Package 'rSDR'

The optimal alpha for rSDR via bootstrap resampling

Description

Perform R bootstrap replications of the cost function in rSDR method and return the corresponding optimal alpha.

Usage

optimal_alpha_boot(alpha.v,X,Y,d,R,maxiter=1000,tol=1e-7)

Arguments

alpha.v	user-supplied alpha sequence. The default is alpha.v=c(0.3,0.4,0.5,0.6,0.7).
X	an $n \times p$ numeric matrix, where $n$ is the number of observations and $p$ is the number of variable.
Y	an $n \times k$ numeric response matrix, where $k (\geq 1)$ is the number of response variables.
d	the number of reduced dimension. The default is d=3.
R	the number of bootstrap replicates.
maxiter	maxiter is the maximum number of iterations allowed for the solver (a non-negative integer). See the Max_Iteration parameter in get.solver.params for details.
tol	tol is used to assess convergence, see the Tolerance parameter in get.solver.params for details.

Value

An object of class "optimal_alpha_boot" is returned. The returned value contains the following components:

opt.alpha

value of alpha that gives minimum f_test.meam.

f_test.mean

The mean of cost function by the alpha sequence - a vector of length length(alpha.v).

f_test.sd

The standard deviation of cost function by the alpha sequence.

f_test

An $R \times$ length(alpha.v) matrix. The cost value for each fold at a given alpha.

d

The value of d as passed to optimal_alpha_boot.

R

The value of R as passed to optimal_alpha_boot.

Examples

library(ManifoldOptim)
library(rSDR)
library(future)
library(future.apply)
utils::data("ionosphere", package = "fdm2id")
X<-as.matrix(ionosphere[,c(1:33)])
Y<-ifelse(ionosphere[,34]=='b',0,1)
Y<-matrix(Y,length(Y),1)
set.seed(2435)
# plan(multisession) will launch parallel workers running in the background
# to save running time. To shut down background workers launched this way, call
# plan(sequential)
# use all local cores except one
# future::plan(future::multisession, workers = future::availableCores() - 1)
# use 2 cores for parallel

future::plan("multisession", workers = 2)
opt_results<-optimal_alpha_boot(alpha.v=c(0.3,0.5,0.7),X=X,Y=Y,d=3,R=5)
opt_results
future::plan(sequential)

---

The optimal alpha for rSDR via cross-validation

Description

Performs k-folds cross-validation for rSDR method and returns the corresponding optimal alpha.

Usage

optimal_alpha_cv(alpha.v,X,Y,d,kfolds=10,maxiter=1000,tol=1e-7)

Arguments

alpha.v	user-supplied alpha sequence. The default is alpha.v=c(0.3,0.4,0.5,0.6,0.7).
X	an $n \times p$ numeric matrix, where $n$ is the number of observations and $p$ is the number of variable.
Y	an $n \times k$ numeric response matrix, where $k (\geq 1)$ is the number of response variables.
d	the number of reduced dimension. The default is d=3.
kfolds	the number of folds - default is 10.
maxiter	maxiter is the maximum number of iterations allowed for the solver (a non-negative integer). See the Max_Iteration parameter in get.solver.params for details.
tol	tol is used to assess convergence, see the Tolerance parameter in get.solver.params for details.

Value

An object of class "optimal_alpha_cv" is returned. The returned value contains the following components:

opt.alpha

value of alpha that gives minimum f_test.meam.

f_test.mean

The mean of cost value by the alpha sequence - a vector of length length(alpha.v).

f_test.sd

The standard deviation of cost value by the alpha sequence - a vector of length length(alpha.v).

f_test

A kfolds $\times$ length(alpha.v) matrix. The cost value for each fold at a given alpha.

d

The value of d as passed to optimal_alpha_cv.

kfolds

The value of kfolds as passed to optimal_alpha_cv.

Examples

library(ManifoldOptim)
library(rSDR)
library(future)
library(future.apply)
utils::data("ionosphere", package = "fdm2id")
X<-as.matrix(ionosphere[,c(1:33)])
Y<-ifelse(ionosphere[,34]=='b',0,1)
Y<-matrix(Y,length(Y),1)
set.seed(2435)
# plan(multisession) will launch parallel workers running in the background
# to save running time. To shut down background workers launched this way, call
# plan(sequential)
# use all local cores except one
# future::plan(future::multisession, workers = future::availableCores() - 1)
# use 2 cores for parallel

future::plan("multisession", workers = 2)
opt_results<-optimal_alpha_cv(alpha.v=c(0.3, 0.5, 0.7),X=X,Y=Y,d=3,kfolds=5)
opt_results

---

Plot for the optimal alpha

Description

Plot for the mean with the standard deviation of cost function and alpha

Usage

plot_alpha(opt_results)

Arguments

opt_results

opt_results is from either optimal_alpha_boot or optimal_alpha_cv

Value

No return value, showing the mean and standard deviation of cost function for each alpha value.

