MoEClust: Gaussian Parsimonious Clustering Models with Gating and Expert Network Covariates and a Noise Component

Introduction

MoEClust is an R package which fits finite Gaussian Mixtures of Experts models using a range of parsimonious covariance parameterisations via the EM/CEM algorithm, i.e. allows incorporation of covariates into the mixing proportions and/or Gaussian densities of finite Gaussian mixture models under the various parsimonious covariance parameterisations in the GPCM family (e.g. mclust). These models were introduced by Murphy and Murphy (2020). The package also facilitates the inclusion of an additional noise component, and allows visualisation of Gaussian mixture of experts models with parsimonious covariance parameterisations using generalised pairs plots.

The most important function in the MoEClust package is: MoE_clust, for fitting the model via the EM/CEM algorithm with gating and/or expert network covariates, supplied via formula interfaces. MoE_compare is provided for conducting model selection between different results from MoE_clust using different covariate combinations &/or initialisation strategies, etc.

MoE_stepwise is provided for conducting a greedy forward stepwise search to identify the optimal model in terms of the number of components, GPCM covariance type, and the subsets of gating/expert network covariates.

MoE_control allows supplying additional arguments to MoE_clust and MoE_stepwise which govern, among other things, controls on the inclusion of an additional noise component and controls on the initialisation of the allocations for the EM/CEM algorithm.

A dedicated plotting function exists for visualising the results using generalised pairs plots, for examining the gating network, and/or log-likelihood, and/or clustering uncertainties, and/or similarity matrix, and/or graphing model selection criteria values. The generalised pairs plots (MoE_gpairs) visualise all pairwise relationships between clustered response variables and associated continuous, categorical, and/or ordinal covariates in the gating &/or expert networks, coloured according to the MAP classification, and also give the marginal distributions of each variable (incl. the covariates) along the diagonal.

An as.Mclust method is provided to coerce the output of class "MoEClust" from MoE_clust to the "Mclust" class, to facilitate use of plotting and other functions for the "Mclust" class within the mclust package. As per mclust, MoEClust also facilitates modelling with an additional noise component (with or without the mixing proportion for the noise component depending on covariates).

Finally, a predict method is provided for predicting the fitted response and probability of cluster membership (and by extension the MAP classification) for new data, in the form of new covariates and new response variables, or new covariates only.

Other functions also exist, e.g. MoE_crit, MoE_dens, MoE_estep, and aitken, which are all used within MoE_clust but are nonetheless made available for standalone use. The package also contains two data sets: CO2data and ais.

This vignette aims to demonstrate the MoEClust models via application to these two well-known univariate and multivariate data sets, respectively.

install.packages('devtools')
devtools::install_github('Keefe-Murphy/MoEClust')

install.packages('MoEClust')

library(MoEClust)

CO2 Data

Load the CO2 data.

data(CO2data)
CO2 <- CO2data$CO2
GNP <- CO2data$GNP

Fit various MoEClust mixture models to cluster the CO2 data, allowing the GNP variable to enter the gating &/or expert networks, or neither, via a formula interface. Also consider models with equal mixing proportions. Note that for models with covariates in the gating network, or models with equal mixing proportions, we don’t need to fit single-component models (though it could be done!) as this would merely duplicate the single-component models within m1 and m3, respectively.

m1 <- MoE_clust(CO2, G=1:3)
m2 <- MoE_clust(CO2, G=2:3, gating= ~ GNP)
m3 <- MoE_clust(CO2, G=1:3, expert= ~ GNP)
m4 <- MoE_clust(CO2, G=2:3, gating= ~ GNP, expert= ~ GNP)
m5 <- MoE_clust(CO2, G=2:3, equalPro=TRUE)
m6 <- MoE_clust(CO2, G=2:3, expert= ~ GNP, equalPro=TRUE)

Choose the best model among these. Specify optimal.only=TRUE so that only the optimal model in each set of models is included in the comparison.

comp <- MoE_compare(m1, m2, m3, m4, m5, m6, optimal.only=TRUE)

Now, see if a better model can be found using greedy forward stepwise selection, with the aid of the MoE_stepwise function, on the same data. By default, this starts from a single-component model and successively adds components and covariates to both networks, while evaluating each action over all possible model types, until a model which is optimal according to some criterion is found.

