| Title: | Split Knockoffs for Structural Sparsity |
|---|---|
| Description: | Split Knockoff is a data adaptive variable selection framework for controlling the (directional) false discovery rate (FDR) in structural sparsity, where variable selection on linear transformation of parameters is of concern. This proposed scheme relaxes the linear subspace constraint to its neighborhood, often known as variable splitting in optimization. Simulation experiments can be reproduced following the Vignette. We include data (both .mat and .csv format) and application with our method of Alzheimer's Disease study in this package. 'Split Knockoffs' is first defined in Cao et al. (2021) <arXiv:2103.16159>. |
| Authors: | Haoxue Wang [aut, cre] (Development of the whole packages), Yang Cao [aut] (Revison of this package), Xinwei Sun [aut] (Original ideas about the package), Yuan Yao [aut] (Testing for the package and management of the development) |
| Maintainer: | Haoxue Wang <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 1.1 |
| Built: | 2024-11-21 03:34:37 UTC |
| Source: | https://github.com/wanghaoxue0/splitknockoff |
Computes a reduced SVD without sign ambiguity
canonicalSVD(X)canonicalSVD(X)
X |
the input matrix |
S
U
V
nu = 10 n = 350 m = 100 A_gamma <- rbind(matrix(0,n,m),-diag(m)/sqrt(nu)) svd.result = canonicalSVD(A_gamma) S <- svd.result$S S <- diag(S) V <- svd.result$Vnu = 10 n = 350 m = 100 A_gamma <- rbind(matrix(0,n,m),-diag(m)/sqrt(nu)) svd.result = canonicalSVD(A_gamma) S <- svd.result$S S <- diag(S) V <- svd.result$V
calculate the hitting time and the sign of respective variable in a path.
hittingpoint(coef, lambda_vec)hittingpoint(coef, lambda_vec)
coef |
the path for one variable |
lambda_vec |
respective value of lambda in the path |
Z: the hitting time
r: the sign of respective variable at the hitting time
normalize columns of a matrix.
normc(X)normc(X)
X |
the input martix |
Y the output matrix
library(mvtnorm) n = 350 p = 100 Sigma = matrix(0, p, p) X <- rmvnorm(n,matrix(0, p, 1), Sigma) X <- normc(X)library(mvtnorm) n = 350 p = 100 Sigma = matrix(0, p, p) X <- rmvnorm(n,matrix(0, p, 1), Sigma) X <- normc(X)
calculate the FDR and Power for simulations.
simu_eval(gamma_true, result, r)simu_eval(gamma_true, result, r)
gamma_true |
true signal of gamma |
result |
the estimated support set of gamma |
r |
the estimated directional effect |
fdr: false discovery rate of the estimated support set
power: power of the estimated support set
the simulation unit for simulation experiments.
simu_unit(n, p, D, A, c, k, option)simu_unit(n, p, D, A, c, k, option)
n |
the sample size |
p |
the dimension of variables |
D |
the linear transform |
A |
SNR |
c |
feature correlation |
k |
number of nonnulls in beta |
option |
option for split knockoffs |
simu_data: a structure contains the following elements
simu_data$fdr_split: a vector recording fdr of split knockoffs w.r.t.nu
simu_data$power_split: a vector recording power of split knockoffs w.r.t.nu
Give the variable splitting design matrix and response vector. It will also create a split knockoff copy if required.
sk.create(X, y, D, nu, option)sk.create(X, y, D, nu, option)
X |
the design matrix |
y |
the response vector |
D |
the linear transform |
nu |
the parameter for variable splitting |
option |
options for creating the Knockoff copy; option$copy true : create a knockoff copy; option$eta the choice of eta for creating the split knockoff copy |
A_beta: the design matrix for beta after variable splitting
A_gamma: the design matrix for gamma after variable splitting
tilde_y: the response vector after variable splitting.
tilde_A_gamma: the knockoff copy of A_beta; will be [] if option$copy = false.
