Description
Mixture-of-Experts Modeling for Complex Non-Normal Distributions.
Description
Provides a unified mixture-of-experts (ME) modeling and estimation framework with several original and flexible ME models to model, cluster and classify heterogeneous data in many complex situations where the data are distributed according to non-normal, possibly skewed distributions, and when they might be corrupted by atypical observations. Mixtures-of-Experts models for complex and non-normal distributions ('meteorits') are originally introduced and written in 'Matlab' by Faicel Chamroukhi. The references are mainly the following ones. The references are mainly the following ones. Chamroukhi F., Same A., Govaert, G. and Aknin P. (2009) <doi:10.1016/j.neunet.2009.06.040>. Chamroukhi F. (2010) <https://chamroukhi.com/FChamroukhi-PhD.pdf>. Chamroukhi F. (2015) <arXiv:1506.06707>. Chamroukhi F. (2015) <https://chamroukhi.com/FChamroukhi-HDR.pdf>. Chamroukhi F. (2016) <doi:10.1109/IJCNN.2016.7727580>. Chamroukhi F. (2016) <doi:10.1016/j.neunet.2016.03.002>. Chamroukhi F. (2017) <doi:10.1016/j.neucom.2017.05.044>.
README.md
MEteorits: Mixtures-of-ExperTs modEling for cOmplex and non-noRmal dIsTributions
MEteorits is an open source toolbox (available in R and Matlab) containing several original and flexible mixtures-of-experts models to model, cluster and classify heteregenous data in many complex situations where the data are distributed according to non-normal and possibly skewed distributions, and when they might be corrupted by atypical observations. The toolbox also contains sparse mixture-of-experts models for high-dimensional data.
Our (dis-)covered meteorits are for instance the following ones:
- NMoE (Normal Mixtures-of-Experts);
- tMoE (t Mixtures-of-Experts);
- SNMoE (Skew-Normal Mixtures-of-Experts);
- StMoE (Skew t Mixtures-of-Experts).
The models and algorithms are developped and written in Matlab by Faicel Chamroukhi, and translated and designed into R packages by Florian Lecocq, Marius Bartcus and Faicel Chamroukhi.
Installation
You can install the development version of MEteorits from GitHub with:
# install.packages("devtools")
devtools::install_github("fchamroukhi/MEteorits")
To build vignettes for examples of usage, type the command below instead:
# install.packages("devtools")
devtools::install_github("fchamroukhi/MEteorits",
build_opts = c("--no-resave-data", "--no-manual"),
build_vignettes = TRUE)
Use the following command to display vignettes:
browseVignettes("meteorits")
Usage
library(meteorits)
NMoE
# Application 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(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 <- sampleUnivNMoE(alphak = alphak, betak = betak,
sigmak = sigmak, 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)
nmoe <- emNMoE(X = x, Y = y, K = K, p = p, q = q, verbose = TRUE)
#> EM NMoE: Iteration: 1 | log-likelihood: -850.659719240158
#> EM NMoE: Iteration: 2 | log-likelihood: -850.524629010475
#> EM NMoE: Iteration: 3 | log-likelihood: -850.430788051698
#> EM NMoE: Iteration: 4 | log-likelihood: -850.283793706938
#> EM NMoE: Iteration: 5 | log-likelihood: -849.97811162098
#> EM NMoE: Iteration: 6 | log-likelihood: -849.309846170774
#> EM NMoE: Iteration: 7 | log-likelihood: -847.853073877546
#> EM NMoE: Iteration: 8 | log-likelihood: -844.760254765814
#> EM NMoE: Iteration: 9 | log-likelihood: -838.538908952736
#> EM NMoE: Iteration: 10 | log-likelihood: -827.124841419721
#> EM NMoE: Iteration: 11 | log-likelihood: -809.002195790739
#> EM NMoE: Iteration: 12 | log-likelihood: -786.082845509062
#> EM NMoE: Iteration: 13 | log-likelihood: -765.697860048611
#> EM NMoE: Iteration: 14 | log-likelihood: -753.84437315637
#> EM NMoE: Iteration: 15 | log-likelihood: -748.545284749922
#> EM NMoE: Iteration: 16 | log-likelihood: -746.181369709665
#> EM NMoE: Iteration: 17 | log-likelihood: -745.062227019926
#> EM NMoE: Iteration: 18 | log-likelihood: -744.517209155278
#> EM NMoE: Iteration: 19 | log-likelihood: -744.248035626126
#> EM NMoE: Iteration: 20 | log-likelihood: -744.113273238347
#> EM NMoE: Iteration: 21 | log-likelihood: -744.04458797388
#> EM NMoE: Iteration: 22 | log-likelihood: -744.008709857418
#> EM NMoE: Iteration: 23 | log-likelihood: -743.989337491229
#> EM NMoE: Iteration: 24 | log-likelihood: -743.978422442498
#> EM NMoE: Iteration: 25 | log-likelihood: -743.971951246252
#> EM NMoE: Iteration: 26 | log-likelihood: -743.967895060795
#> EM NMoE: Iteration: 27 | log-likelihood: -743.965208755974
#> EM NMoE: Iteration: 28 | log-likelihood: -743.963339864259
#> EM NMoE: Iteration: 29 | log-likelihood: -743.961986174011
#> EM NMoE: Iteration: 30 | log-likelihood: -743.960975097926
#> EM NMoE: Iteration: 31 | log-likelihood: -743.960202991077
#> EM NMoE: Iteration: 32 | log-likelihood: -743.959604173327
nmoe$summary()
#> ------------------------------------------
#> Fitted Normal Mixture-of-Experts model
#> ------------------------------------------
#>
#> NMoE model with K = 2 experts:
#>
#> log-likelihood df AIC BIC ICL
#> -743.9596 8 -751.9596 -768.818 -827.3815
#>
#> Clustering table (Number of observations in each expert):
#>
#> 1 2
#> 292 208
#>
#> Regression coefficients:
#>
#> Beta(k = 1) Beta(k = 2)
#> 1 0.01265767 -0.1734812
#> X^1 2.26644322 -2.4105137
#>
#> Variances:
#>
#> Sigma2(k = 1) Sigma2(k = 2)
#> 1.103732 0.8591557
nmoe$plot()
# Application to a real data set
data("tempanomalies")
x <- tempanomalies$Year
y <- tempanomalies$AnnualAnomaly
K <- 2 # Number of regressors/experts
p <- 1 # Order of the polynomial regression (regressors/experts)
q <- 1 # Order of the logistic regression (gating network)
nmoe <- emNMoE(X = x, Y = y, K = K, p = p, q = q, verbose = TRUE)
#> EM NMoE: Iteration: 1 | log-likelihood: 48.3988726040827
#> EM NMoE: Iteration: 2 | log-likelihood: 48.9326207295142
#> EM NMoE: Iteration: 3 | log-likelihood: 50.051039377426
#> EM NMoE: Iteration: 4 | log-likelihood: 52.9250961462781
#> EM NMoE: Iteration: 5 | log-likelihood: 59.1669854674966
#> EM NMoE: Iteration: 6 | log-likelihood: 67.5520185593279
#> EM NMoE: Iteration: 7 | log-likelihood: 73.0997722565129
#> EM NMoE: Iteration: 8 | log-likelihood: 75.5728843281524
#> EM NMoE: Iteration: 9 | log-likelihood: 77.1804335125676
