# A quick tour of mclust

#### Luca Scrucca

#### 29 Apr 2024

# 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 6.1.1) 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)
    ##                    __           __
    ##    ____ ___  _____/ /_  _______/ /_
    ##   / __ `__ \/ ___/ / / / / ___/ __/
    ##  / / / / / / /__/ / /_/ (__  ) /_
    ## /_/ /_/ /_/\___/_/\__,_/____/\__/   version 6.1.1
    ## 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
    clPairs(X, class)

    BIC <- mclustBIC(X)
    plot(BIC)

    summary(BIC)
    ## Best BIC values:
    ##              VVV,3       VVV,4       EVE,6
    ## BIC      -4751.316 -4784.32213 -4785.24591
    ## BIC diff     0.000   -33.00573   -33.92951

    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
    ##       -2303.496 145 29 -4751.316 -4770.169
    ##
    ## Clustering table:
    ##  1  2  3
    ## 81 36 28
    ##
    ## Mixing probabilities:
    ##         1         2         3
    ## 0.5368974 0.2650129 0.1980897
    ##
    ## Means:
    ##              [,1]     [,2]       [,3]
    ## glucose  90.96239 104.5335  229.42136
    ## insulin 357.79083 494.8259 1098.25990
    ## sspg    163.74858 309.5583   81.60001
    ##
    ## Variances:
    ## [,,1]
    ##          glucose    insulin       sspg
    ## glucose 57.18044   75.83206   14.73199
    ## insulin 75.83206 2101.76553  322.82294
    ## sspg    14.73199  322.82294 2416.99074
    ## [,,2]
    ##           glucose   insulin       sspg
    ## glucose  185.0290  1282.340  -509.7313
    ## insulin 1282.3398 14039.283 -2559.0251
    ## sspg    -509.7313 -2559.025 23835.7278
    ## [,,3]
    ##           glucose   insulin       sspg
    ## glucose  5529.250  20389.09  -2486.208
    ## insulin 20389.088  83132.48 -10393.004
    ## sspg    -2486.208 -10393.00   2217.533

    plot(mod1, what = "classification")

    table(class, mod1$classification)
    ##
    ## class       1  2  3
    ##   Chemical  9 26  1
    ##   Normal   72  4  0
    ##   Overt     0  6 27

    plot(mod1, what = "uncertainty")

    ICL <- mclustICL(X)
    summary(ICL)
    ## Best ICL values:
    ##              VVV,3       EVE,6       EVE,7
    ## ICL      -4770.169 -4797.38232 -4797.50566
    ## ICL diff     0.000   -27.21342   -27.33677
    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   361.16739             0.001
    ## 2 vs 3   123.49685             0.001
    ## 3 vs 4    16.76161             0.498

## 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       VVV,4       EVE,6
    ## BIC      -4751.316 -4784.32213 -4785.24591
    ## BIC diff     0.000   -33.00573   -33.92951

    (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,4
    ## BIC      -4760.091 -4775.53693 -4793.26143
    ## BIC diff     0.000   -15.44628   -33.17079

    (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       VVE,3
    ## BIC      -4751.354 -4757.091572 -4775.69587
    ## BIC diff     0.000    -5.737822   -24.34212

Update BIC by merging the best results:

    BIC <- mclustBICupdate(BIC1, BIC2, BIC3)
    summary(BIC)
    ## Best BIC values:
    ##              VVV,3        VVE,4       VVE,3
    ## BIC      -4751.316 -4757.091572 -4775.53693
    ## BIC diff     0.000    -5.775172   -24.22053
    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 model summary:
    ##
    ##  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
    ## Class        setosa versicolor virginica
    ##   setosa         50          0         0
    ##   versicolor      0         47         3
    ##   virginica       0          0        50
    ## Classification error = 0.02
    ## 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 model summary:
    ##
    ##  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
    ## Class         counterfeit genuine
    ##   counterfeit         100       0
    ##   genuine               0     100
    ## Classification error = 0
    ## Brier score          = 0
    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
    ## 0.05185780 0.05058160 0.03559685
    ##
    ## Means:
    ##                1         2         3
    ## glucose 1.042239  3.444948 16.340816
    ## insulin 7.554105 29.047203 63.483315
    ## sspg    7.669033 31.684647  9.926121
    ##
    ## Variances:
    ## [,,1]
    ##          glucose   insulin      sspg
    ## glucose 10.78177  51.28084  51.61617
    ## insulin 51.28084 529.62298 416.38176
    ## sspg    51.61617 416.38176 623.81098
    ## [,,2]
    ##           glucose   insulin      sspg
    ## glucose  65.66172  616.6785  442.0993
    ## insulin 616.67852 7279.0671 3240.3558
    ## sspg    442.09927 3240.3558 7070.4152
    ## [,,3]
    ##           glucose   insulin      sspg
    ## glucose 1045.6542  4178.685  667.2709
    ## insulin 4178.6846 18873.253 2495.0278
    ## sspg     667.2709  2495.028  506.8173
    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%  0.4490043 0.1510533 0.1324862
    ## 97.5% 0.6518326 0.3548749 0.2688038
    ##
    ## Means:
    ## [,,1]
    ##        glucose  insulin     sspg
    ## 2.5%  89.13950 344.9890 150.8405
    ## 97.5% 93.16603 374.7221 181.8322
    ## [,,2]
    ##         glucose  insulin     sspg
    ## 2.5%   98.82567 447.4121 257.9011
    ## 97.5% 112.28459 561.3273 374.6194
    ## [,,3]
    ##        glucose   insulin      sspg
    ## 2.5%  198.5986  969.6231  63.22103
    ## 97.5% 263.2932 1226.2654 101.09078
    ##
    ## Variances:
    ## [,,1]
    ##        glucose  insulin     sspg
    ## 2.5%  38.65508 1234.198 1514.416
    ## 97.5% 79.43401 3287.722 4146.024
    ## [,,2]
    ##         glucose   insulin     sspg
    ## 2.5%   88.35268  3514.662 12583.92
    ## 97.5% 358.15175 31416.557 39228.47
    ## [,,3]
    ##        glucose   insulin     sspg
    ## 2.5%  3377.773  47477.74 1317.041
    ## 97.5% 7379.344 120297.75 3229.747

    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 81
    ##        2 36
    ##        3 28
    ##
    ## Estimated basis vectors:
    ##              Dir1     Dir2
    ## glucose  0.764699  0.86359
    ## insulin -0.643961 -0.22219
    ## sspg     0.023438 -0.45260
    ##
    ##                Dir1      Dir2
    ## Eigenvalues  1.2629   0.35218
    ## Cum. %      78.1939 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 81
    ##        2 36
    ##        3 28
    ##
    ## Estimated basis vectors:
    ##              Dir1     Dir2
    ## glucose  0.764699  0.86359
    ## insulin -0.643961 -0.22219
    ## sspg     0.023438 -0.45260
    ##
    ##                Dir1      Dir2
    ## Eigenvalues  1.2629   0.35218
    ## Cum. %      78.1939 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/archive/2016/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 4.4.0 (2024-04-24)
    ## Platform: x86_64-apple-darwin20
    ## Running under: macOS Ventura 13.6
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    ## Matrix products: default
    ## BLAS:   /Library/Frameworks/R.framework/Versions/4.4-x86_64/Resources/lib/libRblas.0.dylib
    ## LAPACK: /Library/Frameworks/R.framework/Versions/4.4-x86_64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0
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    ## locale:
    ## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
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    ## time zone: Europe/Rome
    ## tzcode source: internal
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    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
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    ## other attached packages:
    ## [1] mclust_6.1.1 knitr_1.45
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] digest_0.6.35     R6_2.5.1          fastmap_1.1.1     xfun_0.43
    ##  [5] cachem_1.0.8      htmltools_0.5.8.1 rmarkdown_2.26    lifecycle_1.0.4
    ##  [9] cli_3.6.2         sass_0.4.9        jquerylib_0.1.4   compiler_4.4.0
    ## [13] highr_0.10        rstudioapi_0.16.0 tools_4.4.0       evaluate_0.23
    ## [17] bslib_0.7.0       yaml_2.3.8        rlang_1.1.3       jsonlite_1.8.8