library(knitr) knitr::opts_chunk$set( fig.align = "center", fig.height = 5.5, fig.width = 6, warning = FALSE, collapse = TRUE, dev.args = list(pointsize = 10), out.width = "90%", par = TRUE ) knit_hooks$set(par = function(before, options, envir) { if (before && options$fig.show != "none") par(family = "sans", mar = c(4.1,4.1,1.1,1.1), mgp = c(3,1,0), tcl = -0.5) })
library(tMoE)
TMoE (t Mixture-of-Experts) provides a flexible and robust modelling framework for heterogenous data with possibly heavy-tailed distributions and corrupted by atypical observations. TMoE consists of a mixture of K t expert regressors network (of degree p) gated by a softmax gating network (of degree q) and is represented by:
alpha
's of the softmax net.beta
's, scale parameters sigma
's, and the degree of freedom
(robustness) parameters nu
's. TMoE thus generalises mixtures of (normal,
t, and) distributions and mixtures of regressions with these distributions. For
example, when $q=0$, we retrieve mixtures of (t-, or normal) regressions, and
when both $p=0$ and $q=0$, it is a mixture of (t-, or normal) distributions.
It also reduces to the standard (normal, t) distribution when we only use a
single expert ($K=1$).Model estimation/learning is performed by a dedicated expectation conditional maximization (ECM) algorithm by maximizing the observed data log-likelihood. We provide simulated examples to illustrate the use of the model in model-based clustering of heterogeneous regression data and in fitting non-linear regression functions.
It was written in R Markdown, using the knitr package for production.
See help(package="tMoE")
for further details and references provided by citation("tMoE")
.
n <- 500 # Size of the sample alphak <- matrix(c(0, 8), ncol = 1) # Parameters of the gating network betak <- matrix(c(0, -2.5, 0, 2.5), ncol = 2) # Regression coefficients of the experts sigmak <- c(0.5, 0.5) # Standard deviations of the experts nuk <- c(5, 7) # Degrees of freedom of the experts network t densities x <- seq.int(from = -1, to = 1, length.out = n) # Inputs (predictors) # Generate sample of size n sample <- sampleUnivTMoE(alphak = alphak, betak = betak, sigmak = sigmak, nuk = nuk, x = x) y <- sample$y
K <- 2 # Number of regressors/experts p <- 1 # Order of the polynomial regression (regressors/experts) q <- 1 # Order of the logistic regression (gating network)
n_tries <- 1 max_iter <- 1500 threshold <- 1e-5 verbose <- TRUE verbose_IRLS <- FALSE
tmoe <- emTMoE(X = x, Y = y, K, p, q, n_tries, max_iter, threshold, verbose, verbose_IRLS)
tmoe$summary()
tmoe$plot(what = "meancurve")
tmoe$plot(what = "confregions")
tmoe$plot(what = "clusters")
tmoe$plot(what = "loglikelihood")
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