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Robust modeling, density estimation and model-based clustering of heterogeneous regression data with possibly skewed and non-normal distributions using skew-t mixture of experts.
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README.md

Overview

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.

Installation

You can install the development version of StMoE from GitHub with:

# install.packages("devtools")
devtools::install_github("fchamroukhi/StMoE")

To build vignettes for examples of usage, type the command below instead:

# install.packages("devtools")
devtools::install_github("fchamroukhi/StMoE", 
                         build_opts = c("--no-resave-data", "--no-manual"), 
                         build_vignettes = TRUE)

Use the following command to display vignettes:

browseVignettes("StMoE")

Usage

library(StMoE)
# Applicartion to a simulated data set

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

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

stmoe <- emStMoE(X = x, Y = y, K = K, p = p, q = q, verbose = TRUE)
#> EM - StMoE: Iteration: 1 | log-likelihood: -313.677036627204
#> EM - StMoE: Iteration: 2 | log-likelihood: -304.743366974552
#> EM - StMoE: Iteration: 3 | log-likelihood: -303.241083338317
#> EM - StMoE: Iteration: 4 | log-likelihood: -302.398166051641
#> EM - StMoE: Iteration: 5 | log-likelihood: -301.932198441693
#> EM - StMoE: Iteration: 6 | log-likelihood: -301.652858521563
#> EM - StMoE: Iteration: 7 | log-likelihood: -301.463184106081
#> EM - StMoE: Iteration: 8 | log-likelihood: -301.316302040905
#> EM - StMoE: Iteration: 9 | log-likelihood: -301.189780597304
#> EM - StMoE: Iteration: 10 | log-likelihood: -301.071905070965
#> EM - StMoE: Iteration: 11 | log-likelihood: -300.955853288454
#> EM - StMoE: Iteration: 12 | log-likelihood: -300.836643184448
#> EM - StMoE: Iteration: 13 | log-likelihood: -300.709435818877
#> EM - StMoE: Iteration: 14 | log-likelihood: -300.569024504568
#> EM - StMoE: Iteration: 15 | log-likelihood: -300.408661189373
#> EM - StMoE: Iteration: 16 | log-likelihood: -300.218849079537
#> EM - StMoE: Iteration: 17 | log-likelihood: -299.985627697807
#> EM - StMoE: Iteration: 18 | log-likelihood: -299.687683044761
#> EM - StMoE: Iteration: 19 | log-likelihood: -299.292130029605
#> EM - StMoE: Iteration: 20 | log-likelihood: -298.748230172636
#> EM - StMoE: Iteration: 21 | log-likelihood: -297.980506294133
#> EM - StMoE: Iteration: 22 | log-likelihood: -296.884602134782
#> EM - StMoE: Iteration: 23 | log-likelihood: -295.336508276659
#> EM - StMoE: Iteration: 24 | log-likelihood: -293.246932579091
#> EM - StMoE: Iteration: 25 | log-likelihood: -290.625743944283
#> EM - StMoE: Iteration: 26 | log-likelihood: -287.611320408979
#> EM - StMoE: Iteration: 27 | log-likelihood: -284.41975156752
#> EM - StMoE: Iteration: 28 | log-likelihood: -281.270955252439
#> EM - StMoE: Iteration: 29 | log-likelihood: -278.324697139084
#> EM - StMoE: Iteration: 30 | log-likelihood: -275.663312584699
#> EM - StMoE: Iteration: 31 | log-likelihood: -273.312782249227
#> EM - StMoE: Iteration: 32 | log-likelihood: -271.247799566446
#> EM - StMoE: Iteration: 33 | log-likelihood: -269.423673029226
#> EM - StMoE: Iteration: 34 | log-likelihood: -267.775445509454
#> EM - StMoE: Iteration: 35 | log-likelihood: -266.219396053972
#> EM - StMoE: Iteration: 36 | log-likelihood: -264.661604129841
#> EM - StMoE: Iteration: 37 | log-likelihood: -263.009700555625
#> EM - StMoE: Iteration: 38 | log-likelihood: -261.196661029592
#> EM - StMoE: Iteration: 39 | log-likelihood: -259.203601716413
#> EM - StMoE: Iteration: 40 | log-likelihood: -257.071233696119
#> EM - StMoE: Iteration: 41 | log-likelihood: -254.879737370432
#> EM - StMoE: Iteration: 42 | log-likelihood: -252.721037210609
#> EM - StMoE: Iteration: 43 | log-likelihood: -250.679002209029
