Fixed-X knockoffs

Description

Creates fixed-X knockoff variables.

Usage

create.fixed(
  X,
  method = c("sdp", "equi"),
  sigma = NULL,
  y = NULL,
  randomize = F
)

Arguments

Details

Fixed-X knockoffs assume a homoscedastic linear regression model for Y|X. Moreover, they only guarantee FDR control when used in combination with statistics satisfying the "sufficiency" property. In particular, the default statistics based on the cross-validated lasso does not satisfy this property and should not be used with fixed-X knockoffs.

Value

An object of class "knockoff.variables". This is a list containing at least the following components:

References

Barber and Candes, Controlling the false discovery rate via knockoffs. Ann. Statist. 43 (2015), no. 5, 2055–2085.

See Also

Other create: create.gaussian(), create.second_order()

Examples

set.seed(2022)
p=50; n=100; k=15
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 5.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=create.fixed)
print(result$selected)

# Advanced usage with custom arguments
knockoffs = function(X) create.fixed(X, method='equi')
result = knockoff.filter(X, y, knockoffs=knockoffs)
print(result$selected) 

Model-X Gaussian knockoffs

Description

Samples multivariate Gaussian model-X knockoff variables.

Usage

create.gaussian(X, mu, Sigma, method = c("asdp", "sdp", "equi"), diag_s = NULL)

Arguments

Value

A n-by-p matrix of knockoff variables.

References

Candes et al., Panning for Gold: Model-free Knockoffs for High-dimensional Controlled Variable Selection, arXiv:1610.02351 (2016). https://web.stanford.edu/group/candes/knockoffs/index.html

See Also

Other create: create.fixed(), create.second_order()

Examples

set.seed(2022)
p=100; n=80; k=15
rho = 0.4
mu = rep(0,p); Sigma = toeplitz(rho^(0:(p-1)))
X = matrix(rnorm(n*p),n) %*% chol(Sigma)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)

# Basic usage with default arguments
knockoffs = function(X) create.gaussian(X, mu, Sigma)
result = knockoff.filter(X, y, knockoffs=knockoffs)
print(result$selected)

# Advanced usage with custom arguments
knockoffs = function(X) create.gaussian(X, mu, Sigma, method='equi')
result = knockoff.filter(X, y, knockoffs=knockoffs)
print(result$selected)

Second-order Gaussian knockoffs

Description

This function samples second-order multivariate Gaussian knockoff variables. First, a multivariate Gaussian distribution is fitted to the observations of X. Then, Gaussian knockoffs are generated according to the estimated model.

Usage

create.second_order(X, method = c("asdp", "equi", "sdp"), shrink = F)

Arguments

Details

If the argument shrink is set to T, a James-Stein-type shrinkage estimator for the covariance matrix is used instead of the traditional maximum-likelihood estimate. This option requires the package corpcor. See cov.shrink for more details.

Even if the argument shrink is set to F, in the case that the estimated covariance matrix is not positive-definite, this function will apply some shrinkage.

Value

A n-by-p matrix of knockoff variables.

References

Candes et al., Panning for Gold: Model-free Knockoffs for High-dimensional Controlled Variable Selection, arXiv:1610.02351 (2016). https://web.stanford.edu/group/candes/knockoffs/index.html

See Also

Other create: create.fixed(), create.gaussian()

Examples

set.seed(2022)
p=100; n=80; k=15
rho = 0.4
Sigma = toeplitz(rho^(0:(p-1)))
X = matrix(rnorm(n*p),n) %*% chol(Sigma)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=create.second_order)
print(result$selected)

# Advanced usage with custom arguments
knockoffs = function(X) create.second_order(X, method='equi')
result = knockoff.filter(X, y, knockoffs=knockoffs)
print(result$selected)   
  

Relaxed optimization for fixed-X and Gaussian knockoffs

Description

This function solves the optimization problem needed to create fixed-X and Gaussian SDP knockoffs on a block-diagonal approximation of the covariance matrix. This will be less powerful than create.solve_sdp, but more computationally efficient.