option <- array(data = NA, dim = length(data), dimnames = NULL) option$q <- 0.2 option$eta <- 0.1 option$method <- 'knockoff' option$normalize <- 'true' option$lambda <- 10.^seq(0, -6, by=-0.01) option$nu <- 10 option$copy <- 'true' option$sign <- 'enabled' option <- option[-1] library(mvtnorm) sigma <-1 p <- 100 D <- diag(p) m <- nrow(D) n <- 350 nu = 10 c = 0.5 Sigma = matrix(0, p, p) for( i in 1: p){ for(j in 1: p){ Sigma[i, j] <- c^(abs(i - j)) } } X <- rmvnorm(n,matrix(0, p, 1), Sigma) beta_true <- matrix(0, p, 1) varepsilon <- rnorm(n) * sqrt(sigma) y <- X %*% beta_true + varepsilon creat.result <- sk.create(X, y, D, nu, option) A_beta <- creat.result$A_beta A_gamma <- creat.result$A_gamma tilde_y <- creat.result$tilde_y tilde_A_gamma <- creat.result$tilde_A_gammaoption <- array(data = NA, dim = length(data), dimnames = NULL) option$q <- 0.2 option$eta <- 0.1 option$method <- 'knockoff' option$normalize <- 'true' option$lambda <- 10.^seq(0, -6, by=-0.01) option$nu <- 10 option$copy <- 'true' option$sign <- 'enabled' option <- option[-1] library(mvtnorm) sigma <-1 p <- 100 D <- diag(p) m <- nrow(D) n <- 350 nu = 10 c = 0.5 Sigma = matrix(0, p, p) for( i in 1: p){ for(j in 1: p){ Sigma[i, j] <- c^(abs(i - j)) } } X <- rmvnorm(n,matrix(0, p, 1), Sigma) beta_true <- matrix(0, p, 1) varepsilon <- rnorm(n) * sqrt(sigma) y <- X %*% beta_true + varepsilon creat.result <- sk.create(X, y, D, nu, option) A_beta <- creat.result$A_beta A_gamma <- creat.result$A_gamma tilde_y <- creat.result$tilde_y tilde_A_gamma <- creat.result$tilde_A_gamma
make SVD as well as orthogonal complements
sk.decompose(X, randomize)sk.decompose(X, randomize)
X |
the input matrix |
randomize |
whether to randomize |
U
S
V
U_perp : orthogonal complement for U
library(mvtnorm) n = 350 p = 100 Sigma = matrix(0, p, p) X <- rmvnorm(n,matrix(0, p, 1), Sigma) decompose.result <- sk.decompose(X) U_perp <- decompose.result$U_perplibrary(mvtnorm) n = 350 p = 100 Sigma = matrix(0, p, p) X <- rmvnorm(n,matrix(0, p, 1), Sigma) decompose.result <- sk.decompose(X) U_perp <- decompose.result$U_perp
the main function, Split Knockoff filter, for variable selection in structural sparsity problem.
sk.filter(X, D, y, option)sk.filter(X, D, y, option)
X |
the design matrix |
D |
the linear transform |
y |
the response vector |
option |
various options for split knockoff filter, the details will be specified in the example |
results: a cell with the selected variable set in each cell w.r.t. nu.
Z: a cell with the feature significance Z in each cell w.r.t. nu.
t_Z: a cell with the knockoff significance tilde_Z in each cell w.r.t. nu.
