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:
alpha
’s of the softmax
net.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")
.
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
stmoe <- emStMoE(X = x, Y = y, K, p, q, n_tries, max_iter,
threshold, verbose, verbose_IRLS)
## EM - StMoE: Iteration: 1 | log-likelihood: -397.948314671532
## EM - StMoE: Iteration: 2 | log-likelihood: -350.93972199686
## EM - StMoE: Iteration: 3 | log-likelihood: -349.240283640744
## EM - StMoE: Iteration: 4 | log-likelihood: -347.770731236186
## EM - StMoE: Iteration: 5 | log-likelihood: -346.431831376919
## EM - StMoE: Iteration: 6 | log-likelihood: -345.159432404738
## EM - StMoE: Iteration: 7 | log-likelihood: -343.932277630414
## EM - StMoE: Iteration: 8 | log-likelihood: -342.746164097378
## EM - StMoE: Iteration: 9 | log-likelihood: -341.61066331776
## EM - StMoE: Iteration: 10 | log-likelihood: -340.529285089636
## EM - StMoE: Iteration: 11 | log-likelihood: -339.50610071294
## EM - StMoE: Iteration: 12 | log-likelihood: -338.546145505096
## EM - StMoE: Iteration: 13 | log-likelihood: -337.6486394321
## EM - StMoE: Iteration: 14 | log-likelihood: -336.812062829723
## EM - StMoE: Iteration: 15 | log-likelihood: -336.037063649317
## EM - StMoE: Iteration: 16 | log-likelihood: -335.323865049103
## EM - StMoE: Iteration: 17 | log-likelihood: -334.669120856753
## EM - StMoE: Iteration: 18 | log-likelihood: -334.065165483472
## EM - StMoE: Iteration: 19 | log-likelihood: -333.501607812029
## EM - StMoE: Iteration: 20 | log-likelihood: -332.965794988019
## EM - StMoE: Iteration: 21 | log-likelihood: -332.434277272932
## EM - StMoE: Iteration: 22 | log-likelihood: -331.878002557026
## EM - StMoE: Iteration: 23 | log-likelihood: -331.253653101185
## EM - StMoE: Iteration: 24 | log-likelihood: -330.504141103484
## EM - StMoE: Iteration: 25 | log-likelihood: -329.559752552549
## EM - StMoE: Iteration: 26 | log-likelihood: -328.349406321036
## EM - StMoE: Iteration: 27 | log-likelihood: -326.804558485056
## EM - StMoE: Iteration: 28 | log-likelihood: -324.878320187264
## EM - StMoE: Iteration: 29 | log-likelihood: -322.572712842797
## EM - StMoE: Iteration: 30 | log-likelihood: -319.948971160812
## EM - StMoE: Iteration: 31 | log-likelihood: -317.120231764782
## EM - StMoE: Iteration: 32 | log-likelihood: -314.228878599706
## EM - StMoE: Iteration: 33 | log-likelihood: -311.416074038568
## EM - StMoE: Iteration: 34 | log-likelihood: -308.789972994702
## EM - StMoE: Iteration: 35 | log-likelihood: -306.420022880788
## EM - StMoE: Iteration: 36 | log-likelihood: -304.330485207363
## EM - StMoE: Iteration: 37 | log-likelihood: -302.520532732544
## EM - StMoE: Iteration: 38 | log-likelihood: -300.967170468015
## EM - StMoE: Iteration: 39 | log-likelihood: -299.645575292584
## EM - StMoE: Iteration: 40 | log-likelihood: -298.523866635305
## EM - StMoE: Iteration: 41 | log-likelihood: -297.57539508599
## EM - StMoE: Iteration: 42 | log-likelihood: -296.771042600862
## EM - StMoE: Iteration: 43 | log-likelihood: -296.087197942475
## EM - StMoE: Iteration: 44 | log-likelihood: -295.505213405586
## EM - StMoE: Iteration: 45 | log-likelihood: -295.007135904701
## EM - StMoE: Iteration: 46 | log-likelihood: -294.579141923347
## EM - StMoE: Iteration: 47 | log-likelihood: -294.210334251161
## EM - StMoE: Iteration: 48 | log-likelihood: -293.891756643337
## EM - StMoE: Iteration: 49 | log-likelihood: -293.616274028923
## EM - StMoE: Iteration: 50 | log-likelihood: -293.37775658278
## EM - StMoE: Iteration: 51 | log-likelihood: -293.171799844088
## EM - StMoE: Iteration: 52 | log-likelihood: -292.994502716825
## EM - StMoE: Iteration: 53 | log-likelihood: -292.841571184246
## EM - StMoE: Iteration: 54 | log-likelihood: -292.709688698353
## EM - StMoE: Iteration: 55 | log-likelihood: -292.595840191259
## EM - StMoE: Iteration: 56 | log-likelihood: -292.497616932657
## EM - StMoE: Iteration: 57 | log-likelihood: -292.412820308307
## EM - StMoE: Iteration: 58 | log-likelihood: -292.339614741704
## EM - StMoE: Iteration: 59 | log-likelihood: -292.276450806372
## EM - StMoE: Iteration: 60 | log-likelihood: -292.221954128326
## EM - StMoE: Iteration: 61 | log-likelihood: -292.175007166992
## EM - StMoE: Iteration: 62 | log-likelihood: -292.135450971376
## EM - StMoE: Iteration: 63 | log-likelihood: -292.102121573629
## EM - StMoE: Iteration: 64 | log-likelihood: -292.074057543758
## EM - StMoE: Iteration: 65 | log-likelihood: -292.050459437671
## EM - StMoE: Iteration: 66 | log-likelihood: -292.030658635995
## EM - StMoE: Iteration: 67 | log-likelihood: -292.014093071031
## EM - StMoE: Iteration: 68 | log-likelihood: -292.000207062251
## EM - StMoE: Iteration: 69 | log-likelihood: -291.988697397527
## EM - StMoE: Iteration: 70 | log-likelihood: -291.979215566862
## EM - StMoE: Iteration: 71 | log-likelihood: -291.97144859128
## EM - StMoE: Iteration: 72 | log-likelihood: -291.965150137581
## EM - StMoE: Iteration: 73 | log-likelihood: -291.960103112887
## EM - StMoE: Iteration: 74 | log-likelihood: -291.956121596583
## EM - StMoE: Iteration: 75 | log-likelihood: -291.953046073114
## EM - StMoE: Iteration: 76 | log-likelihood: -291.950739478271
stmoe$summary()
## ------------------------------------------
## Fitted Skew t Mixture-of-Experts model
## ------------------------------------------
##
## StMoE model with K = 2 experts:
##
## log-likelihood df AIC BIC ICL
## -291.9507 12 -303.9507 -329.2384 -329.3226
##
## Clustering table (Number of observations in each expert):
##
## 1 2
## 246 254
##
## Regression coefficients:
##
## Beta(k = 1) Beta(k = 2)
## 1 -0.04092653 -0.1102991
## X^1 2.56992285 -2.4339405
##
## Variances:
##
## Sigma2(k = 1) Sigma2(k = 2)
## 0.5989221 0.5147709
stmoe <- emStMoE(X = x, Y = y, K, p, q, n_tries, max_iter,
threshold, verbose, verbose_IRLS)
## EM - StMoE: Iteration: 1 | log-likelihood: -600.826947369113
## EM - StMoE: Iteration: 2 | log-likelihood: -589.977357431034
## EM - StMoE: Iteration: 3 | log-likelihood: -587.167104670215
## EM - StMoE: Iteration: 4 | log-likelihood: -586.39063569506
## EM - StMoE: Iteration: 5 | log-likelihood: -585.772911963497
## EM - StMoE: Iteration: 6 | log-likelihood: -583.979686780225
## EM - StMoE: Iteration: 7 | log-likelihood: -577.639617948489
## EM - StMoE: Iteration: 8 | log-likelihood: -570.532765150934
## EM - StMoE: Iteration: 9 | log-likelihood: -565.320213419045
## EM - StMoE: Iteration: 10 | log-likelihood: -562.869076654372
## EM - StMoE: Iteration: 11 | log-likelihood: -562.282402938482
## EM - StMoE: Iteration: 12 | log-likelihood: -562.191915724672
## EM - StMoE: Iteration: 13 | log-likelihood: -562.190451737714
stmoe$summary()
## ------------------------------------------
## Fitted Skew t Mixture-of-Experts model
## ------------------------------------------
##
## StMoE model with K = 4 experts:
##
## log-likelihood df AIC BIC ICL
## -562.1905 30 -592.1905 -635.5457 -635.542
##
## 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 -7.1531379 992.484590 -2096.748499 193.34820147
## X^1 1.2479320 -103.890334 132.327916 -8.94276956
## X^2 -0.1088423 2.432542 -2.040076 0.09275632
##
## Variances:
##
## Sigma2(k = 1) Sigma2(k = 2) Sigma2(k = 3) Sigma2(k = 4)
## 27.15949 454.1761 336.1893 1085.644