| Type: | Package |
| Title: | Tools Developed for Structured Sufficient Dimension Reduction (sSDR) |
| Version: | 1.2.0 |
| Date: | 2016-03-26 |
| Author: | Yang Liu <zjubioly@gmail.com>, Francesca Chiaromonte, Bing Li |
| Maintainer: | Yang Liu <zjubioly@gmail.com> |
| Description: | Performs structured OLS (sOLS) and structured SIR (sSIR). |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| LazyData: | TRUE |
| Depends: | R (≥ 3.0.0), MASS, Matrix |
| NeedsCompilation: | no |
| Packaged: | 2016-03-26 18:07:48 UTC; yangliu |
| Repository: | CRAN |
| Date/Publication: | 2016-03-26 22:02:24 |
Center a vector
Description
Center a vector
Usage
center(v)
Arguments
v |
A vector. |
Details
This function centers any vector and returns a vector with mean zero.
Value
A vector with mean zero.
Examples
data <- gen.data(n=100)
y.centered <- center(data$y)
Covariance matrix
Description
Covariance matrix
Usage
cov.x(X)
Arguments
X |
a n x p matrix of n observations and p predictors. |
Details
This function returns A p x p covariance matrix for any n x p matrix.
Value
A p x p covariance matrix.
Examples
data <- gen.data(n=100)
x.cov <- cov.x(data$X)
Subspace distance
Description
Subspace distance
Usage
disvm(v1, v2)
Arguments
v1 |
A matrix, each column consists of a p-dimensional vector. |
v2 |
A matrix, each column consists of a p-dimensional vector. |
Details
This function computes the distances between two spaces using the formulation in Li, Zha, Chiaromonte (2005), which is the Frobenius norm of the difference between the two orthogonal projection matrices defined by v1 and v2.
Value
A scaler represents the distance between the two spaces spanned by v1 and v2 respectively.
References
Li, B., Zha, H., and Chiaromonte, F. (2005). Contour regression: a general approach to dimension reduction. Annals of Statistics, 33(4):1580-1616.
Examples
v1 <- c(1, 0, 0)
v2 <- c(0, 1, 0)
disvm(v1, v1)
disvm(v1, v2)
Groupwise OLS (gOLS)
Description
Groupwise OLS (gOLS)
Usage
gOLS(X, Y, groups, dims)
Arguments
X |
A covariate matrix of n observations and p predictors. |
Y |
A univariate response. |
groups |
A vector with the number of predictors in each group. |
dims |
A vector with the dimension (at most 1) for each predictor group. |
Details
This function estimates directions for each predictor group using gOLS. Predictors need to be organized in groups within the "X" matrix, as the same order saved in "groups". We only allow continuous covariates in the "X" matrix; while categorical covariates can be handled outside of gOLS, e.g. structured OLS.
Value
gOLS returns a list containning at least the following components: "b_est", the estimated directions for each group with its own dimension using gOLS AFTER normalization; "B", the estimated directions for each group using gOLS BEFORE normalization.
References
Liu, Y., Chiaromonte, F., and Li, B. (2015). Structured Ordinary Least Squares: a sufficient dimension reduction approach for regressions with partitioned predictors and heterogeneous units. Submitted.
Examples
data <- gen.data(n=1000, binary=FALSE) # generate data
dim(data$X) # covariate matrix of 1000 observations and 15 predictors
dim(data$y) # univariate response
groups <- c(5, 10) # two predictor groups and their numbers of predictors
dims <- c(1,1) # dimension of each predictor group
est_gOLS <- gOLS(data$X,data$y,groups,dims)
names(est_gOLS)
Groupwise OLS (gOLS) BIC criterion to estimate dimensions with eigen-decomposition
Description
Groupwise OLS (gOLS) BIC criterion to estimate dimensions with eigen-decomposition
Usage
gOLS.comp.d(X, y, groups)
Arguments
X |
A covariate matrix of n observations and p predictors. |
y |
A univariate response. |
groups |
A vector with the number of predictors in each group. |
Details
This function estimates dimension for each predictor group using eigen-decomposition. Predictors need to be organized in groups within the "X" matrix, as the same order saved in "groups". We only allow continuous covariates in the "X" matrix; while categorical covariates can be handled outside of gOLS, e.g. structured OLS.
