Package 'mixSPE'

Mixtures of skew power exponential or power exponential distributions.

Description

An implementation of skewed and elliptical mixture distributions for use in model-based clustering.

Details

Package:	mixSPE
Type:	Package
Version:	0.9.2
Date:	2023-10-16
License:	GPL (>= 2)

---

Function for model-based clustering with the multivariate power exponential (MPE) or the skew power exponential (MSPE) distribution.

Description

For fitting of a family of 16 mixture models based on mixtures of multivariate skew power exponential distributions with eigen-decomposed covariance structures.

Usage

EMGr(data = NULL, initialization = 10, iModel = "EIIE", G = 2, max.iter = 500,
epsilon = 0.01, label = NULL, modelSet = "all", skewness = FALSE,
keepResults = FALSE, seedno = 1, scale = TRUE)

Arguments

data	A matrix such that rows correspond to observations and columns correspond to variables.
initialization	0 means a k-means start. A single positive number indicates the number of random soft starts in addition to 10 k-means starts, done via short EM runs; the best initialization is followed by a single long EM run until convergence. A single negative number indicates initializing with multiple random soft starts only; this is akin to taking the best initialization from multiple short EM runs for a long EM run until convergence. A z matrix can be provided directly here as well. Finally, a list can be provided with the same format as modelfit$bestmod$gpar. Often, it is helpful to run a long random-starts only run and a long k-means start run, and pick between those two based on BIC. See Dang et al 2023 for an example.
iModel	Initialization model used to generate initial parameter estimates.
G	A sequence of integers corresponding to the number of components to be fitted.
max.iter	Maximum number of GEM iterations allowed
epsilon	Threshold for convergence for the GEM algorithm used in the Aitken's stopping criterion.
label	Used for model-based classification aka semi-supervised classification. This is a vector of group labels with 0 for unlabelled observations.
modelSet	A total of 16 models are provided: "EIIE", "VIIE", "EEIE", "VVIE", "EEEE", "EEVE", "VVEE", "VVVE", "EIIV", "VIIV", "EEIV", "VVIV", "EEEV", "EEVV", "VVEV", "VVVV". modelSet="all" fits all models automatically. Otherwise, a character vector of a subset of these models can be provided.
skewness	If FALSE (default), fits mixtures of multivariate power exponential distributions that cannot model skewness. If TRUE, fits mixtures of multivariate skewed power exponential distributions that can model skewness.
keepResults	Keep results from all models
seedno	Seed number for initialization of k-means or random starts.
scale	If TRUE, scales the data before model fitting. Recommended unless to check parameter recovery.

Details

The component scale matrix is decomposed using an eigen-decomposition:
$\Sigma_g$ = $\lambda_g$ $\Gamma_g$ $\Delta_g$ $\Gamma'_g$
The nomenclature is as follows: a EEVE model denotes a model with equal constants associated with the eigenvalues ($\lambda$) for each group, equal orthogonal matrix of eigenvectors ($\Gamma$), variable diagonal matrices with values proportional to the eigenvalues of each component scale matrix ($\Delta_g$), and equal shape parameter ($\beta$).

Value

allModels	Output for each model run.
bestmod	Output for the best model chosen by the BIC.
loglik	Maximum log likelihood for each model
num.iter	Number of iterations required for convergence for each model
num.par	Number of parameters fit for each model
BIC	BIC for each model
bestBIC	Which model was selected by the BIC in the BIC matrix?

Author(s)

Ryan P. Browne, Utkarsh J. Dang, Michael P. B. Gallaugher, and Paul D. McNicholas

Examples

set.seed(1)
Nobs1 <- 200
Nobs2 <- 250
X1 <- rpe(n = Nobs1, mean = c(0,0), scale = diag(2), beta = 1)
X2 <- rpe(n = Nobs2, mean = c(3,0), scale = diag(2), beta = 2)
x <- as.matrix(rbind(X1, X2))
membership <- c(rep(1, Nobs1), rep(2, Nobs2))
mperun <- EMGr(data=x, initialization=0, iModel="EIIV", G=2:3,
max.iter=500, epsilon=5e-3, label=NULL, modelSet=c("EIIV"),
skewness=FALSE, keepResults=TRUE, seedno=1, scale=FALSE) #use "all" in modelSet for all models
print(mperun)
print(table(membership,mperun$bestmod$map))
msperun <- EMGr(data=x, initialization=0, iModel="EIIV", G=2:3,
max.iter=500, epsilon=5e-3, label=NULL, modelSet=c("EIIV"),
skewness=TRUE, keepResults=TRUE, seedno=1, scale=FALSE) #usually data should be scaled.
#print(msperun)
#print(table(membership,msperun$bestmod$map))

