A-quick-tour-of-SNMoE

Introduction

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:

  • 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, 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").

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
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

Set up SNMoE 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-6
verbose <- TRUE
verbose_IRLS <- FALSE

Estimation

snmoe <- emSNMoE(X = x, Y = y, K, p, q, n_tries, max_iter, 
                 threshold, verbose, verbose_IRLS)
## EM - SNMoE: Iteration: 1 | log-likelihood: -529.700494167196
## EM - SNMoE: Iteration: 2 | log-likelihood: -525.989268269454
## EM - SNMoE: Iteration: 3 | log-likelihood: -525.946377307157
## EM - SNMoE: Iteration: 4 | log-likelihood: -525.934606017225
## EM - SNMoE: Iteration: 5 | log-likelihood: -525.929806778679
## EM - SNMoE: Iteration: 6 | log-likelihood: -525.926673627579
## EM - SNMoE: Iteration: 7 | log-likelihood: -525.924003699033
## EM - SNMoE: Iteration: 8 | log-likelihood: -525.921530110722
## EM - SNMoE: Iteration: 9 | log-likelihood: -525.919204730305
## EM - SNMoE: Iteration: 10 | log-likelihood: -525.917029773346
## EM - SNMoE: Iteration: 11 | log-likelihood: -525.914980143827
## EM - SNMoE: Iteration: 12 | log-likelihood: -525.91304788326
## EM - SNMoE: Iteration: 13 | log-likelihood: -525.911218074198
## EM - SNMoE: Iteration: 14 | log-likelihood: -525.909500410472
## EM - SNMoE: Iteration: 15 | log-likelihood: -525.907884356338
## EM - SNMoE: Iteration: 16 | log-likelihood: -525.90635095873
## EM - SNMoE: Iteration: 17 | log-likelihood: -525.904909482292
## EM - SNMoE: Iteration: 18 | log-likelihood: -525.903555017547
## EM - SNMoE: Iteration: 19 | log-likelihood: -525.902278840505
## EM - SNMoE: Iteration: 20 | log-likelihood: -525.901075974653
## EM - SNMoE: Iteration: 21 | log-likelihood: -525.899928951112
## EM - SNMoE: Iteration: 22 | log-likelihood: -525.898834641916
## EM - SNMoE: Iteration: 23 | log-likelihood: -525.897788324004
## EM - SNMoE: Iteration: 24 | log-likelihood: -525.896806199865
## EM - SNMoE: Iteration: 25 | log-likelihood: -525.89587743652
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## EM - SNMoE: Iteration: 30 | log-likelihood: -525.891816503127
## EM - SNMoE: Iteration: 31 | log-likelihood: -525.891113801274
## EM - SNMoE: Iteration: 32 | log-likelihood: -525.890441671635
## EM - SNMoE: Iteration: 33 | log-likelihood: -525.88979524545
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## EM - SNMoE: Iteration: 35 | log-likelihood: -525.888589714703
## EM - SNMoE: Iteration: 36 | log-likelihood: -525.888029370465
## EM - SNMoE: Iteration: 37 | log-likelihood: -525.887489599761
## EM - SNMoE: Iteration: 38 | log-likelihood: -525.886968850595

Summary

snmoe$summary()
## -----------------------------------------------
## Fitted Skew-Normal Mixture-of-Experts model
## -----------------------------------------------
## 
## SNMoE model with K = 2 experts:
## 
##  log-likelihood df      AIC     BIC       ICL
##        -525.887 10 -535.887 -556.96 -557.0111
## 
## Clustering table (Number of observations in each expert):
## 
##   1   2 
## 249 251 
## 
## Regression coefficients:
## 
##     Beta(k = 1) Beta(k = 2)
## 1     0.9764542   0.9590402
## X^1   2.7009367  -2.8352644
## 
## Variances:
## 
##  Sigma2(k = 1) Sigma2(k = 2)
##      0.4805849     0.4837961

Plots

Mean curve

snmoe$plot(what = "meancurve")

Confidence regions

snmoe$plot(what = "confregions")

Clusters

snmoe$plot(what = "clusters")

Log-likelihood

snmoe$plot(what = "loglikelihood")

Application to a real dataset

Load data

data("tempanomalies")
x <- tempanomalies$Year
y <- tempanomalies$AnnualAnomaly

Set up SNMoE 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-6
verbose <- TRUE
verbose_IRLS <- FALSE