#> EM - StMoE: Iteration: 44 | log-likelihood: -248.81377156785
#> EM - StMoE: Iteration: 45 | log-likelihood: -247.155815597416
#> EM - StMoE: Iteration: 46 | log-likelihood: -245.714964975576
#> EM - StMoE: Iteration: 47 | log-likelihood: -244.481277797412
#> EM - StMoE: Iteration: 48 | log-likelihood: -243.436323002969
#> EM - StMoE: Iteration: 49 | log-likelihood: -242.555006506039
#> EM - StMoE: Iteration: 50 | log-likelihood: -241.819072823087
#> EM - StMoE: Iteration: 51 | log-likelihood: -241.207701943778
#> EM - StMoE: Iteration: 52 | log-likelihood: -240.698844726696
#> EM - StMoE: Iteration: 53 | log-likelihood: -240.275684577036
#> EM - StMoE: Iteration: 54 | log-likelihood: -239.92769041402
#> EM - StMoE: Iteration: 55 | log-likelihood: -239.642524941545
#> EM - StMoE: Iteration: 56 | log-likelihood: -239.407925889792
#> EM - StMoE: Iteration: 57 | log-likelihood: -239.214894050006
#> EM - StMoE: Iteration: 58 | log-likelihood: -239.056455654766
#> EM - StMoE: Iteration: 59 | log-likelihood: -238.927514839681
#> EM - StMoE: Iteration: 60 | log-likelihood: -238.824016678719
#> EM - StMoE: Iteration: 61 | log-likelihood: -238.741434320082
#> EM - StMoE: Iteration: 62 | log-likelihood: -238.676164882533
#> EM - StMoE: Iteration: 63 | log-likelihood: -238.625284613346
#> EM - StMoE: Iteration: 64 | log-likelihood: -238.586424632453
#> EM - StMoE: Iteration: 65 | log-likelihood: -238.557414897791
#> EM - StMoE: Iteration: 66 | log-likelihood: -238.536735631366
#> EM - StMoE: Iteration: 67 | log-likelihood: -238.524181304992
#> EM - StMoE: Iteration: 68 | log-likelihood: -238.518114601757
#> EM - StMoE: Iteration: 69 | log-likelihood: -238.517231398422
#> EM - StMoE: Iteration: 70 | log-likelihood: -238.520493627843
#> EM - StMoE: Iteration: 71 | log-likelihood: -238.52706429015
#> EM - StMoE: Iteration: 72 | log-likelihood: -238.536260790101
#> EM - StMoE: Iteration: 73 | log-likelihood: -238.54752329793
#> EM - StMoE: Iteration: 74 | log-likelihood: -238.560398935216
#> EM - StMoE: Iteration: 75 | log-likelihood: -238.574532328275
#> EM - StMoE: Iteration: 76 | log-likelihood: -238.589477015097
#> EM - StMoE: Iteration: 77 | log-likelihood: -238.605088066555
#> EM - StMoE: Iteration: 78 | log-likelihood: -238.621134173219
#> EM - StMoE: Iteration: 79 | log-likelihood: -238.637426566943
#> EM - StMoE: Iteration: 80 | log-likelihood: -238.653782793141
#> EM - StMoE: Iteration: 81 | log-likelihood: -238.670141096202
#> EM - StMoE: Iteration: 82 | log-likelihood: -238.686345061787
#> EM - StMoE: Iteration: 83 | log-likelihood: -238.702364058959
#> EM - StMoE: Iteration: 84 | log-likelihood: -238.718048958972
#> EM - StMoE: Iteration: 85 | log-likelihood: -238.733817065732
#> EM - StMoE: Iteration: 86 | log-likelihood: -238.749432901382
#> EM - StMoE: Iteration: 87 | log-likelihood: -238.764802587963
#> EM - StMoE: Iteration: 88 | log-likelihood: -238.779855505803
#> EM - StMoE: Iteration: 89 | log-likelihood: -238.79453920768
#> EM - StMoE: Iteration: 90 | log-likelihood: -238.808815635146
#> EM - StMoE: Iteration: 91 | log-likelihood: -238.822658179524
#> EM - StMoE: Iteration: 92 | log-likelihood: -238.836049346667
#> EM - StMoE: Iteration: 93 | log-likelihood: -238.848978046385
#> EM - StMoE: Iteration: 94 | log-likelihood: -238.86144499411
#> EM - StMoE: Iteration: 95 | log-likelihood: -238.873439717377
#> EM - StMoE: Iteration: 96 | log-likelihood: -238.884968650432
#> EM - StMoE: Iteration: 97 | log-likelihood: -238.896036051151
#> EM - StMoE: Iteration: 98 | log-likelihood: -238.906648555212
#> EM - StMoE: Iteration: 99 | log-likelihood: -238.916880446633
#> EM - StMoE: Iteration: 100 | log-likelihood: -238.926655910993
#> EM - StMoE: Iteration: 101 | log-likelihood: -238.935997808103
#> EM - StMoE: Iteration: 102 | log-likelihood: -238.944937477143
#> EM - StMoE: Iteration: 103 | log-likelihood: -238.953573696157
