Package '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

m	Vector of non-negative integers. Each entry gives the number of inner knots for the respective covariate.
q	Vector of positive integers. Each entry gives the spline degree for the respective covariate.
lambda	Positive number as weight for the penalty term.
X	Matrix containing the covariates as columns and the units as rows.
y	Vector of length nrow(X) as the variable of interest.
pen_type	Utilized penalization method. Either "curve" for the curvature penalty or "diff" for the difference penalty. Defaults to "curve".
l	Positive integer vector of length P indicating for the penalty degree. Only required if pen_type = "diff".
alpha_start	Vector of length prod(m+q+1) as starting value for the CG-method. Defaults to zero.
K_max	Positive integer as upper bound for the number of CG-iterations. Defaults to prod(m+q+1).
tolerance	Positive number as error tolerance for the stopping criterion of the CG-method. Defaults to 1e-6.
print_error	Logical, indicating if the iteration error should be printed or not.

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)

---

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

m	Vector of non-negative integers. Each entry gives the number of inner knots for the respective covariate.
q	Vector of positive integers. Each entry gives the spline degree for the respective covariate.
lambda	Positive number as weight for the penalty term.
X	Matrix containing the covariates as columns and the units as rows.
pen_type	Utilized penalization method. Either "curve" for the curvature penalty or "diff" for the difference penalty. Defaults to "curve".
l	Positive integer vector of length P indicating for the penalty degree. Only required if pen_type = "diff".
n_random	Positive integer for the number of random vectors in the trace estimate. Defaults to 5.

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

n	Numer of samples
P	Spatial dimension
split	A value between 0 and 1 for the train / test split.

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)

---

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

G	Positive integer greater than one for the maximum number of grids.
q	Vector of positive integers. Each entry gives the spline degree for the respective covariate.
lambda	Positive number as weight for the penalty term.
X	Matrix containing the covariates as columns and the units as rows.
y	Vector of length nrow(X) as the variable of interest.
w	Damping factor of the Jacobi smoother. Defaults to 0.1.
nu	Two-dimensional vector of non-negative integers. Gives the number of pre- and post-smoothing steps in the multigrid algorithm.
alpha_start	Vector of length prod(m+q+1) as starting value for the MGCG-method. Defaults to zero.
K_max	Positive integer as upper bound for the number of MGCG-iterations. Defaults to prod(m+q+1).
tolerance	Positive number as error tolerance for the stopping criterion of the MGCG-method. Defaults to 1e-6.
print_error	Logical, indicating if the iteration error should be printed or not.
coarse_grid_solver	Utilized coarse grid solver. Either "PCG" for diagonal preconditioned CG or "Cholesky" for Cholesky decomposition. Defaults to "Cholesky".

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

m	Vector of non-negative integers. Each entry gives the number of inner knots for the respective covariate.
q	Vector of positive integers. Each entry gives the spline degree for the respective covariate.
lambda	Positive number as weight for the penalty term.
X	Matrix containing the covariates as columns and the units as rows.
y	Vector of length nrow(X) as the variable of interest.
pen_type	Utilized penalization method. Either "curve" for the curvature penalty or "diff" for the difference penalty. Defaults to "curve".
l	Positive integer vector of length P indicating for the penalty degree. Only required if pen_type = "diff".
alpha_start	Vector of length prod(m+q+1) as starting value for the PCG-method. Defaults to zero.
K_max	Positive integer as upper bound for the number of PCG-iterations. Defaults to prod(m+q+1).
tolerance	Positive number as error tolerance for the stopping criterion of the PCG-method. Defaults to 1e-6.
print_error	Logical, indicating if the iteration error should be printed or not.

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)

---

Predictions from model

Description

Makes predictions of new observations from a fitted spline model.

Usage

predict_smooth(model_smooth, X)

Arguments

model_smooth	A spline model resulting from CG_smooth, PCG_smooth, or MGCG_smooth.
X	Matrix containing the new observations.

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)

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