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: -385.368829345481
## EM - StMoE: Iteration: 2 | log-likelihood: -338.957872929278
## EM - StMoE: Iteration: 3 | log-likelihood: -327.905843918556
## EM - StMoE: Iteration: 4 | log-likelihood: -321.091943693949
## EM - StMoE: Iteration: 5 | log-likelihood: -315.86005566305
## EM - StMoE: Iteration: 6 | log-likelihood: -311.586637947083
## EM - StMoE: Iteration: 7 | log-likelihood: -307.909989823282
## EM - StMoE: Iteration: 8 | log-likelihood: -304.622933118144
## EM - StMoE: Iteration: 9 | log-likelihood: -301.606754948985
## EM - StMoE: Iteration: 10 | log-likelihood: -298.790519771365
## EM - StMoE: Iteration: 11 | log-likelihood: -296.133891470474
## EM - StMoE: Iteration: 12 | log-likelihood: -293.622855137567
## EM - StMoE: Iteration: 13 | log-likelihood: -291.272248882122
## EM - StMoE: Iteration: 14 | log-likelihood: -289.095233130465
## EM - StMoE: Iteration: 15 | log-likelihood: -287.095460150291
## EM - StMoE: Iteration: 16 | log-likelihood: -285.267766070026
## EM - StMoE: Iteration: 17 | log-likelihood: -283.602385978574
## EM - StMoE: Iteration: 18 | log-likelihood: -282.086217862809
## EM - StMoE: Iteration: 19 | log-likelihood: -280.708683619841
## EM - StMoE: Iteration: 20 | log-likelihood: -279.458580113783
## EM - StMoE: Iteration: 21 | log-likelihood: -278.319474845686
## EM - StMoE: Iteration: 22 | log-likelihood: -277.278284811922
## EM - StMoE: Iteration: 23 | log-likelihood: -276.325613980097
## EM - StMoE: Iteration: 24 | log-likelihood: -275.452054838214
## EM - StMoE: Iteration: 25 | log-likelihood: -274.652613602365
## EM - StMoE: Iteration: 26 | log-likelihood: -273.917906535753
## EM - StMoE: Iteration: 27 | log-likelihood: -273.241683367699
## EM - StMoE: Iteration: 28 | log-likelihood: -272.620297909754
## EM - StMoE: Iteration: 29 | log-likelihood: -272.048100302575
## EM - StMoE: Iteration: 30 | log-likelihood: -271.521464819982
## EM - StMoE: Iteration: 31 | log-likelihood: -271.038599059341
## EM - StMoE: Iteration: 32 | log-likelihood: -270.595796976774
## EM - StMoE: Iteration: 33 | log-likelihood: -270.190801323282
## EM - StMoE: Iteration: 34 | log-likelihood: -269.819543204151
## EM - StMoE: Iteration: 35 | log-likelihood: -269.478373231286
## EM - StMoE: Iteration: 36 | log-likelihood: -269.166160200449
## EM - StMoE: Iteration: 37 | log-likelihood: -268.880164029563
## EM - StMoE: Iteration: 38 | log-likelihood: -268.619209103127
## EM - StMoE: Iteration: 39 | log-likelihood: -268.382415131731
## EM - StMoE: Iteration: 40 | log-likelihood: -268.166837431276
## EM - StMoE: Iteration: 41 | log-likelihood: -267.971180122846
## EM - StMoE: Iteration: 42 | log-likelihood: -267.793959798785
## EM - StMoE: Iteration: 43 | log-likelihood: -267.632678733798
## EM - StMoE: Iteration: 44 | log-likelihood: -267.484641759751
## EM - StMoE: Iteration: 45 | log-likelihood: -267.349025284738
