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 5.2.2) 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 5.2.2
## Type 'citation("mclust")' for citing this R package in publications.
data(diabetes)
class = diabetes$class
table(class)
## class
## Chemical Normal Overt
## 36 76 33
X = diabetes[,-1]
head(X)
## glucose insulin sspg
## 1 80 356 124
## 2 97 289 117
## 3 105 319 143
## 4 90 356 199
## 5 90 323 240
## 6 86 381 157
clPairs(X, class)
BIC = mclustBIC(X)
plot(BIC)
summary(BIC)
## Best BIC values:
## VVV,3 VVE,3 EVE,4
## BIC -4760.091 -4775.53693 -4793.26143
## BIC diff 0.000 -15.44628 -33.17079
mod1 = Mclust(X, x = BIC)
summary(mod1, parameters = TRUE)
## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm
## ----------------------------------------------------
##
## Mclust VVV (ellipsoidal, varying volume, shape, and orientation) model with 3 components:
##
## log.likelihood n df BIC ICL
## -2307.883 145 29 -4760.091 -4776.086
##
## Clustering table:
## 1 2 3
## 82 33 30
##
## Mixing probabilities:
## 1 2 3
## 0.5603211 0.2244432 0.2152356
##
## Means:
## [,1] [,2] [,3]
## glucose 91.39558 105.1109 219.21971
## insulin 358.61206 516.2814 1040.59177
## sspg 166.02012 320.2471 98.56807
##
## Variances:
## [,,1]
## glucose insulin sspg
## glucose 61.81664 97.41582 34.42346
## insulin 97.41582 2106.98136 378.95467
## sspg 34.42346 378.95467 2669.14406
## [,,2]
## glucose insulin sspg
## glucose 152.2496 789.1576 -483.0501
## insulin 789.1576 6476.1400 -2752.2840
## sspg -483.0501 -2752.2840 26029.0307
## [,,3]
## glucose insulin sspg
## glucose 6350.858 26190.11 -4448.25
## insulin 26190.111 122126.21 -22772.10
## sspg -4448.250 -22772.10 5913.76
plot(mod1, what = "classification")
table(class, mod1$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 0.949
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
## 0.02000000 0.01422916
unlist(cvMclustDA(mod3, nfold = 10)[2:3])
## error se
## 0 0
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
## 0.05102386 0.04496594 0.03845737
##
## Means:
## 1 2 3
## glucose 1.010982 3.593004 17.66879
## insulin 7.096809 25.821106 76.51611
## sspg 7.287973 33.683088 17.38803
##
## Variances:
## [,,1]
## glucose insulin sspg
## glucose 11.18237 50.24017 52.89064
## insulin 50.24017 471.04816 344.31006
## sspg 52.89064 344.31006 538.77053
## [,,2]
## glucose insulin sspg
## glucose 68.26909 512.317 492.0601
## insulin 512.31699 3863.114 3488.0661
## sspg 492.06008 3488.066 7080.8828
## [,,3]
## glucose insulin sspg
## glucose 1128.046 6071.06 1795.550
## insulin 6071.060 38301.72 11222.865
## sspg 1795.550 11222.87 3251.302
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% 0.4645613 0.1423399 0.1427335
## 97.5% 0.6594747 0.3206742 0.2911989
##
## Means:
## [,,1]
## glucose insulin sspg
## 2.5% 89.32221 344.8416 152.452
## 97.5% 93.49057 371.8961 182.519
## [,,2]
## glucose insulin sspg
## 2.5% 99.32398 474.4882 254.9461
## 97.5% 114.16576 585.6604 390.9073
## [,,3]
## glucose insulin sspg
## 2.5% 186.7694 899.5266 67.23241
## 97.5% 259.0817 1194.0774 132.92073
##
## Variances:
## [,,1]
## glucose insulin sspg
## 2.5% 40.54956 1237.822 1684.985
## 97.5% 83.99202 3021.035 3867.211
## [,,2]
## glucose insulin sspg
## 2.5% 64.92173 2148.504 12667.61
## 97.5% 359.03199 18705.313 40043.75
## [,,3]
## glucose insulin sspg
## 2.5% 3907.581 55081.73 1548.502
## 97.5% 8170.624 196995.19 12712.059
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
## 0.03929136 0.03929136
##
## Means:
## 1 2
## 0.04637595 0.06784880
##
## Variances:
## 1 2
## 0.02390473 0.02390473
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% 0.5500662 0.2970760
## 97.5% 0.7029240 0.4499338
##
## Means:
## 1 2
## 2.5% 4.284893 6.187829
## 97.5% 4.474834 6.446708
##
## Variances:
## 1 2
## 2.5% 0.1394434 0.1394434
## 97.5% 0.2348289 0.2348289
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)