This vignette illustrates the basic usage of the knockoff package with Model-X knockoffs. In this scenario we assume that the distribution of the predictors is known (or that it can be well approximated), but we make no assumptions on the conditional distribution of the response. For simplicity, we will use synthetic data constructed from a linear model such that the response only depends on a small fraction of the variables.
set.seed(1234)# Problem parameters
n = 1000 # number of observations
p = 1000 # number of variables
k = 60 # number of variables with nonzero coefficients
amplitude = 4.5 # signal amplitude (for noise level = 1)
# Generate the variables from a multivariate normal distribution
mu = rep(0,p)
rho = 0.25
Sigma = toeplitz(rho^(0:(p-1)))
X = matrix(rnorm(n*p),n) %*% chol(Sigma)
# Generate the response from a linear model
nonzero = sample(p, k)
beta = amplitude * (1:p %in% nonzero) / sqrt(n)
y.sample = function(X) X %*% beta + rnorm(n)
y = y.sample(X)To begin, we call knockoff.filter with all the default settings.
library(knockoff)
result = knockoff.filter(X, y)We can display the results with
print(result)## Call:
## knockoff.filter(X = X, y = y)
##
## Selected variables:
## [1] 36 37 40 62 87 93 101 139 176 185 204 225 235 276 280 310 311 335 374
## [20] 418 440 441 447 449 470 488 494 530 544 545 560 571 580 681 683 687 691 747
## [39] 757 838 857 874 879 889 929 940 986 988 995The default value for the target false discovery rate is 0.1. In this experiment the false discovery proportion is
fdp = function(selected) sum(beta[selected] == 0) / max(1, length(selected))
fdp(result$selected)## [1] 0.04081633By default, the knockoff filter creates model-X second-order Gaussian knockoffs. This construction estimates from the data the mean \(\mu\) and the covariance \(\Sigma\) of the rows of \(X\), instead of using the true parameters (\(\mu, \Sigma\)) from which the variables were sampled.
The knockoff package also includes other knockoff construction methods, all of which have names prefixed withknockoff.create. In the next snippet, we generate knockoffs using the true model parameters.
gaussian_knockoffs = function(X) create.gaussian(X, mu, Sigma)
result = knockoff.filter(X, y, knockoffs=gaussian_knockoffs)
print(result)## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs)
##
## Selected variables:
## [1] 34 36 37 54 62 87 93 101 139 164 176 185 204 225 235 276 280 305 310
## [20] 311 335 371 374 418 440 441 442 447 467 470 488 494 530 544 545 560 571 580
## [39] 612 681 683 687 691 709 747 756 757 792 838 852 857 874 879 883 889 929 931
## [58] 940 986 988 995Now the false discovery proportion is
fdp(result$selected)## [1] 0.1147541By default, the knockoff filter uses a test statistic based on the lasso. Specifically, it uses the statistic stat.glmnet_coefdiff, which computes \[
W_j = |Z_j| - |\tilde{Z}_j|
\] where \(Z_j\) and \(\tilde{Z}_j\) are the lasso coefficient estimates for the jth variable and its knockoff, respectively. The value of the regularization parameter \(\lambda\) is selected by cross-validation and computed with glmnet.
Several other built-in statistics are available, all of which have names prefixed with stat. For example, we can use statistics based on random forests. In addition to choosing different statistics, we can also vary the target FDR level (e.g. we now increase it to 0.2).