## EM - StMoE: Iteration: 46 | log-likelihood: -267.224969938457
## EM - StMoE: Iteration: 47 | log-likelihood: -267.111038588831
## EM - StMoE: Iteration: 48 | log-likelihood: -267.006140754541
## EM - StMoE: Iteration: 49 | log-likelihood: -266.909638800181
## EM - StMoE: Iteration: 50 | log-likelihood: -266.821844217366
## EM - StMoE: Iteration: 51 | log-likelihood: -266.741675390736
## EM - StMoE: Iteration: 52 | log-likelihood: -266.668039491455
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## EM - StMoE: Iteration: 62 | log-likelihood: -266.19351962907
## EM - StMoE: Iteration: 63 | log-likelihood: -266.164573626451
## EM - StMoE: Iteration: 64 | log-likelihood: -266.137898698178
## EM - StMoE: Iteration: 65 | log-likelihood: -266.112925279816
## EM - StMoE: Iteration: 66 | log-likelihood: -266.08987886246
## EM - StMoE: Iteration: 67 | log-likelihood: -266.068497191113
## EM - StMoE: Iteration: 68 | log-likelihood: -266.048645457238
## EM - StMoE: Iteration: 69 | log-likelihood: -266.030200027439
## EM - StMoE: Iteration: 70 | log-likelihood: -266.01302419053
## EM - StMoE: Iteration: 71 | log-likelihood: -265.997173019411
## EM - StMoE: Iteration: 72 | log-likelihood: -265.982666379885
## EM - StMoE: Iteration: 73 | log-likelihood: -265.969291450179
## EM - StMoE: Iteration: 74 | log-likelihood: -265.95723588626
## EM - StMoE: Iteration: 75 | log-likelihood: -265.945950370812
## EM - StMoE: Iteration: 76 | log-likelihood: -265.935368063594
## EM - StMoE: Iteration: 77 | log-likelihood: -265.925493175691
## EM - StMoE: Iteration: 78 | log-likelihood: -265.916283730973
## EM - StMoE: Iteration: 79 | log-likelihood: -265.907695902565
## EM - StMoE: Iteration: 80 | log-likelihood: -265.899671140248
## EM - StMoE: Iteration: 81 | log-likelihood: -265.892173595027
## EM - StMoE: Iteration: 82 | log-likelihood: -265.885161575097
## EM - StMoE: Iteration: 83 | log-likelihood: -265.878598628939
## EM - StMoE: Iteration: 84 | log-likelihood: -265.872455454284
## EM - StMoE: Iteration: 85 | log-likelihood: -265.866700516292
## EM - StMoE: Iteration: 86 | log-likelihood: -265.861307843709
## EM - StMoE: Iteration: 87 | log-likelihood: -265.856252249653
## EM - StMoE: Iteration: 88 | log-likelihood: -265.851510833699
## EM - StMoE: Iteration: 89 | log-likelihood: -265.847062266813
## EM - StMoE: Iteration: 90 | log-likelihood: -265.842886812221
## EM - StMoE: Iteration: 91 | log-likelihood: -265.838966268996
## EM - StMoE: Iteration: 92 | log-likelihood: -265.835283763561
## EM - StMoE: Iteration: 93 | log-likelihood: -265.831823644097
## EM - StMoE: Iteration: 94 | log-likelihood: -265.828571383852
## EM - StMoE: Iteration: 95 | log-likelihood: -265.825513492698
## EM - StMoE: Iteration: 96 | log-likelihood: -265.822637436224
## EM - StMoE: Iteration: 97 | log-likelihood: -265.819931561722
## EM - StMoE: Iteration: 98 | log-likelihood: -265.817385030429

