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)
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
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 0.01088662
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, 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
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)