1. Mixture-of-Experts Wishart models for covariance data

This R-package moewishart provides maximum likelihood estimation (MLE) and Bayesian estimation for the Wishart mixture model and the Wishart mixture-of-experts (MoE-Wishart) model. It implements four different inference algorithms for the two model:

• mixture model of Wishart distributions:
  • EM algorithm for finding the MLE;
  • Bayesian approach using Gibbs-within-MH sampling algorithm.
• Mixture-of-Expert model, in which the gating probabilities depend on covariates:
  • EM-MoE algorithm for finding the MLE;
  • Bayesian-MoE approach using a Gibbs-within-MH sampling algorithm.

2. Installation

Install the latest development version from GitHub:

#library("devtools")
devtools::install_github("zhizuio/moewishart")

3. Example: data generating process

Data simulation from a MoE-Wishart model:

• Sample size \(n = 200\)
• Dimension of the Wishart distribution \(p = 2\)
• Number of latent components \(K = 3\)
• \(q=3\) gating covariates \(\mathbf X = [x_{ij}] \in \mathbb R^{n\times q}\), \(x_{ij}\sim\text{N}(0,1)\), \(i=1,...,n\), \(j=1,...,q\)
• Fixed covariate effects \(\boldsymbol\beta=[\boldsymbol\beta_1,...,\boldsymbol\beta_K] \in \mathbb R^{q\times K}\), with \(\boldsymbol\beta_{K}=0\)
• Probabilities of subpopulations \(\boldsymbol\pi = [\pi_{ik}] \in \mathbb R^{n\times q}\), \(\pi_{ik} = \exp(\mathbf X_i\boldsymbol\beta_k) / \sum_{l=1}^K\exp(\mathbf X_i\boldsymbol\beta_l)\), \(k=1,...,K\)
• Degrees of freedom \(\boldsymbol\nu = (\nu_1,\nu_2,\nu_3) = (8, 12, 3)\)
• Scale matrices of the Wishart distribution \(\Sigma_1\), \(\Sigma_2\), \(\Sigma_3 \in \mathbb R^{p\times p}\)
• Data \(S_i \sim \pi_{i1}\text{Wishart}(\nu_1, \Sigma_1) + \pi_{i2}\text{Wishart}(\nu_2, \Sigma_2) + \pi_{i3}\text{Wishart}(\nu_3, \Sigma_3)\)

library(moewishart)

n <- 200 # number of subjects
p <- 2 # dimension of covariance matrix
set.seed(123) # fix coefficients of underlying MoE model
Xq <- 3; K = 3
betas <- matrix(runif(Xq * K, -2, 2), nrow = Xq, ncol = K)
betas[, K] <- 0

# simulate data
dat <- simData(n, p,
  Xq = 3, K = 3, betas = betas,
  pis = c(0.35, 0.40, 0.25),
  nus = c(8, 12, 3)
)

4. Model fitting examples

4.1 Bayesian MoE-Wishart model

# fit Bayesian MoE-Wishart model
set.seed(123)
fit <- moewishart(
  dat$S, X = cbind(1, dat$X), K = 3, 
  mh_sigma = c(0.2, 0.1, 0.2), # RW-MH variances (length K)
  mh_beta = c(0.3, 0.3), # RW-MH variances (length K-1)
  niter = 3000, burnin = 1000
)

4.2 Bayesian Wishart mixture model

# fit Bayesian Wishart mixture model
set.seed(123)
fit2 <- mixturewishart(
  dat$S, K = 3, 
  mh_sigma = c(0.2, 0.1, 0.2), # RW-MH variances
  niter = 3000, burnin = 1000
)

4.3 EM for MoE-Wishart

# fit MoE-Wishart model via EM alg.
set.seed(123)
fit3 <- moewishart(
  dat$S, X = cbind(1, dat$X), K = 3, 
  method = "em",
  niter = 3000
)

4.4 EM for Wishart mixture

# fit Wishart mixture model via EM alg.
set.seed(123)
fit4 <- mixturewishart(
  dat$S, K = 3, 
  method = "em",
  niter = 3000
)