Summary

stmoe$summary()
## ------------------------------------------
## Fitted Skew t Mixture-of-Experts model
## ------------------------------------------
## 
## StMoE model with K = 2 experts:
## 
##  log-likelihood df       AIC      BIC       ICL
##       -265.8174 12 -277.8174 -303.105 -303.1061
## 
## Clustering table (Number of observations in each expert):
## 
##   1   2 
## 250 250 
## 
## Regression coefficients:
## 
##     Beta(k = 1) Beta(k = 2)
## 1   -0.03015558  0.04117581
## X^1  2.56978382 -2.62659694
## 
## Variances:
## 
##  Sigma2(k = 1) Sigma2(k = 2)
##      0.4781602     0.2703674

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: -606.771761531912
## EM - StMoE: Iteration: 2 | log-likelihood: -586.561692945793
## EM - StMoE: Iteration: 3 | log-likelihood: -583.447512566881
## EM - StMoE: Iteration: 4 | log-likelihood: -582.444436830449
## EM - StMoE: Iteration: 5 | log-likelihood: -581.81775123626
## EM - StMoE: Iteration: 6 | log-likelihood: -578.755448157513
## EM - StMoE: Iteration: 7 | log-likelihood: -574.084770502747
## EM - StMoE: Iteration: 8 | log-likelihood: -567.483719855793
## EM - StMoE: Iteration: 9 | log-likelihood: -563.392320705151
## EM - StMoE: Iteration: 10 | log-likelihood: -561.070833661223
## EM - StMoE: Iteration: 11 | log-likelihood: -559.986179911805
## EM - StMoE: Iteration: 12 | log-likelihood: -559.394908329518
## EM - StMoE: Iteration: 13 | log-likelihood: -559.031608284086
## EM - StMoE: Iteration: 14 | log-likelihood: -558.851433366803
## EM - StMoE: Iteration: 15 | log-likelihood: -558.744986081847
## EM - StMoE: Iteration: 16 | log-likelihood: -558.653737337899
## EM - StMoE: Iteration: 17 | log-likelihood: -558.563781308371
## EM - StMoE: Iteration: 18 | log-likelihood: -558.475801877821
## EM - StMoE: Iteration: 19 | log-likelihood: -558.394465699026
## EM - StMoE: Iteration: 20 | log-likelihood: -558.997974541465
## EM - StMoE: Iteration: 21 | log-likelihood: -559.037190048569
## EM - StMoE: Iteration: 22 | log-likelihood: -556.107785417442
## EM - StMoE: Iteration: 23 | log-likelihood: -553.601906436936
## EM - StMoE: Iteration: 24 | log-likelihood: -550.826015554533
## EM - StMoE: Iteration: 25 | log-likelihood: -548.047967799554
## EM - StMoE: Iteration: 26 | log-likelihood: -546.505574727569
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## EM - StMoE: Iteration: 28 | log-likelihood: -545.930939419644
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## EM - StMoE: Iteration: 30 | log-likelihood: -545.547761419624
## EM - StMoE: Iteration: 31 | log-likelihood: -545.392888056218
## EM - StMoE: Iteration: 32 | log-likelihood: -545.201933146192
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## EM - StMoE: Iteration: 48 | log-likelihood: -544.278420092602
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## EM - StMoE: Iteration: 154 | log-likelihood: -542.563651834462
## EM - StMoE: Iteration: 155 | log-likelihood: -542.55793815251
## EM - StMoE: Iteration: 156 | log-likelihood: -542.552314444163
## EM - StMoE: Iteration: 157 | log-likelihood: -542.546778469773
## EM - StMoE: Iteration: 158 | log-likelihood: -542.541328068303
## EM - StMoE: Iteration: 159 | log-likelihood: -542.535961153738

Summary

stmoe$summary()
## ------------------------------------------
## Fitted Skew t Mixture-of-Experts model
## ------------------------------------------
## 
## StMoE model with K = 4 experts:
## 
##  log-likelihood df      AIC       BIC      ICL
##        -542.536 30 -572.536 -615.8912 -616.539
## 
## Clustering table (Number of observations in each expert):
## 
##  1  2  3  4 
## 25 40 31 37 
## 
## Regression coefficients:
## 
##     Beta(k = 1) Beta(k = 2)  Beta(k = 3)  Beta(k = 4)
## 1   -0.86071020  1117.69501 -1807.612653 132.89203206
## X^1 -0.70112209  -121.38548   111.287737  -6.69941241
## X^2  0.02430659     2.93944    -1.666764   0.07041107
## 
## Variances:
## 
##  Sigma2(k = 1) Sigma2(k = 2) Sigma2(k = 3) Sigma2(k = 4)
##        6.91373      504.1935      570.7413      1404.837

Plots

Mean curve

stmoe$plot(what = "meancurve")

Confidence regions

stmoe$plot(what = "confregions")

Clusters

stmoe$plot(what = "clusters")

Log-likelihood

stmoe$plot(what = "loglikelihood")