deepgmm {deepgmm}

Fits Deep Gaussian Mixture Models Using Stochastic EM algorithm.

Description

Fits a deep Gaussian mixture model to multivariate data.

Usage

deepgmm(y, layers, k, r,
        it = 250, eps = 0.001, init = "kmeans", init_est = "factanal",
        seed = NULL, scale = TRUE)

Arguments

y

A matrix or a data frame in which the rows correspond to observations and the columns to variables.

layers

The number of layers in the deep Gaussian mixture model. Limited to 1, 2 or 3.

k

A vector of integers of length layers containing the number of groups in the different layers.

r

A vector of integers of length layers containing the dimensions at the different layers. Dimension of the layers must be in decreasing size. See details.

it

Maximum number of EM iterations.

eps

The EM algorithm terminates if the relative increment of the log-likelihood falls below this value.

init

Procedure to obtain an initial partition of the observations. See Details.

init_est

Procedure for computing the initial parameter values for the given initial partition of the data. See Details.

seed

Integer value to be passed to the set.seed function at the biginning of the deepgmm function.

scale

If scale = TRUE, the columns of data, y, will be scaled to zero mean and unit variance.

Details

The deep Gaussian mixture model is an hierarchical model organized in a multilayered architecture where, at each layer, the variables follow a mixture of Gaussian distributions. This set of nested mixtures of linear models provides a globally nonlinear model that can model the data in a very flexible way. In order to avoid overparameterized solutions, dimension reduction by factor models can be applied at each layer of the architecture, thus resulting in deep mixtures of factor analyzers.

The data y must be a matrix or a data frame containing numerical values, with no missing values. The rows must correspond to observations and the columns to variables.

Presently, the maximum number of layers layers implemented is 3.

The ith element of k contain number of groups in the ith layer. Thus the length k must equal to layers.

The parameter vector r contains the latent variable dimension of each layer. Variables at different layers have progressively decreasing dimension, r_1, r_2, ..., r_h, where p > r_1 > r_2 > \dots > r_h \geq 1.

The EM algorithm used by dgmm requires initialization. The initialization is done by first partitioning the dataset, and then estimating the initial values for model parameters based on the partition. There are four options available in dgmm for the initial partitioning of the data; random partitioning (init = "random"), clustering using the k-means algorithm of "Hartigan-Wong" (init = "kmeans"), agglomerative hierarchical clustering (init = "hclass"). and Gaussian mixture model based clustering (init = "mclust").

After the initial partitioning has been chosen, initial values of the parameters in the component analyzers need to be calculated. Curently only one option available. This default option, init_est = "factanal" provides initial estimates of the parameters based on factor analysis.

Value

An object of class "dgmm" containing fitted values. It contains

H

A list in which the ith element is the loading matrix for the ith layer

w

A list containing mixing proportions for each layer. (i.e. the element w[[i]][j] contain the mixing proportion of the jth component in the i layer.)

mu

A list of matrices containing components means in the columns. (i.e. the element mu[[i]][, j] contain the component mean of the jth component in the i layer.)

psi

A list of arrays which contain covariance matrices for the random error components of each component (i.e. the element psi[[i]][j, ,, ] contain the error covariance matrix for the jth component in the i layer.)

lik

The log-likelihood after each EM iteration

bic

The Bayesian information criterion for the model fit

acl

The Akaike information criterion for the model fit

clc

The Classification likelihood information criterion for the model fit

icl.bic

The integrated classification criterion for the model fit

s

Clustering of the observations

seed

Value of the seed used

Author(s)

Cinzia Viroli, Geoffrey J. McLachlan

References

Viroli, C. and McLachlan, G.J. (2019). Deep Gaussian mixture models. Statistics and Computing 29, 43-51.

Examples

layers <- 2
k <- c(3, 4)
r <- c(3, 2)
it <- 50
eps <- 0.001
y <- scale(mtcars)

set.seed(1)
fit <-deepgmm(y = y, layers = layers, k = k, r = r,
                  it = it, eps = eps)
fit

summary(fit)