Type: Package
Title: Methods for Partial Linear Single Index Model
Version: 1.0.4
Depends: R(> 4.2)
Date: 2022-09-24
Author: Shunyao Wu, Qi Zhang, Zhiruo Li, Hua Liang
Maintainer: Shunyao Wu <wushunyao@qdu.edu.cn>
Description: Estimation, hypothesis tests, and variable selection in partially linear single-index models. Please see H. (2010) at <doi:10.1214/10-AOS835> for more details.
Encoding: UTF-8
RoxygenNote: 7.2.1
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Imports: Rcpp (≥ 0.12.11), crayon, methods, stats, purrr
LinkingTo: Rcpp,RcppArmadillo
NeedsCompilation: yes
Packaged: 2022-09-24 11:22:00 UTC; hp
Repository: CRAN
Date/Publication: 2022-09-24 11:40:02 UTC

Minimum Average Variance Estimation

Description

MAVE (Minimum Average Variance Estimation), proposed by Xia et al. (2006) to estimate parameters in PLSiM

Y=\eta(Z^T\alpha)+X^T\beta+\epsilon.

Usage

plsim.MAVE(...)

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

## Default S3 method:
plsim.MAVE(xdat=NULL, zdat, ydat, h=NULL, zeta_i=NULL, maxStep=100,
tol=1e-8, iniMethods="MAVE_ini", ParmaSelMethod="SimpleValidation", TestRatio=0.1, 
K = 3, seed=0, verbose=TRUE, ...)

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 numerical 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 given. plsim.bw is employed to select the optimal bandwidth when h is a vector or NULL.

zeta_i

initial coefficients, optional (default: NULL). It could be obtained by the function plsim.ini. zeta_i[1:ncol(z)] is the initial coefficient vector \alpha_0, and zeta_i[(ncol(z)+1):(ncol(z)+ncol(x))] is the initial coefficient vector \beta_0.

maxStep

the maximum iterations, default: 100.

tol

convergence tolerance, default: 1e-8.

iniMethods

string, optional (default: "SimpleValidation").

ParmaSelMethod

the parameter for the function plsim.bw.

TestRatio

the parameter for the function plsim.bw.

K

the parameter for the function plsim.bw.

seed

int, default: 0.

verbose

bool, default: TRUE. Enable verbose output.

Value

eta

estimated non-parametric part \hat{\eta}(Z^T{\hat{\alpha} }).

zeta

estimated coefficients. zeta[1:ncol(z)] is \hat{\alpha}, and zeta[(ncol(z)+1):(ncol(z)+ncol(x))] is \hat{\beta}.

data

data information including x, z, y, bandwidth h, initial coefficients zetaini and iteration step MaxStep.

y_hat

y's estimates.

mse

mean squares erros between y and y_hat.

variance

variance of y_hat.

r_square

multiple correlation coefficient.

Z_alpha

Z^T{\hat{\alpha}}.

References

Y. Xia, W. Härdle. Semi-parametric estimation of partially linear single-index models. Journal of Multivariate Analysis, 2006, 97(5): 1162-1184.

Examples


# EXAMPLE 1 (INTERFACE=FORMULA)
# To estimate parameters in partially linear single-index model using MAVE.

n = 30
sigma = 0.1

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

beta = matrix(4,1,1)

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)

fit = plsim.MAVE(y~x|z, h=0.1)

# EXAMPLE 2 (INTERFACE=DATA FRAME)
# To estimate parameters in partially linear single-index model using MAVE.

n = 30
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)

fit = plsim.MAVE(xdat=X, zdat=Z, ydat=y, h=0.1)

select bandwidth

Description

Select bandwidth for methods, including MAVE, Profile Least Squares Estimator and Penalized Profile Least Squares Estimator by cross validation or simple validation.

Usage

plsim.bw(...)

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

## Default S3 method:
plsim.bw(xdat, zdat, ydat, zeta_i=NULL, bandwidthList=NULL, 
ParmaSelMethod="CrossValidation", K=5, TestRatio=0.1, TargetMethod='plsimest',
lambda=NULL, l1_ratio=NULL, VarSelMethod = "SCAD", MaxStep = 1L, 
verbose=FALSE, 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).

bandwidthList

vector, candidate bandwidths.

TargetMethod

string, optional (default: "plsimest"). target method to be selected bandwidth for, which could be "MAVE", "plsimest" and "plsim".

ParmaSelMethod

string, optional (default: "CrossValidation"). Method to select bandwidth, which could be Cross Validation ("CrossValidation") and Simple Validation ("SimpleValidation").

