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)
```

**SNMoE** (Skew-Normal Mixtures-of-Experts) provides a flexible modelling
framework for heterogenous data with possibly skewed distributions to
generalize the standard Normal mixture of expert model. **SNMoE** consists of
a mixture of *K* skew-Normal 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 skewness parameters`lambda`

's.**SNMoE**thus generalises mixtures of (normal, skew-normal) distributions and mixtures of regressions with these distributions. For example, when $q=0$, we retrieve mixtures of (skew-normal, or normal) regressions, and when both $p=0$ and $q=0$, it is a mixture of (skew-normal, or normal) distributions. It also reduces to the standard (normal, skew-normal) 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 lambdak <- c(3, 5) # Skewness parameters of the experts sigmak <- c(1, 1) # Standard deviations of the experts x <- seq.int(from = -1, to = 1, length.out = n) # Inputs (predictors) # Generate sample of size n sample <- sampleUnivSNMoE(alphak = alphak, betak = betak, sigmak = sigmak, lambdak = lambdak, 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-6 verbose <- TRUE verbose_IRLS <- FALSE

snmoe <- emSNMoE(X = x, Y = y, K, p, q, n_tries, max_iter, threshold, verbose, verbose_IRLS)

snmoe$summary()

snmoe$plot(what = "meancurve")

snmoe$plot(what = "confregions")

snmoe$plot(what = "clusters")

snmoe$plot(what = "loglikelihood")

data("tempanomalies") x <- tempanomalies$Year y <- tempanomalies$AnnualAnomaly

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-6 verbose <- TRUE verbose_IRLS <- FALSE

snmoe <- emSNMoE(X = x, Y = y, K, p, q, n_tries, max_iter, threshold, verbose, verbose_IRLS)

snmoe$summary()

snmoe$plot(what = "meancurve")

snmoe$plot(what = "confregions")

snmoe$plot(what = "clusters")

snmoe$plot(what = "loglikelihood")

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