kernelPSI: a Post-Selection Inference Framework for Nonlinear Variable Selection

In this vignette, we illustrate on a synthetic dataset how to perform post-selection inference (PSI) for a set of kernels using our R package kernelPSI (Slim et al. 2019). The kernels are selected in a forward fashion according to quadratic kernel association scores, leading to the modeling of the selection event as a set of quadratic constraints. For valid inference, we need to correct for the bias introduced by the prior selection of the kernels. Determining the exact post-selection distribution of our test statistics under the selection event was impossible. To overcome that, we use sampling in order to derive empirical \(p\)-values.

Simulation

We first start by giving the setup for our simulation. We associate \(10\) gaussian kernels to \(10\) independent groups of variables of size \(5\) each. Within each group, the variables are drawn from a multivariate normal distribution with mean \(0\) and a correlation matrix \(V_{ij} = \rho^{\lvert i-j\rvert}\), where \(\rho = 0.6\) and \(i,j\in[1\mathrel{{.}\,{.}} 5]\). The dataset we consider here consists of \(100\) independent samples.

require("MASS")
#> Loading required package: MASS

n_kernels <- 10 # total number of kernels
size_kernels <- 5 # dimensionality of the data associated to each kernel

n <- 100 # sample size
rho <- 0.6 # correlation parameter

# intra-group correlation matrix
corr <- outer(seq_len(size_kernels), seq_len(size_kernels),
              function(i, j) return(rho^(abs(i-j))))

# design matrix
X <- replicate(n_kernels,
               mvrnorm(n, mu = rep(0, size_kernels), Sigma = corr),
               simplify = FALSE)

To each group, we associate a local gaussian kernel \(K\) of variance \(\sigma^2 = 5\), i.e. \(K(x_i, x_j) = \exp(\lvert\lvert x_i-x_j\rvert\rvert^2/\sigma^2)\).

require("kernlab")
#> Loading required package: kernlab
K <- replicate(n_kernels, rbfdot(sigma = 1 / size_kernels))  # full list of Gaussian kernels

# List of Gram matrices
Kmat <- sapply(seq_len(n_kernels),
               function(i) {kMatrix <- kernelMatrix(K[[i]], X[[i]]);
               return(as.kernelMatrix(kMatrix, center = TRUE))},
               simplify = FALSE)

We select the first three kernels as causal kernels, and simulate the outcome \(Y\) in the following way: \(Y\sim 0.1 * U + \epsilon\). \(U\) is the first eigenvector of the similarity matrix of the sum kernel of the first three kernel and \(\epsilon\) is a reduced gaussian random error. For the outcome, we consider the linear kernel.

m_kernels <- 3 # number of causal kernels
theta <- 0.1 # amplitude of size effect

Ksum <- Reduce(`+`, Kmat[seq_len(m_kernels)]) # sum kernel of the causal kernels
decompK <- eigen(Ksum) # eigenvalue decomposition of the sum kernel Ksum

Y <- as.matrix(theta * decompK$values[1] * decompK$vectors[, 1] + rnorm(n), ncol = 1) # response vector
Lmat <- kernelMatrix(new("vanillakernel"), Y) # linear response vector

Kernel selection

The first step in PSI is to select the kernels. For that purpose, we use the function FOHSIC for the fixed variant and adaFOHSIC for the adaptive variant. Afterwards, to generate the list of matrices associated to the quadratic constraint of each selection event, we respectively apply forwardQ and adaQ.

require("kernelPSI")
#> Loading required package: kernelPSI

candidate_kernels <- 3 # number of kernels for the fixed variant
selectFOHSIC <- FOHSIC(Kmat, Lmat, mKernels = candidate_kernels) # fixed variant
constraintFO <- forwardQ(Kmat, selectFOHSIC) # list of quadratic constraints modeling the selection event

If the number of causal kernels is not available beforehand, we resort to the adaptive version:

selectAHSIC <- adaFOHSIC(Kmat, Lmat) # adaptive variant
adaFO <- adaQ(Kmat, selectAHSIC[["selection"]], selectAHSIC[["n"]]) # list of quadratic constraints for the adaptive selection method
adaS <- selectAHSIC$selection[seq_len(selectAHSIC$n)] # indices of selected kernels

Inference

Finally, using the obtained constraints, we can derive PSI significance values for three statistics: the log-likelihood ratio for the ridge prototype, the log-likelihood ratio for the kernel PCA prototype and the HSIC score. The \(p\)-values are computed by comparing the statistics of the original response to those of replicates sampled within the acceptance region of the selection event. Most often, because of the difference in their statistical power, the methods yield different \(p\)-values.

n_replicates <- 5000 # number of replicates (statistical power and validity require a higher number of samples)
burn_in <- 1000 # number of burn-in iterations

# Fixed variant ------------------
# selected methods: 'ridge' for the kernel ridge regression prototype 
# and 'pca' for the kernel principal component regression prototype
kernelPSI(Y, K_select = Kmat[selectFOHSIC], constraintFO, method = c("ridge", "pca"), 
          n_replicates = n_replicates, burn_in = burn_in)
#> $ridge
#> [1] 0.0382
#> 
#> $pca
#> [1] 0.2502

# Adaptive variant ------------------
# selected methods: 'hsic' for the unbiased HSIC estimator
kernelPSI(Y, K_select = Kmat[adaS], constraintFO, method = "hsic", 
          n_replicates = n_replicates, burn_in = burn_in)
#> $hsic
#> [1] 0.0912

kernelPSI: a Post-Selection Inference Framework for Nonlinear Variable Selection

Lotfi Slim

2019-12-07

Simulation

Kernel selection

Inference

References