Model-Based Clustering

Model-based clustering based on parameterized finite Gaussian mixture models. Models are estimated by EM algorithm initialized by hierarchical model-based agglomerative clustering. The optimal model is then selected according to BIC.

Usage

Mclust(data, G = NULL, modelNames = NULL, 
     prior = NULL, 
     control = emControl(), 
     initialization = NULL, 
     warn = mclust.options("warn"), 
     x =  NULL, 
     verbose = interactive(), ...)

Arguments

data

A numeric vector, matrix, or data frame of observations. Categorical variables are not allowed. If a matrix or data frame, rows correspond to observations (\(n\)) and columns correspond to variables (\(d\)).

G

An integer vector specifying the numbers of mixture components (clusters) for which the BIC is to be calculated. The default is G=1:9.

modelNames

A vector of character strings indicating the models to be fitted in the EM phase of clustering. The default is:

for univariate data (\(d = 1\)): c("E", "V")
for multivariate data (\(n > d\)): all the models available in mclust.options("emModelNames")
for multivariate data (\(n <= d\)): the spherical and diagonal models, i.e. c("EII", "VII", "EEI", "EVI", "VEI", "VVI")

The help file for mclustModelNames describes the available models.

prior

The default assumes no prior, but this argument allows specification of a conjugate prior on the means and variances through the function priorControl.
Note that, as described in defaultPrior, in the multivariate case only 10 out of 14 models may be used in conjunction with a prior, i.e. those available in MCLUST up to version 4.4.

control

A list of control parameters for EM. The defaults are set by the call emControl().

initialization

A list containing zero or more of the following components:

hcPairs: A matrix of merge pairs for hierarchical clustering such as produced by function hc.
For multivariate data, the default is to compute a hierarchical agglomerative clustering tree by applying function hc with model specified by mclust.options("hcModelName"), and data transformation set by mclust.options("hcUse").
All the input or a subset as indicated by the subset argument is used for initial clustering.
The hierarchical clustering results are then used to start the EM algorithm from a given partition.
For univariate data, the default is to use quantiles to start the EM algorithm. However, hierarchical clustering could also be used by calling hc with model specified as "V" or "E".
subset: A logical or numeric vector specifying a subset of the data to be used in the initial hierarchical clustering phase. No subset is used unless the number of observations exceeds the value specified by mclust.options("subset"), which by default is set to 2000 (see mclust.options). Note that in this case to guarantee exact reproducibility of results a seed must be specified (see set.seed).
noise: A logical or numeric vector indicating an initial guess as to which observations are noise in the data. If numeric the entries should correspond to row indexes of the data. If supplied, a noise term will be added to the model in the estimation.

warn

A logical value indicating whether or not certain warnings (usually related to singularity) should be issued. The default is controlled by mclust.options.

x

An object of class 'mclustBIC'. If supplied, BIC values for models that have already been computed and are available in x are not recomputed. All arguments, with the exception of data, G and modelName, are ignored and their values are set as specified in the attributes of x. Defaults for G and modelNames are taken from x.

verbose

A logical controlling if a text progress bar is displayed during the fitting procedure. By default is TRUE if the session is interactive, and FALSE otherwise.

...

Catches unused arguments in indirect or list calls via do.call.

Value

An object of class 'Mclust' providing the optimal (according to BIC) mixture model estimation.

The details of the output components are as follows:

call

The matched call

data

The input data matrix.

modelName

A character string denoting the model at which the optimal BIC occurs.

n

The number of observations in the data.

d

The dimension of the data.

G

The optimal number of mixture components.

BIC

All BIC values.

loglik

The log-likelihood corresponding to the optimal BIC.

df

The number of estimated parameters.

bic

BIC value of the selected model.

icl

ICL value of the selected model.

hypvol

The hypervolume parameter for the noise component if required, otherwise set to NULL (see hypvol).

parameters

A list with the following components:

pro: A vector whose kth component is the mixing proportion for the kth component of the mixture model. If missing, equal proportions are assumed.
mean: The mean for each component. If there is more than one component, this is a matrix whose kth column is the mean of the kth component of the mixture model.
variance: A list of variance parameters for the model. The components of this list depend on the model specification. See the help file for mclustVariance for details.

z

A matrix whose [i,k]th entry is the probability that observation i in the test data belongs to the kth class.

classification

The classification corresponding to z, i.e. map(z).

uncertainty

The uncertainty associated with the classification.

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. 289-317.

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.

C. Fraley and A. E. Raftery (2007) Bayesian regularization for normal mixture estimation and model-based clustering. Journal of Classification, 24, 155-181.

