A quick tour of mclust

Luca Scrucca

19 May 2017

Introduction

mclust is a contributed R package for model-based clustering, classification, and density estimation based on finite normal mixture modelling. It provides functions for parameter estimation via the EM algorithm for normal mixture models with a variety of covariance structures, and functions for simulation from these models. Also included are functions that combine model-based hierarchical clustering, EM for mixture estimation and the Bayesian Information Criterion (BIC) in comprehensive strategies for clustering, density estimation and discriminant analysis. Additional functionalities are available for displaying and visualizing fitted models along with clustering, classification, and density estimation results.

This document gives a quick tour of mclust (version 5.3) functionalities. It was written in R Markdown, using the knitr package for production. See help(package="mclust") for further details and references provided by citation("mclust").

library(mclust)

Clustering

data(diabetes)
class = diabetes$class
table(class)
## class
## Chemical   Normal    Overt 
##       36       76       33
X = diabetes[,-1]
head(X)
##   glucose insulin sspg
## 1      80     356  124
## 2      97     289  117
## 3     105     319  143
## 4      90     356  199
## 5      90     323  240
## 6      86     381  157
clPairs(X, class)


BIC = mclustBIC(X)
plot(BIC)

summary(BIC)
## Best BIC values:
##              VVV,3       VVE,3       EVE,4
## BIC      -4760.091 -4775.53693 -4793.26143
## BIC diff     0.000   -15.44628   -33.17079

mod1 = Mclust(X, x = BIC)
summary(mod1, parameters = TRUE)
## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm 
## ----------------------------------------------------
## 
## Mclust VVV (ellipsoidal, varying volume, shape, and orientation) model with 3 components:
## 
##  log.likelihood   n df       BIC       ICL
##       -2307.883 145 29 -4760.091 -4776.086
## 
## Clustering table:
##  1  2  3 
## 82 33 30 
## 
## Mixing probabilities:
##         1         2         3 
## 0.5603211 0.2244432 0.2152356 
## 
## Means:
##              [,1]     [,2]       [,3]
## glucose  91.39558 105.1109  219.21971
## insulin 358.61206 516.2814 1040.59177
## sspg    166.02012 320.2471   98.56807
## 
## Variances:
## [,,1]
##          glucose    insulin       sspg
## glucose 61.81664   97.41582   34.42346
## insulin 97.41582 2106.98136  378.95467
## sspg    34.42346  378.95467 2669.14406
## [,,2]
##           glucose    insulin       sspg
## glucose  152.2496   789.1576  -483.0501
## insulin  789.1576  6476.1400 -2752.2840
## sspg    -483.0501 -2752.2840 26029.0307
## [,,3]
##           glucose   insulin      sspg
## glucose  6350.858  26190.11  -4448.25
## insulin 26190.111 122126.21 -22772.10
## sspg    -4448.250 -22772.10   5913.76

plot(mod1, what = "classification")

table(class, mod1$classification)
##           
## class       1  2  3
##   Chemical  8 26  2
##   Normal   74  2  0
##   Overt     0  5 28

par(mfrow = c(2,2))
plot(mod1, what = "uncertainty", dimens = c(2,1), main = "")
plot(mod1, what = "uncertainty", dimens = c(3,1), main = "")
plot(mod1, what = "uncertainty", dimens = c(2,3), main = "")
par(mfrow = c(1,1))


ICL = mclustICL(X)
summary(ICL)
## Best ICL values:
##              VVV,3       VVE,3       EVE,4
## ICL      -4776.086 -4793.27143 -4809.16868
## ICL diff     0.000   -17.18553   -33.08278
plot(ICL)


LRT = mclustBootstrapLRT(X, modelName = "VVV")
LRT
## Bootstrap sequential LRT for the number of mixture components
## -------------------------------------------------------------
## Model        = VVV 
## Replications = 999 
##                LRTS bootstrap p-value
## 1 vs 2   361.186445             0.001
## 2 vs 3   114.703559             0.001
## 3 vs 4     7.437806             0.925

