Package 'MixGHD' reference manual

Title:	Model Based Clustering, Classification and Discriminant Analysis Using the Mixture of Generalized Hyperbolic Distributions
Description:	Carries out model-based clustering, classification and discriminant analysis using five different models. The models are all based on the generalized hyperbolic distribution. The first model 'MGHD' (Browne and McNicholas (2015) <doi:10.1002/cjs.11246>) is the classical mixture of generalized hyperbolic distributions. The 'MGHFA' (Tortora et al. (2016) <doi:10.1007/s11634-015-0204-z>) is the mixture of generalized hyperbolic factor analyzers for high dimensional data sets. The 'MSGHD' is the mixture of multiple scaled generalized hyperbolic distributions, the 'cMSGHD' is a 'MSGHD' with convex contour plots and the 'MCGHD', mixture of coalesced generalized hyperbolic distributions is a new more flexible model (Tortora et al. (2019)<doi:10.1007/s00357-019-09319-3>. The paper related to the software can be found at <doi:10.18637/jss.v098.i03>.
Authors:	Cristina Tortora [aut, cre, cph], Aisha ElSherbiny [com], Ryan P. Browne [aut, cph], Brian C. Franczak [aut, cph], and Paul D. McNicholas [aut, cph], and Donald D. Amos [ctb].
Maintainer:	Cristina Tortora <[email protected]>
License:	GPL (>= 2)
Version:	2.3.7
Built:	2024-12-06 06:48:49 UTC
Source:	CRAN

Adjusted Rand Index.

Description

Compares two classifications using the adjusted Rand index (ARI).

Usage

ARI(x=NULL, y=NULL)
ARI(x=NULL, y=NULL)

Arguments

`x`	A n dimensional vector of class labels.
`y`	A n dimensional vector of class labels. .

Details

The ARI has expected value 0 in case of random partition, it is equal to one in case of perfect agreement..

Value

The adjusted Rand index value

Author(s)

Cristina Tortora Maintainer: Cristina Tortora <[email protected]>

References

L. Hubert and P. Arabie (1985) Comparing Partitions, Journal of the Classification 2:193-218.

Examples

##loading banknote data
data(banknote)

##model estimation
res=MGHD(data=banknote[,2:7],  G=2   )

#result
ARI(res@map, banknote[,1])

##loading banknote data
data(banknote)

##model estimation
res=MGHD(data=banknote[,2:7],  G=2   )

#result
ARI(res@map, banknote[,1])

Swiss Banknote data

Description

The data set contain 6 measures of 100 genuine and 100 counterfeit Swiss franc banknotes.

Usage

data(banknote)data(banknote)

Format

A data frame with the following variables:

Status: the status of the banknote: genuine or counterfeit
Length: Length of bill (mm)
Left: Width of left edge (mm)
Right: Width of right edge (mm)
Bottom: Bottom margin width (mm)
Top: Top margin width (mm)
Diagonal: Length of diagonal (mm)

References

Flury, B. and Riedwyl, H. (1988). Multivariate Statistics: A practical approach. London: Chapman & Hall, Tables 1.1 and 1.2, pp. 5-8

The data set contain the ratio of retained earnings (RE) to total assets, and the ratio of earnings before interests and taxes (EBIT) to total assets of 66 American firms recorded in the form of ratios. Half of the selected firms had filed for bankruptcy.

Usage

data(bankruptcy)data(bankruptcy)

Format

A data frame with the following variables:

Y: the status of the firm: 0 bankruptcy or 1 financially sound.
RE: ratio
EBIT: ratio

References

Altman E.I. (1968). Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. J Finance 23(4): 589-609

Convex mixture of multiple scaled generalized hyperbolic distributions (cMSGHD).

Description

Carries out model-based clustering using the convex mixture of multiple scaled generalized hyperbolic distributions. The cMSGHD only allows conves level sets.

Usage

cMSGHD(data=NULL,gpar0=NULL,G=2,max.iter=100,label=NULL,eps=1e-2,
	method="km",scale=TRUE,nr=10, modelSel="AIC")
cMSGHD(data=NULL,gpar0=NULL,G=2,max.iter=100,label=NULL,eps=1e-2,
	method="km",scale=TRUE,nr=10, modelSel="AIC")