K

int, optional (default: 5). The number of folds for Cross Validation.

TestRatio

double, optional (default: 0.1). The ratio of test data for Simple Validation.

zeta_i

initial coefficients. It could be obtained by the function plsim.ini. zeta_i[1:ncol(z)] is the initial coefficient vector \alpha_0, and zeta_i[(ncol(z)+1):(ncol(z)+ncol(x))] is the initial coefficient vector \beta_0.

lambda

the parameter for the function plsim.vs.soft, default: NULL.

l1_ratio

the parameter for the function plsim.vs.soft, default: NULL.

VarSelMethod

the parameter for the function plsim.vs.soft, default : "SCAD".

MaxStep

the parameter for the function plsim.vs.soft, default: 1.

verbose

the parameter for the function plsim.vs.soft, default: FALSE.

seed

int, default: 0.

Value

bandwidthBest

selected bandwidth

mse

mean square errors corresponding to the bandwidthList

Examples


# EXAMPLE 1 (INTERFACE=FORMULA)
# To select bandwidth by cross validation and simple validation. 

n = 50
sigma = 0.1

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

beta = matrix(4,1,1)

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)

# Select bandwidth for profile least squares estimator by cross validation
res_plsimest_cross = plsim.bw(y~x|z,bandwidthList=c(0.02,0.04,0.06,0.08,0.10))

# Select bandwidth for profile least squares estimator by simple validation
res_plsimest_simple = plsim.bw(y~x|z,bandwidthList=c(0.02,0.04,0.06,0.08,0.10),
                            ParmaSelMethod="SimpleValidation")

# Select bandwidth for penalized profile least squares estimator by simple validation
res_plsim_simple = plsim.bw(y~x|z,bandwidthList=c(0.02,0.04,0.06,0.08,0.10),
                         ParmaSelMethod="SimpleValidation",TargetMethod="plsim",lambda=0.01)


# EXAMPLE 2 (INTERFACE=DATA FRAME)
# To select bandwidth by cross validation and simple validation. 

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)

# Select bandwidth for profile least squares estimator by cross validation
res_plsimest_cross = plsim.bw(xdat=X,zdat=Z,ydat=y,bandwidthList=c(0.02,0.04,0.06,0.08,0.10))

# Select bandwidth for profile least squares estimator by simple validation
res_plsimest_simple = plsim.bw(xdat=X,zdat=Z,ydat=y,bandwidthList=c(0.02,0.04,0.06,0.08,0.10),
                            ParmaSelMethod="SimpleValidation")

# Select bandwidth for penalized profile least squares estimator by simple validation
res_plsim_simple = plsim.bw(xdat=X,zdat=Z,ydat=y,bandwidthList=c(0.02,0.04,0.06,0.08,0.10),
                         ParmaSelMethod="SimpleValidation",TargetMethod="plsim",lambda=0.01)

Profile Least Squares Estimator

Description

PLS was proposed by Liang et al. (2010) to estimate parameters in PLSiM

Y = \eta(Z^T\alpha) + X^T\beta + \epsilon.

Usage

plsim.est(...)

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

## Default S3 method:
plsim.est(xdat=NULL, zdat, ydat, h=NULL, zetaini=NULL, MaxStep = 200L,
ParmaSelMethod="SimpleValidation", TestRatio=0.1, K = 3, seed=0, verbose=TRUE, ...)

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 \alpha_0, and zetaini[(ncol(z)+1):(ncol(z)+ncol(x))] is the initial coefficient vector \beta_0.

MaxStep

the maximum iterations, optional (default=200).

ParmaSelMethod

the parameter for the function plsim.bw.

TestRatio

the parameter for the function plsim.bw.

K

the parameter for the function plsim.bw.

seed

int, default: 0.

verbose

bool, default: TRUE. Enable verbose output.

Value

eta

estimated non-parametric part \hat{\eta}(Z^T{\hat{\alpha} }).

zeta

estimated coefficients. zeta[1:ncol(z)] is \hat{\alpha}, and zeta[(ncol(z)+1):(ncol(z)+ncol(x))] is \hat{\beta}.

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 and flag SiMflag. SiMflag is TRUE when x is NULL, otherwise SiMflag is FALSE.

Z_alpha

Z^T{\hat{\alpha}}.

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 (PLSiM). 

