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:
alpha
’s of the softmax net.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$y
snmoe <- emSNMoE(X = x, Y = y, K, p, q, n_tries, max_iter,
threshold, verbose, verbose_IRLS)
## EM - SNMoE: Iteration: 1 | log-likelihood: -656.585355371133
## EM - SNMoE: Iteration: 2 | log-likelihood: -555.851354544233
## EM - SNMoE: Iteration: 3 | log-likelihood: -550.40397615086
## EM - SNMoE: Iteration: 4 | log-likelihood: -548.256269199245
## EM - SNMoE: Iteration: 5 | log-likelihood: -546.79761691946
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## EM - SNMoE: Iteration: 107 | log-likelihood: -522.814582615201
## EM - SNMoE: Iteration: 108 | log-likelihood: -522.814077443441
snmoe$summary()
## -----------------------------------------------
## Fitted Skew-Normal Mixture-of-Experts model
## -----------------------------------------------
##
## SNMoE model with K = 2 experts:
##
## log-likelihood df AIC BIC ICL
## -522.8141 10 -532.8141 -553.8871 -553.8946
##
## Clustering table (Number of observations in each expert):
##
## 1 2
## 249 251
##
## Regression coefficients:
##
## Beta(k = 1) Beta(k = 2)
## 1 0.1643449 1.030347
## X^1 2.7624124 -2.713522
##
## Variances:
##
## Sigma2(k = 1) Sigma2(k = 2)
## 1.130847 0.6125231
snmoe <- emSNMoE(X = x, Y = y, K, p, q, n_tries, max_iter,
threshold, verbose, verbose_IRLS)
## EM - SNMoE: Iteration: 1 | log-likelihood: 61.8107860891512
## EM - SNMoE: Iteration: 2 | log-likelihood: 87.653433029568
## EM - SNMoE: Iteration: 3 | log-likelihood: 88.8833709628339
## EM - SNMoE: Iteration: 4 | log-likelihood: 89.2593088472611
## EM - SNMoE: Iteration: 5 | log-likelihood: 89.5213488387829
## EM - SNMoE: Iteration: 6 | log-likelihood: 89.7275294002096
## EM - SNMoE: Iteration: 7 | log-likelihood: 89.8478728949139
## EM - SNMoE: Iteration: 8 | log-likelihood: 89.9133766120692
## EM - SNMoE: Iteration: 9 | log-likelihood: 89.9496067667942
## EM - SNMoE: Iteration: 10 | log-likelihood: 89.972407680618
## EM - SNMoE: Iteration: 11 | log-likelihood: 89.9899191229059
## EM - SNMoE: Iteration: 12 | log-likelihood: 90.004975878789
## EM - SNMoE: Iteration: 13 | log-likelihood: 90.0183693239978
## EM - SNMoE: Iteration: 14 | log-likelihood: 90.0306979505493
## EM - SNMoE: Iteration: 15 | log-likelihood: 90.0423279583216
## EM - SNMoE: Iteration: 16 | log-likelihood: 90.0534008186115
## EM - SNMoE: Iteration: 17 | log-likelihood: 90.0639652852913
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## EM - SNMoE: Iteration: 165 | log-likelihood: 90.821219422153
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## EM - SNMoE: Iteration: 167 | log-likelihood: 90.8226555970784
## EM - SNMoE: Iteration: 168 | log-likelihood: 90.8233188520901
## EM - SNMoE: Iteration: 169 | log-likelihood: 90.8239503919047
## EM - SNMoE: Iteration: 170 | log-likelihood: 90.8245515685013
## EM - SNMoE: Iteration: 171 | log-likelihood: 90.8251239839533
## EM - SNMoE: Iteration: 172 | log-likelihood: 90.8256690003121
## EM - SNMoE: Iteration: 173 | log-likelihood: 90.8261878084732
## EM - SNMoE: Iteration: 174 | log-likelihood: 90.8267438933268
## EM - SNMoE: Iteration: 175 | log-likelihood: 90.8272119529069
## EM - SNMoE: Iteration: 176 | log-likelihood: 90.8276542638926
## EM - SNMoE: Iteration: 177 | log-likelihood: 90.8280107080239
## EM - SNMoE: Iteration: 178 | log-likelihood: 90.8284939649173
## EM - SNMoE: Iteration: 179 | log-likelihood: 90.8288527776589
## EM - SNMoE: Iteration: 180 | log-likelihood: 90.8292220806304
## EM - SNMoE: Iteration: 181 | log-likelihood: 90.8295633025039
## EM - SNMoE: Iteration: 182 | log-likelihood: 90.8298873107783
## EM - SNMoE: Iteration: 183 | log-likelihood: 90.8302047282032
## EM - SNMoE: Iteration: 184 | log-likelihood: 90.8304964386323
## EM - SNMoE: Iteration: 185 | log-likelihood: 90.8307733087526
## EM - SNMoE: Iteration: 186 | log-likelihood: 90.8310360569017
## EM - SNMoE: Iteration: 187 | log-likelihood: 90.8312853671496
## EM - SNMoE: Iteration: 188 | log-likelihood: 90.8315218931287
## EM - SNMoE: Iteration: 189 | log-likelihood: 90.8317462589012
## EM - SNMoE: Iteration: 190 | log-likelihood: 90.8319590599853
## EM - SNMoE: Iteration: 191 | log-likelihood: 90.8321608643629
## EM - SNMoE: Iteration: 192 | log-likelihood: 90.83235221346
## EM - SNMoE: Iteration: 193 | log-likelihood: 90.8325336231233
## EM - SNMoE: Iteration: 194 | log-likelihood: 90.8327055941627
## EM - SNMoE: Iteration: 195 | log-likelihood: 90.8329125906151
## EM - SNMoE: Iteration: 196 | log-likelihood: 90.8330665084005
## EM - SNMoE: Iteration: 197 | log-likelihood: 90.8331681272115
## EM - SNMoE: Iteration: 198 | log-likelihood: 90.8333513395607
## EM - SNMoE: Iteration: 199 | log-likelihood: 90.8334823518607
## EM - SNMoE: Iteration: 200 | log-likelihood: 90.8335619566083
snmoe$summary()
## -----------------------------------------------
## Fitted Skew-Normal Mixture-of-Experts model
## -----------------------------------------------
##
## SNMoE model with K = 2 experts:
##
## log-likelihood df AIC BIC ICL
## 90.83356 10 80.83356 66.27029 66.16225
##
## Clustering table (Number of observations in each expert):
##
## 1 2
## 69 67
##
## Regression coefficients:
##
## Beta(k = 1) Beta(k = 2)
## 1 -14.428214634 -32.6510585
## X^1 0.007366202 0.0166957
##
## Variances:
##
## Sigma2(k = 1) Sigma2(k = 2)
## 0.01816954 0.03712389