Arguments

`data`	A n x p matrix or data frame such that rows correspond to observations and columns correspond to variables.
`gpar0`	(optional) A list containing the initial parameters of the mixture model. See the 'Details' section.
`G`	The range of values for the number of clusters.
`max.iter`	(optional) A numerical parameter giving the maximum number of iterations each EM algorithm is allowed to use.
`label`	( optional) A n dimensional vector, if label[i]=k then observation i belongs to group k, if NULL then the data has no known groups.
`eps`	(optional) A number specifying the epsilon value for the convergence criteria used in the EM algorithms. For each algorithm, the criterion is based on the difference between the log-likelihood at an iteration and an asymptotic estimate of the log-likelihood at that iteration. This asymptotic estimate is based on the Aitken acceleration.
`method`	( optional) A string indicating the initialization criterion, if not specified kmeans clustering is used. Alternative methods are: hierarchical "hierarchical",random "random", kmedoids "kmedoids", and model based "modelBased"
`scale`	( optional) A logical value indicating whether or not the data should be scaled, true by default.
`nr`	( optional) A number indicating the number of starting value when random is used, 10 by default.
`modelSel`	( optional) A string indicating the model selection criterion, if not specified AIC is used. Alternative methods are: BIC,ICL, and AIC3

Details

The arguments gpar0, if specified, is a list structure containing at least one p dimensional vector mu, alpha and phi, a pxp matrix gamma, and a px2 matrix cpl containing the vector omega and the vector lambda.

Value

A S4 object of class MixGHD with slots:

`index`	Value of the index used for model selection (AIC or ICL or BIC or AIC3) for each G,the index used is specified by the user, if not specified AIC is used.
`BIC`	Bayesian information criterion.
`ICL`	Integrated completed likelihood.
`AIC`	Akaike information criterion.
`AIC3`	Akaike information criterion 3.
`gpar`	A list of the model parameters
`loglik`	The log-likelihood values.
`map`	A vector of integers indicating the maximum a posteriori classifications for the best model.
`z`	A matrix giving the raw values upon which map is based.

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

C. Tortora, B.C. Franczak, R.P. Browne, and P.D. McNicholas (2019). A Mixture of Coalesced Generalized Hyperbolic Distributions. Journal of Classification 36(1) 26-57.\ C. Tortora, R. P. Browne, A. ElSherbiny, B. C. Franczak, and P. D. McNicholas (2021). Model-Based Clustering, Classification, and Discriminant Analysis using the Generalized Hyperbolic Distribution: MixGHD R package, Journal of Statistical Software 98(3) 1–24, <doi:10.18637/jss.v098.i03>.

Examples


##Generate random data
set.seed(3)

mu1 <- mu2 <- c(0,0)
Sigma1 <- matrix(c(1,0.85,0.85,1),2,2)
Sigma2 <- matrix(c(1,-0.85,-0.85,1),2,2)

X1 <- mvrnorm(n=150,mu=mu1,Sigma=Sigma1)
X2 <- mvrnorm(n=150,mu=mu2,Sigma=Sigma2)

X <- rbind(X1,X2)

##model estimation
em=cMSGHD(X,G=2,max.iter=30,method="random",nr=2)



#result
plot(em)##Generate random data
set.seed(3)

mu1 <- mu2 <- c(0,0)
Sigma1 <- matrix(c(1,0.85,0.85,1),2,2)
Sigma2 <- matrix(c(1,-0.85,-0.85,1),2,2)

X1 <- mvrnorm(n=150,mu=mu1,Sigma=Sigma1)
X2 <- mvrnorm(n=150,mu=mu2,Sigma=Sigma2)

X <- rbind(X1,X2)

##model estimation
em=cMSGHD(X,G=2,max.iter=30,method="random",nr=2)



#result
plot(em)

Coefficients for objects of class MixGHD

Description

Coefficents of the estimated model.

Usage

	## S4 method for signature 'MixGHD'
coef(object)

## S4 method for signature 'MixGHD'
coef(object)

Arguments

object

An S4 object of class MixGHD.

Value

The coefficents of the estimated model

Author(s)

Cristina Tortora Maintainer: Cristina Tortora <[email protected]>

Examples

##loading bankruptcy data
data(bankruptcy)

##model estimation
res=MCGHD(data=bankruptcy[,2:3],G=2,method="kmedoids",max.iter=30)
#rcoefficients of the model
coef(res)

##loading bankruptcy data
data(bankruptcy)

##model estimation
res=MCGHD(data=bankruptcy[,2:3],G=2,method="kmedoids",max.iter=30)
#rcoefficients of the model
coef(res)

Contour plot

Description

Contour plot for a given set of parameters.

Usage

contourpl(input)
contourpl(input)

Arguments

input

An S4 object of class MixGHD.

Value

The contour plot

Author(s)

Cristina Tortora Maintainer: Cristina Tortora <[email protected]>

Examples

##loading bankruptcy data
data(bankruptcy)

##model estimation
res=MCGHD(data=bankruptcy[,2:3],G=2,method="kmedoids",max.iter=30)
#result
contourpl(res)

##loading bankruptcy data
data(bankruptcy)

##model estimation
res=MCGHD(data=bankruptcy[,2:3],G=2,method="kmedoids",max.iter=30)
#result
contourpl(res)

Discriminant analysis using the mixture of generalized hyperbolic distributions.

