Type:	Package
Title:	Slab and Shrinkage Linear Regression Estimation
Version:	0.1.0
Description:	Implements a suite of shrinkage estimators for multivariate linear regression to improve estimation stability and predictive accuracy. Provides methods including the Stein estimator, Diagonal Shrinkage, the general Shrinkage estimator (solving a Sylvester equation), and Slab Regression (Simple and Generalized). These methods address Stein's paradox by introducing structured bias to reduce variance without requiring cross-validation, except for Shrinkage Ridge Regression where the intensity is chosen by minimizing an explicit Mean Squared Error (MSE) criterion. Methods are based on paper https://openaccess.city.ac.uk/id/eprint/35005/.
License:	GPL (≥ 3)
Encoding:	UTF-8
URL:	https://ziwei-chenchen.github.io/savvySh/
RoxygenNote:	7.3.2
Depends:	R (≥ 3.6.0)
Imports:	Matrix, glmnet, MASS, expm, mnormt, stats
Suggests:	knitr, rmarkdown, testthat (≥ 3.0.0)
VignetteBuilder:	knitr
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-02-24 17:49:54 UTC; lanceyun
Author:	Ziwei Chen [aut, cre], Vali Asimit [aut], Marina Anca Cidota [aut], Jennifer Asimit [aut]
Maintainer:	Ziwei Chen <Ziwei.Chen.3@citystgeorges.ac.uk>
Repository:	CRAN
Date/Publication:	2026-03-03 10:00:07 UTC

Extract Coefficients for a Slab and Shrinkage Linear Regression Model

Description

Extracts the regression coefficients from a savvySh_model object. You may specify one or more shrinkage estimators through the estimator parameter. If no estimator is specified, the function returns coefficients for all available estimators as a named list.

Usage

## S3 method for class 'savvySh_model'
coef(object, estimator = NULL, ...)

Arguments

Details

This function internally calls predict.savvySh_model with type = "coefficients" to retrieve the desired coefficient estimates. If multiple estimators are requested (or if none is specified, in which case all are returned), the output is a named list in which each element is a numeric vector of coefficients. The coefficient vectors are named according to whether an intercept is present (for Linear shrinkage, no intercept). If a single estimator is specified, a single named numeric vector is returned.

Value

A named numeric vector of regression coefficients if a single estimator is specified, or a named list of such vectors if multiple estimators are requested.

Author(s)

Ziwei Chen, Vali Asimit, Marina Anca Cidota, Jennifer Asimit
Maintainer: Ziwei Chen <ziwei.chen.3@citystgeorges.ac.uk>

See Also

predict.savvySh_model for generating predictions, savvySh for fitting slab and shrinkage linear models.

Examples

# Generate simulated data for example
set.seed(123)
x <- matrix(rnorm(100 * 5), 100, 5)
y <- rnorm(100)

# Fit a Multiplicative shrinkage model
fit <- savvySh(x, y, model_class = "Multiplicative", include_Sh = TRUE)

# Extract coefficients for all available estimators (St and DSh by default)
all_coefs <- coef(fit)

# Extract coefficients for a specific estimator
st_coefs <- coef(fit, estimator = "St")
print(st_coefs)

Predict Method for Slab and Shrinkage Linear Regression Models

Description

Generate predictions (fitted values) or extract regression coefficients from a savvySh_model object returned by savvySh. This function allows you to specify one or more shrinkage estimators (via the estimator parameter) available in the model. If no estimator is specified, all available estimators are used and their results are returned in a named list.

Usage

## S3 method for class 'savvySh_model'
predict(
  object,
  newx = NULL,
  type = c("response", "coefficients"),
  estimator = NULL,
  ...
)

Arguments

Details

The behavior depends on the value of type:

"response":

Generates predicted values using the coefficient estimates from the specified shrinkage estimator(s) for new data supplied via newx.

"coefficients":

Extracts the regression coefficient vector(s) corresponding to the specified estimator(s). Coefficient names are assigned based on whether an intercept is present (for Linear shrinkage, no intercept).

If no estimator is specified, the function returns results for all available estimators as a named list. If a single estimator is specified (or only one is provided in the vector), the result is returned as a numeric vector (for coefficients) or a numeric vector of predictions (for response).

Value

If type = "response", the function returns:

• A numeric vector of predicted values if exactly one estimator is specified;

• Otherwise, a named list of numeric vectors, one for each specified estimator.

If type = "coefficients", the function returns:

• A named numeric vector of regression coefficients if exactly one estimator is specified;

• Otherwise, a named list of numeric vectors corresponding to each specified estimator.

