Author: Luca Scrucca
Date: 2025-10-31

# A quick tour of mclust

#### Luca Scrucca

#### 31 Oct 2025

# 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 6.1.2) 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)
    ##                    __           __
    ##    ____ ___  _____/ /_  _______/ /_
    ##   / __ `__ \/ ___/ / / / / ___/ __/
    ##  / / / / / / /__/ / /_/ (__  ) /_
    ## /_/ /_/ /_/\___/_/\__,_/____/\__/   version 6.1.2
    ## Type 'citation("mclust")' for citing this R package in publications.

# Clustering

    data(diabetes)
    #-- data corrections (see help("diabetes", package = "mclust")) --#
    names(diabetes) <- c("class", "fpg", "glucose", "insulin")
    diabetes$glucose[104] <- 455
    #-----------------------------------------------------------------#
    class <- diabetes$class
    table(class)
    ## class
    ## Chemical   Normal    Overt
    ##       36       76       33
    X <- diabetes[,-1]
    head(X)
    ##   fpg glucose insulin
    ## 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,5       VVE,6
    ## BIC      -4734.561 -4750.75832 -4761.35626
    ## BIC diff     0.000   -16.19685   -26.79479

    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
    ##       -2295.118 145 29 -4734.561 -4749.16
    ##
    ## Clustering table:
    ##  1  2  3
    ## 82 35 28
    ##
    ## Mixing probabilities:
    ##         1         2         3
    ## 0.5553630 0.2479432 0.1966939
    ##
    ## Means:
    ##              [,1]     [,2]       [,3]
    ## fpg      91.35843 104.6401  230.32908
    ## glucose 357.86717 515.7332 1103.21235
    ## insulin 165.85059 314.5807   81.40487
    ##
    ## Variances:
    ## [,,1]
    ##              fpg   glucose    insulin
    ## fpg     61.41309   93.8443   33.84212
    ## glucose 93.84430 2051.4193  365.21412
    ## insulin 33.84212  365.2141 2643.75977
    ## [,,2]
    ##               fpg    glucose    insulin
    ## fpg      194.3494   940.1594  -546.2879
    ## glucose  940.1594  7271.8069 -2775.9918
    ## insulin -546.2879 -2775.9918 24757.4385
    ## [,,3]
    ##               fpg   glucose    insulin
    ## fpg      5451.219  19936.61  -2473.121
    ## glucose 19936.610  80705.15 -10272.620
    ## insulin -2473.121 -10272.62   2243.084

    plot(mod1, what = "classification")

    table(class, mod1$classification)
    ##
    ## class       1  2  3
    ##   Chemical  8 27  1
    ##   Normal   74  2  0
    ##   Overt     0  6 27

    plot(mod1, what = "uncertainty")

    ICL <- mclustICL(X)
    summary(ICL)
    ## Best ICL values:
    ##             VVV,3       VVE,4       EVE,5
    ## ICL      -4749.16 -4778.76438 -4781.49369
    ## ICL diff     0.00   -29.60428   -32.33359
    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   369.98537             0.001
    ## 2 vs 3   121.10652             0.001
    ## 3 vs 4    19.90071             0.283

## Initialisation

EM algorithm is used by mclust for maximum likelihood estimation. Initialisation of EM is performed using the partitions obtained from agglomerative hierarchical clustering. For details see `help(mclustBIC)` or `help(Mclust)`, and `help(hc)`.

    (hc1 <- hc(X, modelName = "VVV", use = "SVD"))
    ## Call:
    ## hc(data = X, modelName = "VVV", use = "SVD")
    ##
    ## Model-Based Agglomerative Hierarchical Clustering
    ## Model name        = VVV
    ## Use               = SVD
    ## Number of objects = 145
    BIC1 <- mclustBIC(X, initialization = list(hcPairs = hc1)) # default
    summary(BIC1)
    ## Best BIC values:
    ##              VVV,3       VVE,5       VVE,6
    ## BIC      -4734.561 -4750.75832 -4761.35626
    ## BIC diff     0.000   -16.19685   -26.79479

    (hc2 <- hc(X, modelName = "VVV", use = "VARS"))
    ## Call:
    ## hc(data = X, modelName = "VVV", use = "VARS")
    ##
    ## Model-Based Agglomerative Hierarchical Clustering
    ## Model name        = VVV
    ## Use               = VARS
    ## Number of objects = 145
    BIC2 <- mclustBIC(X, initialization = list(hcPairs = hc2))
    summary(BIC2)
    ## Best BIC values:
    ##             VVV,3       VVE,3      EVE,5
    ## BIC      -4734.53 -4759.36113 -4765.3420
    ## BIC diff     0.00   -24.83078   -30.8116

