dppmix (version 0.1.1)

dppmix_mvnorm: Fit a determinantal point process multivariate normal mixture model.

Description

Discover clusters in multidimensional data using a multivariate normal mixture model with a determinantal point process prior.

Usage

dppmix_mvnorm(
  X,
  hparams = NULL,
  store = NULL,
  control = NULL,
  fixed = NULL,
  verbose = TRUE
)

Arguments

X

N x J data matrix of N observations and J features

hparams

a list of hyperparameter values: delta, a0, b0, theta, sigma_prop_mu

store

a vector of character strings specifying additional vars of interest; a value of NA indicates that samples of all parameters in the model will be stored

control

a list of control parameters: niter, burnin, thin

fixed

a list of fixed parameter values

verbose

whether to emit verbose message

Value

a dppmix_mcmc object containing posterior samples of the parameters

Details

A determinantal point process (DPP) prior is a repulsive prior. Compare to mixture models using independent priors, a DPP mixutre model will often discover a parsimonious set of mixture components (clusters).

Model fitting is done by sampling parameters from the posterior distribution using a reversible jump Markov chain Monte Carlo sampling approach.

Given \(X = [x_i]\), where each \(x_i\) is a D-dimensional real vector, we seek the posterior distribution the latent variable \(z = [z_i]\), where each \(z_i\) is an integer representing cluster membership.

$$ x_i \mid z_i \sim Normal(\mu_k, \Sigma_k) $$ $$ z_i \sim Categorical(w) $$ $$ w \sim Dirichlet([\delta ... \delta]) $$ $$ \mu_k \sim DPP(C) $$

where \(C\) is the covariance function that evaluates the distances among the data points:

$$ C(x_1, x_2) = exp( - \sum_d \frac{ (x_1 - x_2)^2 }{ \theta^2 } ) $$

We also define \(\Sigma_k = E_k \Lambda_k E_k^\top\), where \(E_k\) is an orthonormal matrix whose column represents eigenvectors. We further assume that \(E_k = E\) is fixed across all cluster components so that \(E\) can be estimated as the eigenvectors of the covariance matrix of the data matrix \(X\). Finally, we put a prior on the entries of the \(\Lambda_k\) diagonal matrix:

$$ \lambda_{kd}^{-1} \sim Gamma( a_0, b_0 ) $$

Hence, the hyperameters of the model include: delta, a0, b0, theta, as well as sampling hyperparameter sigma_pro_mu, which controls the spread of the Gaussian proposal distribution for the random-walk Metropolis-Hastings update of the \(\mu\) parameter.

The parameters (and their dimensions) in the model include: K, z (N x 1), w (K x 1), lambda (K x J), mu (K x J), Sigma (J x J x K). If any parameter is fixed, then K must be fixed as well.

References

Yanxun Xu, Peter Mueller, Donatello Telesca. Bayesian Inference for Latent Biologic Structure with Determinantal Point Processes. Biometrics. 2016;72(3):955-64.

Examples

# NOT RUN {
set.seed(1)
ns <- c(3, 3)
means <- list(c(-6, -3), c(0, 4))
d <- rmvnorm_clusters(ns, means)

mcmc <- dppmix_mvnorm(d$X, verbose=FALSE)
res <- estimate(mcmc)
table(d$cl, res$z)

# }