Classification

EDDA

data(iris)
class = iris$Species
table(class)
## class
##     setosa versicolor  virginica 
##         50         50         50
X = iris[,1:4]
head(X)
##   Sepal.Length Sepal.Width Petal.Length Petal.Width
## 1          5.1         3.5          1.4         0.2
## 2          4.9         3.0          1.4         0.2
## 3          4.7         3.2          1.3         0.2
## 4          4.6         3.1          1.5         0.2
## 5          5.0         3.6          1.4         0.2
## 6          5.4         3.9          1.7         0.4
mod2 = MclustDA(X, class, modelType = "EDDA")
summary(mod2)
## ------------------------------------------------
## Gaussian finite mixture model for classification 
## ------------------------------------------------
## 
## EDDA model summary:
## 
##  log.likelihood   n df       BIC
##       -187.7097 150 36 -555.8024
##             
## Classes       n Model G
##   setosa     50   VEV 1
##   versicolor 50   VEV 1
##   virginica  50   VEV 1
## 
## Training classification summary:
## 
##             Predicted
## Class        setosa versicolor virginica
##   setosa         50          0         0
##   versicolor      0         47         3
##   virginica       0          0        50
## 
## Training error = 0.02
plot(mod2, what = "scatterplot")

plot(mod2, what = "classification")

MclustDA

data(banknote)
class = banknote$Status
table(class)
## class
## counterfeit     genuine 
##         100         100
X = banknote[,-1]
head(X)
##   Length  Left Right Bottom  Top Diagonal
## 1  214.8 131.0 131.1    9.0  9.7    141.0
## 2  214.6 129.7 129.7    8.1  9.5    141.7
## 3  214.8 129.7 129.7    8.7  9.6    142.2
## 4  214.8 129.7 129.6    7.5 10.4    142.0
## 5  215.0 129.6 129.7   10.4  7.7    141.8
## 6  215.7 130.8 130.5    9.0 10.1    141.4
mod3 = MclustDA(X, class)
summary(mod3)
## ------------------------------------------------
## Gaussian finite mixture model for classification 
## ------------------------------------------------
## 
## MclustDA model summary:
## 
##  log.likelihood   n df       BIC
##       -646.0798 200 66 -1641.849
##              
## Classes         n Model G
##   counterfeit 100   EVE 2
##   genuine     100   XXX 1
## 
## Training classification summary:
## 
##              Predicted
## Class         counterfeit genuine
##   counterfeit         100       0
##   genuine               0     100
## 
## Training error = 0
plot(mod3, what = "scatterplot")

plot(mod3, what = "classification")

Cross-validation error

unlist(cvMclustDA(mod2, nfold = 10)[2:3])
##      error         se 
## 0.02666667 0.01088662
unlist(cvMclustDA(mod3, nfold = 10)[2:3])
## error    se 
## 0.005 0.005

Density estimation

Univariate

data(acidity)
mod4 = densityMclust(acidity)
summary(mod4)
## -------------------------------------------------------
## Density estimation via Gaussian finite mixture modeling 
## -------------------------------------------------------
## 
## Mclust E (univariate, equal variance) model with 2 components:
## 
##  log.likelihood   n df       BIC       ICL
##       -185.9493 155  4 -392.0723 -398.5554
## 
## Clustering table:
##  1  2 
## 98 57
plot(mod4, what = "BIC")

plot(mod4, what = "density", data = acidity, breaks = 15)

plot(mod4, what = "diagnostic", type = "cdf")

plot(mod4, what = "diagnostic", type = "qq")

Multivariate

data(faithful)
mod5 = densityMclust(faithful)
summary(mod5)
## -------------------------------------------------------
## Density estimation via Gaussian finite mixture modeling 
## -------------------------------------------------------
## 
## Mclust EEE (ellipsoidal, equal volume, shape and orientation) model with 3 components:
## 
##  log.likelihood   n df       BIC       ICL
##       -1126.361 272 11 -2314.386 -2360.865
## 
## Clustering table:
##   1   2   3 
## 130  97  45
plot(mod5, what = "BIC")

plot(mod5, what = "density")

plot(mod5, what = "density", type = "image", col = "dodgerblue3", grid = 100)

plot(mod5, what = "density", type = "persp")

