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 5.2) 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").
## Package 'mclust' version 5.2
## Type 'citation("mclust")' for citing this R package in publications.
Clustering
data(diabetes)
class = diabetes$class
table(class)
## class
## Chemical Normal Overt
## 36 76 33
X = diabetes[,-1]
head(X)
## glucose insulin sspg
## 1 80 356 124
## 2 97 289 117
## 3 105 319 143
## 4 90 356 199
## 5 90 323 240
## 6 86 381 157
BIC = mclustBIC(X)
plot(BIC)
## Best BIC values:
## VVV,3 VVE,3 EVE,4
## BIC -4760.091 -4775.53693 -4793.26143
## BIC diff 0.000 -15.44628 -33.17079
mod1 = Mclust(X, x = BIC)
summary(mod1, parameters = TRUE)
## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm
## ----------------------------------------------------
##
## Mclust VVV (ellipsoidal, varying volume, shape, and orientation) model with 3 components:
##
## log.likelihood n df BIC ICL
## -2307.883 145 29 -4760.091 -4776.086
##
## Clustering table:
## 1 2 3
## 82 33 30
##
## Mixing probabilities:
## 1 2 3
## 0.5603211 0.2244432 0.2152356
##
## Means:
## [,1] [,2] [,3]
## glucose 91.39558 105.1109 219.21971
## insulin 358.61206 516.2814 1040.59177
## sspg 166.02012 320.2471 98.56807
##
## Variances:
## [,,1]
## glucose insulin sspg
## glucose 61.81664 97.41582 34.42346
## insulin 97.41582 2106.98136 378.95467
## sspg 34.42346 378.95467 2669.14406
## [,,2]
## glucose insulin sspg
## glucose 152.2496 789.1576 -483.0501
## insulin 789.1576 6476.1400 -2752.2840
## sspg -483.0501 -2752.2840 26029.0307
## [,,3]
## glucose insulin sspg
## glucose 6350.858 26190.11 -4448.25
## insulin 26190.111 122126.21 -22772.10
## sspg -4448.250 -22772.10 5913.76
plot(mod1, what = "classification")
table(class, mod1$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
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 0.937
Classification
EDDA
data(iris)
class = iris$Species
table(class)
## class
## setosa versicolor virginica
## 50 50 50
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## 1 5.1 3.5 1.4 0.2
## 2 4.9 3.0 1.4 0.2
## 3 4.7 3.2 1.3 0.2
## 4 4.6 3.1 1.5 0.2
## 5 5.0 3.6 1.4 0.2
## 6 5.4 3.9 1.7 0.4
mod2 = MclustDA(X, class, modelType = "EDDA")
summary(mod2)
## ------------------------------------------------
## Gaussian finite mixture model for classification
## ------------------------------------------------
##
## EDDA model summary:
##
## log.likelihood n df BIC
## -187.7097 150 36 -555.8024
##
## Classes n Model G
## setosa 50 VEV 1
## versicolor 50 VEV 1
## virginica 50 VEV 1
##
## Training classification summary:
##
## Predicted
## Class setosa versicolor virginica
## setosa 50 0 0
## versicolor 0 47 3
## virginica 0 0 50
##
## Training error = 0.02
plot(mod2, what = "scatterplot")
plot(mod2, what = "classification")
MclustDA
data(banknote)
class = banknote$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
## 0.0200000 0.0101835
unlist(cvMclustDA(mod3, nfold = 10)[2:3])
## error se
## 0 0
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
## 0.05044120 0.04609791 0.03861612
##
## Means:
## 1 2 3
## glucose 0.9814553 3.810418 17.52635
## insulin 7.2242640 26.377341 77.14079
## sspg 7.2672391 33.470876 18.01375
##
## Variances:
## [,,1]
## glucose insulin sspg
## glucose 11.18394 51.25614 54.12083
## insulin 51.25614 474.26446 360.32711
## sspg 54.12083 360.32711 593.38836
## [,,2]
## glucose insulin sspg
## glucose 68.04309 505.0524 510.2231
## insulin 505.05239 3832.2323 3531.4074
## sspg 510.22312 3531.4074 7313.1560
## [,,3]
## glucose insulin sspg
## glucose 1124.859 5998.939 1757.518
## insulin 5998.939 37341.597 10916.903
## sspg 1757.518 10916.903 3156.220
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% 0.4553951 0.1432279 0.1392891
## 97.5% 0.6500130 0.3280024 0.2941657
##
## Means:
## [,,1]
## glucose insulin sspg
## 2.5% 89.54115 345.4198 152.5297
## 97.5% 93.28483 373.3272 180.6123
## [,,2]
## glucose insulin sspg
## 2.5% 98.75531 473.2203 256.6425
## 97.5% 114.04552 581.5940 389.0336
## [,,3]
## glucose insulin sspg
## 2.5% 184.7340 896.6154 66.0608
## 97.5% 255.4083 1194.7110 135.6818
##
## Variances:
## [,,1]
## glucose insulin sspg
## 2.5% 40.23261 1182.801 1651.989
## 97.5% 83.38416 2992.432 3962.934
## [,,2]
## glucose insulin sspg
## 2.5% 64.34646 2130.144 13122.82
## 97.5% 357.79201 18440.141 41237.96
## [,,3]
## glucose insulin sspg
## 2.5% 3735.010 53033.88 1488.352
## 97.5% 8152.571 194695.18 12332.496
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
## 0.04118511 0.04118511
##
## Means:
## 1 2
## 0.04591982 0.06642681
##
## Variances:
## 1 2
## 0.02387204 0.02387204
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% 0.5438347 0.2983043
## 97.5% 0.7016957 0.4561653
##
## Means:
## 1 2
## 2.5% 4.286159 6.185099
## 97.5% 4.461763 6.450079
##
## Variances:
## 1 2
## 2.5% 0.1432436 0.1432436
## 97.5% 0.2366966 0.2366966
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)