#> EM NMoE: Iteration: 10 | log-likelihood: 78.8228583260898
#> EM NMoE: Iteration: 11 | log-likelihood: 80.7994256495649
#> EM NMoE: Iteration: 12 | log-likelihood: 83.4327216902578
#> EM NMoE: Iteration: 13 | log-likelihood: 87.167207159755
#> EM NMoE: Iteration: 14 | log-likelihood: 91.7548816275664
#> EM NMoE: Iteration: 15 | log-likelihood: 94.8386054468416
#> EM NMoE: Iteration: 16 | log-likelihood: 95.8702965168198
#> EM NMoE: Iteration: 17 | log-likelihood: 96.201217475001
#> EM NMoE: Iteration: 18 | log-likelihood: 96.3427273583883
#> EM NMoE: Iteration: 19 | log-likelihood: 96.4312445403178
#> EM NMoE: Iteration: 20 | log-likelihood: 96.5068035716238
#> EM NMoE: Iteration: 21 | log-likelihood: 96.5827848006443
#> EM NMoE: Iteration: 22 | log-likelihood: 96.664497621724
#> EM NMoE: Iteration: 23 | log-likelihood: 96.7544065779447
#> EM NMoE: Iteration: 24 | log-likelihood: 96.8535649805854
#> EM NMoE: Iteration: 25 | log-likelihood: 96.9618980067147
#> EM NMoE: Iteration: 26 | log-likelihood: 97.0781807281132
#> EM NMoE: Iteration: 27 | log-likelihood: 97.2000668915646
#> EM NMoE: Iteration: 28 | log-likelihood: 97.3243471857001
#> EM NMoE: Iteration: 29 | log-likelihood: 97.4475005220902
#> EM NMoE: Iteration: 30 | log-likelihood: 97.566473896656
#> EM NMoE: Iteration: 31 | log-likelihood: 97.6794841146006
#> EM NMoE: Iteration: 32 | log-likelihood: 97.7865826549208
#> EM NMoE: Iteration: 33 | log-likelihood: 97.8897593890552
#> EM NMoE: Iteration: 34 | log-likelihood: 97.9924846700633
#> EM NMoE: Iteration: 35 | log-likelihood: 98.0988320818964
#> EM NMoE: Iteration: 36 | log-likelihood: 98.2124589670307
#> EM NMoE: Iteration: 37 | log-likelihood: 98.3358032691223
#> EM NMoE: Iteration: 38 | log-likelihood: 98.4698046243747
#> EM NMoE: Iteration: 39 | log-likelihood: 98.6142554980094
#> EM NMoE: Iteration: 40 | log-likelihood: 98.7685998935106
#> EM NMoE: Iteration: 41 | log-likelihood: 98.9327260646186
#> EM NMoE: Iteration: 42 | log-likelihood: 99.1075255399307
#> EM NMoE: Iteration: 43 | log-likelihood: 99.2951330061669
#> EM NMoE: Iteration: 44 | log-likelihood: 99.4990978545361
#> EM NMoE: Iteration: 45 | log-likelihood: 99.724781385219
#> EM NMoE: Iteration: 46 | log-likelihood: 99.9802114334364
#> EM NMoE: Iteration: 47 | log-likelihood: 100.277506353508
#> EM NMoE: Iteration: 48 | log-likelihood: 100.634603770888
#> EM NMoE: Iteration: 49 | log-likelihood: 101.074685777405
#> EM NMoE: Iteration: 50 | log-likelihood: 101.609342261681
#> EM NMoE: Iteration: 51 | log-likelihood: 102.167518045425
#> EM NMoE: Iteration: 52 | log-likelihood: 102.591482251134
#> EM NMoE: Iteration: 53 | log-likelihood: 102.692086561759
#> EM NMoE: Iteration: 54 | log-likelihood: 102.721983731666
#> EM NMoE: Iteration: 55 | log-likelihood: 102.721991417921
nmoe$summary()
#> ------------------------------------------
#> Fitted Normal Mixture-of-Experts model
#> ------------------------------------------
#>
#> NMoE model with K = 2 experts:
#>
#> log-likelihood df AIC BIC ICL
#> 102.722 8 94.72199 83.07137 83.17998
#>
#> Clustering table (Number of observations in each expert):
#>
#> 1 2
#> 52 84
#>
#> Regression coefficients:
#>
#> Beta(k = 1) Beta(k = 2)
#> 1 -42.36252836 -12.667270814
#> X^1 0.02149289 0.006474796
#>
#> Variances:
#>
#> Sigma2(k = 1) Sigma2(k = 2)
#> 0.01193084 0.01352335
nmoe$plot()
TMoE
# Application to a simulated data set
n <- 500 # Size of the sample
alphak <- matrix(c(0, 8), ncol = 1) # Parameters of the gating network
betak <- matrix(c(0, -2.5, 0, 2.5), ncol = 2) # Regression coefficients of the experts
sigmak <- c(0.5, 0.5) # Standard deviations of the experts
nuk <- c(5, 7) # Degrees of freedom of the experts network t densities
x <- seq.int(from = -1, to = 1, length.out = n) # Inputs (predictors)
# Generate sample of size n
sample <- sampleUnivTMoE(alphak = alphak, betak = betak, sigmak = sigmak,
nuk = nuk, x = x)
y <- sample$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)
tmoe <- emTMoE(X = x, Y = y, K = K, p = p, q = q, verbose = TRUE)
#> EM - tMoE: Iteration: 1 | log-likelihood: -552.125213974242
#> EM - tMoE: Iteration: 2 | log-likelihood: -547.987183857056
#> EM - tMoE: Iteration: 3 | log-likelihood: -546.40733469181
#> EM - tMoE: Iteration: 4 | log-likelihood: -544.898386695277
#> EM - tMoE: Iteration: 5 | log-likelihood: -543.502686575021
#> EM - tMoE: Iteration: 6 | log-likelihood: -542.283105674398
#> EM - tMoE: Iteration: 7 | log-likelihood: -541.266467232123
#> EM - tMoE: Iteration: 8 | log-likelihood: -540.450661063362
#> EM - tMoE: Iteration: 9 | log-likelihood: -539.815711994686
#> EM - tMoE: Iteration: 10 | log-likelihood: -539.333458769544
#> EM - tMoE: Iteration: 11 | log-likelihood: -538.974215771526
#> EM - tMoE: Iteration: 12 | log-likelihood: -538.710672092328
#> EM - tMoE: Iteration: 13 | log-likelihood: -538.519646653311
#> EM - tMoE: Iteration: 14 | log-likelihood: -538.38248504553
#> EM - tMoE: Iteration: 15 | log-likelihood: -538.284724625379
#> EM - tMoE: Iteration: 16 | log-likelihood: -538.215449987784
#> EM - tMoE: Iteration: 17 | log-likelihood: -538.166584335222
#> EM - tMoE: Iteration: 18 | log-likelihood: -538.132238929576
#> EM - tMoE: Iteration: 19 | log-likelihood: -538.108167974741
#> EM - tMoE: Iteration: 20 | log-likelihood: -538.09133618607
#> EM - tMoE: Iteration: 21 | log-likelihood: -538.07958783267
#> EM - tMoE: Iteration: 22 | log-likelihood: -538.071399628517
#> EM - tMoE: Iteration: 23 | log-likelihood: -538.065699459315
#> EM - tMoE: Iteration: 24 | log-likelihood: -538.061735113966
#> EM - tMoE: Iteration: 25 | log-likelihood: -538.058980140461
#> EM - tMoE: Iteration: 26 | log-likelihood: -538.05706681974
#> EM - tMoE: Iteration: 27 | log-likelihood: -538.055738714103
#> EM - tMoE: Iteration: 28 | log-likelihood: -538.054817220152
#> EM - tMoE: Iteration: 29 | log-likelihood: -538.054178073834
#> EM - tMoE: Iteration: 30 | log-likelihood: -538.053734891082
tmoe$summary()
#> -------------------------------------
#> Fitted t Mixture-of-Experts model
#> -------------------------------------
#>
#> tMoE model with K = 2 experts:
#>
#> log-likelihood df AIC BIC ICL
#> -538.0537 10 -548.0537 -569.1268 -569.1248
#>
#> Clustering table (Number of observations in each expert):
