mgss: A Matrix-Free Multigrid Preconditioner for Spline Smoothing

Description

Data smoothing with penalized splines is a popular method and is well established for one- or two-dimensional covariates. The extension to multiple covariates is straightforward but suffers from exponentially increasing memory requirements and computational complexity. This toolbox provides a matrix-free implementation of a conjugate gradient (CG) method for the regularized least squares problem resulting from tensor product B-spline smoothing with multivariate and scattered data. It further provides matrix-free preconditioned versions of the CG-algorithm where the user can choose between a simpler diagonal preconditioner and an advanced geometric multigrid preconditioner. The main advantage is that all algorithms are performed matrix-free and therefore require only a small amount of memory. For further detail see Siebenborn & Wagner (2021).

Author(s)

Maintainer: Martin Siebenborn martin.siebenborn@uni-hamburg.de [copyright holder]

Authors:

• Julian Wagner juwagn89@gmail.com [copyright holder]

See Also

Useful links:

• Report bugs at https://github.com/SplineSmoothing/MGSS

High-dimensional spline smoothing using a matrix-free CG-method.

Description

Fits a smooth spline to a set of given observations using penalized splines with curvature or difference penalty and multiple covariates. The underlying linear system is solved with a matrix-free conjugated gradient (CG) method.

Usage

CG_smooth(
  m,
  q,
  lambda,
  X,
  y,
  pen_type = "curve",
  l = NULL,
  alpha_start = NULL,
  K_max = NULL,
  tolerance = 1e-06,
  print_error = TRUE
)

Arguments

Value

Returns a list containing the input m, q, and Omega. Further gives the fitted spline coefficients alpha, the fitted values fitted_values, the residuals residuals, the root mean squared error rmse and the R-squared value R_squared.

Examples

data <- generate_test_data(100, 2)
X <- data$X_train
y <- data$y_train
CG_smooth(m = c(7,7), q = c(3,3), lambda = 0.1, X = X, y = y)

High-dimensional spline smoothing using a matrix-free multigrid preconditioned CG-method.

Description

Fits a smooth spline to a set of given observations using penalized splines with curvature penalty and multiple covariates. The underlying linear system is solved with a matrix-free preconditioned conjugated gradient method using a geometric multigrid method as preconditioner.

Usage

MGCG_smooth(
  G,
  q,
  lambda,
  X,
  y,
  w = 0.1,
  nu = c(3, 1),
  alpha_start = NULL,
  K_max = NULL,
  tolerance = 1e-06,
  print_error = TRUE,
  coarse_grid_solver = "Cholesky"
)

Arguments

Value

Returns a list containing the input m = 2^G-1, q, and Omega. Further gives the fitted spline coefficients alpha, the fitted values fitted_values, the residuals residuals, the root mean squared error rmse and the R-squared value R_squared.

References

Siebenborn, M. and Wagner, J. (2019) A Multigrid Preconditioner for Tensor Product Spline Smoothing. arXiv:1901.00654

Examples

data <- generate_test_data(100, 2)
X <- data$X_train
y <- data$y_train
MGCG_smooth(G = 3, q = c(3,3), lambda = 0.1, w = 0.8, X = X, y = y)

High-dimensional spline smoothing using a matrix-free PCG-method.

Description

Fits a smooth spline to a set of given observations using penalized splines with curvature or difference penalty and multiple covariates. The underlying linear system is solved with a matrix-free preconditioned conjugated gradient (PCG) method using a diagonal preconditioner.

Usage

PCG_smooth(
  m,
  q,
  lambda,
  X,
  y,
  pen_type = "curve",
  l = NULL,
  alpha_start = NULL,
  K_max = NULL,
  tolerance = 1e-06,
  print_error = TRUE
)

Arguments

Value

Returns a list containing the input m, q, and Omega. Further gives the fitted spline coefficients alpha, the fitted values fitted_values, the residuals residuals, the root mean squared error rmse and the R-squared value R_squared.

Examples

data <- generate_test_data(100, 2)
X <- data$X_train
y <- data$y_train
PCG_smooth(m = c(7,7), q = c(3,3), lambda = 0.1, X = X, y = y)

Trace estimation of the hat matrix.

Description

Estimates the trace of the (unknown) hat-matrix by stochastic estimation in a matrix-free manner.

Usage

estimate_trace(m, q, lambda, X, pen_type = "curve", l = NULL, n_random = 5)

Arguments

Value

An estimate of the trace of the hat-matrix.

Examples

data <- generate_test_data(100, 2)
X <- data$X_train
estimate_trace(m = c(7,7), q = c(2,2), lambda = 0.1, X = X)

Generate multi-dimensional test data for spline smoothing.

Description

Generate a P-dimensional test data set based on a sigmoid function.

Usage

generate_test_data(n, P, split = 0.8)

Arguments

Value

A list of the covarite matrices for the train and test data X_train and X_test and of the variable of interest y_train and y_test.

Examples

generate_test_data(100, 2)

Predictions from model

Description

Makes predictions of new observations from a fitted spline model.

Usage

predict_smooth(model_smooth, X)

Arguments

Value

Vector of length nrow(X) of predictions.

Examples

data <- generate_test_data(100, 2)
X <- data$X_train
y <- data$y_train
result <- PCG_smooth(m = c(7,7), q = c(3,3), lambda = 0.1, X = X, y = y, print_error = FALSE)
X_test <- data$X_test
predict_smooth(model_smooth = result, X = X_test)