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)

fit = plsim.est(y~x|z)
summary(fit)

# 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)

fit = plsim.est(y~x|z1+z2)
summary(fit)
print(fit)

# EXAMPLE 2 (INTERFACE=DATA FRAME) 
# To estimate parameters of partially linear single-index model (PLSiM).  

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)

fit = plsim.est(xdat=X,zdat=Z,ydat=y)
summary(fit)
print(fit)

Initialize coefficients

Description

Xia et al.'s MAVE method is used to obtain initialized coefficients \alpha_0 and \beta_0 for PLSiM

Y = \eta(Z^T\alpha) + X^T\beta + \epsilon

.

Usage

plsim.ini(...)

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

## Default S3 method:
plsim.ini(xdat, zdat, ydat, Method="MAVE_ini", verbose = TRUE, ...)

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).

Method

string, optional (default="MAVE_ini").

verbose

bool, default: TRUE. Enable verbose output.

Value

zeta_i

initial coefficients. zeta_i[1:ncol(z)] is the initial coefficient vector \alpha_0, and zeta_i[(ncol(z)+1):(ncol(z)+ncol(x))] is the initial coefficient vector \beta_0.

References

Y. Xia, W. Härdle. Semi-parametric estimation of partially linear single-index models. Journal of Multivariate Analysis, 2006, 97(5): 1162-1184.

Examples


# EXAMPLE 1 (INTERFACE=FORMULA)
# To obtain initial values by using MAVE methods for partially
# linear single-index model.

n = 50
sigma = 0.1

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

beta = matrix(4,1,1)

# Case1: 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)

zeta_i = plsim.ini(y~x|z)

# 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)

zeta_i = plsim.ini(y~x|z1+z2)


# EXAMPLE 2 (INTERFACE=DATA FRAME)
# To obtain initial values by using MAVE methods for partially
# linear single-index model.

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)

zeta_i = plsim.ini(xdat=X, zdat=Z, ydat=y)

Select lambda for Penalized Profile Least Squares Estimator

Description

Use AIC or BIC to choose the regularization parameters for Penalized Profile least squares (PPLS) estimation.

Usage

plsim.lam(...)

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

## Default S3 method:
plsim.lam(xdat=NULL, ydat, zdat, h, zetaini=NULL, penalty="SCAD", 
lambdaList=NULL, l1_ratio_List=NULL, lambda_selector="BIC", verbose=TRUE, 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

bandwidth.

zetaini

initial coefficients, optional (default: NULL). It could be obtained by the function plsim.ini. zetaini[1:ncol(z)] is the initial coefficient vector \alpha_0, and zetaini[(ncol(z)+1):(ncol(z)+ncol(x))] is the initial coefficient vector \beta_0.

penalty

string, optional (default="SCAD"). It could be "SCAD", "LASSO" or "ElasticNet".

lambdaList

candidates for lambda selection. lambda is a constant that multiplies the penalty term. If lambdaList is NULL, function plsim.lam will automatically set it.

l1_ratio_List

candidates for l1_ratio selection. l1_ratio is a constant that balances the importances of L1 norm and L2 norm for "ElasticNet". If l1_ratio_List is NULL, function plsim.lam ranges from 0 to 1 with an increment 0.1.

lambda_selector

the criterion to select lambda (and l1_ratio), default: "BIC".

verbose

bool, default: TRUE. Enable verbose output.

seed

int, default: 0.

Value

goodness_best

the AIC (or BIC) statistics with lambda_best.

lambda_best

lambda selected by AIC or BIC.

l1_ratio_best

l1_ratio selected by AIC or BIC.

lambdaList

lambdaList automatically selected when inputting NULL.

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 select the regularization parameters based on AIC.

n = 50
sigma = 0.1

alpha = matrix(1,2,1)
alpha = alpha/norm(alpha,"2")
beta = matrix(4,1,1)

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)


fit_plsimest = plsim.est(y~x|z)

# Select the regularization parameters by AIC
res = plsim.lam(y~x|z,h=fit_plsimest$data$h,zetaini = fit_plsimest$zeta,
             lambda_selector='AIC')


# EXAMPLE 2 (INTERFACE=DATA FRAME)
# To select the regularization parameters based on AIC.

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)

fit_plsimest = plsim.est(xdat=X,zdat=Z,ydat=y)

# Select the regularization parameters by AIC
res2 = plsim.lam(xdat=X,ydat=y,zdat=Z,h=fit_plsimest$data$h,
              zetaini = fit_plsimest$zeta, lambda_selector='AIC')

Testing nonparametric component

Description

Study the hypothesis test:

H_0:\eta(u) = \theta_0+\theta_1u \ \mbox{ versus }\quad H_1:\ \eta(u)\ne \theta_0 + \theta_1u \ \mbox{for \ some \ } u

where \theta_0 and \theta_1 are unknown constant parameters.