Value
gOLS.comp.d returns a list containning at least the following components: "d", the estimated dimension (at most 1) for each predictor group; "crit", the BIC criterion from each iteration.
References
Liu, Y., Chiaromonte, F., and Li, B. (2015). Structured Ordinary Least Squares: a sufficient dimension reduction approach for regressions with partitioned predictors and heterogeneous units. Submitted.
Examples
data <- gen.data(n=1000, binary=FALSE) # generate data
dim(data$X) # covariate matrix of 1000 observations and 15 predictors
dim(data$y) # univariate response
groups <- c(5, 10) # two predictor groups and their numbers of predictors
dim_gOLS<-gOLS.comp.d(data$X,data$y,groups)
names(dim_gOLS)
Groupwise SIR (gSIR) for binary response
Description
Groupwise SIR (gSIR) for binary response
Usage
gSIR(X, Y, groups, dims)
Arguments
X |
A covariate matrix of n observations and p predictors. |
Y |
A binary response. |
groups |
A vector with the number of predictors in each group. |
dims |
A vector with the dimension (at most 1) for each predictor group. |
Details
This function estimates directions for each predictor group using gSIR. Predictors need to be organized in groups within the "X" matrix, as the same order saved in "groups". We only allow continuous covariates in the "X" matrix; while categorical covariates can be handled outside of gSIR, e.g. structured SIR.
Value
gSIR returns a list containning at least the following components: "b_est", the estimated directions for each group with its own dimension using gSIR AFTER normalization; "B", the estimated directions for each group using gSIR BEFORE normalization.
References
Guo, Z., Li, L., Lu, W., and Li, B. (2014). Groupwise dimension reduction via envelope method. Journal of the American Statistical Association, accepted.
Examples
data <- gen.data(n=1000, binary=TRUE) # generate data
dim(data$X) # covariate matrix of 1000 observations and 15 predictors
length(data$y) # binary response
groups <- c(5, 10) # two predictor groups and their numbers of predictors
dims <- c(1,1) # dimension of each predictor group
est_gSIR<-gSIR(data$X,data$y,groups,dims)
names(est_gSIR)
Groupwise SIR (gSIR) BIC criterion to estimate dimensions with eigen-decomposition (binary response)
Description
Groupwise SIR (gSIR) BIC criterion to estimate dimensions with eigen-decomposition (binary response)
Usage
gSIR.comp.d(X, y, groups)
Arguments
X |
A covariate matrix of n observations and p predictors. |
y |
A binary response. |
groups |
A vector with the number of predictors in each group. |
Details
This function estimates dimension for each predictor group using eigen-decomposition. Predictors need to be organized in groups within the "X" matrix, as the same order saved in "groups". We only allow continuous covariates in the "X" matrix; while categorical covariates can be handled outside of gSIR, e.g. structured SIR.
Value
gSIR.comp.d returns a list containning at least the following components: "d", the estimated dimension (at most 1) for each predictor group; "crit", the BIC criterion from each iteration.
References
Liu, Y. (2015). Approaches to reduce and integrate data in structured and high-dimensional regression problems in Genomics. Ph.D. Dissertation, The Pennsylvania State University, University Park, Department of Statistics.