Usage

create.solve_asdp(
  Sigma,
  max.size = 500,
  gaptol = 1e-06,
  maxit = 1000,
  verbose = FALSE
)

Arguments

Details

Solves the following two-step semidefinite program:

(step 1)

\mathrm{maximize} \; \mathrm{sum}(s) \quad \mathrm{subject} \; \mathrm{to:} \; 0 \leq s \leq 1, \; 2 \Sigma_{\mathrm{approx}} - \mathrm{diag}(s) \geq 0

(step 2)

\mathrm{maximize} \; \gamma \quad \mathrm{subject} \; \mathrm{to:} \; \mathrm{diag}(\gamma s) \leq 2 \Sigma

Each smaller SDP is solved using the interior-point method implemented in dsdp.

The parameter max.size controls the size of the largest semidefinite program that needs to be solved. A larger value of max.size will increase the computation cost, while yielding a solution closer to that of the original semidefinite program.

If the matrix Sigma supplied by the user is a non-scaled covariance matrix (i.e. its diagonal entries are not all equal to 1), then the appropriate scaling is applied before solving the SDP defined above. The result is then scaled back before being returned, as to match the original scaling of the covariance matrix supplied by the user.

Value

The solution s to the semidefinite program defined above.

See Also

Other optimization: create.solve_equi(), create.solve_sdp()

Optimization for equi-correlated fixed-X and Gaussian knockoffs

Description

This function solves a very simple optimization problem needed to create fixed-X and Gaussian SDP knockoffs on the full the covariance matrix. This may be significantly less powerful than create.solve_sdp.

Usage

create.solve_equi(Sigma)

Arguments

Details

Computes the closed-form solution to the semidefinite programming problem:

\mathrm{maximize} \; s \quad \mathrm{subject} \; \mathrm{to:} \; 0 \leq s \leq 1, \; 2\Sigma - sI \geq 0

used to generate equi-correlated knockoffs.

The closed form-solution to this problem is s = 2\lambda_{\mathrm{min}}(\Sigma) \land 1.

Value

The solution s to the optimization problem defined above.

See Also

Other optimization: create.solve_asdp(), create.solve_sdp()

Optimization for fixed-X and Gaussian knockoffs

Description

This function solves the optimization problem needed to create fixed-X and Gaussian SDP knockoffs on the full covariance matrix. This will be more powerful than create.solve_asdp, but more computationally expensive.

Usage

create.solve_sdp(Sigma, gaptol = 1e-06, maxit = 1000, verbose = FALSE)

Arguments

Details

Solves the semidefinite programming problem:

\mathrm{maximize} \; \mathrm{sum}(s) \quad \mathrm{subject} \; \mathrm{to} 0 \leq s \leq 1, \; 2\Sigma - \mathrm{diag}(s) \geq 0

This problem is solved using the interior-point method implemented in dsdp.

If the matrix Sigma supplied by the user is a non-scaled covariance matrix (i.e. its diagonal entries are not all equal to 1), then the appropriate scaling is applied before solving the SDP defined above. The result is then scaled back before being returned, as to match the original scaling of the covariance matrix supplied by the user.

Value

The solution s to the semidefinite programming problem defined above.

See Also

Other optimization: create.solve_asdp(), create.solve_equi()

Vectorize a matrix into the SCS format

Description

Vectorize a matrix into the SCS format

Usage

create.vectorize_matrix(M)

Create equicorrelated fixed-X knockoffs.

Description

Create equicorrelated fixed-X knockoffs.

Usage

create_equicorrelated(X, randomize)

Create SDP fixed-X knockoffs.

Description

Create SDP fixed-X knockoffs.

Usage

create_sdp(X, randomize)

Compute the SVD of X and construct an orthogonal matrix U_perp such that U_perp * U = 0.

Description

Compute the SVD of X and construct an orthogonal matrix U_perp such that U_perp * U = 0.

Usage

decompose(X, randomize)

Approximate a covariance matrix by a block diagonal matrix with blocks of approximately equal size using Ward's method for hierarchical clustering

Description

Approximate a covariance matrix by a block diagonal matrix with blocks of approximately equal size using Ward's method for hierarchical clustering

Usage

divide.sdp(Sigma, max.size)

Forward selection

Description

Perform forward variable selection with or without OMP

Usage

fs(X, y, omp = FALSE)

Arguments

Value

vector with jth component the variable added at step j

knockoff: A package for controlled variable selection

Description

This package implements the Knockoff Filter, which is a powerful and versatile tool for controlled variable selection.