option <- list(data = NA, dim = length(data), dimnames = NULL) # the target (directional) FDR control option$q <- 0.2 # choice on threshold, the other choice is 'knockoff+' option$method <- 'knockoff' # degree of separation between original design and its split knockoff copy # in the range of [0, 2], the less the more separated option$eta <- 0.1 # whether to normalize the dataset option$normalize <- 'true' # choice on the set of regularization parameters for split LASSO path option$lambda <- 10.^seq(0, -6, by=-0.01) # choice of nu for split knockoffs option$nu <- 10 # choice on whether to estimate the directional effect, 'disabled'/'enabled' option$sign <- 'enabled' option <- option[-1] # Settings on simulation parameters k <- 20 # sparsity level A <- 1 # magnitude n <- 350 # sample size p <- 100 # dimension of variables c <- 0.5 # feature correlation sigma <-1 # noise level # generate D D <- diag(p) m <- nrow(D) # generate X Sigma = matrix(0, p, p) for( i in 1: p){ for(j in 1: p){ Sigma[i, j] <- c^(abs(i - j)) } } library(mvtnorm) set.seed(100) X <- rmvnorm(n,matrix(0, p, 1), Sigma) # generate beta and gamma beta_true <- matrix(0, p, 1) for( i in 1: k){ beta_true[i, 1] = A if ( i%%3 == 1){ beta_true[i, 1] = -A } } gamma_true <- D %*% beta_true S0 <- which(gamma_true!=0) # generate varepsilon set.seed(1) # generate noise and y varepsilon <- rnorm(n) * sqrt(sigma) y <- X %*% beta_true + varepsilon filter_result <- sk.filter(X, D, y, option) Z_path <- filter_result$Z t_Z_path <- filter_result$t_Zoption <- list(data = NA, dim = length(data), dimnames = NULL) # the target (directional) FDR control option$q <- 0.2 # choice on threshold, the other choice is 'knockoff+' option$method <- 'knockoff' # degree of separation between original design and its split knockoff copy # in the range of [0, 2], the less the more separated option$eta <- 0.1 # whether to normalize the dataset option$normalize <- 'true' # choice on the set of regularization parameters for split LASSO path option$lambda <- 10.^seq(0, -6, by=-0.01) # choice of nu for split knockoffs option$nu <- 10 # choice on whether to estimate the directional effect, 'disabled'/'enabled' option$sign <- 'enabled' option <- option[-1] # Settings on simulation parameters k <- 20 # sparsity level A <- 1 # magnitude n <- 350 # sample size p <- 100 # dimension of variables c <- 0.5 # feature correlation sigma <-1 # noise level # generate D D <- diag(p) m <- nrow(D) # generate X Sigma = matrix(0, p, p) for( i in 1: p){ for(j in 1: p){ Sigma[i, j] <- c^(abs(i - j)) } } library(mvtnorm) set.seed(100) X <- rmvnorm(n,matrix(0, p, 1), Sigma) # generate beta and gamma beta_true <- matrix(0, p, 1) for( i in 1: k){ beta_true[i, 1] = A if ( i%%3 == 1){ beta_true[i, 1] = -A } } gamma_true <- D %*% beta_true S0 <- which(gamma_true!=0) # generate varepsilon set.seed(1) # generate noise and y varepsilon <- rnorm(n) * sqrt(sigma) y <- X %*% beta_true + varepsilon filter_result <- sk.filter(X, D, y, option) Z_path <- filter_result$Z t_Z_path <- filter_result$t_Z
split knockoff selector given W statistics
sk.select(W, q, option)sk.select(W, q, option)
W |
statistics W_j for testing null hypothesis |
q |
target FDR |
option |
option$method can be 'knockoff' or 'knockoff+' |
S array of selected variable indices
generate the split knockoff statistics W for a split LASSO path, take the directional effect into account
W_sign(X, D, y, nu, option)W_sign(X, D, y, nu, option)
X |
the design matrix |
D |
the linear transform |
y |
the response vector |
nu |
the parameter for variable splitting |
option |
options for creating the Knockoff statistics option$eta specify the choice of eta for creating the knockoff copy; option$lambda specify the choice of lambda for the split LASSO path |
W: the knockoff statistics
Z: feature significance
r: the sign estimator
t_Z: knockoff significance
generate the split knockoff statistics W for a split LASSO path, only consider the support set estimation
W_support(X, D, y, nu, option)W_support(X, D, y, nu, option)
X |
the design matrix |
D |
the linear transform |
y |
the response vector |
nu |
the parameter for variable splitting |
option |
options for creating the Split Knockoff statistics; option$eta specify the choice of eta for creating the knockoff copy; option$lambda specify the choice of lambda for the split LASSO path |
W: the knockoff statistics
Z: feature significance
t_Z: knockoff significance