#>
#> 1 2
#> 249 251
#>
#> Regression coefficients:
#>
#> Beta(k = 1) Beta(k = 2)
#> 1 0.1725939 -0.08414846
#> X^1 2.7387008 -2.33997997
#>
#> Variances:
#>
#> Sigma2(k = 1) Sigma2(k = 2)
#> 0.2727009 0.4847398
tmoe$plot()
# Application 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)
tmoe <- emTMoE(X = x, Y = y, K = K, p = p, q = q, verbose = TRUE)
#> EM - tMoE: Iteration: 1 | log-likelihood: -605.266571357791
#> EM - tMoE: Iteration: 2 | log-likelihood: -599.044701698548
#> EM - tMoE: Iteration: 3 | log-likelihood: -595.501279714269
#> EM - tMoE: Iteration: 4 | log-likelihood: -593.009530361222
#> EM - tMoE: Iteration: 5 | log-likelihood: -590.714969153092
#> EM - tMoE: Iteration: 6 | log-likelihood: -587.897449166264
#> EM - tMoE: Iteration: 7 | log-likelihood: -583.582012360803
#> EM - tMoE: Iteration: 8 | log-likelihood: -578.122132426342
#> EM - tMoE: Iteration: 9 | log-likelihood: -573.081475929554
#> EM - tMoE: Iteration: 10 | log-likelihood: -570.74014908355
#> EM - tMoE: Iteration: 11 | log-likelihood: -569.7657737772
#> EM - tMoE: Iteration: 12 | log-likelihood: -568.885074316649
#> EM - tMoE: Iteration: 13 | log-likelihood: -568.011955227929
#> EM - tMoE: Iteration: 14 | log-likelihood: -567.159312820848
#> EM - tMoE: Iteration: 15 | log-likelihood: -566.350991948378
#> EM - tMoE: Iteration: 16 | log-likelihood: -565.616862268021
#> EM - tMoE: Iteration: 17 | log-likelihood: -564.990448386782
#> EM - tMoE: Iteration: 18 | log-likelihood: -564.496384022067
#> EM - tMoE: Iteration: 19 | log-likelihood: -564.13571445338
#> EM - tMoE: Iteration: 20 | log-likelihood: -563.887578265863
#> EM - tMoE: Iteration: 21 | log-likelihood: -563.72301337972
#> EM - tMoE: Iteration: 22 | log-likelihood: -563.61586828125
#> EM - tMoE: Iteration: 23 | log-likelihood: -563.546554999698
#> EM - tMoE: Iteration: 24 | log-likelihood: -563.501679965445
#> EM - tMoE: Iteration: 25 | log-likelihood: -563.472480239373
#> EM - tMoE: Iteration: 26 | log-likelihood: -563.453334332534
#> EM - tMoE: Iteration: 27 | log-likelihood: -563.440660583559
#> EM - tMoE: Iteration: 28 | log-likelihood: -563.43217720637
#> EM - tMoE: Iteration: 29 | log-likelihood: -563.426425658754
#> EM - tMoE: Iteration: 30 | log-likelihood: -563.422468915477
#> EM - tMoE: Iteration: 31 | log-likelihood: -563.41970146878
#> EM - tMoE: Iteration: 32 | log-likelihood: -563.417729585165
#> EM - tMoE: Iteration: 33 | log-likelihood: -563.416295552506
#> EM - tMoE: Iteration: 34 | log-likelihood: -563.415229512982
#> EM - tMoE: Iteration: 35 | log-likelihood: -563.414418669214
#> EM - tMoE: Iteration: 36 | log-likelihood: -563.413787491396
#> EM - tMoE: Iteration: 37 | log-likelihood: -563.413284930069
tmoe$summary()
#> -------------------------------------
#> Fitted t Mixture-of-Experts model
#> -------------------------------------
#>
#> tMoE model with K = 4 experts:
#>
#> log-likelihood df AIC BIC ICL
#> -563.4133 26 -589.4133 -626.9878 -626.9753
#>
#> Clustering table (Number of observations in each expert):
#>
#> 1 2 3 4
#> 28 36 32 37
#>
#> Regression coefficients:
#>
#> Beta(k = 1) Beta(k = 2) Beta(k = 3) Beta(k = 4)
#> 1 -1.037712416 1774.38349 -1434.398457 292.6068438
#> X^1 -0.111685768 -189.85966 84.930824 -12.1664690
#> X^2 -0.007693142 4.74843 -1.205771 0.1248612
#>
#> Variances:
#>
#> Sigma2(k = 1) Sigma2(k = 2) Sigma2(k = 3) Sigma2(k = 4)
#> 1.585304 30.88009 588.3835 572.0153
tmoe$plot()
SNMoE
# Application 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
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
K <- 2 # Number of regressors/experts
p <- 1 # Order of the polynomial regression (regressors/experts)
q <- 1 # Order of the logistic regression (gating network)
snmoe <- emSNMoE(X = x, Y = y, K = K, p = p, q = q, verbose = TRUE)
#> EM - SNMoE: Iteration: 1 | log-likelihood: -624.23502477139
#> EM - SNMoE: Iteration: 2 | log-likelihood: -506.408654239152
#> EM - SNMoE: Iteration: 3 | log-likelihood: -501.732792195309
#> EM - SNMoE: Iteration: 4 | log-likelihood: -500.859509544961
#> EM - SNMoE: Iteration: 5 | log-likelihood: -500.597208229948
#> EM - SNMoE: Iteration: 6 | log-likelihood: -500.433113585124
#> EM - SNMoE: Iteration: 7 | log-likelihood: -500.281152659166
#> EM - SNMoE: Iteration: 8 | log-likelihood: -500.133110076618
#> EM - SNMoE: Iteration: 9 | log-likelihood: -499.99191573382
#> EM - SNMoE: Iteration: 10 | log-likelihood: -499.859975065605
#> EM - SNMoE: Iteration: 11 | log-likelihood: -499.738162701963
#> EM - SNMoE: Iteration: 12 | log-likelihood: -499.626441124172
#> EM - SNMoE: Iteration: 13 | log-likelihood: -499.524203019735
#> EM - SNMoE: Iteration: 14 | log-likelihood: -499.430767562131
#> EM - SNMoE: Iteration: 15 | log-likelihood: -499.345391721334
#> EM - SNMoE: Iteration: 16 | log-likelihood: -499.267327179952
#> EM - SNMoE: Iteration: 17 | log-likelihood: -499.195801598347
#> EM - SNMoE: Iteration: 18 | log-likelihood: -499.130253804223
#> EM - SNMoE: Iteration: 19 | log-likelihood: -499.070132602352
#> EM - SNMoE: Iteration: 20 | log-likelihood: -499.014947707728
#> EM - SNMoE: Iteration: 21 | log-likelihood: -498.964221084507
#> EM - SNMoE: Iteration: 22 | log-likelihood: -498.917518912217
#> EM - SNMoE: Iteration: 23 | log-likelihood: -498.874538031141
#> EM - SNMoE: Iteration: 24 | log-likelihood: -498.834888076035
#> EM - SNMoE: Iteration: 25 | log-likelihood: -498.7983021104
#> EM - SNMoE: Iteration: 26 | log-likelihood: -498.764480641263
#> EM - SNMoE: Iteration: 27 | log-likelihood: -498.733202141712
#> EM - SNMoE: Iteration: 28 | log-likelihood: -498.704238852666
#> EM - SNMoE: Iteration: 29 | log-likelihood: -498.677392482473
#> EM - SNMoE: Iteration: 30 | log-likelihood: -498.652491791682
#> EM - SNMoE: Iteration: 31 | log-likelihood: -498.629390755752
#> EM - SNMoE: Iteration: 32 | log-likelihood: -498.607918269276
#> EM - SNMoE: Iteration: 33 | log-likelihood: -498.587910911632
#> EM - SNMoE: Iteration: 34 | log-likelihood: -498.569277705623
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#> EM - SNMoE: Iteration: 46 | log-likelihood: -498.421074031024
#> EM - SNMoE: Iteration: 47 | log-likelihood: -498.41311296707
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#> EM - SNMoE: Iteration: 50 | log-likelihood: -498.391844793288