Outline

The procedure is based on the contruction of artificial 'knockoff copies' of the variables present in the given statistical model. Then, it selects those variables that are clearly better than their corresponding knockoffs, based on some measure of variable importance. A wide range of statistics and machine learning tools can be exploited to estimate the importance of each variable, while guaranteeing finite-sample control of the false discovery rate (FDR).

The Knockoff Filter controls the FDR in either of two statistical scenarios:

• The "model-X" scenario: the response Y can depend on the variables X=(X_1,\ldots,X_p) in an arbitrary and unknown fashion, but the distribution of X must be known. In thise case there are no constraints on the dimensions n and p of the problem.

• The "fixed-X" scenario: the response Y depends upon X through a homoscedastic Gaussian linear model and the problem is low-dimensional (n \geq p). In this case, no modeling assumptions on X are required.

For more information, see the website below and the accompanying paper.

https://web.stanford.edu/group/candes/knockoffs/index.html

The Knockoff Filter

Description

This function runs the Knockoffs procedure from start to finish, selecting variables relevant for predicting the outcome of interest.

Usage

knockoff.filter(
  X,
  y,
  knockoffs = create.second_order,
  statistic = stat.glmnet_coefdiff,
  fdr = 0.1,
  offset = 1
)

Arguments

Details

This function creates the knockoffs, computes the importance statistics, and selects variables. It is the main entry point for the knockoff package.

The parameter knockoffs controls how knockoff variables are created. By default, the model-X scenario is assumed and a multivariate normal distribution is fitted to the original variables X. The estimated mean vector and the covariance matrix are used to generate second-order approximate Gaussian knockoffs. In general, the function knockoffs should take a n-by-p matrix of observed variables X as input and return a n-by-p matrix of knockoffs. Two default functions for creating knockoffs are provided with this package.

In the model-X scenario, under the assumption that the rows of X are distributed as a multivariate Gaussian with known parameters, then the function create.gaussian can be used to generate Gaussian knockoffs, as shown in the examples below.

In the fixed-X scenario, one can create the knockoffs using the function create.fixed. This requires n \geq p and it assumes that the response Y follows a homoscedastic linear regression model.

For more information about creating knockoffs, type ??create.

The default importance statistic is stat.glmnet_coefdiff. For a complete list of the statistics provided with this package, type ??stat.

It is possible to provide custom functions for the knockoff constructions or the importance statistics. Some examples can be found in the vignette.

Value

An object of class "knockoff.result". This object is a list containing at least the following components:

References

Candes et al., Panning for Gold: Model-free Knockoffs for High-dimensional Controlled Variable Selection, arXiv:1610.02351 (2016). https://web.stanford.edu/group/candes/knockoffs/index.html

Barber and Candes, Controlling the false discovery rate via knockoffs. Ann. Statist. 43 (2015), no. 5, 2055–2085.

Examples

set.seed(2022)
p=100; n=80; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)

# Basic usage with default arguments
result = knockoff.filter(X, y)
print(result$selected)

# Advanced usage with custom arguments
knockoffs = function(X) create.gaussian(X, mu, Sigma)
k_stat = function(X, Xk, y) stat.glmnet_coefdiff(X, Xk, y, nfolds=5)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Threshold for the knockoff filter

Description

Computes the threshold for the knockoff filter.

Usage

knockoff.threshold(W, fdr = 0.1, offset = 1)

Arguments

Value

The threshold for variable selection.

Maximum lambda in lasso model

Description

Computes the earliest (largest) lambda's for which predictors enter the lasso model.

Usage

lasso_max_lambda(X, y, method = c("glmnet", "lars"), ...)

Arguments

Value

vector of maximum lambda's

Merge consecutive clusters of correlated variables while ensuring that no cluster has size larger than max.size

Description

Merge consecutive clusters of correlated variables while ensuring that no cluster has size larger than max.size

Usage

## S3 method for class 'clusters'
merge(clusters, max.size)

Print results for the knockoff filter

Description

Prints the list of variables selected by the knockoff filter and the corresponding function call.

