# A-quick-tour-of-StMoE

# Introduction

StMoE (Skew-t Mixture-of-Experts) provides a flexible and robust modelling framework for heterogenous data with possibly skewed, heavy-tailed distributions and corrupted by atypical observations. StMoE consists of a mixture of K skew-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, the skewness parameters `lambda`’s and the degree of freedom parameters `nu`’s. StMoE thus generalises mixtures of (normal, skew-normal, t, and skew-t) distributions and mixtures of regressions with these distributions. For example, when \(q=0\), we retrieve mixtures of (skew-t, t-, skew-normal, or normal) regressions, and when both \(p=0\) and \(q=0\), it is a mixture of (skew-t, t-, skew-normal, or normal) distributions. It also reduces to the standard (normal, skew-normal, t, and skew-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")`.

# Application to a simulated dataset

## Generate sample

    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
    lambdak <- c(3, 5) # Skewness parameters 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 <- sampleUnivStMoE(alphak = alphak, betak = betak, sigmak = sigmak,
                              lambdak = lambdak, nuk = nuk, x = x)
    y <- sample$y

## Set up StMoE model parameters

    K <- 2 # Number of regressors/experts
    p <- 1 # Order of the polynomial regression (regressors/experts)
    q <- 1 # Order of the logistic regression (gating network)

## Set up EM parameters

    n_tries <- 1
    max_iter <- 1500
    threshold <- 1e-5
    verbose <- TRUE
    verbose_IRLS <- FALSE

## Estimation

    stmoe <- emStMoE(X = x, Y = y, K, p, q, n_tries, max_iter,
                     threshold, verbose, verbose_IRLS)
    ## EM - StMoE: Iteration: 1 | log-likelihood: -358.341886247016
    ## EM - StMoE: Iteration: 2 | log-likelihood: -322.659951557355
    ## EM - StMoE: Iteration: 3 | log-likelihood: -320.693470431902
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    ## EM - StMoE: Iteration: 180 | log-likelihood: -253.963358402817

## Summary

    stmoe$summary()
    ## ------------------------------------------
    ## Fitted Skew t Mixture-of-Experts model
    ## ------------------------------------------
    ##
    ## StMoE model with K = 2 experts:
    ##
    ##  log-likelihood df       AIC      BIC      ICL
    ##       -253.9634 12 -265.9634 -291.251 -291.283
    ##
    ## Clustering table (Number of observations in each expert):
    ##
    ##   1   2
    ## 254 246
    ##
    ## Regression coefficients:
    ##
    ##     Beta(k = 1) Beta(k = 2)
    ## 1   -0.00186094 -0.08398201
    ## X^1  2.57239459 -2.55715595
    ##
    ## Variances:
    ##
    ##  Sigma2(k = 1) Sigma2(k = 2)
    ##      0.4328539     0.6125303

## Plots

### Mean curve

    stmoe$plot(what = "meancurve")

### Confidence regions

    stmoe$plot(what = "confregions")

### Clusters

    stmoe$plot(what = "clusters")

### Log-likelihood

    stmoe$plot(what = "loglikelihood")

# Application to real dataset

## Load data

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

## Set up StMoE model parameters

    K <- 4 # Number of regressors/experts
    p <- 2 # Order of the polynomial regression (regressors/experts)
    q <- 1 # Order of the logistic regression (gating network)

## Set up EM parameters

    n_tries <- 1
    max_iter <- 1500
    threshold <- 1e-5
    verbose <- TRUE
    verbose_IRLS <- FALSE

## Estimation

    stmoe <- emStMoE(X = x, Y = y, K, p, q, n_tries, max_iter,
                     threshold, verbose, verbose_IRLS)
    ## EM - StMoE: Iteration: 1 | log-likelihood: -598.565702033975
    ## EM - StMoE: Iteration: 2 | log-likelihood: -586.594322769658
    ## EM - StMoE: Iteration: 3 | log-likelihood: -585.173591847768
    ## EM - StMoE: Iteration: 4 | log-likelihood: -582.369299935532
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    ## EM - StMoE: Iteration: 60 | log-likelihood: -559.462898651175
    ## EM - StMoE: Iteration: 61 | log-likelihood: -559.442679098028
    ## EM - StMoE: Iteration: 62 | log-likelihood: -559.436665069673
    ## EM - StMoE: Iteration: 63 | log-likelihood: -559.435030878558

## Summary

    stmoe$summary()
    ## ------------------------------------------
    ## Fitted Skew t Mixture-of-Experts model
    ## ------------------------------------------
    ##
    ## StMoE model with K = 4 experts:
    ##
    ##  log-likelihood df      AIC       BIC       ICL
    ##        -559.435 30 -589.435 -632.7903 -632.7874
    ##
    ## Clustering table (Number of observations in each expert):
    ##
    ##  1  2  3  4
    ## 28 37 31 37
    ##
    ## Regression coefficients:
    ##
    ##     Beta(k = 1) Beta(k = 2)  Beta(k = 3) Beta(k = 4)
    ## 1    -3.6985620   992.62382 -1889.618688 292.2075328
    ## X^1   0.9395230  -103.99905   117.331380 -12.1454859
    ## X^2  -0.0859998     2.43552    -1.775234   0.1246457
    ##
    ## Variances:
    ##
    ##  Sigma2(k = 1) Sigma2(k = 2) Sigma2(k = 3) Sigma2(k = 4)
    ##       15.06473      453.8693      491.3541      575.0638

## Plots

### Mean curve

    stmoe$plot(what = "meancurve")

### Confidence regions

    stmoe$plot(what = "confregions")

### Clusters

    stmoe$plot(what = "clusters")

### Log-likelihood

    stmoe$plot(what = "loglikelihood")