    (hc3 <- hc(X, modelName = "EEE", use = "SVD"))
    ## Call:
    ## hc(data = X, modelName = "EEE", use = "SVD")
    ##
    ## Model-Based Agglomerative Hierarchical Clustering
    ## Model name        = EEE
    ## Use               = SVD
    ## Number of objects = 145
    BIC3 <- mclustBIC(X, initialization = list(hcPairs = hc3))
    summary(BIC3)
    ## Best BIC values:
    ##             VVV,3        VVE,4       VVV,4
    ## BIC      -4734.53 -4740.185182 -4755.40561
    ## BIC diff     0.00    -5.654734   -20.87517

Update BIC by merging the best results:

    BIC <- mclustBICupdate(BIC1, BIC2, BIC3)
    summary(BIC)
    ## Best BIC values:
    ##             VVV,3        VVE,4       VVE,5
    ## BIC      -4734.53 -4740.185182 -4750.75832
    ## BIC diff     0.00    -5.654828   -16.22797
    plot(BIC)

Univariate fit using random starting points obtained by creating random agglomerations (see `help(hcRandomPairs)`) and merging best results:

    data(galaxies, package = "MASS")
    galaxies <- galaxies / 1000
    BIC <- NULL
    for(j in 1:20)
    {
      rBIC <- mclustBIC(galaxies, verbose = FALSE,
                        initialization = list(hcPairs = hcRandomPairs(galaxies)))
      BIC <- mclustBICupdate(BIC, rBIC)
    }
    summary(BIC)
    ## Best BIC values:
    ##                V,3         V,4        V,5
    ## BIC      -441.6122 -443.399746 -446.34966
    ## BIC diff    0.0000   -1.787536   -4.73745
    plot(BIC)

    mod <- Mclust(galaxies, x = BIC)
    summary(mod)
    ## ----------------------------------------------------
    ## Gaussian finite mixture model fitted by EM algorithm
    ## ----------------------------------------------------
    ##
    ## Mclust V (univariate, unequal variance) model with 3 components:
    ##
    ##  log-likelihood  n df       BIC       ICL
    ##       -203.1792 82  8 -441.6122 -441.6126
    ##
    ## Clustering table:
    ##  1  2  3
    ##  3  7 72

# 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
    ## ------------------------------------------------
    ##  log-likelihood   n df       BIC
    ##       -187.7097 150 38 -565.8236
    ##
    ## Classes       n     % Model G
    ##   setosa     50 33.33   VEV 1
    ##   versicolor 50 33.33   VEV 1
    ##   virginica  50 33.33   VEV 1
    ##
    ## Training confusion matrix
    ## ------------------------------------------------
    ##             Predicted
    ## Classes      setosa versicolor virginica
    ##   setosa         50          0         0
    ##   versicolor      0         47         3
    ##   virginica       0          0        50
    ##
    ## Classification error = 0.0200
    ## Brier score          = 0.0127
    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
    ## ------------------------------------------------
    ##  log-likelihood   n df       BIC
    ##       -646.0801 200 67 -1647.147
    ##
    ## Classes         n  % Model G
    ##   counterfeit 100 50   EVE 2
    ##   genuine     100 50   XXX 1
    ##
    ## Training confusion matrix
    ## ------------------------------------------------
    ##              Predicted
    ## Classes       counterfeit genuine
    ##   counterfeit         100       0
    ##   genuine               0     100
    ##
    ## Classification error = 0.0000
    ## Brier score          = 0.0000
    plot(mod3, what = "scatterplot")

    plot(mod3, what = "classification")

## Cross-validation error

    cv <- cvMclustDA(mod2, nfold = 10)
    str(cv)
    ## List of 6
    ##  $ classification: Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...
    ##  $ z             : num [1:150, 1:3] 1 1 1 1 1 1 1 1 1 1 ...
    ##   ..- attr(*, "dimnames")=List of 2
    ##   .. ..$ : NULL
    ##   .. ..$ : chr [1:3] "setosa" "versicolor" "virginica"
    ##  $ ce            : num 0.0267
    ##  $ se.ce         : num 0.0109
    ##  $ brier         : num 0.0208
    ##  $ se.brier      : num 0.00738
    unlist(cv[3:6])
    ##          ce       se.ce       brier    se.brier
    ## 0.026666667 0.010886621 0.020795887 0.007383247
    cv <- cvMclustDA(mod3, nfold = 10)
    str(cv)
    ## List of 6
    ##  $ classification: Factor w/ 2 levels "counterfeit",..: 2 2 2 2 2 2 2 2 2 2 ...
    ##  $ z             : num [1:200, 1:2] 1.56e-06 3.50e-19 5.41e-28 3.33e-20 2.42e-29 ...
    ##   ..- attr(*, "dimnames")=List of 2
    ##   .. ..$ : NULL
    ##   .. ..$ : chr [1:2] "counterfeit" "genuine"
    ##  $ ce            : num 0.005
    ##  $ se.ce         : num 0.005
    ##  $ brier         : num 0.00514
    ##  $ se.brier      : num 0.00498
    unlist(cv[3:6])
    ##          ce       se.ce       brier    se.brier
    ## 0.005000000 0.005000000 0.005135796 0.004980123