Usage

## S3 method for class 'knockoff.result'
print(x, ...)

Arguments

Stability selection

Description

Perform variable selection with stability selection

Usage

stability_selection_importance(X, y, ...)

Arguments

Value

vector with jth component the selection probability of variable j

Importance statistics based on forward selection

Description

Computes the statistic

W_j = \max(Z_j, Z_{j+p}) \cdot \mathrm{sgn}(Z_j - Z_{j+p}),

where Z_1,\dots,Z_{2p} give the reverse order in which the 2p variables (the originals and the knockoffs) enter the forward selection model. See the Details for information about forward selection.

Usage

stat.forward_selection(X, X_k, y, omp = F)

Arguments

Details

In forward selection, the variables are chosen iteratively to maximize the inner product with the residual from the previous step. The initial residual is always y. In standard forward selection (stat.forward_selection), the next residual is the remainder after regressing on the selected variable; when orthogonal matching pursuit is used, the next residual is the remainder after regressing on all the previously selected variables.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.glmnet_coefdiff(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff_bin(), stat.lasso_coefdiff(), stat.lasso_lambdadiff_bin(), stat.lasso_lambdadiff(), stat.random_forest(), stat.sqrt_lasso(), stat.stability_selection()

Examples

set.seed(2022)
p=100; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs,
                           statistic=stat.forward_selection)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.forward_selection
k_stat = function(X, X_k, y) foo(X, X_k, y, omp=TRUE)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based on a GLM with cross-validation

Description

Fits a generalized linear model via penalized maximum likelihood and cross-validation. Then, compute the difference statistic

W_j = |Z_j| - |\tilde{Z}_j|

where Z_j and \tilde{Z}_j are the 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.

Usage

stat.glmnet_coefdiff(X, X_k, y, family = "gaussian", cores = 2, ...)

Arguments

Details

This function uses the glmnet package to fit a generalized linear model via penalized maximum likelihood.

The statistics W_j are constructed by taking the difference between the coefficient of the j-th variable and its knockoff.

By default, the value of the regularization parameter is chosen by 10-fold cross-validation.

The default response family is 'gaussian', for a linear regression model. Different response families (e.g. 'binomial') can be specified by passing an optional parameter 'family'.

The optional nlambda parameter can be used to control the granularity of the grid of \lambda's. The default value of nlambda is 500, where p is the number of columns of X.

If the family is 'binomial' and a lambda sequence is not provided by the user, this function generates it on a log-linear scale before calling 'glmnet'.

For a complete list of the available additional arguments, see cv.glmnet and glmnet.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff_bin(), stat.lasso_coefdiff(), stat.lasso_lambdadiff_bin(), stat.lasso_lambdadiff(), stat.random_forest(), stat.sqrt_lasso(), stat.stability_selection()

Examples

set.seed(2022)
p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs, 
                           statistic=stat.glmnet_coefdiff)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.glmnet_coefdiff
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based on a GLM

Description

Fits a generalized linear model via penalized maximum likelihood and computes the difference statistic

W_j = Z_j - \tilde{Z}_j

where Z_j and \tilde{Z}_j are the maximum values of the regularization parameter \lambda at which the jth variable and its knockoff enter the model, respectively.

Usage

stat.glmnet_lambdadiff(X, X_k, y, family = "gaussian", ...)

Arguments

Details

This function uses glmnet to compute the regularization path on a fine grid of \lambda's.

The nlambda parameter can be used to control the granularity of the grid of \lambda's. The default value of nlambda is 500.

If the family is 'binomial' and a lambda sequence is not provided by the user, this function generates it on a log-linear scale before calling 'glmnet'.

The default response family is 'gaussian', for a linear regression model. Different response families (e.g. 'binomial') can be specified by passing an optional parameter 'family'.