#> EM - StMoE: Iteration: 104 | log-likelihood: -238.961810452354
#> EM - StMoE: Iteration: 105 | log-likelihood: -238.969666380971
#> EM - StMoE: Iteration: 106 | log-likelihood: -238.977158583039
#> EM - StMoE: Iteration: 107 | log-likelihood: -238.984303049676
#> EM - StMoE: Iteration: 108 | log-likelihood: -238.991114897365
#> EM - StMoE: Iteration: 109 | log-likelihood: -238.997608506568
#> EM - StMoE: Iteration: 110 | log-likelihood: -239.003797609158
#> EM - StMoE: Iteration: 111 | log-likelihood: -239.009695347534
#> EM - StMoE: Iteration: 112 | log-likelihood: -239.015314317262
#> EM - StMoE: Iteration: 113 | log-likelihood: -239.020666599624
#> EM - StMoE: Iteration: 114 | log-likelihood: -239.025763787696
#> EM - StMoE: Iteration: 115 | log-likelihood: -239.030617008102
#> EM - StMoE: Iteration: 116 | log-likelihood: -239.035236939822
#> EM - StMoE: Iteration: 117 | log-likelihood: -239.039633830913
#> EM - StMoE: Iteration: 118 | log-likelihood: -239.043817513777
#> EM - StMoE: Iteration: 119 | log-likelihood: -239.047797419374
#> EM - StMoE: Iteration: 120 | log-likelihood: -239.051582590672
#> EM - StMoE: Iteration: 121 | log-likelihood: -239.055181695541
#> EM - StMoE: Iteration: 122 | log-likelihood: -239.058603039229
#> EM - StMoE: Iteration: 123 | log-likelihood: -239.061854576498
#> EM - StMoE: Iteration: 124 | log-likelihood: -239.064943923499
#> EM - StMoE: Iteration: 125 | log-likelihood: -239.067878369414
#> EM - StMoE: Iteration: 126 | log-likelihood: -239.070664887875
#> EM - StMoE: Iteration: 127 | log-likelihood: -239.073310148188
#> EM - StMoE: Iteration: 128 | log-likelihood: -239.075820526336
#> EM - StMoE: Iteration: 129 | log-likelihood: -239.078202458469
#> EM - StMoE: Iteration: 130 | log-likelihood: -239.08046194937
#> EM - StMoE: Iteration: 131 | log-likelihood: -239.082604373028
#> EM - StMoE: Iteration: 132 | log-likelihood: -239.084634900991
#> EM - StMoE: Iteration: 133 | log-likelihood: -239.086558499205
#> EM - StMoE: Iteration: 134 | log-likelihood: -239.088379928592
#> EM - StMoE: Iteration: 135 | log-likelihood: -239.090103747864
#> EM - StMoE: Iteration: 136 | log-likelihood: -239.091734317747
#> EM - StMoE: Iteration: 137 | log-likelihood: -239.093275806105
#> EM - StMoE: Iteration: 138 | log-likelihood: -239.094732193658
#> EM - StMoE: Iteration: 139 | log-likelihood: -239.096107280062
#> EM - StMoE: Iteration: 140 | log-likelihood: -239.097404690216
#> EM - StMoE: Iteration: 141 | log-likelihood: -239.098627880668
#> EM - StMoE: Iteration: 142 | log-likelihood: -239.099780146058
#> EM - StMoE: Iteration: 143 | log-likelihood: -239.100864625507
#> EM - StMoE: Iteration: 144 | log-likelihood: -239.101884308933
#> EM - StMoE: Iteration: 145 | log-likelihood: -239.102842043246
#> EM - StMoE: Iteration: 146 | log-likelihood: -239.103740538392
#> EM - StMoE: Iteration: 147 | log-likelihood: -239.104582373239
#> EM - StMoE: Iteration: 148 | log-likelihood: -239.105370001276
#> EM - StMoE: Iteration: 149 | log-likelihood: -239.106105756137
#> EM - StMoE: Iteration: 150 | log-likelihood: -239.106791856921
#> EM - StMoE: Iteration: 151 | log-likelihood: -239.107430413325
#> EM - StMoE: Iteration: 152 | log-likelihood: -239.108023430572
#> EM - StMoE: Iteration: 153 | log-likelihood: -239.108572814152
#> EM - StMoE: Iteration: 154 | log-likelihood: -239.109080374363
#> EM - StMoE: Iteration: 155 | log-likelihood: -239.109547830665
#> EM - StMoE: Iteration: 156 | log-likelihood: -239.109976815845
#> EM - StMoE: Iteration: 157 | log-likelihood: -239.110368879998
#> EM - StMoE: Iteration: 158 | log-likelihood: -239.110725494333
#> EM - StMoE: Iteration: 159 | log-likelihood: -239.111048054801
#> EM - StMoE: Iteration: 160 | log-likelihood: -239.111337885551
#> EM - StMoE: Iteration: 161 | log-likelihood: -239.111596242234
#> EM - StMoE: Iteration: 162 | log-likelihood: -239.11182431513