Examples
data <- gen.data(n=1000, binary=TRUE) # generate data
dim(data$X) # covariate matrix of 1000 observations and 15 predictors
length(data$y) # univariate response
groups <- c(5, 10) # two predictor groups and their numbers of predictors
dim_gSIR<-gSIR.comp.d(data$X,data$y,groups)
names(dim_gSIR)
Simulate data
Description
Simulate data
Usage
gen.data(n, rho = 0.5, theta = 1, binary = FALSE)
Arguments
n |
Sample size. |
rho |
Pairwise correlation between covariates. |
theta |
Standard deviation of the random error. |
binary |
If TRUE, generate binary responses; otherwise, by default, create continuous responses. |
Details
This function simulates data as presented in Liu (2015).
Value
gen.data returns a list containning at least the following components: "X", a covariate matrix of n observations and p predictors; "y", a univariate response; "b.true", the actual coefficients for each predictor group.
References
Liu, Y. (2015). Approaches to reduce and integrate data in structured and high-dimensional regression problems in Genomics. Ph.D. Dissertation, The Pennsylvania State University, University Park, Department of Statistics.
Examples
data <- gen.data(n=100)
names(data)
Power of a matrix
Description
Power of a matrix
Usage
matpower(X, alpha)
Arguments
X |
A p x p square matrix. |
alpha |
A scaler determining the order of the power. |
Details
This function calculates the power of a square matrix.
Value
A p x p square matrix.
Examples
data <- gen.data(n=100)
cov.squared <- matpower(cov.x(data$X), 2)
Normalize a vector
Description
Normalize a vector
Usage
norm1(v)
Arguments
v |
A vector. |
Details
This function normalizes any non-zero vector and returns a vector with the norm equal to 1.
Value
A vector with norm 1.
Examples
data <- gen.data(n=100)
y.norm1 <- norm1(data$y)
Gram-Schmidt orthonormalization
Description
Gram-Schmidt orthonormalization
Usage
orthnormal(X)
Arguments
X |
a n x p matrix of n observations and p predictors. |
Details
This function orthonormalizes any n x p matrix.
Value
A n x p matrix of n observations and p predictors.
Examples
data <- gen.data(n=100)
x.orth <- orthnormal(data$X)
Structured OLS (sOLS) outer level BIC criterion to estimate dimension with eigen-decomposition
Description
Structured OLS (sOLS) outer level BIC criterion to estimate dimension with eigen-decomposition
Usage
sOLS.comp.d(X, sizes)
Arguments
X |
A matrix containing directions estimated from all subpopulations. |
sizes |
A vector with the sample sizes of all subpopulation. |
Details
This function estimates dimension across the subpopulations using eigen-decomposition. The order of the subpopulations in the "sizes" vector should match the one in the "X" matrix. Also, this function returns the linearly independent directions among all subpopulations.
Value
sOLS.comp.d returns a list containning at least the following components: "d", the dimension estimated across subpopulations; "u", the "d" linearly independent directions among the matrix X.
References
Liu, Y., Chiaromonte, F., and Li, B. (2015). Structured Ordinary Least Squares: a sufficient dimension reduction approach for regressions with partitioned predictors and heterogeneous units. Submitted.
Examples
v1 <- c(1, 1, 0, 0)
v2 <- c(0, 1, 1, 0)
v3 <- c(0, 0, 1, 1)
v4 <- c(1, 1, 1, 1)
m1 <- cbind(v1, v2)
sizes1 <- c(100, 200)
sOLS.comp.d(m1, sizes1)
m2 <- cbind(v1, v2, v3)
sizes2 <- c(100, 200, 500)
sOLS.comp.d(m2, sizes2)
m3 <- cbind(v1, v3, v4)
sizes3 <- c(100, 500, 1000)
sOLS.comp.d(m3, sizes3)
Matrix standardization
Description
Matrix standardization
Usage
standmat(x)
Arguments
x |
A n x p matrix of n observations and p predictors. |
Details
This function standardizes a matrix treating each row as a random vector in an iid sample. It returns a n x p matrix with column-mean zero and identity-covariance matrix.
Value
A n x p matrix of n observations and p predictors.
Examples
data <- gen.data(n=100)
x.std <- standmat(data$X)