Usage

plsim.npTest(fit)

Arguments

fit

the result of function plsim.est or plsim.vs.soft.

Value

A list with class "htest" containing the following components

statistic

the value of the test statistic.

p.value

the p-value for the test

method

a character string indicating what type of test was performed

data.name

a character string giving the name of input

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


n = 50
sigma = 0.1

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

beta = matrix(4,1,1)

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)

# Obtain parameters in PLSiM using Profile Least Squares Estimator
fit_plsimest = plsim.est(x, z, y)

res_npTest_plsimest = plsim.npTest(fit_plsimest)

# Obtain parameters in PLSiM using Penalized Profile Least Squares Estimator
# with lambda set as 0.01
fit_plsim = plsim.vs.soft(x,z,y,lambda = 0.01)

res_npTest_plsim = plsim.npTest(fit_plsim)

Testing Parametric Components

Description

Test whether some elements of \alpha and \beta are zero, that is,

H_0: \alpha_{i_1}=\cdots=\alpha_{i_k}=0 \ \mbox{ and } \beta_{j_1}=\cdots=\beta_{j_l}=0

versus

H_1: \mbox{not \ all }\ \alpha_{i_1},\cdots,\alpha_{i_k} \ \mbox{ and } \beta_{j_1}, \cdots,\beta_{j_l} \ \mbox{ are equal to }\ 0.

Usage

plsim.pTest(fit, parameterSelected = NULL, TargetMethod = "plsimest")

Arguments

fit

the result of function plsim.est or plsim.vs.soft.

parameterSelected

select some coefficients for testing, default: NULL.

TargetMethod

default: "plsim.est".

Value

A list with class "htest" containing the following components

statistic

the value of the test statistic.

parameter

the degree of freedom for the test

p.value

the p-value for the test

method

a character string indicating what type of test was performed

data.name

a character string giving the name of input

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


n = 50
sigma = 0.1

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

beta = matrix(4,1,1)

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)

# Obtain parameters in PLSiM using Profile Least Squares Estimator
fit_plsimest = plsim.est(x, z, y)

# Test whether the parameters of parametric part estimated by plsimest
# are zero
res_pTest_plsimest = plsim.pTest(fit_plsimest)

# Test whether the second parameter of parametric part estimated by plsimest
# is zero
res_pTest_plsimest_ = plsim.pTest(fit_plsimest,parameterSelected = c(2))

# Obtain parameters in PLSiM using Penalized Profile Least Squares Estimator
# with lambda set as 0.01
fit_plsim = plsim.vs.soft(x,z,y,lambda = 0.01)

# Test whether the parameters of parametric part estimated by plsim are zero
res_pTest_plsim = plsim.pTest(fit_plsim,TargetMethod = "plsim")

# Test whether the second parameter of parametric part estimated by plsim is zero
res_pTest_plsim_ = plsim.pTest(fit_plsim,parameterSelected = c(2),TargetMethod = "plsim")

Variable Selection for Partial Linear Single Index Models

Description

Variable Selection based on AIC, BIC, SCAD, LASSO and Elastic Net. The methods based on SCAD, LASSO and Elastic Net are implemented with Penalized Profile Least Squares Estimator, while AIC and BIC are implemented with Stepwise Regression.

Usage

plsim.vs.hard(...)

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

## Default S3 method:
plsim.vs.hard(xdat=NULL, zdat, ydat, h=NULL, zeta_i=NULL, 
lambdaList=NULL, l1RatioList=NULL, lambda_selector="BIC", threshold=0.05,
Method="SCAD", verbose=TRUE, ParmaSelMethod="SimpleValidation", 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 numerical 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.

zeta_i

initial coefficients, optional (default: NULL). It could be obtained by the function plsim.ini. zeta_i[1:ncol(z)] is the initial coefficient vector \alpha_0, and zeta_i[(ncol(z)+1):(ncol(z)+ncol(x))] is the initial coefficient vector \beta_0.

verbose

bool, default: TRUE. Enable verbose output.

Method

variable selection method, default: "SCAD". It could be "SCAD", "LASSO", "ElasticNet", "AIC" or "BIC".

lambdaList

the parameter for the function plsim.lam, default: "NULL".

l1RatioList

the parameter for the function plsim.lam, default: "NULL".

lambda_selector

the parameter for the function plsim.lam, default: "BIC".

threshold

the threshold to select important variable according to the estimated coefficients.

ParmaSelMethod

the parameter for the function plsim.bw.

seed

int, default: 0.