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#> EM - SNMoE: Iteration: 80 | log-likelihood: -498.294116793465
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#> EM - SNMoE: Iteration: 103 | log-likelihood: -498.271755604265
#> EM - SNMoE: Iteration: 104 | log-likelihood: -498.271148680279
#> EM - SNMoE: Iteration: 105 | log-likelihood: -498.270561891543
#> EM - SNMoE: Iteration: 106 | log-likelihood: -498.269995057457
#> EM - SNMoE: Iteration: 107 | log-likelihood: -498.269443948584
#> EM - SNMoE: Iteration: 108 | log-likelihood: -498.268911556956
#> EM - SNMoE: Iteration: 109 | log-likelihood: -498.268399011873
#> EM - SNMoE: Iteration: 110 | log-likelihood: -498.267901191242
snmoe$summary()
#> -----------------------------------------------
#> Fitted Skew-Normal Mixture-of-Experts model
#> -----------------------------------------------
#>
#> SNMoE model with K = 2 experts:
#>
#> log-likelihood df AIC BIC ICL
#> -498.2679 10 -508.2679 -529.3409 -529.3804
#>
#> Clustering table (Number of observations in each expert):
#>
#> 1 2
#> 249 251
#>
#> Regression coefficients:
#>
#> Beta(k = 1) Beta(k = 2)
#> 1 0.9709634 1.021977
#> X^1 2.6703213 -2.736127
#>
#> Variances:
#>
#> Sigma2(k = 1) Sigma2(k = 2)
#> 0.4324076 0.4345685
snmoe$plot()
# Application to a real data set
data("tempanomalies")
x <- tempanomalies$Year
y <- tempanomalies$AnnualAnomaly
K <- 2 # Number of regressors/experts
p <- 1 # Order of the polynomial regression (regressors/experts)
q <- 1 # Order of the logistic regression (gating network)
snmoe <- emSNMoE(X = x, Y = y, K = K, p = p, q = q, verbose = TRUE)
#> EM - SNMoE: Iteration: 1 | log-likelihood: 62.292138332677
#> EM - SNMoE: Iteration: 2 | log-likelihood: 87.9145425373437
#> EM - SNMoE: Iteration: 3 | log-likelihood: 89.0086739618696
#> EM - SNMoE: Iteration: 4 | log-likelihood: 89.3937914630249
#> EM - SNMoE: Iteration: 5 | log-likelihood: 89.6535757640902
#> EM - SNMoE: Iteration: 6 | log-likelihood: 89.8306729565452
#> EM - SNMoE: Iteration: 7 | log-likelihood: 89.9327809726066
#> EM - SNMoE: Iteration: 8 | log-likelihood: 89.9900101339138
#> EM - SNMoE: Iteration: 9 | log-likelihood: 90.0239748570575
#> EM - SNMoE: Iteration: 10 | log-likelihood: 90.0475270018937
#> EM - SNMoE: Iteration: 11 | log-likelihood: 90.0668977115359
#> EM - SNMoE: Iteration: 12 | log-likelihood: 90.0840572217388
#> EM - SNMoE: Iteration: 13 | log-likelihood: 90.0996801870063
#> EM - SNMoE: Iteration: 14 | log-likelihood: 90.114322139282
#> EM - SNMoE: Iteration: 15 | log-likelihood: 90.1282969436904
#> EM - SNMoE: Iteration: 16 | log-likelihood: 90.1417415969409
#> EM - SNMoE: Iteration: 17 | log-likelihood: 90.1547229730222
#> EM - SNMoE: Iteration: 18 | log-likelihood: 90.1672833715162
#> EM - SNMoE: Iteration: 19 | log-likelihood: 90.1794404195207
#> EM - SNMoE: Iteration: 20 | log-likelihood: 90.1912239923447
#> EM - SNMoE: Iteration: 21 | log-likelihood: 90.2026508363525
#> EM - SNMoE: Iteration: 22 | log-likelihood: 90.2137384324658
#> EM - SNMoE: Iteration: 23 | log-likelihood: 90.2245065555815
#> EM - SNMoE: Iteration: 24 | log-likelihood: 90.2349750729752
#> EM - SNMoE: Iteration: 25 | log-likelihood: 90.2452036384708
#> EM - SNMoE: Iteration: 26 | log-likelihood: 90.255194718755
#> EM - SNMoE: Iteration: 27 | log-likelihood: 90.2649308422496
#> EM - SNMoE: Iteration: 28 | log-likelihood: 90.2744302834456
#> EM - SNMoE: Iteration: 29 | log-likelihood: 90.2837106908074
#> EM - SNMoE: Iteration: 30 | log-likelihood: 90.2928990112787
#> EM - SNMoE: Iteration: 31 | log-likelihood: 90.3016166594444
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#> EM - SNMoE: Iteration: 33 | log-likelihood: 90.318900466279
#> EM - SNMoE: Iteration: 34 | log-likelihood: 90.3274306744404
#> EM - SNMoE: Iteration: 35 | log-likelihood: 90.3358671627185
#> EM - SNMoE: Iteration: 36 | log-likelihood: 90.3443859982734
#> EM - SNMoE: Iteration: 37 | log-likelihood: 90.3526959594068
#> EM - SNMoE: Iteration: 38 | log-likelihood: 90.3609977875098
#> EM - SNMoE: Iteration: 39 | log-likelihood: 90.3692807155406
#> EM - SNMoE: Iteration: 40 | log-likelihood: 90.3774653899762
#> EM - SNMoE: Iteration: 41 | log-likelihood: 90.3856183967777
#> EM - SNMoE: Iteration: 42 | log-likelihood: 90.393720562392
#> EM - SNMoE: Iteration: 43 | log-likelihood: 90.4017868140956
#> EM - SNMoE: Iteration: 44 | log-likelihood: 90.4098062308868
#> EM - SNMoE: Iteration: 45 | log-likelihood: 90.4177820881849
#> EM - SNMoE: Iteration: 46 | log-likelihood: 90.4257087087515
#> EM - SNMoE: Iteration: 47 | log-likelihood: 90.4335795968909
#> EM - SNMoE: Iteration: 48 | log-likelihood: 90.4413904823497
#> EM - SNMoE: Iteration: 49 | log-likelihood: 90.4491411451583
#> EM - SNMoE: Iteration: 50 | log-likelihood: 90.4568177335901
#> EM - SNMoE: Iteration: 51 | log-likelihood: 90.4644226598137
#> EM - SNMoE: Iteration: 52 | log-likelihood: 90.4719514584366
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#> EM - SNMoE: Iteration: 54 | log-likelihood: 90.4867728675382
#> EM - SNMoE: Iteration: 55 | log-likelihood: 90.4940408877717
#> EM - SNMoE: Iteration: 56 | log-likelihood: 90.5012509412612
#> EM - SNMoE: Iteration: 57 | log-likelihood: 90.5083684584508
#> EM - SNMoE: Iteration: 58 | log-likelihood: 90.5154189149307
#> EM - SNMoE: Iteration: 59 | log-likelihood: 90.5223930629885
#> EM - SNMoE: Iteration: 60 | log-likelihood: 90.5292946631728
#> EM - SNMoE: Iteration: 61 | log-likelihood: 90.5361177953662
#> EM - SNMoE: Iteration: 62 | log-likelihood: 90.5428708142414
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#> EM - SNMoE: Iteration: 83 | log-likelihood: 90.667130410317
#> EM - SNMoE: Iteration: 84 | log-likelihood: 90.6721658072011
#> EM - SNMoE: Iteration: 85 | log-likelihood: 90.6771148463352
#> EM - SNMoE: Iteration: 86 | log-likelihood: 90.681975635394
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#> EM - SNMoE: Iteration: 95 | log-likelihood: 90.7215671817578
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#> EM - SNMoE: Iteration: 97 | log-likelihood: 90.7291598798633
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#> EM - SNMoE: Iteration: 99 | log-likelihood: 90.7361862704723
