Generate randomly sampled data including noisy observations of latent variables, where proxies differ in their relevance (noise level)

Generate a data set including latent features Z, observed features X (which may include noisy or noiseless observations of the latent features in Z), an observed response y which is a linear model of features from Z and X as well as independent mean zero noise, and mu (the responses from y without the added noise). Data is generated in the same way as in the simulations from Section 5.3 of Faletto and Bien (2022).

Usage

genClusteredDataWeighted(
  n,
  p,
  k_unclustered,
  cluster_size,
  n_strong_cluster_vars,
  n_clusters = 1,
  sig_clusters = 1,
  rho_high = 0.9,
  rho_low = 0.5,
  beta_latent = 1.5,
  beta_unclustered = 1,
  snr = as.numeric(NA),
  sigma_eps_sq = as.numeric(NA)
)

Arguments

n

Integer or numeric; the number of observations to generate. (The generated X and Z will have n rows, and the generated y and mu will have length n.)

p

Integer or numeric; the number of features to generate. The generated X will have p columns.

k_unclustered

Integer or numeric; the number of features in X that will have nonzero coefficients in the true model for y among those features not generated from the n_clusters latent variables (called "weak signal" features in the simulations from Faletto and Bien 2022). The coefficients on these features will be determined by beta_unclustered.

cluster_size

Integer or numeric; for each of the n_clusters latent variables, X will contain cluster_size noisy proxies that are correlated with the latent variable.

n_strong_cluster_vars

Integer or numeric; among the cluster_size proxies in each cluster, the first n_strong_cluster_vars will have a high covariance (rho_high) with the latent variable and the next cluster_size - n_strong_cluster_vars will have a low covariance (rho_low) with the latent variable.

n_clusters

Integer or numeric; the number of latent variables to generate, each of which will be associated with an observed cluster in X. Must be at least 1. Default is 1.

sig_clusters

Integer or numeric; the number of generated latent features that will have nonzero coefficients in the true model for y (all of them will have coefficient beta_latent). Must be less than or equal to n_clusters. Default is 1.

rho_high

Integer or numeric; the covariance of the "strong proxies" in each cluster with the latent variable (and each other). Note that the correlation between the "strong proxy" features in the cluster will be rho_high/var. rho_high cannot equal 0 and must be at least as large as rho_low. Default is 0.9.

rho_low

Integer or numeric; the covariance of the "weak proxies" in each cluster with the latent variable (and each other). Note that the correlation between the "weak proxy" features in the cluster will be rho_low/var. rho_low cannot equal 0 and must be no larger than rho_high. Default is 0.5.

beta_latent

Integer or numeric; the coefficient used for all sig_clusters latent variables that have nonzero coefficients in the true model for y. Can't equal 0. Default is 1.5.

beta_unclustered

Integer or numeric; the maximum coefficient in the model for y among the k_unclustered features in X not generated from the latent variables. The coefficients of the features will be beta_unclustered/sqrt(1:k_unclustered). Can't equal 0. Default is 1.

snr

Integer or numeric; the signal-to-noise ratio of the response y. If sigma_eps_sq is not specified, the variance of the noise in y will be calculated using the formula sigma_eps_sq = sum(mu^2)/(n * snr). Only one of snr and sigma_eps_sq must be specified. Default is NA.

sigma_eps_sq

Integer or numeric; the variance on the noise added to y. Only one of snr and sigma_eps_sq must be specified. Default is NA.

var

Integer or numeric; the variance of all of the observed features in X (both the proxies for the latent variables and the k_unclustered other features). Can't equal 0. Default is 1.

Value

A list of the following elements.

X

An n x p numeric matrix of n observations from a p-dimensional multivariate normal distribution generated using the specified parameters. The first n_clusters times cluster_size features will be the clusters of features correlated with the n_clusters latent variables. The next k_unclustered features will be the "weak signal" features, and the remaining p - n_clusters*cluster_size - k_unclustered features will be the unclustered noise features.

y

A length n numeric vector; the response generated from X, the latent features from Z, and the coefficient vector, along with additive noise.

Z

The latent features; either a numeric vector (if n_clusters > 1) or a numeric matrix (if n_clusters > 1). Note that (X, Z) is multivariate Gaussian.

mu

A length n numeric vector; the expected response given X, Z, and the true coefficient vector (equal to y minus the added noise).

References

Faletto, G., & Bien, J. (2022). Cluster Stability Selection. arXiv preprint arXiv:2201.00494. https://arxiv.org/abs/2201.00494.

Author

Gregory Faletto, Jacob Bien