(mod1 <- MoE_stepwise(CO2, GNP))
#> ---------------------------------------------------------------------
#> Comparison of Gaussian Parsimonious Clustering Models with Covariates
#> Data: CO2
#> Ranking Criterion: BIC
#> Optimal Only: TRUE
#> ---------------------------------------------------------------------
#> 
#>  rank MoENames modelNames G df iters      bic      icl      aic  loglik gating
#>     1   Step_4          E 3  7    21   -155.2 -159.062 -145.875 -65.937   None
#>     2   Step_3          V 2  7     7 -157.205 -160.039  -147.88  -66.94   None
#>     3   Step_2          E 2  4    19 -163.164 -163.911 -157.835 -74.917   None
#>     4   Step_1          E 1  2     1 -163.905 -163.905  -161.24  -78.62   None
#>  expert algo equalPro
#>    ~GNP   EM     TRUE
#>    ~GNP   EM    FALSE
#>    None   EM    FALSE
#>    None   EM

Next, conduct another stepwise search considering models with a noise component. This notably starts from a model with only a uniform noise component and then proceeds as above to add Gaussian components and covariates one at a time, accepting each action that increases a chosen criterion ("bic", by default).

(mod2 <- MoE_stepwise(CO2, GNP, noise=TRUE))
#> ---------------------------------------------------------------------
#> Comparison of Gaussian Parsimonious Clustering Models with Covariates
#> Data: CO2
#> Ranking Criterion: BIC
#> Optimal Only: TRUE
#> ---------------------------------------------------------------------
#> 
#>  rank MoENames modelNames G df iters      bic      icl      aic  loglik gating
#>     1   Step_2          E 1  4    22 -160.781 -173.158 -155.453 -73.726   None
#>     2   Step_1            0  1     1 -165.503 -165.503 -164.171 -81.086   None
#>  expert algo  noise
#>    None   EM hypvol
#>    None   EM hypvol

Finally, compare all sets of results to choose the optimal model.

(best <- MoE_compare(mod1, mod2, comp, pick=1)$optimal)
#> Call:    MoE_stepwise(data = CO2, network.data = GNP)
#> 
#> Best Model (according to BIC):
#> univariate, equal variance (E), with 3 components
#> 
#> Equal Mixing Proportions
#> BIC = -155.2 | ICL = -159.062 | AIC = -145.875
#> Including expert network covariates:
#>  Expert: ~GNP

(summ <- summary(best, classification=TRUE, parameters=FALSE, networks=FALSE))
#> ------------------------------------------------------
#> Gaussian Parsimonious Clustering Model with Covariates
#> Data: CO2
#> ------------------------------------------------------
#> 
#> MoEClust: E (univariate, equal variance), with 3 components
#> 
#> Gating Network Covariates:  None
#> Expert Network Covariates:  ~GNP
#> Equal Mixing Proportions:   TRUE
#> Noise Component:            FALSE
#> 
#>  log.likelihood  n d df iters    BIC      ICL      AIC Algo
#>         -65.937 28 1  7    21 -155.2 -159.062 -145.875   EM
#> 
#> Clustering table :
#>  1  2  3 
#>  8 10 10

Visualise the results for the optimal model using a generalised pairs plot.

plot(best, what="gpairs", jitter=FALSE)

Visualise the density of the mixture distribution.

Convert from the "MoEClust" class to the "Mclust" class in order to further visualise the results. Examine the "classification" and "uncertainty" options.

mod <- as.Mclust(comp$optimal)
plot(mod, what="classification")

plot(mod, what="uncertainty")

Predictions can also be made from MoEClust models: the response, probability of cluster membership, and the MAP classification can be predicted for the fitted data or for new data (in the form of new covariates and new response variables, or new covariates only). Let’s predict the response variable using the optimal model fit above to the CO2 data.

as.vector(predict(comp$optimal)$y)

#>  [1] 14.258797  3.901356 20.461833  9.057538  8.292203 14.981863  6.849704
#>  [8]  6.679846  9.695510 10.632816  9.451615  9.831188  6.255028  9.590701
#> [15]  7.237534  5.289565  9.782900  6.588562  9.531164 17.968480  8.514569
#> [22]  6.936316  6.725192  6.275709  5.546887  3.319349  9.910969 10.736908