Bootstrap inference

boot1 = MclustBootstrap(mod1, nboot = 999, type = "bs")
summary(boot1, what = "se")
## ----------------------------------------------------------
## Resampling standard errors 
## ----------------------------------------------------------
## Model                      = VVV 
## Num. of mixture components = 3 
## Replications               = 999 
## Type                       = nonparametric bootstrap 
## 
## Mixing probabilities:
##          1          2          3 
## 0.05120824 0.04508324 0.03806893 
## 
## Means:
##                 1        2        3
## glucose 0.9945279  3.43943 17.08839
## insulin 7.2001106 24.33858 76.34847
## sspg    7.0504801 32.38694 17.41000
## 
## Variances:
## [,,1]
##          glucose   insulin      sspg
## glucose 11.44945  51.21326  50.58981
## insulin 51.21326 480.89378 344.97604
## sspg    50.58981 344.97604 535.16081
## [,,2]
##           glucose  insulin      sspg
## glucose  61.86552  465.327  470.7474
## insulin 465.32701 3500.540 3281.7808
## sspg    470.74740 3281.781 6842.7519
## [,,3]
##          glucose   insulin      sspg
## glucose 1091.339  5851.474  1729.598
## insulin 5851.474 36739.917 10812.417
## sspg    1729.598 10812.417  3178.887
summary(boot1, what = "ci")
## ----------------------------------------------------------
## Resampling confidence intervals 
## ----------------------------------------------------------
## Model                      = VVV 
## Num. of mixture components = 3 
## Replications               = 999 
## Type                       = nonparametric bootstrap 
## Confidence level           = 0.95 
## 
## Mixing probabilities:
##               1         2         3
## 2.5%  0.4577087 0.1444094 0.1459768
## 97.5% 0.6600427 0.3190719 0.2862708
## 
## Means:
## [,,1]
##        glucose  insulin     sspg
## 2.5%  89.28151 343.8986 153.3424
## 97.5% 93.24221 372.1487 181.0177
## [,,2]
##         glucose  insulin     sspg
## 2.5%   98.98576 476.3710 262.7668
## 97.5% 112.84124 568.4816 384.8814
## [,,3]
##        glucose  insulin    sspg
## 2.5%  186.0369  892.382  67.654
## 97.5% 253.0063 1197.211 134.809
## 
## Variances:
## [,,1]
##        glucose  insulin     sspg
## 2.5%  39.20559 1242.131 1656.011
## 97.5% 84.47626 3105.682 3760.276
## [,,2]
##         glucose   insulin     sspg
## 2.5%   66.48473  1888.358 12567.49
## 97.5% 310.32133 16468.899 39481.98
## [,,3]
##        glucose  insulin      sspg
## 2.5%  4001.842  57628.5  1579.465
## 97.5% 8290.156 192388.8 12097.907

boot4 = MclustBootstrap(mod4, nboot = 999, type = "bs")
summary(boot4, what = "se")
## ----------------------------------------------------------
## Resampling standard errors 
## ----------------------------------------------------------
## Model                      = E 
## Num. of mixture components = 2 
## Replications               = 999 
## Type                       = nonparametric bootstrap 
## 
## Mixing probabilities:
##         1         2 
## 0.0408117 0.0408117 
## 
## Means:
##          1          2 
## 0.04512188 0.06994303 
## 
## Variances:
##          1          2 
## 0.02343116 0.02343116
summary(boot4, what = "ci")
## ----------------------------------------------------------
## Resampling confidence intervals 
## ----------------------------------------------------------
## Model                      = E 
## Num. of mixture components = 2 
## Replications               = 999 
## Type                       = nonparametric bootstrap 
## Confidence level           = 0.95 
## 
## Mixing probabilities:
##               1         2
## 2.5%  0.5424041 0.2999348
## 97.5% 0.7000652 0.4575959
## 
## Means:
##              1        2
## 2.5%  4.285316 6.179847
## 97.5% 4.459470 6.454325
## 
## Variances:
##               1         2
## 2.5%  0.1419211 0.1419211
## 97.5% 0.2290251 0.2290251