stmoe$summary()
#> ------------------------------------------
#> Fitted Skew t Mixture-of-Experts model
#> ------------------------------------------
#> 
#> StMoE model with K = 2 experts:
#> 
#>  log-likelihood df       AIC       BIC       ICL
#>       -239.1118 12 -251.1118 -276.3995 -276.7281
#> 
#> Clustering table (Number of observations in each expert):
#> 
#>   1   2 
#> 249 251 
#> 
#> Regression coefficients:
#> 
#>     Beta(k = 1) Beta(k = 2)
#> 1   -0.08583445   -0.139524
#> X^1  2.45592534   -2.465143
#> 
#> Variances:
#> 
#>  Sigma2(k = 1) Sigma2(k = 2)
#>      0.4514526     0.5186537

stmoe$plot()

# Applicartion to a real data set

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)

stmoe <- emStMoE(X = x, Y = y, K = K, p = p, q = q, verbose = TRUE)
#> EM - StMoE: Iteration: 1 | log-likelihood: -599.121493144955
#> EM - StMoE: Iteration: 2 | log-likelihood: -589.783485660297
#> EM - StMoE: Iteration: 3 | log-likelihood: -587.281458998556
#> EM - StMoE: Iteration: 4 | log-likelihood: -585.537728626033
#> EM - StMoE: Iteration: 5 | log-likelihood: -584.033642364304
#> EM - StMoE: Iteration: 6 | log-likelihood: -582.48206640408
#> EM - StMoE: Iteration: 7 | log-likelihood: -580.356388983666
#> EM - StMoE: Iteration: 8 | log-likelihood: -577.103667178712
#> EM - StMoE: Iteration: 9 | log-likelihood: -573.388628316526
#> EM - StMoE: Iteration: 10 | log-likelihood: -570.250947412588
#> EM - StMoE: Iteration: 11 | log-likelihood: -568.1468459503
#> EM - StMoE: Iteration: 12 | log-likelihood: -566.756839289315
#> EM - StMoE: Iteration: 13 | log-likelihood: -565.707664444501
#> EM - StMoE: Iteration: 14 | log-likelihood: -564.903164719276
#> EM - StMoE: Iteration: 15 | log-likelihood: -564.355421686011
#> EM - StMoE: Iteration: 16 | log-likelihood: -563.982704775745
#> EM - StMoE: Iteration: 17 | log-likelihood: -563.701762917998
#> EM - StMoE: Iteration: 18 | log-likelihood: -563.474918508736
#> EM - StMoE: Iteration: 19 | log-likelihood: -563.285736927829
#> EM - StMoE: Iteration: 20 | log-likelihood: -563.124474080127
#> EM - StMoE: Iteration: 21 | log-likelihood: -562.984492716253
#> EM - StMoE: Iteration: 22 | log-likelihood: -562.860312916241
#> EM - StMoE: Iteration: 23 | log-likelihood: -562.745326816856
#> EM - StMoE: Iteration: 24 | log-likelihood: -562.634346131802
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#> EM - StMoE: Iteration: 369 | log-likelihood: -554.175987317441
#> EM - StMoE: Iteration: 370 | log-likelihood: -554.175262338154
#> EM - StMoE: Iteration: 371 | log-likelihood: -554.174541883094
#> EM - StMoE: Iteration: 372 | log-likelihood: -554.173825914512
#> EM - StMoE: Iteration: 373 | log-likelihood: -554.173114363734
#> EM - StMoE: Iteration: 374 | log-likelihood: -554.17240727309
#> EM - StMoE: Iteration: 375 | log-likelihood: -554.171704527303
#> EM - StMoE: Iteration: 376 | log-likelihood: -554.171006095513
#> EM - StMoE: Iteration: 377 | log-likelihood: -554.170311946685
#> EM - StMoE: Iteration: 378 | log-likelihood: -554.1696220425
#> EM - StMoE: Iteration: 379 | log-likelihood: -554.168936345233
#> EM - StMoE: Iteration: 380 | log-likelihood: -554.168254816978
#> EM - StMoE: Iteration: 381 | log-likelihood: -554.167577420304
