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