# 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
    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
    ##       -1126.326 272 11 -2314.316 -2357.824
    plot(mod5, what = "BIC")

    plot(mod5, what = "density", type = "hdr", data = faithful, points.cex = 0.5)

    plot(mod5, what = "density", type = "persp")

# 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
    ## 0.04962408 0.04679956 0.03568767
    ##
    ## Means:
    ##                1         2        3
    ## fpg     1.029210  3.431536 16.45148
    ## glucose 7.348229 23.102112 63.57099
    ## insulin 6.890494 32.494958 10.35781
    ##
    ## Variances:
    ## [,,1]
    ##              fpg   glucose   insulin
    ## fpg     11.22247  51.32446  54.33822
    ## glucose 51.32446 491.07894 370.15616
    ## insulin 54.33822 370.15616 559.97118
    ## [,,2]
    ##              fpg   glucose   insulin
    ## fpg      63.0556  414.6464  448.4747
    ## glucose 414.6464 3039.4632 3016.3035
    ## insulin 448.4747 3016.3035 7049.5005
    ## [,,3]
    ##              fpg   glucose   insulin
    ## fpg     1003.569  3970.504  680.3030
    ## glucose 3970.504 17898.128 2492.8805
    ## insulin  680.303  2492.881  588.4059
    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%  0.4609834 0.1469903 0.1311157
    ## 97.5% 0.6607129 0.3354713 0.2701468
    ##
    ## Means:
    ## [,,1]
    ##            fpg  glucose  insulin
    ## 2.5%  89.41502 344.5513 153.5648
    ## 97.5% 93.43859 373.0637 180.4059
    ## [,,2]
    ##             fpg  glucose  insulin
    ## 2.5%   98.15373 475.4215 258.5614
    ## 97.5% 111.47178 567.0759 386.4861
    ## [,,3]
    ##            fpg  glucose   insulin
    ## 2.5%  198.4823  977.025  63.04963
    ## 97.5% 263.9015 1228.576 103.91638
    ##
    ## Variances:
    ## [,,1]
    ##            fpg  glucose  insulin
    ## 2.5%  39.89112 1208.988 1722.407
    ## 97.5% 83.56151 3139.945 3959.596
    ## [,,2]
    ##             fpg   glucose  insulin
    ## 2.5%   89.47421  2338.032 13035.79
    ## 97.5% 336.72367 14195.303 39635.30
    ## [,,3]
    ##            fpg   glucose  insulin
    ## 2.5%  3347.581  47557.08 1320.557
    ## 97.5% 7267.330 117148.25 3615.588

    plot(boot1, what = "pro")
    plot(boot1, what = "mean")

    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
    ## 0.04130937 0.04130937
    ##
    ## Means:
    ##          1          2
    ## 0.04669993 0.06719883
    ##
    ## Variances:
    ##          1          2
    ## 0.02376885 0.02376885
    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%  0.5364895 0.3004131
    ## 97.5% 0.6995869 0.4635105
    ##
    ## Means:
    ##              1        2
    ## 2.5%  4.279055 6.184439
    ## 97.5% 4.461108 6.449465
    ##
    ## Variances:
    ##               1         2
    ## 2.5%  0.1395796 0.1395796
    ## 97.5% 0.2317769 0.2317769

    plot(boot4, what = "pro")
    plot(boot4, what = "mean")

# 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 35
    ##        3 28
    ##
    ## Estimated basis vectors:
    ##              Dir1     Dir2
    ## fpg      0.884389  0.92601
    ## glucose -0.466586 -0.20579
    ## insulin -0.012419 -0.31647
    ##
    ##               Dir1      Dir2
    ## Eigenvalues  1.371   0.41864
    ## Cum. %      76.607 100.00000
    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 35
    ##        3 28
    ##
    ## Estimated basis vectors:
    ##              Dir1     Dir2
    ## fpg      0.884389  0.92601
    ## glucose -0.466586 -0.20579
    ## insulin -0.012419 -0.31647
    ##
    ##               Dir1      Dir2
    ## Eigenvalues  1.371   0.41864
    ## Cum. %      76.607 100.00000
    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
    ## Sepal.Length  0.20874 -0.006532
    ## Sepal.Width   0.38620 -0.586611
    ## Petal.Length -0.55401  0.252562
    ## Petal.Width  -0.70735 -0.769453
    ##
    ##                Dir1       Dir2
    ## Eigenvalues  1.8813   0.098592
    ## Cum. %      95.0204 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
    ## Length   -0.07016 -0.25690
    ## Left     -0.36888 -0.19963
    ## Right     0.29525 -0.10111
    ## Bottom    0.54683  0.46254
    ## Top       0.55720  0.41370
    ## Diagonal -0.40290  0.70628
    ##
    ##                Dir1     Dir2
    ## Eigenvalues  1.7188   1.0607
    ## Cum. %      61.8373 100.0000
    plot(mod3dr, what = "scatterplot")

