hierbasis: Univariate Nonparametric Regression Estimation via a...

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

Description

Univariate Nonparametric Regression Estimation via a Hierarchical Penalty

Usage

hierbasis(x, y, nbasis = length(y), max.lambda = NULL,
  lam.min.ratio = 1e-04, nlam = 50, m.const = 3, type = c("gaussian",
  "binomial"), weights = NULL, basis.type = c("poly", "trig", "wave"))

Arguments

x	A univariate vector representing the predictor.
y	A univariate vector respresenting the response.
nbasis	The number of basis functions to be used in the basis expansion of x. Default: nbasis = length(y).
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.
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.
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

Solves the univariate minimization problem (see Haris et al. 2016)

argmin_{β} (1/2n) || y - Ψ_K β ||^2_2 + λΩ(β; m),

where β is a real-valued vector of length K = nbasis, Ψ_K is a n\times K real-valued matrix whose columns correspond to the basis expansion of x (ordered in increasing) complexity. The penalty function Ω is defined as

Ω(β; m) = ∑^K_{k = 1} w_{k, m} || Ψ_{k:K} β_{k:K} ||_2,

for Ψ_{k:K} denotes the submatrix of Ψ_K corresponding to the final columns k, k + 1, ..., K, and β_{k:K} the subvector of β corresponding to the final elements k, k + 1, ..., K. The weights w_{k, m} in Ω are given by the m - 1 degree polynomial

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

, controlling the level of decay on higher order estimates of β.

Value

Returns an object of class hierbasis with elements:

x, y	The original x and y used as inputs.
m.const	The parameter m.const used for smoothing.
nbasis	The maximum number of basis functions used for computing the basis expansion of x.
basis.expansion	The n * nbasis-matrix containing the basis expansion of x, with column indices corresponding to increasing basis function complexity.
basis.expansion.means	The nbasis-vector containing the columnwise means of the basis expansion.
ybar	Mean of the response vector.
lambdas	Sequence of tuning parameters lambda used for penalizing the fits.
intercept	The nlam-vector of estimated intercepts β_0.
beta	The nbasis * nlam-matrix of estimated linear coefficients.
fitted.values	The nbasis * nlam-matrix of fitted responses.
weights	The weights used for smoothing/penalizing the basis functions.
active	The size of the active set (number of nonzero β).
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 HierBasis 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.