#> EM - SNMoE: Iteration: 100 | log-likelihood: 90.7395673235959
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#> EM - SNMoE: Iteration: 116 | log-likelihood: 90.7809308289703
#> EM - SNMoE: Iteration: 117 | log-likelihood: 90.7827763625305
#> EM - SNMoE: Iteration: 118 | log-likelihood: 90.7845439068696
#> EM - SNMoE: Iteration: 119 | log-likelihood: 90.7862393745289
#> EM - SNMoE: Iteration: 120 | log-likelihood: 90.7878675208113
#> EM - SNMoE: Iteration: 121 | log-likelihood: 90.7894302838505
#> EM - SNMoE: Iteration: 122 | log-likelihood: 90.7909296453262
#> EM - SNMoE: Iteration: 123 | log-likelihood: 90.7923678843603
#> EM - SNMoE: Iteration: 124 | log-likelihood: 90.7938593408333
#> EM - SNMoE: Iteration: 125 | log-likelihood: 90.7951720999987
#> EM - SNMoE: Iteration: 126 | log-likelihood: 90.7964313348435
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#> EM - SNMoE: Iteration: 128 | log-likelihood: 90.7987952062374
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#> EM - SNMoE: Iteration: 134 | log-likelihood: 90.8047809346196
#> EM - SNMoE: Iteration: 135 | log-likelihood: 90.8056288697112
#> EM - SNMoE: Iteration: 136 | log-likelihood: 90.8064365102508
#> EM - SNMoE: Iteration: 137 | log-likelihood: 90.8072083975976
#> EM - SNMoE: Iteration: 138 | log-likelihood: 90.8079459623392
#> EM - SNMoE: Iteration: 139 | log-likelihood: 90.8086504624255
#> EM - SNMoE: Iteration: 140 | log-likelihood: 90.8093228982677
#> EM - SNMoE: Iteration: 141 | log-likelihood: 90.8099654751608
#> EM - SNMoE: Iteration: 142 | log-likelihood: 90.8106446102449
#> EM - SNMoE: Iteration: 143 | log-likelihood: 90.8112306380695
#> EM - SNMoE: Iteration: 144 | log-likelihood: 90.8117910134191
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#> EM - SNMoE: Iteration: 146 | log-likelihood: 90.8128353064792
#> EM - SNMoE: Iteration: 147 | log-likelihood: 90.813320758887
#> EM - SNMoE: Iteration: 148 | log-likelihood: 90.8137841377893
#> EM - SNMoE: Iteration: 149 | log-likelihood: 90.8142266707801
#> EM - SNMoE: Iteration: 150 | log-likelihood: 90.8146593536573
#> EM - SNMoE: Iteration: 151 | log-likelihood: 90.8150625412625
#> EM - SNMoE: Iteration: 152 | log-likelihood: 90.8154579266665
#> EM - SNMoE: Iteration: 153 | log-likelihood: 90.8158258545648
#> EM - SNMoE: Iteration: 154 | log-likelihood: 90.8161773354621
#> EM - SNMoE: Iteration: 155 | log-likelihood: 90.8165130292432
#> EM - SNMoE: Iteration: 156 | log-likelihood: 90.8168339082089
#> EM - SNMoE: Iteration: 157 | log-likelihood: 90.8171408598141
#> EM - SNMoE: Iteration: 158 | log-likelihood: 90.8174343507612
#> EM - SNMoE: Iteration: 159 | log-likelihood: 90.8177150405841
#> EM - SNMoE: Iteration: 160 | log-likelihood: 90.8179829474516
#> EM - SNMoE: Iteration: 161 | log-likelihood: 90.8182386605315
#> EM - SNMoE: Iteration: 162 | log-likelihood: 90.8185290095565
#> EM - SNMoE: Iteration: 163 | log-likelihood: 90.818718303285
#> EM - SNMoE: Iteration: 164 | log-likelihood: 90.8189892103138
#> EM - SNMoE: Iteration: 165 | log-likelihood: 90.8192039239698
#> EM - SNMoE: Iteration: 166 | log-likelihood: 90.8193641478361
#> EM - SNMoE: Iteration: 167 | log-likelihood: 90.8196070442012
#> EM - SNMoE: Iteration: 168 | log-likelihood: 90.8197949331449
#> EM - SNMoE: Iteration: 169 | log-likelihood: 90.81997483171
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#> EM - SNMoE: Iteration: 171 | log-likelihood: 90.8203136768737
#> EM - SNMoE: Iteration: 172 | log-likelihood: 90.8204744695155
#> EM - SNMoE: Iteration: 173 | log-likelihood: 90.820628259348
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#> EM - SNMoE: Iteration: 175 | log-likelihood: 90.8209167839697
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#> EM - SNMoE: Iteration: 178 | log-likelihood: 90.8213073953412
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#> EM - SNMoE: Iteration: 180 | log-likelihood: 90.8215444184688
#> EM - SNMoE: Iteration: 181 | log-likelihood: 90.821656408958
#> EM - SNMoE: Iteration: 182 | log-likelihood: 90.8217642942328
#> EM - SNMoE: Iteration: 183 | log-likelihood: 90.8218682729254
#> EM - SNMoE: Iteration: 184 | log-likelihood: 90.8219689323786
#> EM - SNMoE: Iteration: 185 | log-likelihood: 90.8220661409908
#> EM - SNMoE: Iteration: 186 | log-likelihood: 90.8221825393501
#> EM - SNMoE: Iteration: 187 | log-likelihood: 90.8222726222892
snmoe$summary()
#> -----------------------------------------------
#> Fitted Skew-Normal Mixture-of-Experts model
#> -----------------------------------------------
#>
#> SNMoE model with K = 2 experts:
#>
#> log-likelihood df AIC BIC ICL
#> 90.82227 10 80.82227 66.259 66.16274
#>
#> Clustering table (Number of observations in each expert):
#>
#> 1 2
#> 69 67
#>
#> Regression coefficients:
#>
#> Beta(k = 1) Beta(k = 2)
#> 1 -14.217412214 -32.63731250
#> X^1 0.007303448 0.01668922
#>
#> Variances:
#>
#> Sigma2(k = 1) Sigma2(k = 2)
#> 0.01492812 0.03739716
snmoe$plot()
StMoE
# Applicartion to a simulated data set
n <- 500 # Size of the sample
alphak <- matrix(c(0, 8), ncol = 1) # Parameters of the gating network
betak <- matrix(c(0, -2.5, 0, 2.5), ncol = 2) # Regression coefficients of the experts
sigmak <- c(0.5, 0.5) # Standard deviations of the experts
lambdak <- c(3, 5) # Skewness parameters of the experts
nuk <- c(5, 7) # Degrees of freedom of the experts network t densities
x <- seq.int(from = -1, to = 1, length.out = n) # Inputs (predictors)
# Generate sample of size n
sample <- sampleUnivStMoE(alphak = alphak, betak = betak,
sigmak = sigmak, lambdak = lambdak,
nuk = nuk, x = x)
y <- sample$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: -440.883552325807
#> EM - StMoE: Iteration: 2 | log-likelihood: -375.041451068938
#> EM - StMoE: Iteration: 3 | log-likelihood: -362.870294323756
#> EM - StMoE: Iteration: 4 | log-likelihood: -353.094409813433
#> EM - StMoE: Iteration: 5 | log-likelihood: -345.084704974844
#> EM - StMoE: Iteration: 6 | log-likelihood: -338.385721447189
#> EM - StMoE: Iteration: 7 | log-likelihood: -332.711434207247
#> EM - StMoE: Iteration: 8 | log-likelihood: -327.869250919329
#> EM - StMoE: Iteration: 9 | log-likelihood: -323.722486281363
#> EM - StMoE: Iteration: 10 | log-likelihood: -320.174322569213
