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 5.1) 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)
## Package 'mclust' version 5.1
## Type 'citation("mclust")' for citing this R package in publications.

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)
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
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             0.938

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 0.01474055
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)
summary(boot1, what = "se")
## ----------------------------------------------------------
## Bootstrap standard errors
## ----------------------------------------------------------
## Model                      = VVV 
## Num. of mixture components = 3 
## Replications               = 999 
## Type                       = nonparametric bootstrap 
## 
## Mixing probabilities:
##          1          2          3 
## 0.05309343 0.04753583 0.03907416 
## 
## Means:
##                 1         2        3
## glucose 0.9989911  3.692174 17.07895
## insulin 7.3967616 26.389773 74.46090
## sspg    7.3802168 32.877103 16.76334
## 
## Variances:
## [,,1]
##          glucose   insulin      sspg
## glucose 11.57170  50.78437  54.24866
## insulin 50.78437 471.12349 360.33105
## sspg    54.24866 360.33105 562.12023
## [,,2]
##           glucose   insulin      sspg
## glucose  65.25921  489.3145  500.0332
## insulin 489.31450 3707.7378 3427.8499
## sspg    500.03316 3427.8499 7300.0158
## [,,3]
##          glucose   insulin      sspg
## glucose 1118.441  5842.177  1721.268
## insulin 5842.177 36019.344 10573.329
## sspg    1721.268 10573.329  3076.612
summary(boot1, what = "ci")
## ----------------------------------------------------------
## Bootstrap 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%  0.4533405 0.1419648 0.1437303
## 97.5% 0.6611373 0.3269809 0.2906524
## 
## Means:
## [,,1]
##        glucose  insulin     sspg
## 2.5%  89.42882 344.1348 151.6999
## 97.5% 93.37312 374.1732 181.4697
## [,,2]
##         glucose  insulin     sspg
## 2.5%   98.76934 471.4933 260.3137
## 97.5% 113.48587 573.3321 386.6452
## [,,3]
##        glucose   insulin      sspg
## 2.5%  186.9106  893.0594  67.61361
## 97.5% 253.3291 1189.6889 129.91618
## 
## Variances:
## [,,1]
##        glucose  insulin     sspg
## 2.5%  38.39105 1220.120 1617.351
## 97.5% 84.19771 3041.229 3886.482
## [,,2]
##         glucose   insulin     sspg
## 2.5%   63.64167  1999.143 12561.21
## 97.5% 313.35916 17963.183 39614.25
## [,,3]
##        glucose   insulin      sspg
## 2.5%  3969.459  57416.47  1540.643
## 97.5% 8293.247 186260.91 12103.955
boot4 = MclustBootstrap(mod4)
summary(boot4, what = "se")
## ----------------------------------------------------------
## Bootstrap standard errors
## ----------------------------------------------------------
## Model                      = E 
## Num. of mixture components = 2 
## Replications               = 999 
## Type                       = nonparametric bootstrap 
## 
## Mixing probabilities:
##          1          2 
## 0.03850512 0.03850512 
## 
## Means:
##          1          2 
## 0.04477045 0.06796375 
## 
## Variances:
##          1          2 
## 0.02405869 0.02405869
summary(boot4, what = "ci")
## ----------------------------------------------------------
## Bootstrap confidence intervals
## ----------------------------------------------------------
## Model                      = E 
## Num. of mixture components = 2 
## Replications               = 999 
## Type                       = nonparametric bootstrap 
## Confidence level           = 0.95 
## 
## Mixing probabilities:
##               1         2
## 2.5%  0.5520673 0.2981968
## 97.5% 0.7018032 0.4479327
## 
## Means:
##              1        2
## 2.5%  4.281040 6.191263
## 97.5% 4.464921 6.448773
## 
## Variances:
##               1         2
## 2.5%  0.1398440 0.1398440
## 97.5% 0.2345903 0.2345903

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)