For a complete list of the available additional arguments, see glmnet.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_coefdiff(), stat.lasso_coefdiff_bin(), stat.lasso_coefdiff(), stat.lasso_lambdadiff_bin(), stat.lasso_lambdadiff(), stat.random_forest(), stat.sqrt_lasso(), stat.stability_selection()

Examples

set.seed(2022)
p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs, 
                           statistic=stat.glmnet_lambdadiff)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.glmnet_lambdadiff
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

GLM statistics for knockoff

Description

Computes the signed maximum statistic

W_j = \max(Z_j, \tilde{Z}_j) \cdot \mathrm{sgn}(Z_j - \tilde{Z}_j),

where Z_j and \tilde{Z}_j are the maximum values of \lambda at which the jth variable and its knockoff, respectively, enter the generalized linear model.

Usage

stat.glmnet_lambdasmax(X, X_k, y, family = "gaussian", ...)

Arguments

Details

This function uses glmnet to compute the regularization path on a fine grid of \lambda's.

The additional nlambda parameter can be used to control the granularity of the grid of \lambda values. The default value of nlambda is 500.

If the family is 'binomial' and a lambda sequence is not provided by the user, this function generates it on a log-linear scale before calling 'glmnet'.

For a complete list of the available additional arguments, see glmnet.

Value

A vector of statistics W of length p.

Examples

p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoff=knockoffs,
                           statistic=stat.glmnet_lambdasmax)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.glmnet_lambdasmax
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based the lasso with cross-validation

Description

Fits a linear regression model via penalized maximum likelihood and cross-validation. Then, compute the difference statistic

W_j = |Z_j| - |\tilde{Z}_j|

where Z_j and \tilde{Z}_j are the 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.

Usage

stat.lasso_coefdiff(X, X_k, y, cores = 2, ...)

Arguments

Details

This function uses the glmnet package to fit the lasso path and is a wrapper around the more general stat.glmnet_coefdiff.

The statistics W_j are constructed by taking the difference between the coefficient of the j-th variable and its knockoff.

By default, the value of the regularization parameter is chosen by 10-fold cross-validation.

The optional nlambda parameter can be used to control the granularity of the grid of \lambda's. The default value of nlambda is 500, where p is the number of columns of X.

Unless a lambda sequence is provided by the user, this function generates it on a log-linear scale before calling 'glmnet' (default 'nlambda': 500).

For a complete list of the available additional arguments, see cv.glmnet and glmnet.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_coefdiff(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff_bin(), stat.lasso_lambdadiff_bin(), stat.lasso_lambdadiff(), stat.random_forest(), stat.sqrt_lasso(), stat.stability_selection()

Examples

set.seed(2022)
p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs, 
                           statistic=stat.lasso_coefdiff)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.lasso_coefdiff
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based on regularized logistic regression with cross-validation

Description

Fits a logistic regression model via penalized maximum likelihood and cross-validation. Then, compute the difference statistic

W_j = |Z_j| - |\tilde{Z}_j|

where Z_j and \tilde{Z}_j are the 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.

Usage

stat.lasso_coefdiff_bin(X, X_k, y, cores = 2, ...)

Arguments

Details

This function uses the glmnet package to fit the penalized logistic regression path and is a wrapper around the more general stat.glmnet_coefdiff.

The statistics W_j are constructed by taking the difference between the coefficient of the j-th variable and its knockoff.

By default, the value of the regularization parameter is chosen by 10-fold cross-validation.

The optional nlambda parameter can be used to control the granularity of the grid of \lambda's. The default value of nlambda is 500, where p is the number of columns of X.

For a complete list of the available additional arguments, see cv.glmnet and glmnet.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_coefdiff(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff(), stat.lasso_lambdadiff_bin(), stat.lasso_lambdadiff(), stat.random_forest(), stat.sqrt_lasso(), stat.stability_selection()

Examples

set.seed(2022)
p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
pr = 1/(1+exp(-X %*% beta))
y = rbinom(n,1,pr)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs, 
                           statistic=stat.lasso_coefdiff_bin)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.lasso_coefdiff_bin
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based on the lasso

Description

Fit the lasso path and computes the difference statistic

W_j = Z_j - \tilde{Z}_j

where Z_j and \tilde{Z}_j are the maximum values of the regularization parameter \lambda at which the jth variable and its knockoff enter the penalized linear regression model, respectively.