Examples

library(ManifoldOptim)
library(rSDR)
utils::data("ionosphere", package = "fdm2id")
X<-as.matrix(ionosphere[,c(1:33)])
Y<-ifelse(ionosphere[,34]=='b',0,1)
Y<-matrix(Y,length(Y),1)
set.seed(2435)
# plan(multisession) will launch parallel workers running in the background
# to save running time. To shut down background workers launched this way, call
# plan(sequential)
# use all local cores except one
# future::plan(future::multisession, workers = future::availableCores() - 1)
# use 2 cores for parallel

future::plan("multisession", workers = 2)
opt_results<-optimal_alpha_cv(alpha.v=c(0.3, 0.5, 0.7),X=X,Y=Y,d=3,kfolds=10)
plot_alpha(opt_results=opt_results)

---

Projected data plotting

Description

Function for plotting of projected_data from rSDR results.

Usage

plot_rSDR(projected_data,Y,Y.name,colors=NULL)

Arguments

projected_data	projected data from rSDR results.
Y	an $n \times 1$ numeric matrix.
Y.name	label for y-axis
colors	Assign specific colors to each level of the response variable.

Value

No return value, visualizing reduced-dimensional data using 1D, 2D, or 3D projections. When the reduced dimension exceeds three, pairwise scatter plots are automatically generated.

Examples

library(ManifoldOptim)
library(rSDR)
utils::data("ionosphere", package = "fdm2id")
X<-as.matrix(ionosphere[,c(1:33)])
Y<-ifelse(ionosphere[,34]=='b',0,1)
Y<-matrix(Y,length(Y),1)
ionosphere$V35<-factor(ionosphere$V35,levels=c('b','g'),labels=c('Bad','Good'))
set.seed(2435)

sdr_result<-rSDR(X=X, Y=Y, d=3, alpha=0.3,maxiter=1000,tol=1e-7)
plot_rSDR(projected_data=sdr_result$projected_data,Y=ionosphere$V35,
Y.name='group',colors=c("#374E55FF", "#DF8F44FF"))

---

Robust Sufficient Dimension Reduction

Description

Robust Sufficient Dimension Reduction with alpha-Distance Covariance and Stiefel Manifold Learning for supervised dimension reduction.

Usage

rSDR(X, Y, d, alpha=0.5,maxiter=1000,tol=1e-7)

Arguments

X	an $n \times p$ numeric matrix, where $n$ is the number of observations and $p$ is the number of variable.
Y	an $n \times k$ numeric response matrix, where $k (\geq 1)$ is the number of response variables.
d	the number of reduced dimension.
alpha	this parameter represents the exponent applied to the Euclidean distance in the computation of distance covariance. When alpha=1, it corresponds to the classical distance covariance. When 0 < alpha < 1, it is a more robust version by reducing the influence of large values in the distance matrices.
maxiter	maxiter is the maximum number of iterations allowed for the solver (a non-negative integer). See the Max_Iteration parameter in get.solver.params for details.
tol	tol is used to assess convergence, see the Tolerance parameter in get.solver.params for details.

Value

The returned value is an object of class "rSDR", containing the following components:

projected_data

an $n \times d$ matrix representing the projected data using the rSDR method.

beta

a $p \times d$ matrix.Solve $\boldsymbol{\beta}$ by $\textbf{C}=\Sigma_x^{1/2} \boldsymbol{\beta}$

C_value

an optimal of C is obtained by maximizing the target function using ManifoldOptim method.

f_value

The value of cost function f is defined as the negative of the target function.

References

Hsin-Hsiung Huang, Feng Yu & Teng Zhang (19 Feb 2024): Robust sufficient dimension reduction via alpha-distance covariance, Journal of Nonparametric Statistics, DOI:10.1080/10485252.2024.2313137

Examples

library(ManifoldOptim)
library(rSDR)
utils::data("ionosphere", package = "fdm2id")
X<-as.matrix(ionosphere[,c(1:33)])
Y<-ifelse(ionosphere[,34]=='b',0,1)
Y<-matrix(Y,length(Y),1)
set.seed(2435)

sdr_result<-rSDR(X=X, Y=Y, d=3, alpha=0.3,maxiter=1000,tol=1e-7)

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