result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs, statistic = stat.random_forest, fdr=0.2)
print(result)## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs,
## statistic = stat.random_forest, fdr = 0.2)
##
## Selected variables:
## [1] 36 37 62 146 156 176 204 210 220 225 280 310 311 374 440 441 447 450 488
## [20] 494 530 545 560 571 683 689 691 709 747 784 879 889 940 988 995fdp(result$selected)## [1] 0.2285714In addition to using the predefined test statistics, it is also possible to use your own custom test statistics. To illustrate this functionality, we implement one of the simplest test statistics from the original knockoff filter paper, namely \[ W_j = \left|X_j^\top \cdot y\right| - \left|\tilde{X}_j^\top \cdot y\right|. \]
my_knockoff_stat = function(X, X_k, y) {
abs(t(X) %*% y) - abs(t(X_k) %*% y)
}
result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs, statistic = my_knockoff_stat)
print(result)## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs,
## statistic = my_knockoff_stat)
##
## Selected variables:
## [1] 37 225 310 373 374 440 447 494 691 857 889 929 940 988fdp(result$selected)## [1] 0.07142857As another example, we show how to customize the grid of \(\lambda\)’s used to compute the lasso path in the default test statistic.
my_lasso_stat = function(...) stat.glmnet_coefdiff(..., nlambda=100)
result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs, statistic = my_lasso_stat)
print(result)## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs,
## statistic = my_lasso_stat)
##
## Selected variables:
## [1] 36 37 40 54 62 87 93 101 139 164 176 185 204 225 235 276 280 305 310
## [20] 311 335 374 418 440 441 442 447 449 467 470 488 494 530 544 545 560 571 580
## [39] 612 681 683 687 691 709 747 757 838 841 852 857 874 879 883 889 929 931 940
## [58] 986 988 995fdp(result$selected)## [1] 0.08333333The nlambda parameter is passed by stat.glmnet_coefdiff to the glmnet, which is used to compute the lasso path. For more information about this and other parameters, see the documentation for stat.glmnet_coefdiff or glmnet.glmnet.
In addition to using the predefined procedures for construction knockoff variables, it is also possible to create your own knockoffs. To illustrate this functionality, we implement a simple wrapper for the construction of second-order Model-X knockoffs.
create_knockoffs = function(X) {
create.second_order(X, shrink=T)
}
result = knockoff.filter(X, y, knockoffs=create_knockoffs)
print(result)## Call:
## knockoff.filter(X = X, y = y, knockoffs = create_knockoffs)
##
## Selected variables:
## [1] 36 37 40 54 93 101 139 176 185 204 225 235 276 280 305 310 311 335 374
## [20] 418 440 441 447 449 467 488 494 530 544 545 560 571 580 681 683 687 691 747
## [39] 757 792 838 841 852 857 874 879 883 889 929 940 986 988 995fdp(result$selected)## [1] 0.0754717The knockoff package supports two main styles of knockoff variables, semidefinite programming (SDP) knockoffs (the default) and equi-correlated knockoffs. Though more computationally expensive, the SDP knockoffs are statistically superior by having higher power. To create SDP knockoffs, this package relies on the R library [Rdsdp][Rdsdp] to efficiently solve the semidefinite program. In high-dimensional settings, this program becomes computationally intractable. A solution is then offered by approximate SDP (ASDP) knockoffs, which address this issue by solving a simpler relaxed problem based on a block-diagonal approximation of the covariance matrix. By default, the knockoff filter uses SDP knockoffs if \(p<500\) and ASDP knockoffs otherwise.
In this example we generate second-order Gaussian knockoffs using the estimated model parameters and the full SDP construction. Then, we run the knockoff filter as usual.
gaussian_knockoffs = function(X) create.second_order(X, method='sdp', shrink=T)
result = knockoff.filter(X, y, knockoffs = gaussian_knockoffs)
print(result)## Call:
## knockoff.filter(X = X, y = y, knockoffs = gaussian_knockoffs)
##
## Selected variables:
## [1] 36 37 40 54 62 93 101 139 164 176 185 204 225 235 276 280 310 311 335
## [20] 374 418 440 441 447 449 467 488 494 530 544 545 560 571 580 681 683 687 691
## [39] 747 757 838 841 857 874 879 889 929 940 988 995fdp(result$selected)## [1] 0.06If you want to look inside the knockoff filter, see the advanced vignette. If you want to see how to use knockoffs for Fixed-X variables, see the Fixed-X vignette.