Value

alpha_varSel

selected variables in z.

beta_varSel

selected variables in x.

fit_plsimest

fit_plsimest is not NULL when h is a vector or NULL. For each bandwidth, plsim.est is employed to integrate selected variabels. Finally, the optimal fitted model will be selected according to BIC.

Examples


# EXAMPLE 1 (INTERFACE=FORMULA)
# To select variables with Penalized Profile Least Squares Estimation based on 
# the penalty LASSO.

n = 50
dx = 10
dz = 5
sigma = 0.2
alpha = matrix(c(1,3,1.5,0.5,0),dz,1)
alpha = alpha/norm(alpha,"2")
beta = matrix(c(3,2,0,0,0,1.5,0,0.2,0.3,0.15),dx,1)

A = sqrt(3)/2-1.645/sqrt(12)
B = sqrt(3)/2+1.645/sqrt(12)
z = matrix(runif(n*dz),n,dz)
x = matrix(runif(n*dx),n,dx)
y = sin( (z%*%alpha - A) * 3.1415926 * (B-A) ) + x%*%beta + sigma*matrix(rnorm(n),n,1)

# Variable Selectioin Based on LASSO
res_varSel_LASSO = plsim.vs.hard(y~x|z,h=0.1,Method="LASSO")


# EXAMPLE 2 (INTERFACE=DATA FRAME)
# To select variables with Penalized Profile Least Squares Estimation based on 
# the penalty LASSO.

n = 50
dx = 10
dz = 5
sigma = 0.2
alpha = matrix(c(1,3,1.5,0.5,0),dz,1)
alpha = alpha/norm(alpha,"2")
beta = matrix(c(3,2,0,0,0,1.5,0,0.2,0.3,0.15),dx,1)

A = sqrt(3)/2-1.645/sqrt(12)
B = sqrt(3)/2+1.645/sqrt(12)
z = matrix(runif(n*dz),n,dz)
x = matrix(runif(n*dx),n,dx)
y = sin( (z%*%alpha - A) * 3.1415926 * (B-A) ) + x%*%beta + sigma*matrix(rnorm(n),n,1)

Z = data.frame(z)
X = data.frame(x)

# Variable Selectioin Based on LASSO
res_varSel_LASSO = plsim.vs.hard(xdat=X,zdat=Z,ydat=y,h=0.1,Method="LASSO")

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 = \eta(Z^T\alpha) + X^T\beta + \epsilon

.

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 \alpha_0, and zetaini[(ncol(z)+1):(ncol(z)+ncol(x))] is the initial coefficient vector {\beta}_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{\eta}(Z^T{\hat{\alpha} }).

zeta

estimated coefficients. zeta[1:ncol(z)] is \hat{\alpha}, and zeta[(ncol(z)+1):(ncol(z)+ncol(x))] is \hat{\beta}.

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{\alpha}}.

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)

Predict according to the Estimated Parameters

Description

Predict Y based on new observations.

Usage

## S3 method for class 'pls'
predict(object, x_test = NULL, z_test, ...)

Arguments

object

fitted partially linear single-index model, which could be obtained by

x_test

input matrix (linear covariates of test set).

z_test

input matrix (nonlinear covariates of test set).

...

additional arguments.

plsim.MAVE, or plsim.est, or plsim.vs.soft.

Value

y_hat

prediction.

Examples


n = 50
sigma = 0.1

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

beta = matrix(4, 1, 1)

x = matrix(1, n, 1)
x_test = matrix(1,n,1)

z = matrix(runif(n*2), n, 2)
z_test = matrix(runif(n*2), n, 2)

y = 4*((z%*%alpha-1/sqrt(2))^2) + x%*%beta + sigma*matrix(rnorm(n),n,1)
y_test = 4*((z_test%*%alpha-1/sqrt(2))^2) + x_test%*%beta + sigma*matrix(rnorm(n),n,1)


# Obtain parameters in PLSiM using Profile Least Squares Estimator
fit_plsimest = plsim.est(x, z, y)

preds_plsimest = predict(fit_plsimest, x_test, z_test)

# Print the MSE of the Profile Least Squares Estimator method
print( sum( (preds_plsimest-y_test)^2)/nrow(y_test) )

# Obtain parameters in PLSiM using Penalized Profile Least Squares Estimator
fit_plsim = plsim.vs.soft(x, z, y,lambda = 0.01)

preds_plsim = predict(fit_plsim, x_test, z_test)

# Print the MSE of the Penalized Profile Least Squares Estimator method
print( sum( (preds_plsim-y_test)^2)/nrow(y_test) )