Usage

stat.lasso_lambdadiff(X, X_k, y, ...)

Arguments

Details

This function uses glmnet to compute the lasso path on a fine grid of \lambda's and is a wrapper around the more general stat.glmnet_lambdadiff.

The nlambda parameter can be used to control the granularity of the grid of \lambda's. The default value of nlambda is 500.

Unless a lambda sequence is provided by the user, this function generates it on a log-linear scale before calling glmnet (default 'nlambda': 500).

For a complete list of the available additional arguments, see glmnet or lars.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_coefdiff(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff_bin(), stat.lasso_coefdiff(), stat.lasso_lambdadiff_bin(), stat.random_forest(), stat.sqrt_lasso(), stat.stability_selection()

Examples

set.seed(2022)
p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs, 
                           statistic=stat.lasso_lambdadiff)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.lasso_lambdadiff
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based on regularized logistic regression

Description

Fit the lasso path and computes the difference statistic

W_j = Z_j - \tilde{Z}_j

where Z_j and \tilde{Z}_j are the maximum values of the regularization parameter \lambda at which the jth variable and its knockoff enter the penalized logistic regression model, respectively.

Usage

stat.lasso_lambdadiff_bin(X, X_k, y, ...)

Arguments

Details

This function uses glmnet to compute the lasso path on a fine grid of \lambda's.

The nlambda parameter can be used to control the granularity of the grid of \lambda's. The default value of nlambda is 500.

This function is a wrapper around the more general stat.glmnet_lambdadiff.

For a complete list of the available additional arguments, see glmnet or lars.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_coefdiff(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff_bin(), stat.lasso_coefdiff(), stat.lasso_lambdadiff(), stat.random_forest(), stat.sqrt_lasso(), stat.stability_selection()

Examples

set.seed(2022)
p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
pr = 1/(1+exp(-X %*% beta))
y = rbinom(n,1,pr)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs, 
                           statistic=stat.lasso_lambdadiff_bin)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.lasso_lambdadiff_bin
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Penalized linear regression statistics for knockoff

Description

Computes the signed maximum statistic

W_j = \max(Z_j, \tilde{Z}_j) \cdot \mathrm{sgn}(Z_j - \tilde{Z}_j),

where Z_j and \tilde{Z}_j are the maximum values of \lambda at which the jth variable and its knockoff, respectively, enter the penalized linear regression model.

Usage

stat.lasso_lambdasmax(X, X_k, y, ...)

Arguments

Details

This function uses glmnet to compute the regularization path on a fine grid of \lambda's.

The additional nlambda parameter can be used to control the granularity of the grid of \lambda values. The default value of nlambda is 500.

Unless a lambda sequence is provided by the user, this function generates it on a log-linear scale before calling glmnet (default 'nlambda': 500).

This function is a wrapper around the more general stat.glmnet_lambdadiff.

For a complete list of the available additional arguments, see glmnet.

Value

A vector of statistics W of length p.

Examples

p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoff=knockoffs,
                           statistic=stat.lasso_lambdasmax)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.lasso_lambdasmax
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Penalized logistic regression statistics for knockoff

Description

Computes the signed maximum statistic

W_j = \max(Z_j, \tilde{Z}_j) \cdot \mathrm{sgn}(Z_j - \tilde{Z}_j),

where Z_j and \tilde{Z}_j are the maximum values of \lambda at which the jth variable and its knockoff, respectively, enter the penalized logistic regression model.

Usage

stat.lasso_lambdasmax_bin(X, X_k, y, ...)

Arguments

Details

This function uses glmnet to compute the regularization path on a fine grid of \lambda's.

The additional nlambda parameter can be used to control the granularity of the grid of \lambda values. The default value of nlambda is 500.

This function is a wrapper around the more general stat.glmnet_lambdadiff.

For a complete list of the available additional arguments, see glmnet.

Value

A vector of statistics W of length p.

