(ppls, Ztest)$ypred

# Plot fitted values for 2 components
plot(X, y, lwd = 3, xlim = range(Xtest))
lines(Xtest, new.ppls[, 2], col = "blue")

set.seed(42)
X <- matrix(rnorm(20 * 100), ncol = 20)
y <- rnorm(100)

# Example with no penalty
result <- penalized.pls.cv(X, y, lambda = c(0, 1, 10), ncomp = 5)
result$lambda.opt
result$ncomp.opt
result$min.ppls

# Using jackknife estimation after CV
jack <- jack.ppls(result)
coef(jack)

set.seed(123)
X <- matrix(rnorm(20 * 50), nrow = 50)
y <- rnorm(50)
M <- diag(ncol(X))  # No penalty
coef <- penalized.pls.default(scale(X, TRUE, FALSE), scale(y, TRUE, FALSE),
  M, ncomp = 3)$coefficients
coef[, 1]  # coefficients for 1st component

set.seed(123)
X <- matrix(rnorm(100 * 10), nrow = 100)
y <- rnorm(100)
K <- X %*% t(X)
coef <- penalized.pls.kernel(X, y, M = NULL, ncomp = 2)$coefficients
head(coef[, 1])  # coefficients for 1st component

set.seed(321)
X <- matrix(rnorm(40 * 30), ncol = 40)
y <- rnorm(30)

# Define 4 blocks of 10 variables each
blocks <- rep(1:4, each = 10)
result <- penalized.pls.select(X, y, M = NULL, ncomp = 2, blocks = blocks)
result$coefficients[, 1]  # Coefficients for first component

Cross-Validation for Penalized PLS with Spline-Transformed Predictors

Description

Performs cross-validation to select the optimal number of components and penalization parameter for a penalized partial least squares model (PPLS) fitted to spline-transformed predictors.

Usage

ppls.splines.cv(
  X,
  y,
  lambda = 1,
  ncomp = NULL,
  degree = 3,
  order = 2,
  nknot = NULL,
  k = 5,
  kernel = FALSE,
  scale = FALSE,
  reduce.knots = FALSE,
  select = FALSE
)

Arguments

Details

This function performs the following steps for each cross-validation fold:

1. Transforms predictors using B-spline basis functions via X2s.

2. Computes the penalty matrix using Penalty.matrix.

3. Fits a penalized PLS model using penalized.pls with the given lambda and number of components.

4. Evaluates prediction performance on the test fold using new.penalized.pls.

The optimal parameters are those minimizing the average squared prediction error across all folds.

Value

A list with the following components:

error.cv

Matrix of prediction errors: rows = lambda values, columns = components.

min.ppls

The minimum cross-validated error.

lambda.opt

Optimal lambda value.

ncomp.opt

Optimal number of components.

References

N. Kraemer, A.-L. Boulesteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94(1), 60–69. doi:10.1016/j.chemolab.2008.06.009

See Also

X2s, Penalty.matrix, penalized.pls, penalized.pls.cv

Examples

# Simulated data
set.seed(123)
X <- matrix(rnorm(30 * 100), ncol = 30)
y <- rnorm(100)

# Run CV with 3 lambdas and max 4 components
result <- ppls.splines.cv(X, y, lambda = c(1, 10, 100), ncomp = 4)
result$lambda.opt
result$ncomp.opt

Simulate Data for Penalized Partial Least Squares (PPLS)

Description

Generates a training and test dataset with non-linear relationships between predictors and response, as used in PPLS simulation studies.

Usage

sim.data.ppls(ntrain, ntest, stnr, p, a = NULL, b = NULL)

Arguments

Details

The function simulates a response variable y as a combination of additive linear and sinusoidal effects of the first 5 predictors:

f(x) = \sum_{j=1}^{5} a_j x_j + \sin(6 b_j x_j)

The response y is then generated by adding Gaussian noise scaled to match the specified signal-to-noise ratio (stnr).

Remaining variables (p - 5) are included as noise variables, making the dataset suitable to evaluate selection or regularization methods.

Value

A list with the following components:

Xtrain

ntrain x p matrix of training predictors (uniform in [-1, 1]).

ytrain

Numeric vector of training responses.

Xtest

ntest x p matrix of test predictors.

ytest

Numeric vector of test responses.

sigma

Standard deviation of the added noise.

a

Linear coefficients used in the simulation.

b

Nonlinear sine coefficients used in the simulation.

See Also

ppls.splines.cv, graphic.ppls.splines

Examples

set.seed(123)
sim <- sim.data.ppls(ntrain = 100, ntest = 100, stnr = 3, p = 10)
str(sim)
plot(sim$Xtrain[, 1], sim$ytrain, main = "Effect of x1 on y")

t-Test for Penalized PLS Regression Coefficients

Description

Computes two-sided t-tests and p-values for the regression coefficients of a penalized PLS model based on jackknife estimation.

Usage

ttest.ppls(
  ppls.object,
  ncomp = ppls.object$ncomp.opt,
  index.lambda = ppls.object$index.lambda
)

Arguments

Details

This function calls jack.ppls to estimate:

• The mean of the jackknife coefficients (point estimates),

• The covariance matrix (for standard errors),

• The degrees of freedom, equal to k - 1, where k is the number of cross-validation folds.

It then performs standard two-sided t-tests:

t_j = \frac{\hat{\beta}_j}{\text{SE}_j}, \quad \text{df} = k - 1

and computes associated p-values.

These p-values can be used for variable selection or inference, although they are based on cross-validation folds and should be interpreted with caution.

Value

A list with:

tvalues

Numeric vector of t-statistics.

pvalues

Numeric vector of two-sided p-values.

See Also

jack.ppls, coef.mypls, vcov.mypls

Examples

set.seed(123)
X <- matrix(rnorm(20 * 100), ncol = 20)
y <- rnorm(100)
result <- penalized.pls.cv(X, y, lambda = c(0, 1), ncomp = 3)
tstats <- ttest.ppls(result)
print(tstats$pvalues)

Variance-Covariance Matrix for Penalized PLS Coefficients

Description

Returns the estimated variance-covariance matrix of the regression coefficients from a jackknife-based PPLS model.

Usage

## S3 method for class 'mypls'
vcov(object, ...)

Arguments

Details

The function retrieves the covariance matrix stored in the object$covariance field. If this field is NULL, a warning is issued and the function returns NULL.

This method can be used in conjunction with coef.mypls and ttest.ppls to conduct inference on the coefficients of a penalized PLS model.

Value

A numeric matrix representing the variance-covariance matrix of the regression coefficients, or NULL if unavailable.

See Also

coef.mypls, jack.ppls, ttest.ppls

Examples

set.seed(42)
X <- matrix(rnorm(30 * 100), ncol = 30)
y <- rnorm(100)

ppls.cv <- penalized.pls.cv(X, y, lambda = c(1, 10), ncomp = 3)
myjack <- jack.ppls(ppls.cv)
Sigma <- vcov(myjack)
Sigma[1:5, 1:5]