#> EM - StMoE: Iteration: 11 | log-likelihood: -317.145317584615
#> EM - StMoE: Iteration: 12 | log-likelihood: -314.564125534049
#> EM - StMoE: Iteration: 13 | log-likelihood: -312.365799387958
#> EM - StMoE: Iteration: 14 | log-likelihood: -310.494816434848
#> EM - StMoE: Iteration: 15 | log-likelihood: -308.90734115717
#> EM - StMoE: Iteration: 16 | log-likelihood: -307.562662018129
#> EM - StMoE: Iteration: 17 | log-likelihood: -306.431863696511
#> EM - StMoE: Iteration: 18 | log-likelihood: -305.491159408921
#> EM - StMoE: Iteration: 19 | log-likelihood: -304.713201844797
#> EM - StMoE: Iteration: 20 | log-likelihood: -304.076291722324
#> EM - StMoE: Iteration: 21 | log-likelihood: -303.556264095577
#> EM - StMoE: Iteration: 22 | log-likelihood: -303.133105733088
#> EM - StMoE: Iteration: 23 | log-likelihood: -302.788962131677
#> EM - StMoE: Iteration: 24 | log-likelihood: -302.510214757931
#> EM - StMoE: Iteration: 25 | log-likelihood: -302.284355856085
#> EM - StMoE: Iteration: 26 | log-likelihood: -302.102709492201
#> EM - StMoE: Iteration: 27 | log-likelihood: -301.952229413618
#> EM - StMoE: Iteration: 28 | log-likelihood: -301.828306594261
#> EM - StMoE: Iteration: 29 | log-likelihood: -301.727844905847
#> EM - StMoE: Iteration: 30 | log-likelihood: -301.648397641675
#> EM - StMoE: Iteration: 31 | log-likelihood: -301.587551566133
#> EM - StMoE: Iteration: 32 | log-likelihood: -301.543221465035
#> EM - StMoE: Iteration: 33 | log-likelihood: -301.513271473494
#> EM - StMoE: Iteration: 34 | log-likelihood: -301.493057761561
#> EM - StMoE: Iteration: 35 | log-likelihood: -301.479539851439
#> EM - StMoE: Iteration: 36 | log-likelihood: -301.472848664551
#> EM - StMoE: Iteration: 37 | log-likelihood: -301.470938151252
#> EM - StMoE: Iteration: 38 | log-likelihood: -301.473915142384
#> EM - StMoE: Iteration: 39 | log-likelihood: -301.482189626534
#> EM - StMoE: Iteration: 40 | log-likelihood: -301.492724445284
#> EM - StMoE: Iteration: 41 | log-likelihood: -301.504430808141
#> EM - StMoE: Iteration: 42 | log-likelihood: -301.520362479513
#> EM - StMoE: Iteration: 43 | log-likelihood: -301.537844646926
#> EM - StMoE: Iteration: 44 | log-likelihood: -301.558557865998
#> EM - StMoE: Iteration: 45 | log-likelihood: -301.580326531981
#> EM - StMoE: Iteration: 46 | log-likelihood: -301.6019097139
#> EM - StMoE: Iteration: 47 | log-likelihood: -301.622978880623
#> EM - StMoE: Iteration: 48 | log-likelihood: -301.643425685362
#> EM - StMoE: Iteration: 49 | log-likelihood: -301.663097097178
#> EM - StMoE: Iteration: 50 | log-likelihood: -301.682215354801
#> EM - StMoE: Iteration: 51 | log-likelihood: -301.70079024187
#> EM - StMoE: Iteration: 52 | log-likelihood: -301.719757725239
#> EM - StMoE: Iteration: 53 | log-likelihood: -301.739381113304
#> EM - StMoE: Iteration: 54 | log-likelihood: -301.759540649247
#> EM - StMoE: Iteration: 55 | log-likelihood: -301.779249480904
#> EM - StMoE: Iteration: 56 | log-likelihood: -301.797911724126
#> EM - StMoE: Iteration: 57 | log-likelihood: -301.81623977499
#> EM - StMoE: Iteration: 58 | log-likelihood: -301.833272923682
#> EM - StMoE: Iteration: 59 | log-likelihood: -301.849360112204
#> EM - StMoE: Iteration: 60 | log-likelihood: -301.864650313966
#> EM - StMoE: Iteration: 61 | log-likelihood: -301.879219103222
#> EM - StMoE: Iteration: 62 | log-likelihood: -301.893072095918
#> EM - StMoE: Iteration: 63 | log-likelihood: -301.906573460693
#> EM - StMoE: Iteration: 64 | log-likelihood: -301.919762960663
#> EM - StMoE: Iteration: 65 | log-likelihood: -301.932522405111
#> EM - StMoE: Iteration: 66 | log-likelihood: -301.945156344736
#> EM - StMoE: Iteration: 67 | log-likelihood: -301.957387126573
#> EM - StMoE: Iteration: 68 | log-likelihood: -301.969115554813
#> EM - StMoE: Iteration: 69 | log-likelihood: -301.98037510572
#> EM - StMoE: Iteration: 70 | log-likelihood: -301.991617754184
#> EM - StMoE: Iteration: 71 | log-likelihood: -302.002560717356
#> EM - StMoE: Iteration: 72 | log-likelihood: -302.013087192795
#> EM - StMoE: Iteration: 73 | log-likelihood: -302.023158356658
#> EM - StMoE: Iteration: 74 | log-likelihood: -302.033353078519
#> EM - StMoE: Iteration: 75 | log-likelihood: -302.043205904383
#> EM - StMoE: Iteration: 76 | log-likelihood: -302.052643511154
#> EM - StMoE: Iteration: 77 | log-likelihood: -302.061810797848
#> EM - StMoE: Iteration: 78 | log-likelihood: -302.070902764504
#> EM - StMoE: Iteration: 79 | log-likelihood: -302.07973925995
#> EM - StMoE: Iteration: 80 | log-likelihood: -302.088242899838
#> EM - StMoE: Iteration: 81 | log-likelihood: -302.096388313632
#> EM - StMoE: Iteration: 82 | log-likelihood: -302.104175694891
#> EM - StMoE: Iteration: 83 | log-likelihood: -302.111617068742
#> EM - StMoE: Iteration: 84 | log-likelihood: -302.118925441493
#> EM - StMoE: Iteration: 85 | log-likelihood: -302.125457707218
#> EM - StMoE: Iteration: 86 | log-likelihood: -302.13197004521
#> EM - StMoE: Iteration: 87 | log-likelihood: -302.138129224314
#> EM - StMoE: Iteration: 88 | log-likelihood: -302.144160520975
#> EM - StMoE: Iteration: 89 | log-likelihood: -302.149916599427
#> EM - StMoE: Iteration: 90 | log-likelihood: -302.155426446231
#> EM - StMoE: Iteration: 91 | log-likelihood: -302.160714471348
#> EM - StMoE: Iteration: 92 | log-likelihood: -302.165802591342
#> EM - StMoE: Iteration: 93 | log-likelihood: -302.170710814011
#> EM - StMoE: Iteration: 94 | log-likelihood: -302.175457223439
#> EM - StMoE: Iteration: 95 | log-likelihood: -302.180057795817
#> EM - StMoE: Iteration: 96 | log-likelihood: -302.184526236955
#> EM - StMoE: Iteration: 97 | log-likelihood: -302.188117253055
#> EM - StMoE: Iteration: 98 | log-likelihood: -302.191944047832
#> EM - StMoE: Iteration: 99 | log-likelihood: -302.195764719166
#> EM - StMoE: Iteration: 100 | log-likelihood: -302.199416754305
#> EM - StMoE: Iteration: 101 | log-likelihood: -302.202913987336
#> EM - StMoE: Iteration: 102 | log-likelihood: -302.206266279947
#> EM - StMoE: Iteration: 103 | log-likelihood: -302.209481287433
#> EM - StMoE: Iteration: 104 | log-likelihood: -302.21256540455
#> EM - StMoE: Iteration: 105 | log-likelihood: -302.215524201826
#> EM - StMoE: Iteration: 106 | log-likelihood: -302.21844028189