Author(s)

Ziwei Chen, Vali Asimit, Marina Anca Cidota, Jennifer Asimit
Maintainer: Ziwei Chen <ziwei.chen.3@citystgeorges.ac.uk>

See Also

savvySh for fitting slab and shrinkage linear models, coef.savvySh_model for direct coefficient extraction.

Examples

# Generate simulated data
set.seed(123)
x <- matrix(rnorm(100 * 5), 100, 5)
y <- rnorm(100)

# Fit a Multiplicative shrinkage model
fit <- savvySh(x, y, model_class = "Multiplicative")

# Generate predictions for new data
new_x <- matrix(rnorm(10 * 5), 10, 5)
preds <- predict(fit, newx = new_x, type = "response")

# Extract coefficients for specific estimators
coefs_st <- predict(fit, type = "coefficients", estimator = "St")

Print a Slab and Shrinkage Model Summary

Description

Displays a concise summary of a fitted savvySh_model object, including the original function call, the chosen model class, the number of non-zero coefficients per estimator, and the optimal lambda value (if applicable). Additionally, it prints the coefficients for the specified estimator(s) with user-specified precision.

Usage

## S3 method for class 'savvySh_model'
print(x, digits = max(3, getOption("digits") - 3), estimator = NULL, ...)

Arguments

Details

This print method provides a quick diagnostic of the fitted model by showing:

Summary Metrics

A table that includes, for each estimator, the number of non-zero coefficients and the optimal lambda (if applicable).

Coefficients

For each selected estimator, the coefficients are printed with appropriate names: if an intercept is present, it is labeled (Intercept) and the remaining coefficients are labeled according to the predictor names.

If the user does not specify an estimator using the estimator argument, the function prints information for all available estimators stored in the model. If one or more estimators are specified, only those are printed, after verifying that they exist in x$coefficients.

The method invisibly returns a summary data.frame containing key metrics for each estimator.

Value

Invisibly returns a data.frame summarizing each selected estimator's name, number of non-zero coefficients, and the final optimal_lambda (if any).

Author(s)

Ziwei Chen, Vali Asimit, Marina Anca Cidota, Jennifer Asimit
Maintainer: Ziwei Chen <ziwei.chen.3@citystgeorges.ac.uk>

See Also

savvySh for fitting slab and shrinkage linear models, coef.savvySh_model and predict.savvySh_model for extracting coefficients and generating predictions.

Examples

# Generate simulated data for demonstration
set.seed(123)
x <- matrix(rnorm(100 * 5), 100, 5)
y <- rnorm(100)

# Fit a Multiplicative shrinkage model
fit <- savvySh(x, y, model_class = "Multiplicative", include_Sh = TRUE)

# Default print: shows summary metrics and coefficients for all estimators
print(fit)

# Print with specific digits and only for one estimator
print(fit, digits = 4, estimator = "St")

Slab and Shrinkage Linear Regression Estimation

Description

This function estimates coefficients in a linear regression model using several shrinkage methods, including Multiplicative Shrinkage, Slab Regression, Linear shrinkage, and Shrinkage Ridge Regression. Each method gives estimators that balance bias and variance by applying shrinkage to the ordinary least squares (OLS) solution. The shrinkage estimators are computed based on different assumptions about the data.

Usage

savvySh(x, y, model_class = c("Multiplicative", "Slab", "Linear", "ShrinkageRR"),
               v = 1, lambda_vals = NULL, nlambda = 100, folds = 10,
               foldid = FALSE, include_Sh = FALSE, exclude = NULL)

Arguments

Details

The Slab and Shrinkage Linear Regression Estimation methodology provides four classes of shrinkage estimators that reduce variance in the OLS solution by introducing a small, structured bias. These methods handle overfitting, collinearity, and high-dimensional scenarios by controlling how and where the coefficients are shrunk. Each class offers a distinct strategy for controlling instability and improving mean squared error (MSE) in linear models, tailored for different modeling contexts specified in the model_class argument. Note that if the user provides more than one option in model_class, only the first option is used, and a warning is issued.

Model Classes:

Multiplicative Shrinkage:

This class includes three estimators that use the OLS coefficients as a starting point and apply multiplicative adjustments:

St -

Stein estimator, which shrinks all coefficients toward zero by a single global factor. This aims to reduce MSE while keeping the overall bias fairly uniform across coefficients.

DSh -

Diagonal Shrinkage, assigning an individual factor to each coefficient based on its variance. This yields more targeted shrinkage than the global approach and often achieves a lower MSE.

