plsim.vs.soft: Penalized Profile Least Squares Estimator

In PLSiMCpp: Methods for Partial Linear Single Index Model

Penalized Profile Least Squares Estimator

Description

PPLS along with introducing penalty terms so as to simultaneously estimate parameters and select important variables in PLSiM

Y = η(Z^Tα) + X^Tβ + ε

.

Usage

plsim.vs.soft(...)

## S3 method for class 'formula'
plsim.vs.soft(formula, data, ...)

## Default S3 method:
plsim.vs.soft(xdat=NULL, zdat, ydat, h=NULL, zetaini=NULL, 
lambda=0.01, l1_ratio=NULL, MaxStep = 1L, penalty = "SCAD", verbose=TRUE, 
ParmaSelMethod="SimpleValidation", TestRatio=0.1, K = 3, seed=0, ...)

Arguments

...	additional arguments.
formula	a symbolic description of the model to be fitted.
data	an optional data frame, list or environment containing the variables in the model.
xdat	input matrix (linear covariates). The model reduces to a single index model when x is NULL.
zdat	input matrix (nonlinear covariates). z should not be NULL.
ydat	input vector (response variable).
h	a value or a vector for bandwidth. If h is NULL, a default vector c(0.01,0.02,0.05,0.1,0.5) will be set for it. plsim.bw is employed to select the optimal bandwidth when h is a vector or NULL.
zetaini	initial coefficients, optional (default: NULL). It could be obtained by the function plsim.ini. zetaini[1:ncol(z)] is the initial coefficient vector α_0, and zetaini[(ncol(z)+1):(ncol(z)+ncol(x))] is the initial coefficient vector {β}_0.
MaxStep	int, optional (default=1). Hard limit on iterations within solver.
lambda	double. Constant that multiplies the penalty term.
l1_ratio	double, default=NULL. It should be set with a value from the range [0,1] when you choose "ElasticNet" for the parameter penalty.
penalty	string, optional (default="SCAD"). It could be "SCAD", "LASSO" and "ElasticNet".
verbose	bool, default: TRUE. Enable verbose output.
ParmaSelMethod	the parameter for the function plsim.bw.
TestRatio	the parameter for the function plsim.bw.
K	the parameter for the function plsim.vs.soft.
seed	int, default: 0.

Value

eta	estimated non-parametric part \hat{η}(Z^T{\hat{α} }).
zeta	estimated coefficients. zeta[1:ncol(z)] is \hat{α}, and zeta[(ncol(z)+1):(ncol(z)+ncol(x))] is \hat{β}.
y_hat	y's estimates.
mse	mean squared errors between y and y_hat.
data	data information including x, z, y, bandwidth h, initial coefficients zetaini, iteration step MaxStep, flag SiMflag, penalty, lambda and l1_ratio. SiMflag is TRUE when x is NULL, otherwise SiMflag is FALSE.
Z_alpha	Z^T{\hat{α}}.
r_square	multiple correlation coefficient.
variance	variance of y_hat.
stdzeta	standard error of zeta.

References

H. Liang, X. Liu, R. Li, C. L. Tsai. Estimation and testing for partially linear single-index models. Annals of statistics, 2010, 38(6): 3811.

Examples

 
# EXAMPLE 1 (INTERFACE=FORMULA)
# To estimate parameters of partially linear single-index model and select 
# variables using different penalization methods such as SCAD, LASSO, ElasticNet.

n = 50
sigma = 0.1

alpha = matrix(1,2,1)
alpha = alpha/norm(alpha,"2")

beta = matrix(4,1,1)

# Case 1: Matrix Input
x = matrix(1,n,1)
z = matrix(runif(n*2),n,2)
y = 4*((z%*%alpha-1/sqrt(2))^2) + x%*%beta + sigma*matrix(rnorm(n),n,1)

# Compute the penalized profile least-squares estimator with the SCAD penalty
fit_scad = plsim.vs.soft(y~x|z,lambda = 0.01)
summary(fit_scad)

# Compute the penalized profile least-squares estimator with the LASSO penalty
fit_lasso = plsim.vs.soft(y~x|z,lambda = 1e-3, penalty = "LASSO")
summary(fit_lasso)

# Compute the penalized profile least-squares estimator with the ElasticNet penalty
fit_enet = plsim.vs.soft(y~x|z,lambda = 1e-3, penalty = "ElasticNet")
summary(fit_enet)

# Case 2: Vector Input
x = rep(1,n)
z1 = runif(n)
z2 = runif(n) 
y = 4*((z%*%alpha-1/sqrt(2))^2) + x%*%beta + sigma*matrix(rnorm(n),n,1)

# Compute the penalized profile least-squares estimator with the SCAD penalty
fit_scad = plsim.vs.soft(y~x|z1+z2,lambda = 0.01)
summary(fit_scad)

# Compute the penalized profile least-squares estimator with the LASSO penalty
fit_lasso = plsim.vs.soft(y~x|z1+z2,lambda = 1e-3, penalty = "LASSO")
summary(fit_lasso)

# Compute the penalized profile least-squares estimator with the ElasticNet penalty
fit_enet = plsim.vs.soft(y~x|z1+z2,lambda = 1e-3, penalty = "ElasticNet")
summary(fit_enet)

# EXAMPLE 2 (INTERFACE=DATA FRAME)
# To estimate parameters of partially linear single-index model and select 
# variables using different penalization methods such as SCAD, LASSO, ElasticNet.

n = 50
sigma = 0.1

alpha = matrix(1,2,1)
alpha = alpha/norm(alpha,"2")

beta = matrix(4,1,1)

x = rep(1,n)
z1 = runif(n)
z2 = runif(n) 
X = data.frame(x)
Z = data.frame(z1,z2)

x = data.matrix(X)
z = data.matrix(Z)
y = 4*((z%*%alpha-1/sqrt(2))^2) + x%*%beta + sigma*matrix(rnorm(n),n,1)

# Compute the penalized profile least-squares estimator with the SCAD penalty
fit_scad = plsim.vs.soft(xdat=X,zdat=Z,ydat=y,lambda = 0.01)
summary(fit_scad)

# Compute the penalized profile least-squares estimator with the LASSO penalty
fit_lasso = plsim.vs.soft(xdat=X,zdat=Z,ydat=y,lambda = 1e-3, penalty = "LASSO")
summary(fit_lasso)

# Compute the penalized profile least-squares estimator with the ElasticNet penalty
fit_enet = plsim.vs.soft(xdat=X,zdat=Z,ydat=y,lambda = 1e-3, penalty = "ElasticNet")
summary(fit_enet)

PLSiMCpp documentation built on Sept. 24, 2022, 5:05 p.m.