#> EM - StMoE: Iteration: 107 | log-likelihood: -302.221347360257
#> EM - StMoE: Iteration: 108 | log-likelihood: -302.224198487454
#> EM - StMoE: Iteration: 109 | log-likelihood: -302.226968823771
#> EM - StMoE: Iteration: 110 | log-likelihood: -302.22964637945
#> EM - StMoE: Iteration: 111 | log-likelihood: -302.232226540156
#> EM - StMoE: Iteration: 112 | log-likelihood: -302.234708840608
#> EM - StMoE: Iteration: 113 | log-likelihood: -302.237095073969
#> EM - StMoE: Iteration: 114 | log-likelihood: -302.239388193955
#> EM - StMoE: Iteration: 115 | log-likelihood: -302.241591686042
#> EM - StMoE: Iteration: 116 | log-likelihood: -302.243709214893
#> EM - StMoE: Iteration: 117 | log-likelihood: -302.245744433168
#> EM - StMoE: Iteration: 118 | log-likelihood: -302.247700883553
#> EM - StMoE: Iteration: 119 | log-likelihood: -302.24958195369
#> EM - StMoE: Iteration: 120 | log-likelihood: -302.251390860385
#> EM - StMoE: Iteration: 121 | log-likelihood: -302.253130649339
#> EM - StMoE: Iteration: 122 | log-likelihood: -302.254804202543
#> EM - StMoE: Iteration: 123 | log-likelihood: -302.256414248943
#> EM - StMoE: Iteration: 124 | log-likelihood: -302.257963375992
#> EM - StMoE: Iteration: 125 | log-likelihood: -302.25945404088
#> EM - StMoE: Iteration: 126 | log-likelihood: -302.260888580875
#> EM - StMoE: Iteration: 127 | log-likelihood: -302.262269222598
#> EM - StMoE: Iteration: 128 | log-likelihood: -302.263598090208
#> EM - StMoE: Iteration: 129 | log-likelihood: -302.264877212588
#> EM - StMoE: Iteration: 130 | log-likelihood: -302.266108529644
#> EM - StMoE: Iteration: 131 | log-likelihood: -302.26729389785
#> EM - StMoE: Iteration: 132 | log-likelihood: -302.268435095155
#> EM - StMoE: Iteration: 133 | log-likelihood: -302.269533825348
#> EM - StMoE: Iteration: 134 | log-likelihood: -302.270591721985
#> EM - StMoE: Iteration: 135 | log-likelihood: -302.271610351937
#> EM - StMoE: Iteration: 136 | log-likelihood: -302.272591218634
#> EM - StMoE: Iteration: 137 | log-likelihood: -302.273535765038
#> EM - StMoE: Iteration: 138 | log-likelihood: -302.274445376409
#> EM - StMoE: Iteration: 139 | log-likelihood: -302.276048594329
#> EM - StMoE: Iteration: 140 | log-likelihood: -302.277295858768
#> EM - StMoE: Iteration: 141 | log-likelihood: -302.278329251585
#> EM - StMoE: Iteration: 142 | log-likelihood: -302.27922826416
#> EM - StMoE: Iteration: 143 | log-likelihood: -302.280037968947
#> EM - StMoE: Iteration: 144 | log-likelihood: -302.280784391465
#> EM - StMoE: Iteration: 145 | log-likelihood: -302.28148293074
#> EM - StMoE: Iteration: 146 | log-likelihood: -302.282142994437
#> EM - StMoE: Iteration: 147 | log-likelihood: -302.282770566065
#> EM - StMoE: Iteration: 148 | log-likelihood: -302.283369637877
#> EM - StMoE: Iteration: 149 | log-likelihood: -302.283943018484
#> EM - StMoE: Iteration: 150 | log-likelihood: -302.284492793652
#> EM - StMoE: Iteration: 151 | log-likelihood: -302.285020593249
#> EM - StMoE: Iteration: 152 | log-likelihood: -302.285527748826
#> EM - StMoE: Iteration: 153 | log-likelihood: -302.286015388739
#> EM - StMoE: Iteration: 154 | log-likelihood: -302.286484497098
#> EM - StMoE: Iteration: 155 | log-likelihood: -302.286935951393
#> EM - StMoE: Iteration: 156 | log-likelihood: -302.287370547269
#> EM - StMoE: Iteration: 157 | log-likelihood: -302.287789015395
#> EM - StMoE: Iteration: 158 | log-likelihood: -302.288192033309
#> EM - StMoE: Iteration: 159 | log-likelihood: -302.288580234002
#> EM - StMoE: Iteration: 160 | log-likelihood: -302.288954212308
#> EM - StMoE: Iteration: 161 | log-likelihood: -302.28931452978
#> EM - StMoE: Iteration: 162 | log-likelihood: -302.289661718499
#> EM - StMoE: Iteration: 163 | log-likelihood: -302.289996284107
#> EM - StMoE: Iteration: 164 | log-likelihood: -302.290318708259
#> EM - StMoE: Iteration: 165 | log-likelihood: -302.290629450648
#> EM - StMoE: Iteration: 166 | log-likelihood: -302.290928950694
stmoe$summary()
#> ------------------------------------------
#> Fitted Skew t Mixture-of-Experts model
#> ------------------------------------------
#>
#> StMoE model with K = 2 experts:
#>
#> log-likelihood df AIC BIC ICL
#> -302.2909 12 -314.2909 -339.5786 -339.576
#>
#> Clustering table (Number of observations in each expert):
#>
#> 1 2
#> 249 251
#>
#> Regression coefficients:
#>
#> Beta(k = 1) Beta(k = 2)
#> 1 0.06643398 -0.02736487
#> X^1 2.57061178 -2.64710637
#>
#> Variances:
#>
#> Sigma2(k = 1) Sigma2(k = 2)
#> 0.1031365 0.6024446
stmoe$plot()
#> Warning in sqrt(stat$Vary): production de NaN
#> Warning in sqrt(stat$Vary): production de NaN
# 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: -599.970868260844
#> EM - StMoE: Iteration: 2 | log-likelihood: -584.761753783483
#> EM - StMoE: Iteration: 3 | log-likelihood: -584.739108683906
#> EM - StMoE: Iteration: 4 | log-likelihood: -583.152667091195
#> EM - StMoE: Iteration: 5 | log-likelihood: -577.01758775371
#> EM - StMoE: Iteration: 6 | log-likelihood: -570.545320839571
#> EM - StMoE: Iteration: 7 | log-likelihood: -566.102216660141
#> EM - StMoE: Iteration: 8 | log-likelihood: -564.005208184307
#> EM - StMoE: Iteration: 9 | log-likelihood: -563.473769874151
#> EM - StMoE: Iteration: 10 | log-likelihood: -563.371015164243
#> EM - StMoE: Iteration: 11 | log-likelihood: -563.325639042004
#> EM - StMoE: Iteration: 12 | log-likelihood: -563.260956368371
#> EM - StMoE: Iteration: 13 | log-likelihood: -563.160103139318
#> EM - StMoE: Iteration: 14 | log-likelihood: -563.036208961213
#> EM - StMoE: Iteration: 15 | log-likelihood: -562.929568309038
#> EM - StMoE: Iteration: 16 | log-likelihood: -562.873926288988
#> EM - StMoE: Iteration: 17 | log-likelihood: -562.863056340075
#> EM - StMoE: Iteration: 18 | log-likelihood: -562.872058840866
#> EM - StMoE: Iteration: 19 | log-likelihood: -562.884298969276
#> EM - StMoE: Iteration: 20 | log-likelihood: -562.892257042228
#> EM - StMoE: Iteration: 21 | log-likelihood: -562.893400819209
#> EM - StMoE: Iteration: 22 | log-likelihood: -562.886326886739
#> EM - StMoE: Iteration: 23 | log-likelihood: -562.869734547993
#> EM - StMoE: Iteration: 24 | log-likelihood: -562.843373338597
#> EM - StMoE: Iteration: 25 | log-likelihood: -562.806530223593
#> EM - StMoE: Iteration: 26 | log-likelihood: -562.759209213378
