XXt Computation

Description

For an input matrix X, This function computes UU' = tcrossprod(U) where U = scale(X) in a memory efficient way.

Usage

XXt.compute(
  X,
  meanX = NULL,
  varX = NULL,
  ncores = 4,
  check = TRUE,
  chunksize = 1000
)

Arguments

Value

a nrow(X) x nrow(X) dense symmetric matrix.

Author(s)

Somak Dutta

Maintainer: Somak Dutta <somakd@iastate.edu>

Examples

require("Matrix")
set.seed(123)
x <- rsparsematrix(100,100,density = 0.3)
norm(XXt.compute(x) - tcrossprod(scale(x))) # (4.783951e-13 on my machine)

Model selection using extended BIC

Description

This function performs model selection using extended BIC and the ridge partial correlation coefficients

Usage

eBIC(rpc.obj)

Arguments

Value

BIC.PATH vector of eBIC of each model, starting with the intercept only model.

model.best sorted indices (in increased order) of the model with the smallest EBIC

Author(s)

An Nguyen

Somak Dutta

Maintainer: Somak Dutta <somakd@iastate.edu>

Examples

n <- 50; p <- 400;
trueidx <- 1:3
truebeta <- c(4,5,6)
X <- matrix(rnorm(n*p), n, p) # n x p covariate matrix
y <- 0.5 + X[,trueidx] %*% truebeta + rnorm(n)
res <- rpc(X,y, lambda = 0.1, ncores = 1)
eBIC(res) # model.best: model selected by eBIC

Ridge Partial Correlations

Description

The sample ridge partial correlations (RPC) can be used for a simple, efficient, and scalable variables screening method in ultra-high dimensional linear regression. These coefficients incorporate both the ridge regularized estimates of the regression coefficients and the ridge regularized partial variances of the predictor variables, providing the screening method with sure screening property without strong assumptions on the marginal correlations. For more details, please see Wang et al. (2025).

This function computes the ridge partial correlations for each variable in the covariate matrix.

Usage

rpc(X, y, lambda = 1, XXt = NULL, ncores = 1, RAM = 6)

Arguments

Details

Consider the linear regression model:

y = \beta_0 + X\beta + \epsilon,

where X is the n \times p design matrix, \beta is a p-dimensional vector, and \epsilon is the n-dimensional vector of iid residual errors with mean 0 and variance \sigma^2, \epsilon is independent of X.

Assuming the design matrix X is centered and scaled, and letting \tilde{Y} = Y - \bar{Y}_n, the (sample) partial correlation between the i^{th} predictor and Y given the remaining predictors is -P_{i+1, 1} / \sqrt{P_{1, 1}P_{i+1, i+1}}, where P is the (p+1) \times (p+1) joint precision matrix of the response and the covariates. In cases when p > n, P is not invertible. Hence, we consider the ridge regularized version of P, given by R, where:

R := (n-1)\begin{bmatrix}\tilde{Y}^T\tilde{Y} & \tilde{Y}^TX\\ X^T\tilde{Y} & X^TX + \lambda I_p \end{bmatrix}^{-1}.

The ridge partial correlations between y and X_{i} is then defined in terms of elements of R as follows:

\rho_{i, \lambda} = - \dfrac{r_{i+1,1}}{\sqrt{r_{1,1} r_{i+1,i+1}}}

where r_{i,j} is the (i.j)th entry in R.

For variable screening, one may choose the K variables with largest absolute RPC values, for some K > 0 (e.g., K = n or n/\log p). Alternatively, the extended BIC criteria may be used. See eBIC function in this package.

N.B. Further hyper-threading speed-up can be obtained by loading the MatrixExtra package.

Value

an object containing the following items:
- rpc: vector of rpc coefficients.
- X: the original matrix of covariates.
- XXt: the matrix UU', where U is the centered-scaled X.
- y: vector of centered and scaled response variable.
- lambda: the original lambda used.

Author(s)

Somak Dutta

An Nguyen

Maintainer: Somak Dutta <somakd@iastate.edu>

See Also

[eBIC()] for model selecting, [XXt.compute()] for computing crossproduct.

Examples

## Toy example:
n <- 50; p <- 400;
trueidx <- 1:3 ## First three variables are important
truebeta <- c(4,5,6)
X <- matrix(rnorm(n*p), n, p) ## n x p covariate matrix
y <- 0.5 + X[,trueidx] %*% truebeta + rnorm(n) ## Response
res <- rpc(X,y, lambda = 0.1, ncores = 1)
order(abs(res$rpc),decreasing = TRUE)[1:10] # Top 10 variables
## Run another case with the same X and y, but pass XXt to make it faster
res2 <- rpc(X,y, lambda = 0.001,XXt = res$XXt , ncores = 1)
order(abs(res2$rpc),decreasing = TRUE)[1:10] # Top 10 variables