Estimation

snmoe <- emSNMoE(X = x, Y = y, K, p, q, n_tries, max_iter, 
                 threshold, verbose, verbose_IRLS)
## EM - SNMoE: Iteration: 1 | log-likelihood: 74.2853260853685
## EM - SNMoE: Iteration: 2 | log-likelihood: 87.568312175146
## EM - SNMoE: Iteration: 3 | log-likelihood: 88.8289010766741
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## EM - SNMoE: Iteration: 223 | log-likelihood: 90.813386033468
## EM - SNMoE: Iteration: 224 | log-likelihood: 90.8139983802686
## EM - SNMoE: Iteration: 225 | log-likelihood: 90.8145821597711
## EM - SNMoE: Iteration: 226 | log-likelihood: 90.8152022831961
## EM - SNMoE: Iteration: 227 | log-likelihood: 90.8157309720549
## EM - SNMoE: Iteration: 228 | log-likelihood: 90.8161705836726
## EM - SNMoE: Iteration: 229 | log-likelihood: 90.8167240417126
## EM - SNMoE: Iteration: 230 | log-likelihood: 90.8171774584058
## EM - SNMoE: Iteration: 231 | log-likelihood: 90.8175992118328
## EM - SNMoE: Iteration: 232 | log-likelihood: 90.8180204165044
## EM - SNMoE: Iteration: 233 | log-likelihood: 90.818411859425
## EM - SNMoE: Iteration: 234 | log-likelihood: 90.818783831325
## EM - SNMoE: Iteration: 235 | log-likelihood: 90.8191378598692
## EM - SNMoE: Iteration: 236 | log-likelihood: 90.8194843905164
## EM - SNMoE: Iteration: 237 | log-likelihood: 90.8198036557484
## EM - SNMoE: Iteration: 238 | log-likelihood: 90.8201074058724
## EM - SNMoE: Iteration: 239 | log-likelihood: 90.8203967679444
## EM - SNMoE: Iteration: 240 | log-likelihood: 90.820672093918
## EM - SNMoE: Iteration: 241 | log-likelihood: 90.8209340781917
## EM - SNMoE: Iteration: 242 | log-likelihood: 90.8211838280595
## EM - SNMoE: Iteration: 243 | log-likelihood: 90.8214211451648
## EM - SNMoE: Iteration: 244 | log-likelihood: 90.8216468557392
## EM - SNMoE: Iteration: 245 | log-likelihood: 90.8218615614055
## EM - SNMoE: Iteration: 246 | log-likelihood: 90.8220659512825
## EM - SNMoE: Iteration: 247 | log-likelihood: 90.822305394879
## EM - SNMoE: Iteration: 248 | log-likelihood: 90.8224897091046
## EM - SNMoE: Iteration: 249 | log-likelihood: 90.822620057321
## EM - SNMoE: Iteration: 250 | log-likelihood: 90.8228330264267
## EM - SNMoE: Iteration: 251 | log-likelihood: 90.8229922331042
## EM - SNMoE: Iteration: 252 | log-likelihood: 90.8231437957965
## EM - SNMoE: Iteration: 253 | log-likelihood: 90.8232421928194
## EM - SNMoE: Iteration: 254 | log-likelihood: 90.823425321987
## EM - SNMoE: Iteration: 255 | log-likelihood: 90.8235553214655
## EM - SNMoE: Iteration: 256 | log-likelihood: 90.8236791210491
## EM - SNMoE: Iteration: 257 | log-likelihood: 90.8237973291899
## EM - SNMoE: Iteration: 258 | log-likelihood: 90.8239099293079
## EM - SNMoE: Iteration: 259 | log-likelihood: 90.824017181937
## EM - SNMoE: Iteration: 260 | log-likelihood: 90.8241193126624
## EM - SNMoE: Iteration: 261 | log-likelihood: 90.8242165097087
## EM - SNMoE: Iteration: 262 | log-likelihood: 90.824308934124
## EM - SNMoE: Iteration: 263 | log-likelihood: 90.8243974718816

Summary

snmoe$summary()
## -----------------------------------------------
## Fitted Skew-Normal Mixture-of-Experts model
## -----------------------------------------------
## 
## SNMoE model with K = 2 experts:
## 
##  log-likelihood df     AIC      BIC      ICL
##         90.8244 10 80.8244 66.26112 66.16136
## 
## Clustering table (Number of observations in each expert):
## 
##  1  2 
## 69 67 
## 
## Regression coefficients:
## 
##       Beta(k = 1)  Beta(k = 2)
## 1   -14.225630550 -32.64233201
## X^1   0.007305018   0.01669158
## 
## Variances:
## 
##  Sigma2(k = 1) Sigma2(k = 2)
##     0.01466509    0.03729136

Plots

Mean curve

snmoe$plot(what = "meancurve")

Confidence regions

snmoe$plot(what = "confregions")

Clusters

snmoe$plot(what = "clusters")

Log-likelihood

snmoe$plot(what = "loglikelihood")