#> EM - StMoE: Iteration: 27 | log-likelihood: -562.701759556614
#> EM - StMoE: Iteration: 28 | log-likelihood: -562.637851562422
#> EM - StMoE: Iteration: 29 | log-likelihood: -562.578679951567
#> EM - StMoE: Iteration: 30 | log-likelihood: -562.544706088763
#> EM - StMoE: Iteration: 31 | log-likelihood: -562.547659760017
#> EM - StMoE: Iteration: 32 | log-likelihood: -562.573594841724
#> EM - StMoE: Iteration: 33 | log-likelihood: -562.606224655412
#> EM - StMoE: Iteration: 34 | log-likelihood: -562.639170956927
#> EM - StMoE: Iteration: 35 | log-likelihood: -562.670887429489
#> EM - StMoE: Iteration: 36 | log-likelihood: -562.700977889776
#> EM - StMoE: Iteration: 37 | log-likelihood: -562.729333904045
#> EM - StMoE: Iteration: 38 | log-likelihood: -562.75594302018
#> EM - StMoE: Iteration: 39 | log-likelihood: -562.780915483106
#> EM - StMoE: Iteration: 40 | log-likelihood: -562.804273764516
#> EM - StMoE: Iteration: 41 | log-likelihood: -562.826081748726
#> EM - StMoE: Iteration: 42 | log-likelihood: -562.846465069854
#> EM - StMoE: Iteration: 43 | log-likelihood: -562.865494990344
#> EM - StMoE: Iteration: 44 | log-likelihood: -562.883363535599
#> EM - StMoE: Iteration: 45 | log-likelihood: -562.899766649106
#> EM - StMoE: Iteration: 46 | log-likelihood: -562.915105887419
#> EM - StMoE: Iteration: 47 | log-likelihood: -562.929369415829
#> EM - StMoE: Iteration: 48 | log-likelihood: -562.942618350082
#> EM - StMoE: Iteration: 49 | log-likelihood: -562.954914681938
#> EM - StMoE: Iteration: 50 | log-likelihood: -562.966324704433
#> EM - StMoE: Iteration: 51 | log-likelihood: -562.976892924208
#> EM - StMoE: Iteration: 52 | log-likelihood: -562.986679129858
#> EM - StMoE: Iteration: 53 | log-likelihood: -562.995698141401
#> EM - StMoE: Iteration: 54 | log-likelihood: -563.004199322622
#> EM - StMoE: Iteration: 55 | log-likelihood: -563.011948719677
#> EM - StMoE: Iteration: 56 | log-likelihood: -563.019092394262
#> EM - StMoE: Iteration: 57 | log-likelihood: -563.025788220585
#> EM - StMoE: Iteration: 58 | log-likelihood: -563.032130750582
#> EM - StMoE: Iteration: 59 | log-likelihood: -563.038101658285
#> EM - StMoE: Iteration: 60 | log-likelihood: -563.043686700587
#> EM - StMoE: Iteration: 61 | log-likelihood: -563.048913316641
#> EM - StMoE: Iteration: 62 | log-likelihood: -563.053800034428
#> EM - StMoE: Iteration: 63 | log-likelihood: -563.058367081312
#> EM - StMoE: Iteration: 64 | log-likelihood: -563.062634411041
#> EM - StMoE: Iteration: 65 | log-likelihood: -563.066621029848
#> EM - StMoE: Iteration: 66 | log-likelihood: -563.070344865861
#> EM - StMoE: Iteration: 67 | log-likelihood: -563.073822774497
#> EM - StMoE: Iteration: 68 | log-likelihood: -563.077068884576
#> EM - StMoE: Iteration: 69 | log-likelihood: -563.080101318079
#> EM - StMoE: Iteration: 70 | log-likelihood: -563.082932976016
#> EM - StMoE: Iteration: 71 | log-likelihood: -563.085576456654
#> EM - StMoE: Iteration: 72 | log-likelihood: -563.088043769262
#> EM - StMoE: Iteration: 73 | log-likelihood: -563.090354748117
#> EM - StMoE: Iteration: 74 | log-likelihood: -563.092543476789
#> EM - StMoE: Iteration: 75 | log-likelihood: -563.094432674549
#> EM - StMoE: Iteration: 76 | log-likelihood: -563.09630477819
#> EM - StMoE: Iteration: 77 | log-likelihood: -563.098074257544
#> EM - StMoE: Iteration: 78 | log-likelihood: -563.099724441976
#> EM - StMoE: Iteration: 79 | log-likelihood: -563.101258071476
#> EM - StMoE: Iteration: 80 | log-likelihood: -563.102682505525
#> EM - StMoE: Iteration: 81 | log-likelihood: -563.104005588245
#> EM - StMoE: Iteration: 82 | log-likelihood: -563.105234621045
#> EM - StMoE: Iteration: 83 | log-likelihood: -563.106376189189
#> EM - StMoE: Iteration: 84 | log-likelihood: -563.107436197855
#> EM - StMoE: Iteration: 85 | log-likelihood: -563.108419942485
#> EM - StMoE: Iteration: 86 | log-likelihood: -563.109332171131
#> EM - StMoE: Iteration: 87 | log-likelihood: -563.110177132063
#> EM - StMoE: Iteration: 88 | log-likelihood: -563.11095860863
#> EM - StMoE: Iteration: 89 | log-likelihood: -563.11167994535
#> EM - StMoE: Iteration: 90 | log-likelihood: -563.112344067246
#> EM - StMoE: Iteration: 91 | log-likelihood: -563.112953493273
#> EM - StMoE: Iteration: 92 | log-likelihood: -563.113510345457
stmoe$summary()
#> ------------------------------------------
#> Fitted Skew t Mixture-of-Experts model
#> ------------------------------------------
#>
#> StMoE model with K = 4 experts:
#>
#> log-likelihood df AIC BIC ICL
#> -563.1135 30 -593.1135 -636.4687 -636.4969
#>
#> Clustering table (Number of observations in each expert):
#>
#> 1 2 3 4
#> 28 37 31 37
#>
#> Regression coefficients:
#>
#> Beta(k = 1) Beta(k = 2) Beta(k = 3) Beta(k = 4)
#> 1 -3.52358475 996.077085 -1616.483001 134.35786999
#> X^1 0.88184631 -104.419255 95.549943 -6.74970173
#> X^2 -0.08184845 2.446371 -1.386852 0.07092188
#>
#> Variances:
#>
#> Sigma2(k = 1) Sigma2(k = 2) Sigma2(k = 3) Sigma2(k = 4)
#> 14.09186 448.3051 1404.488 1385.116
stmoe$plot()
References
Chamroukhi, F. 2017. “Skew-T Mixture of Experts.” Neurocomputing - Elsevier 266: 390–408. https://chamroukhi.com/papers/STMoE.pdf.
Chamroukhi, F. 2016a. “Robust Mixture of Experts Modeling Using the T-Distribution.” Neural Networks - Elsevier 79: 20–36. https://chamroukhi.com/papers/TMoE.pdf.
Chamroukhi, F. 2016b. “Skew-Normal Mixture of Experts.” In The International Joint Conference on Neural Networks (IJCNN). Vancouver, Canada. https://chamroukhi.com/papers/Chamroukhi-SNMoE-IJCNN2016.pdf.
Chamroukhi, F. 2015a. “Non-Normal Mixtures of Experts.” http://arxiv.org/pdf/1506.06707.pdf.
Chamroukhi, F. 2015b. “Statistical Learning of Latent Data Models for Complex Data Analysis.” Habilitation Thesis (HDR), Université de Toulon. https://chamroukhi.com/FChamroukhi-HDR.pdf.
Chamroukhi, F. 2010. “Hidden Process Regression for Curve Modeling, Classification and Tracking.” Ph.D. Thesis, Université de Technologie de Compiègne. https://chamroukhi.com/FChamroukhi-PhD.pdf.
Chamroukhi, F., A. Samé, G. Govaert, and P. Aknin. 2009. “Time Series Modeling by a Regression Approach Based on a Latent Process.” Neural Networks 22 (5-6): 593–602. https://chamroukhi.com/papers/Chamroukhi_Neural_Networks_2009.pdf.