    plot(mod3dr, what = "boundaries", ngrid = 200)

# Using colorblind-friendly palettes

Most of the graphs produced by mclust use colors that by default are defined in the following options:

    mclust.options("bicPlotColors")
    ##       EII       VII       EEI       EVI       VEI       VVI       EEE       VEE
    ##    "gray"   "black" "#218B21" "#41884F" "#508476" "#58819C" "#597DC3" "#5178EA"
    ##       EVE       VVE       EEV       VEV       EVV       VVV         E         V
    ## "#716EE7" "#9B60B8" "#B2508B" "#C03F60" "#C82A36" "#CC0000"    "gray"   "black"
    mclust.options("classPlotColors")
    ##  [1] "dodgerblue2"    "red3"           "green3"         "slateblue"
    ##  [5] "darkorange"     "skyblue1"       "violetred4"     "forestgreen"
    ##  [9] "steelblue4"     "slategrey"      "brown"          "black"
    ## [13] "darkseagreen"   "darkgoldenrod3" "olivedrab"      "royalblue"
    ## [17] "tomato4"        "cyan2"          "springgreen2"

The first option controls colors used for plotting BIC, ICL, etc. curves, whereas the second option is used to assign colors for indicating clusters or classes when plotting data.

Starting with R version 4.0, the function can be used for retrieving colors from some pre-defined palettes. For instance

    palette.colors(palette = "Okabe-Ito")

returns a color-blind-friendly palette for individuals suffering from protanopia or deuteranopia, the two most common forms of inherited color blindness. For earlier versions of R such palette can be defined as:

    cbPalette <- c("#000000", "#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2",
    "#D55E00", "#CC79A7", "#999999")

and then assigned to the mclust options as follows:

    bicPlotColors <- mclust.options("bicPlotColors")
    bicPlotColors[1:14] <- c(cbPalette, cbPalette[1:5])
    mclust.options("bicPlotColors" = bicPlotColors)
    mclust.options("classPlotColors" = cbPalette[-1])

    clPairs(iris[,-5], iris$Species)

    mod <- Mclust(iris[,-5])
    plot(mod, what = "BIC")

    plot(mod, what = "classification")

If needed, users can easily define their own palettes following the same procedure outlined above.

# References

Scrucca L., Fraley C., Murphy T. B. and Raftery A. E. (2023) Model-Based Clustering, Classification, and Density Estimation Using mclust in R. Chapman & Hall/CRC, ISBN: 978-1032234953, https://mclust-org.github.io/book/

Scrucca L., Fop M., Murphy T. B. and Raftery A. E. (2016) mclust 5: clustering, classification and density estimation using Gaussian finite mixture models, The R Journal, 8/1, pp. 205-233. https://journal.r-project.org/articles/RJ-2016-021/RJ-2016-021.pdf

Fraley C. and Raftery A. E. (2002) Model-based clustering, discriminant analysis and density estimation, Journal of the American Statistical Association, 97/458, pp. 611-631.

    sessionInfo()
    ## R version 4.5.0 (2025-04-11)
    ## Platform: aarch64-apple-darwin20
    ## Running under: macOS Sequoia 15.6.1
    ##
    ## Matrix products: default
    ## BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib
    ## LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
    ##
    ## locale:
    ## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
    ##
    ## time zone: Europe/Rome
    ## tzcode source: internal
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] mclust_6.1.2 knitr_1.50
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] digest_0.6.37     R6_2.6.1          fastmap_1.2.0     xfun_0.53
    ##  [5] cachem_1.1.0      htmltools_0.5.8.1 rmarkdown_2.30    lifecycle_1.0.4
    ##  [9] cli_3.6.5         sass_0.4.10       jquerylib_0.1.4   compiler_4.5.0
    ## [13] rstudioapi_0.17.1 tools_4.5.0       evaluate_1.0.5    bslib_0.9.0
    ## [17] yaml_2.3.10       rlang_1.1.6       jsonlite_2.0.0