Dimension reduction

Clustering

mod1dr = MclustDR(mod1)
summary(mod1dr)
## -----------------------------------------------------------------
## Dimension reduction for model-based clustering and classification 
## -----------------------------------------------------------------
## 
## Mixture model type: Mclust (VVV, 3)
##         
## Clusters  n
##        1 82
##        2 33
##        3 30
## 
## Estimated basis vectors:
##              Dir1     Dir2       Dir3
## glucose -0.986054  0.24922  0.9588647
## insulin  0.157645 -0.11513 -0.2837395
## sspg    -0.053353 -0.96158 -0.0083946
## 
##                Dir1     Dir2      Dir3
## Eigenvalues  1.3749  0.77725   0.65829
## Cum. %      48.9207 76.57662 100.00000
plot(mod1dr, what = "pairs")

plot(mod1dr, what = "boundaries", ngrid = 200)


mod1dr = MclustDR(mod1, lambda = 1)
summary(mod1dr)
## -----------------------------------------------------------------
## Dimension reduction for model-based clustering and classification 
## -----------------------------------------------------------------
## 
## Mixture model type: Mclust (VVV, 3)
##         
## Clusters  n
##        1 82
##        2 33
##        3 30
## 
## Estimated basis vectors:
##             Dir1     Dir2
## glucose  0.81116  0.92578
## insulin -0.56210 -0.19371
## sspg    -0.16147 -0.32467
## 
##                Dir1     Dir2
## Eigenvalues  1.0574   0.3968
## Cum. %      72.7144 100.0000
plot(mod1dr, what = "scatterplot")

plot(mod1dr, what = "boundaries", ngrid = 200)

Classification

mod2dr = MclustDR(mod2)
summary(mod2dr)
## -----------------------------------------------------------------
## Dimension reduction for model-based clustering and classification 
## -----------------------------------------------------------------
## 
## Mixture model type: EDDA 
##             
## Classes       n Model G
##   setosa     50   VEV 1
##   versicolor 50   VEV 1
##   virginica  50   VEV 1
## 
## Estimated basis vectors:
##                  Dir1      Dir2     Dir3     Dir4
## Sepal.Length  0.17425 -0.193663  0.64081 -0.46231
## Sepal.Width   0.45292  0.066561  0.34852  0.57110
## Petal.Length -0.61629 -0.311030 -0.42366  0.46256
## Petal.Width  -0.62024  0.928076  0.53703 -0.49613
## 
##                 Dir1     Dir2      Dir3       Dir4
## Eigenvalues  0.94747  0.68835  0.076141   0.052607
## Cum. %      53.69408 92.70374 97.018700 100.000000
plot(mod2dr, what = "scatterplot")

plot(mod2dr, what = "boundaries", ngrid = 200)


mod3dr = MclustDR(mod3)
summary(mod3dr)
## -----------------------------------------------------------------
## Dimension reduction for model-based clustering and classification 
## -----------------------------------------------------------------
## 
## Mixture model type: MclustDA 
##              
## Classes         n Model G
##   counterfeit 100   EVE 2
##   genuine     100   XXX 1
## 
## Estimated basis vectors:
##              Dir1      Dir2     Dir3      Dir4      Dir5      Dir6
## Length   -0.10027 -0.327553  0.79718 -0.033721 -0.317043  0.084618
## Left     -0.21760 -0.305350 -0.30266 -0.893676  0.371043 -0.565611
## Right     0.29180 -0.018877 -0.49600  0.406605 -0.861020  0.481331
## Bottom    0.57603  0.445501  0.12002 -0.034570  0.004359 -0.078688
## Top       0.57555  0.385645  0.10093 -0.103629  0.136005  0.625416
## Diagonal -0.44088  0.672251 -0.04781 -0.151473 -0.044035  0.209542
## 
##                 Dir1     Dir2     Dir3     Dir4      Dir5       Dir6
## Eigenvalues  0.87241  0.55372  0.48603  0.13301  0.053113   0.027239
## Cum. %      41.04429 67.09530 89.96182 96.21965 98.718473 100.000000
plot(mod3dr, what = "scatterplot")

plot(mod3dr, what = "boundaries", ngrid = 200)