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(meteorits)
```

**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:

- The gating network parameters
`alpha`

's of the softmax net. - The experts network parameters: The location parameters (regression
coefficients)
`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="meteorits")`

for further details and references provided by
`citation("meteorits")`

.

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")

library(MASS) data("mcycle") x <- mcycle$times y <- mcycle$accel

K <- 4 # Number of regressors/experts p <- 2 # 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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