Sh -

Shrinkage estimator that solves a Sylvester equation for a full (non-diagonal) shrinkage matrix. It is more flexible but also more computationally demanding. Included only if include_Sh = TRUE.

Slab Regression:

Slab Regression applies an adaptive quadratic penalty term to the OLS objective:

SR -

Simple Slab Regression, which modifies the OLS objective by adding a penalty in a fixed direction (often the constant vector). This penalty is controlled by v and does not require cross-validation. It can be viewed as a special case of the generalized lasso but focuses on smooth (quadratic) rather than \ell_1 regularization.

GSR -

Generalized Slab Regression, extending SR by allowing shrinkage along multiple directions. Typically, these directions correspond to the eigenvectors of the design covariance matrix, effectively shrinking principal components.

Linear Shrinkage:

The Linear Shrinkage (LSh) estimator forms a convex combination of the OLS estimator (through the origin) and a target estimator that assumes uncorrelated predictors (diagonal approximation of the covariance). This approach is simpler than a full matrix method and is well-suited for standardized data where the intercept is not needed.

Shrinkage Ridge Regression:

The Shrinkage Ridge Regression (SRR) extends standard RR by shrinking the design covariance matrix toward a spherical target (i.e., a diagonal matrix with equal entries). This additional regularization stabilizes the eigenvalues and yields more robust coefficient estimates, particularly when the predictors lie close to a low-dimensional subspace.

Value

A list containing the following elements:

Author(s)

Ziwei Chen, Vali Asimit, Marina Anca Cidota, Jennifer Asimit
Maintainer: Ziwei Chen <ziwei.chen.3@citystgeorges.ac.uk>

References

Asimit, V., Cidota, M. A., Chen, Z., & Asimit, J. (2025). Slab and Shrinkage Linear Regression Estimation. Retrieved from https://openaccess.city.ac.uk/id/eprint/35005/

Examples

# 1. Simple Multiplicative Shrinkage example
set.seed(123)
x <- matrix(rnorm(100 * 5), 100, 5)
y <- rnorm(100)
fit_mult <- savvySh(x, y, model_class = "Multiplicative", include_Sh = TRUE)
print(fit_mult)

# 2. Slab Regression example
fit_slab <- savvySh(x, y, model_class = "Slab", v = 2)
coef(fit_slab, estimator = "GSR")

# 3. Linear Shrinkage (standardized data recommended)
x_centered <- scale(x, center = TRUE, scale = FALSE)
y_centered <- scale(y, center = TRUE, scale = FALSE)
fit_linear <- savvySh(x_centered, y_centered, model_class = "Linear")

# 4. Shrinkage Ridge Regression
fit_srr <- savvySh(x, y, model_class = "ShrinkageRR")
predict(fit_srr, newx = matrix(rnorm(10 * 5), 10, 5), type = "response")

Summarize a Slab and Shrinkage Linear Regression Model

Description

Provides a comprehensive summary for one or more shrinkage estimators contained within a savvySh_model object produced by savvySh. The summary includes estimated coefficients, confidence intervals, residual statistics, R-squared measures, F-statistics, and information criteria (AIC, BIC) for each specified estimator.

Usage

## S3 method for class 'savvySh_model'
summary(object, estimator = NULL, ...)

Arguments

Details

For each estimator present in the savvySh_model object (or for the user-specified subset), this function computes:

• A summary of the residual distribution (quantiles).

• A coefficient table including estimates, standard errors, t-values, p-values, and confidence intervals.

• Residual standard error and degrees of freedom.

• R-squared and adjusted R-squared measures.

• F-statistic (and its p-value) for testing overall regression significance.

• Information criteria (AIC, BIC) and deviance for model fit.

These results are printed in sequence for the selected estimator(s). If no estimator is specified, summaries for all available estimators are printed.

Value

Invisibly returns a data.frame summarizing key metrics for each estimator (including estimator name, number of non-zero coefficients, and optimal lambda if available).

Author(s)

Ziwei Chen, Vali Asimit, Marina Anca Cidota, Jennifer Asimit
Maintainer: Ziwei Chen <ziwei.chen.3@citystgeorges.ac.uk>

See Also

savvySh for fitting slab and shrinkage linear models, predict.savvySh_model for generating predictions, coef.savvySh_model for extracting coefficients directly.

Examples

# Generate simulated data for demonstration
set.seed(123)
x <- matrix(rnorm(100 * 5), 100, 5)
y <- rnorm(100)

# Fit a Slab Regression model
fit <- savvySh(x, y, model_class = "Slab")

# Print a detailed summary for all estimators (SR and GSR)
summary(fit)

# Summarize only a specific estimator
summary(fit, estimator = "GSR")