Now let’s build a model on some of the CO2 data and retain the indices of the withheld observations:

ind <- sample(1:nrow(CO2data), 2)
res <- MoE_clust(CO2data[-ind,]$CO2, G=3, expert= ~ GNP, 
                 equalPro=TRUE, network.data=CO2data[-ind,])

Now we can make predictions on the withheld data, either by using the withheld covariates only, or by also using the withheld response variables. Note that newdata can be either a list with component(s) new.x (and optionally new.y) or a single matrix/data.frame with the appropriate columns.

# Using new covariates only
predict(res, newdata = CO2data[ind,], use.y = FALSE)[1:3]

# Using both new covariates & new response variables
predict(res, newdata = CO2data[ind,])[1:3]         

#> y :
#>         CO2
#> 1  7.054007
#> 2 10.383095
#> 
#> classification :
#> 1 2 
#> 1 1 
#> 
#> z :
#>    Cluster1  Cluster2  Cluster3
#> 1 0.3333333 0.3333333 0.3333333
#> 2 0.3333333 0.3333333 0.3333333
#> y :
#>         CO2
#> 1  8.602999
#> 2 10.202224
#> 
#> classification :
#> 1 2 
#> 2 3 
#> 
#> z :
#>       Cluster1    Cluster2  Cluster3
#> 1 2.882222e-07 0.549832047 0.4501677
#> 2 2.787651e-08 0.005682698 0.9943173

AIS Data

Load the Australian Institute of Sports data.

data(ais)
hema <- ais[,3:7]

Examine the various additional options around initialisation of the algorithm:

?MoE_control

Fit a parsimonious Gaussian mixture of experts MoEClust model to the hematological variables within the AIS data, supplying sex in the expert network and BMI in the gating network via formula interfaces. Include an additional noise component by specifying it’s prior mixing proportion tau0. Toggle between allowing the mixing proportion for the noise component depend on the gating concomitant or not via the noise.gate argument. This time, allow the printing of messages to the screen.

mod <- MoE_clust(hema, G=1:3, expert= ~ sex, gating= ~ BMI, 
                 network.data=ais, tau0=0.1, noise.gate=FALSE)

Visualise the results for the optimal model using a generalised pairs plot.

plot(mod, what="gpairs")

Replace the scatterplots in response vs. response panels with bivariate density contours and add corresponding parametric marginal density estimates along the diagonal. Note that this is liable to be slow for models with expert network covariates.

plot(mod, what="gpairs", response.type="density", show.dens=TRUE, buffer=0.1)

Visualise the clustering uncertainty for the optimal model using a generalised pairs plot.

plot(mod, what="gpairs", response.type="uncertainty")

Instead visualise the clustering uncertainty in the form of an ordered profile plot (type="barplot" can also be specified here).

plot(mod, what="uncertainty", type="profile")

Plot the BIC of the visited models.

Plot the gating network of the optimal model against the gating concomitant BMI. Note the flat horizontal line of grey circles corresponding to the noise component due to the specification of noise.gate=FALSE in the original function call.

plot(mod, what="gating", x.axis=ais$BMI, xlab="BMI")

For the optimal model, plot the log-likelihood vs. the number of EM iterations.

plot(mod, what="loglik")

Produce further visualisations for the Gaussian components with the aid of the lattice library.

require("lattice")
z <- factor(mod$classification[mod$classification > 0], 
            labels=paste0("Cluster", seq_len(mod$G)))
splom(~ hema | ais$sex, groups=z, xlab=NULL)

splom(~ hema | z, groups=ais$sex, xlab=NULL)

References

Murphy, K. and Murphy, T. B. (2020). Gaussian parsimonious clustering models with covariates and a noise component. Advances in Data Analysis and Classification, 14(2): 293–325. <doi:10.1007/s11634-019-00373-8>.

Fraley, C. and Raftery, A. E. (2002). Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association, 97(458): 611–631.

Scrucca L., Fop M., Murphy T. B. and Raftery A. E. (2016). mclust 5: clustering, classification and density estimation using Gaussian finite mixture models. The R Journal, 8(1): 289–317.