Generate randomly sampled data including noisy observations of latent variables

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 Faletto and Bien (2022).

Usage

genClusteredData(
  n,
  p,
  k_unclustered,
  cluster_size,
  n_clusters = 1,
  sig_clusters = 1,
  rho = 0.9,
  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. Must be at least 1.

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. Must be at least 2.

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

Integer or numeric; the correlation of the proxies in each cluster with the latent variable. Must be greater than 0. Default is 0.9.

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.

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.

itemmuA 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