Examples

mod1 <- Mclust(iris[,1:4])
summary(mod1)
#> ---------------------------------------------------- 
#> Gaussian finite mixture model fitted by EM algorithm 
#> ---------------------------------------------------- 
#> 
#> Mclust VEV (ellipsoidal, equal shape) model with 2 components: 
#> 
#>  log-likelihood   n df       BIC       ICL
#>        -215.726 150 26 -561.7285 -561.7289
#> 
#> Clustering table:
#>   1   2 
#>  50 100 

mod2 <- Mclust(iris[,1:4], G = 3)
summary(mod2, parameters = TRUE)
#> ---------------------------------------------------- 
#> Gaussian finite mixture model fitted by EM algorithm 
#> ---------------------------------------------------- 
#> 
#> Mclust VEV (ellipsoidal, equal shape) model with 3 components: 
#> 
#>  log-likelihood   n df       BIC       ICL
#>        -186.074 150 38 -562.5522 -566.4673
#> 
#> Clustering table:
#>  1  2  3 
#> 50 45 55 
#> 
#> Mixing probabilities:
#>         1         2         3 
#> 0.3333333 0.3005423 0.3661243 
#> 
#> Means:
#>               [,1]     [,2]     [,3]
#> Sepal.Length 5.006 5.915044 6.546807
#> Sepal.Width  3.428 2.777451 2.949613
#> Petal.Length 1.462 4.204002 5.482252
#> Petal.Width  0.246 1.298935 1.985523
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.13320850  0.10938369  0.019191764 0.011585649
#> Sepal.Width    0.10938369  0.15495369  0.012096999 0.010010130
#> Petal.Length   0.01919176  0.01209700  0.028275400 0.005818274
#> Petal.Width    0.01158565  0.01001013  0.005818274 0.010695632
#> [,,2]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.22572159  0.07613348   0.14689934  0.04335826
#> Sepal.Width    0.07613348  0.08024338   0.07372331  0.03435893
#> Petal.Length   0.14689934  0.07372331   0.16613979  0.04953078
#> Petal.Width    0.04335826  0.03435893   0.04953078  0.03338619
#> [,,3]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.42943106  0.10784274   0.33452389  0.06538369
#> Sepal.Width    0.10784274  0.11596343   0.08905176  0.06134034
#> Petal.Length   0.33452389  0.08905176   0.36422115  0.08706895
#> Petal.Width    0.06538369  0.06134034   0.08706895  0.08663823

# Using prior
mod3 <- Mclust(iris[,1:4], prior = priorControl())
#> Warning: The presence of BIC values equal to NA is likely due to one or more of the mixture proportions being estimated as zero, so that the model estimated reduces to one with a smaller number of components.
summary(mod3)
#> ---------------------------------------------------- 
#> Gaussian finite mixture model fitted by EM algorithm 
#> ---------------------------------------------------- 
#> 
#> Mclust VEV (ellipsoidal, equal shape) model with 2 components: 
#> 
#> Prior: defaultPrior() 
#> 
#>  log-likelihood   n df       BIC      ICL
#>       -225.2685 150 26 -580.8136 -580.814
#> 
#> Clustering table:
#>   1   2 
#>  50 100 

mod4 <- Mclust(iris[,1:4], prior = priorControl(functionName="defaultPrior", shrinkage=0.1))
#> Warning: The presence of BIC values equal to NA is likely due to one or more of the mixture proportions being estimated as zero, so that the model estimated reduces to one with a smaller number of components.
summary(mod4)
#> ---------------------------------------------------- 
#> Gaussian finite mixture model fitted by EM algorithm 
#> ---------------------------------------------------- 
#> 
#> Mclust VEV (ellipsoidal, equal shape) model with 2 components: 
#> 
#> Prior: defaultPrior(shrinkage = 0.1) 
#> 
#>  log-likelihood   n df       BIC       ICL
#>       -227.6729 150 26 -585.6223 -585.6227
#> 
#> Clustering table:
#>   1   2 
#>  50 100 

# Clustering of faithful data with some artificial noise added 
nNoise <- 100
set.seed(0) # to make it reproducible
Noise <- apply(faithful, 2, function(x) 
              runif(nNoise, min = min(x)-.1, max = max(x)+.1))
data <- rbind(faithful, Noise)
plot(faithful)
points(Noise, pch = 20, cex = 0.5, col = "lightgrey")

set.seed(0)
NoiseInit <- sample(c(TRUE,FALSE), size = nrow(faithful)+nNoise, 
          replace = TRUE, prob = c(3,1)/4)
mod5 <- Mclust(data, initialization = list(noise = NoiseInit))
summary(mod5, parameter = TRUE)
#> ---------------------------------------------------- 
#> Gaussian finite mixture model fitted by EM algorithm 
#> ---------------------------------------------------- 
#> 
#> Mclust VVE (ellipsoidal, equal orientation) model with 2 components and a noise
#> term: 
#> 
#>  log-likelihood   n df       BIC       ICL
#>       -1746.033 372 12 -3563.093 -3674.077
#> 
#> Clustering table:
#>   1   2   0 
#> 180  88 104 
#> 
#> Mixing probabilities:
#>         1         2         0 
#> 0.4301382 0.2112698 0.3585921 
#> 
#> Means:
#>                [,1]      [,2]
#> eruptions  4.342387  1.978433
#> waiting   80.218154 54.005828
#> 
#> Variances:
#> [,,1]
#>           eruptions    waiting
#> eruptions 0.1152778  0.4225761
#> waiting   0.4225761 30.6256680
#> [,,2]
#>            eruptions    waiting
#> eruptions 0.03873859  0.3776241
#> waiting   0.37762409 27.3035511
#> 
#> Hypervolume of noise component:
#> 191.8869 
plot(mod5, what = "classification")

Usage

Arguments

Value

References

See also

Examples