set.seed(1)
data(iris)
membership <- as.numeric(factor(iris[, "Species"]))
label <- membership
label[sample(x = 1:length(membership),size = ceiling(0.6*length(membership)),replace = FALSE)] <- 0
#40% supervision (known groups) and 60% unlabeled.
dat <- data.matrix(iris[, 1:4])
semisup_class_skewed = EMGr(data=dat, initialization=10, iModel="EIIV",
G=3, max.iter=500, epsilon=5e-3, label=label, modelSet=c("VVVE"),
skewness=TRUE, keepResults=TRUE, seedno=5, scale=TRUE)
#table(membership,semisup_class_skewed$bestmod$map)

---

Print a summary of the model fit.

Description

Print a summary of the model fit including the number of components and the scale structure selected by the BIC and the ICL.

Usage

## S3 method for class 'spemix'
print(x, ...)

Arguments

x	An object of class "spemix".
...	Ignore this

Value

Print function.

Author(s)

Utkarsh J. Dang, Michael P. B. Gallaugher, Ryan P. Browne, and Paul D. McNicholas

---

Simulate data from the multivariate power exponential distribution.

Description

Simulate data from the multivariate power exponential distribution given the mean, scale matrix, and the shape parameter.

Usage

rpe(n = NULL, beta = NULL, mean = NULL, scale = NULL)

Arguments

n	Number of observations to simulate.
beta	A positive shape parameter $\beta$ that determines the kurtosis of the distribution.
mean	A $p$-dimensional vector. $\mu$.
scale	A $p$-dimensional square scale matrix $\Sigma$.

Value

A matrix with rows representing the $p$-dimensional observations.

Author(s)

Utkarsh J. Dang, Ryan P. Browne, and Paul D. McNicholas

References

For simulating from the MPE distribution, a modified version of the function rmvpowerexp from package MNM (Nordhausen and Oja, 2011) is used. The function was modified due to a typo in the rmvpowerexp code, as mentioned in the publication (Dang et al., 2015). This program utilizes the stochastic representation of the MPE distribution (Gómez et al., 1998) to generate data. Dang, Utkarsh J., Ryan P. Browne, and Paul D. McNicholas. "Mixtures of multivariate power exponential distributions." Biometrics 71, no. 4 (2015): 1081-1089. Gómez, E., M. A. Gomez-Viilegas, and J. M. Marin. "A multivariate generalization of the power exponential family of distributions." Communications in Statistics-Theory and Methods 27, no. 3 (1998): 589-600. Nordhausen, Klaus, and Hannu Oja. "Multivariate L1 methods: the package MNM." Journal of Statistical Software 43, no. 5 (2011): 1-28.

Examples

dat <- rpe(n = 1000, beta = 2, mean = rep(0,5), scale = diag(5))
dat <- rpe(n = 1000, beta = 0.8, mean = rep(0,5), scale = diag(5))

---

Simulate data from the multivariate skew power exponential distribution.

Description

Simulate data from the multivariate power exponential distribution given the location, scale matrix, shape, and skewness parameter.

Usage

rspe(n, location = rep(0, nrow(scale)), scale = diag(length(location)),
beta = 1, psi = c(0, 0))

Arguments

n	Number of observations to simulate.
location	A $p$-dimensional vector. $\mu$.
scale	A $p$-dimensional square scale matrix $\Sigma$.
beta	A positive shape parameter $\beta$ that determines the kurtosis of the distribution.
psi	A $p$-dimensional vector determining skewness. $\mu$.

Details

Based on a Metropolis-Hastings rule.

Value

A matrix with rows representing the $p$-dimensional observations.

Author(s)

Utkarsh J. Dang, Ryan P. Browne, and Paul D. McNicholas

Examples

dat <- rspe(n = 1000, beta = 0.75, location = c(0,0), scale =
matrix(c(1,0.7,0.7,1),2,2), psi = c(5,5))

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