#> EM - StMoE: Iteration: 382 | log-likelihood: -554.166904118305
#> EM - StMoE: Iteration: 383 | log-likelihood: -554.166234874563
#> EM - StMoE: Iteration: 384 | log-likelihood: -554.165569631099
#> EM - StMoE: Iteration: 385 | log-likelihood: -554.164908426218
#> EM - StMoE: Iteration: 386 | log-likelihood: -554.164251155308
#> EM - StMoE: Iteration: 387 | log-likelihood: -554.163597791943
#> EM - StMoE: Iteration: 388 | log-likelihood: -554.1629483099
#> EM - StMoE: Iteration: 389 | log-likelihood: -554.16230267809
#> EM - StMoE: Iteration: 390 | log-likelihood: -554.161660863568
#> EM - StMoE: Iteration: 391 | log-likelihood: -554.161022833898
#> EM - StMoE: Iteration: 392 | log-likelihood: -554.160388557162
#> EM - StMoE: Iteration: 393 | log-likelihood: -554.159758001963
#> EM - StMoE: Iteration: 394 | log-likelihood: -554.159131137463
#> EM - StMoE: Iteration: 395 | log-likelihood: -554.158507933354
#> EM - StMoE: Iteration: 396 | log-likelihood: -554.157888359915
#> EM - StMoE: Iteration: 397 | log-likelihood: -554.15727238804
#> EM - StMoE: Iteration: 398 | log-likelihood: -554.156659989254
#> EM - StMoE: Iteration: 399 | log-likelihood: -554.156051135749
#> EM - StMoE: Iteration: 400 | log-likelihood: -554.155445799997
#> EM - StMoE: Iteration: 401 | log-likelihood: -554.154843956275
#> EM - StMoE: Iteration: 402 | log-likelihood: -554.154245579133
#> EM - StMoE: Iteration: 403 | log-likelihood: -554.153650644084
#> EM - StMoE: Iteration: 404 | log-likelihood: -554.153059127724
#> EM - StMoE: Iteration: 405 | log-likelihood: -554.152471007861
#> EM - StMoE: Iteration: 406 | log-likelihood: -554.151886250764
#> EM - StMoE: Iteration: 407 | log-likelihood: -554.151304894285
#> EM - StMoE: Iteration: 408 | log-likelihood: -554.15072686049
#> EM - StMoE: Iteration: 409 | log-likelihood: -554.150152141038
#> EM - StMoE: Iteration: 410 | log-likelihood: -554.149580731155
#> EM - StMoE: Iteration: 411 | log-likelihood: -554.149012623907
#> EM - StMoE: Iteration: 412 | log-likelihood: -554.148447819637
#> EM - StMoE: Iteration: 413 | log-likelihood: -554.147886321985
#> EM - StMoE: Iteration: 414 | log-likelihood: -554.147328143312
#> EM - StMoE: Iteration: 415 | log-likelihood: -554.146773308
#> EM - StMoE: Iteration: 416 | log-likelihood: -554.146221857517

stmoe$summary()
#> ------------------------------------------
#> Fitted Skew t Mixture-of-Experts model
#> ------------------------------------------
#> 
#> StMoE model with K = 4 experts:
#> 
#>  log-likelihood df       AIC       BIC       ICL
#>       -554.1462 30 -584.1462 -627.5015 -628.8784
#> 
#> Clustering table (Number of observations in each expert):
#> 
#>  1  2  3  4 
#> 23 42 31 37 
#> 
#> Regression coefficients:
#> 
#>     Beta(k = 1) Beta(k = 2) Beta(k = 3) Beta(k = 4)
#> 1    -6.4614746  940.710496 -1820.86686 293.2486558
#> X^1   1.3976239  -98.407019   112.27249 -12.1880556
#> X^2  -0.1281445    2.288136    -1.68456   0.1250796
#> 
#> Variances:
#> 
#>  Sigma2(k = 1) Sigma2(k = 2) Sigma2(k = 3) Sigma2(k = 4)
#>       1.738587      395.9625      556.8862      569.4436

stmoe$plot()

#> Warning in sqrt(stat$Vary): NaNs produced

#> Warning in sqrt(stat$Vary): NaNs produced

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