pugmm {PUGMM}

Parsimonious Ultrametric Gaussian Mixture Models

Description

Model-based clustering via Parsimonious Ultrametric Gaussian Mixture Models. Hierarchical relationships among variables within and between clusters are inspected. The grouped coordinate ascent algorithm is used for the parameter estimation. The optimal model is selected according to BIC.

Usage

pugmm(
  X,
  G = NULL,
  m = NULL,
  normalization = NULL,
  model = NULL,
  maxiter = 500,
  tol = 1e-06,
  stop = "aitken",
  rndstart = 1,
  initG = "kmeans",
  initm = "ucms",
  gaussian = "mclust",
  parallel = FALSE
)

Arguments

X

(n \times p) numeric matrix or data frame, where n and p represent the number of units and variables, respectively. Categorical variables are not allowed.

G

Integer (vector) specifying the number of mixture components (default: G = 1:5).

m

Integer (vector) specifying the number of variable groups (default: m = 1:5).

normalization

Character string specifying the data transformation. If NULL, no transformation is applied to the data matrix (default). Other options are: "standard" for the standardization; "center" for centering the data; "range" for the MinMax transformation; "SVD" for the Singular Value Decomposition transformation.

model

Vector of character strings indicating the model names to be fitted. If NULL, all the possible models are fitted (default). See the possible models using available_models().

maxiter

Integer value specifying the maximum number of iterations of the algorithm (default: maxiter = 500).

tol

Numeric value specifying the tolerance for the convergence criterion (default: tol = 1e-6).

stop

Character string specifying the convergence criteria. If "aitken", the Aitken acceleration-based stopping rule is used (default); if "relative", the relative log-likelihood in two sequential iterations is evaluated.

rndstart

Integer value specifying the number of random starts (default: rndstart = 1).

initG

Character string specifying the method for the initialization of the unit-component membership. If "kmeans", k-means via RcppArmadillo is used (default). Other options are: "random" for random assignment; "kmeansf" for fuzzy c-means (via the function fcm of the package ppclust).

initm

Character string specifying the method for the initialization of the variable-group membership. If "ucms", the multivariate model to be used for obtaining the variable-group membership estimated is the same model.name used for estimating the Parsimonious Ultrametric Gaussian Mixture Model (default); if "random", a random assignment is performed.

gaussian

Character string specifying the way to compute the log-likelihood. If "mclust", dmvnorm of mclust is used (default); if "canonical", the log-likelihood computation is based upon the canonical representation of an extended ultrametric covariance matrix.

parallel

A logical value, specifying whether the models should be run in parallel.

Details

The grouped coordinate ascent algorithm used for the estimation of PUGMMs parameters was demonstrated to be equivalent to an Expectation-Maximization algorithm in the GMM framework (Hathaway, 1986).

Value

An object of class pugmm containing the results of the optimal - according to BIC - Parsimonious Ultrametric Gaussian Mixture Model estimation.

call Matched call.

X Input data matrix.

G Number of components of the best model.

m Number of variable groups of the best model.

label Integer vector of dimension n, taking values in \{1, \ldots, G\}. It identifies the unit classification according to the maximum a posteriori of the best model.

pp Numeric vector of dimension G containing the prior probabilities for the best model.

mu (G \times p) numeric matrix containing the component mean vectors (by row) for the best model.

sigma List of dimension G containing the (p \times p) numeric component extended ultrametric covariance matrices for the best model.

V List of dimension G containing the (p \times m) binary variable-group membership matrices for the best model.

Sv List of dimension G containing the (m \times m) numeric diagonal matrices of the group variances for the best model.

Sw List of dimension G containing the (m \times m) numeric diagonal matrices of the within-group covariances for the best model.

Sb List of dimension G containing the (m \times m) numeric hallow matrices of the between-group covariances for the best model.

post (n \times G) numeric matrix containing the posterior probabilities for the best model.

pm Number of parameters of the best model.

pm.cov Number of covariance parameters of the best model.

pm.free Number of free parameters of the best model (pm - (constraints on V + count.constr.SwSb + count.constr.SvSw)).

count.constr.SwSb Number of times the constraint between Sw and Sb has been turned on for the best model.

count.constr.SvSw Number of times the constraint between Sv and Sw has been turned on for the best model.

BIC BIC values for all the fitted models. If BIC is NA, the model has not been computed since its structure is equal to another model, while if BIC is -Inf the solution has a number of clusters < G.

bic BIC value of the best model.

loglik Log-likelihood of the best model.

loop Random start corresponding to the selected solution of the best model.

iter Number of iterations needed to estimate the best model.

model.name Character string denoting the PUGMM model name of the best model among the ones fitted.

messages Messages.

References

Cavicchia, C., Vichi, M., Zaccaria, G. (2024) Parsimonious ultrametric Gaussian mixture models. Statistics and Computing, 34, 108.

Cavicchia, C., Vichi, M., Zaccaria, G. (2022) Gaussian mixture model with an extended ultrametric covariance structure. Advances in Data Analysis and Classification, 16(2), 399-427.

Hathaway, R. (1986) Another interpretation of the EM algorithm for mixture distributions. Statistics and Probability Letters, 4(2), 53-56.

See Also

pugmm_available_models(), plot.pugmm()

Examples

data(penguins)
x <- scale(penguins[, 2:5])
pugmm.penguins <- pugmm(x, 3, 1)
table(penguins$species, pugmm.penguins$label)

pugmm.penguins <- pugmm(x)
pugmm.penguins$G
pugmm.penguins$m
pugmm.penguins$model.name