additivehierbasis: Multivariate Nonparametric Regression Estimation via a Sparse...

In dfleis/hierbasis2: Rebuilding Nonparametric Regression and Sparse Additive Modeling

Description

The main function for fitting sparse additive models via the additive hierbasis estimator

Usage

additivehierbasis(X, y, nbasis = 10, max.lambda = NULL,
  lam.min.ratio = 1e-04, nlam = 50, beta.mat = NULL, alpha = NULL,
  m.const = 3, max.iter = 100, tol = 1e-04, type = c("gaussian",
  "binomial"), weights = NULL, basis.type = c("poly", "trig", "wave"))

Arguments

X	An n x p matrix of covariates.
y	A univariate vector respresenting the response.
nbasis	The number of basis functions to be used in the basis expansion of x. Default: nbasis = 10.
max.lambda	The largest value of λ penalizing the hierarchical penalty. Default: max.lambda = NULL (computed data-adaptively).
lam.min.ratio	The ratio of the smallest value of λ to the largest λ. Default: lam.min.ratio = 1e-04.
nlam	The number λ's to compute the hierbasis estimators (computed on a base-10 log scale). Default: nlam = 50.
beta.mat	An initial estimate of the parameter beta, a ncol(x)-by-nbasis matrix. If NULL (default) the inital estimate is set to the zero matrix.
alpha	A scalar between 0 and 1 controlling the balance between the sparsity penalty and the hierarchical penalty. Default is 0.5.
m.const	Smoothing parameter controlling the degree of polynomial smoothing/decay performed by the weights in the penalty Ω.Default: m.const = 3. Only relevant if weights is left unspecified.
max.iter	Maximum number of iterations for block coordinate descent.
tol	Tolerance/stopping precision for the block coordinate descent algorithm.
type	Specifies either Gaussian regression ("gaussian") with a continuous response vector or binomial regression ("binomial") with binary response. Default type is assumed to be Gaussian.
weights	(New parameter for hierbasis2) Permits user-specified weights w_{k, m}. If left unspecified (weights = NULL) then the default is set to w_{k, m} = k^m - (k - 1)^m.
basis.type	(New parameter for hierbasis2) Desired basis functions for the basis expansion of x. Specifies either a polynomial expansion basis.type = "poly", trigonometric expansion basis.type = "trig", or wavelet expansion basis.type = "wave". Default set to a polynomial expansion.

Details

Solve the multivariate minimization problem (see Haris at al. (2016) for details) for β:

argmin_{β_1,…, β_p} (1/(2n)) || y - ∑_jΨ^{(j)}_K β_j ||^2_2 + λ^2 (1 - α) ∑_j || Ψ^{(j)}_K β_j ||_2 + λ α ∑_j Ω_j( β_j; m ) ,

where β_j is a vector of length K = nbasis and the summation is over the index j = 1, ..., p for p covariatres. The penalty function Ω_j(β_j; m) is given by

∑_k w_{k, m}β_{j, [k:K]},

where β_{j, [k:K]} is the vector of coefficients for the j-th predictor and represents beta[k:K] for the corresponding vector beta, summing over k = 1, ..., K. Finally, the weights w_{k, m} are given (by default)

w_{k, m} = k^m - (k - 1)^m,

where m denotes the 'smoothness level'. For details see Haris et al. (2016).

Value

An object of class additivehierbasis with the following elements

X, y	The original X and y used as inputs.
beta	The (nbasis * p)-by-nlam-matrix of the estimated linear coefficients beta

.

beta.array	The nbasis * p * nlam-array of the estimated linear coefficients.
intercept	The nlam-vector of estimated intercepts β_0.
fitted.values	The n * nlam matrix of fitted values.
basis.expansion	The n * nbasis * p-array containing the basis expansion of X. Second-dimension indices correspond to increasing basis function complexity and third-dimension indices correspond to each of the p covariates of X.
basis.expansion.means	The nbasis * p-vector containing the second-dimension-wise means of the basis expansion.
ybar	Mean of the response vector.
lambdas	Sequence of tuning parameters lambda used for penalizing the fits.
fitted.values	The nbasis * nlam-matrix of fitted responses.
alpha	The scalar controlling the balance between the sparsity-inducing penalty and the hierarchical-sparsity-inducing penalty.
m.const	The m.const value used for defining 'order' of smoothness.
nbasis	The maximum number of basis functions used for computing the basis expansion of x.
max.iter	Maximum number of iterations used for the block coordinate descent algorithm.
tol	Tolerance/stopping precision used for block coordinate descent.
weights	The weights used for smoothing/penalizing the basis functions.
active	The size of the active set (number of nonzero β) per tuning parameter λ.
active.mat	The size of the active set per predictor (rowwise), per tuning parameter λ (columnwise).
type	The specified family 'gaussian' or 'binomial'.
basis.type	Specified basis expansion family, polynomial, trigonometric, or wavelet.

Author(s)

Annik Gougeon, David Fleischer ([e-mail]).

References

Haris, A., Shojaie, A. and Simon, N. (2016). Nonparametric Regression with Adaptive Smoothness via a Convex Hierarchical Penalty. Available on request by authors.

See Also

The original AdditiveHierBasis function, as implemented by Haris et al. (2016) can be found via https://github.com/asadharis/HierBasis/.

dfleis/hierbasis2 documentation built on May 17, 2019, 7:03 p.m.