Examples

p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
pr = 1/(1+exp(-X %*% beta))
y = rbinom(n,1,pr)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoff=knockoffs,
                           statistic=stat.lasso_lambdasmax_bin)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.lasso_lambdasmax_bin
k_stat = function(X, X_k, y) foo(X, X_k, y, nlambda=200)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based on random forests

Description

Computes the difference statistic

W_j = |Z_j| - |\tilde{Z}_j|

where Z_j and \tilde{Z}_j are the random forest feature importances of the jth variable and its knockoff, respectively.

Usage

stat.random_forest(X, X_k, y, ...)

Arguments

Details

This function uses the ranger package to compute variable importance measures. The importance of a variable is measured as the total decrease in node impurities from splitting on that variable, averaged over all trees. For regression, the node impurity is measured by residual sum of squares. For classification, it is measured by the Gini index.

For a complete list of the available additional arguments, see ranger.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_coefdiff(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff_bin(), stat.lasso_coefdiff(), stat.lasso_lambdadiff_bin(), stat.lasso_lambdadiff(), stat.sqrt_lasso(), stat.stability_selection()

Examples

set.seed(2022)
p=200; n=100; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs, 
                           statistic=stat.random_forest)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.random_forest
k_stat = function(X, X_k, y) foo(X, X_k, y, nodesize=5)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based on the square-root lasso

Description

Computes the signed maximum statistic

W_j = \max(Z_j, \tilde{Z}_j) \cdot \mathrm{sgn}(Z_j - \tilde{Z}_j),

where Z_j and \tilde{Z}_j are the maximum values of \lambda at which the jth variable and its knockoff, respectively, enter the SQRT lasso model.

Usage

stat.sqrt_lasso(X, X_k, y, ...)

Arguments

Details

With default parameters, this function uses the package RPtests to run the SQRT lasso. By specifying the appropriate optional parameters, one can use different Lasso variants including Dantzig Selector, LAD Lasso, SQRT Lasso and Lq Lasso for estimating high dimensional sparse linear models.

For a complete list of the available additional arguments, see sqrt_lasso.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_coefdiff(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff_bin(), stat.lasso_coefdiff(), stat.lasso_lambdadiff_bin(), stat.lasso_lambdadiff(), stat.random_forest(), stat.stability_selection()

Examples

set.seed(2022)
p=50; n=50; k=10
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=stat.sqrt_lasso)
print(result$selected)

# Advanced usage with custom arguments
foo = stat.sqrt_lasso
k_stat = function(X, X_k, y) foo(X, X_k, y, q=0.5)
result = knockoff.filter(X, y, knockoffs=knockoffs, statistic=k_stat)
print(result$selected)

Importance statistics based on stability selection

Description

Computes the difference statistic

W_j = |Z_j| - |\tilde{Z}_j|

where Z_j and \tilde{Z}_j are measure the importance of the jth variable and its knockoff, respectively, based on the stability of their selection upon subsampling of the data.

Usage

stat.stability_selection(X, X_k, y, fitfun = stabs::lars.lasso, ...)

Arguments

Details

This function uses the stabs package to compute variable selection stability. The selection stability of the j-th variable is defined as its probability of being selected upon random subsampling of the data. The default method for selecting variables in each subsampled dataset is lars.lasso.

For a complete list of the available additional arguments, see stabsel.

Value

A vector of statistics W of length p.

See Also

Other statistics: stat.forward_selection(), stat.glmnet_coefdiff(), stat.glmnet_lambdadiff(), stat.lasso_coefdiff_bin(), stat.lasso_coefdiff(), stat.lasso_lambdadiff_bin(), stat.lasso_lambdadiff(), stat.random_forest(), stat.sqrt_lasso()

Examples

set.seed(2022)
p=50; n=50; k=15
mu = rep(0,p); Sigma = diag(p)
X = matrix(rnorm(n*p),n)
nonzero = sample(p, k)
beta = 3.5 * (1:p %in% nonzero)
y = X %*% beta + rnorm(n)
knockoffs = function(X) create.gaussian(X, mu, Sigma)

# Basic usage with default arguments
result = knockoff.filter(X, y, knockoffs=knockoffs,
                         statistic=stat.stability_selection)
print(result$selected)

Verify dependencies for chosen statistics

Description

Verify dependencies for chosen statistics

Usage

verify_stat_depends(statistic)

Arguments