Description

Carries out model-based discriminant analysis using 5 different models: the mixture of multiple scaled generalized hyperbolic distributions (MGHD), the mixture of generalized hyperbolic factor analyzers (MGHFA), the mixture of multiple scaled generalized hyperbolic distributions (MSGHD),the mixture of convex multiple scaled generalized hyperbolic distributions (cMSGHD) and the mixture of coaelesed generalized hyperbolic distributions (MCGHD).

Usage

DA(train,trainL,test,testL,method="MGHD",starting="km",max.iter=100,
	eps=1e-2,q=2,scale=TRUE)DA(train,trainL,test,testL,method="MGHD",starting="km",max.iter=100,
	eps=1e-2,q=2,scale=TRUE)

Arguments

`train`	A n1 x p matrix or data frame such that rows correspond to observations and columns correspond to variables of the training data set.
`trainL`	A n1 dimensional vector of membership for the units of the training set. If trainL[i]=k then observation belongs to group k.
`test`	A n2 x p matrix or data frame such that rows correspond to observations and columns correspond to variables of the test data set.
`testL`	A n2 dimensional vector of membership for the units of the test set. If testL[i]=k then observation belongs to group k.
`method`	( optional) A string indicating the method to be used form discriminant analysis , if not specified MGHD is used. Alternative methods are: MGHFA, MSGHD, cMSGHD, MCGHD.
`starting`	( optional) A string indicating the initialization criterion, if not specified kmeans clustering is used. Alternative methods are: hierarchical "hierarchical",random "random", kmedoids "kmedoids", and model based "modelBased"
`max.iter`	(optional) A numerical parameter giving the maximum number of iterations each EM algorithm is allowed to use.
`eps`	(optional) A number specifying the epsilon value for the convergence criteria used in the EM algorithms. For each algorithm, the criterion is based on the difference between the log-likelihood at an iteration and an asymptotic estimate of the log-likelihood at that iteration. This asymptotic estimate is based on the Aitken acceleration.
`q`	(optional) used only if MGHFA method is selected. A numerical parameter giving the number of factors.
`scale`	( optional) A logical value indicating whether or not the data should be scaled, true by default.

Value

A list with components

`model`	An S4 object of class `MixGHD` with the model parameters.
`testMembership`	A vector of integers indicating the membership of the units in the test set
`ARItest`	A value indicating the adjusted rand index for the test set.
`ARItrain`	A value indicating the adjusted rand index for the train set.

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

R.P. Browne, and P.D. McNicholas (2015). A Mixture of Generalized Hyperbolic Distributions. Canadian Journal of Statistics, 43.2 176-198.
C. Tortora, B.C. Franczak, R.P. Browne, and P.D. McNicholas (2019). A Mixture of Coalesced Generalized Hyperbolic Distributions. Journal of Classification (to appear).
C.Tortora, P.D. McNicholas, and R.P. Browne (2016). Mixtures of Generalized Hyperbolic Factor Analyzers. Advanced in data analysis and classification 10(4) p.423-440.

Examples

##loading banknote data
data(banknote)
banknote[,1]=as.numeric(factor(banknote[,1]))


##divide the data in training set and test set
train=banknote[c(1:74,126:200),]
test=banknote[75:125,]

##model estimation
 model=DA(train[,2:7],train[,1],test[,2:7],test[,1],method="MGHD",max.iter=20)

#result
model$ARItest##loading banknote data
data(banknote)
banknote[,1]=as.numeric(factor(banknote[,1]))


##divide the data in training set and test set
train=banknote[c(1:74,126:200),]
test=banknote[75:125,]

##model estimation
 model=DA(train[,2:7],train[,1],test[,2:7],test[,1],method="MGHD",max.iter=20)

#result
model$ARItest

Density of a coalesced generalized hyperbolic distribution (MSGHD).

Description

Compute the density of a p dimensional coalesced generalized hyperbolic distribution.

Usage

dCGHD(data,p,mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),lambda=1,omega=1,
  omegav=rep(1,p),lambdav=rep(1,p),wg=0.5,gam=NULL,phi=NULL)dCGHD(data,p,mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),lambda=1,omega=1,
  omegav=rep(1,p),lambdav=rep(1,p),wg=0.5,gam=NULL,phi=NULL)

Arguments

`data`	n x p data set
`p`	number of variables.
`mu`	(optional) the p dimensional mean
`alpha`	(optional) the p dimensional skewness parameter alpha
`sigma`	(optional) the p x p dimensional scale matrix
`lambda`	(optional) the 1 dimensional index parameter lambda
`omega`	(optional) the 1 dimensional concentration parameter omega
`omegav`	(optional) the p dimensional concentration parameter omega
`lambdav`	(optional) the p dimensional index parameter lambda
`wg`	(optional) weight
`gam`	(optional) the pxp gamma matrix
`phi`	(optional) the p dimensional vector phi

Details

The default values are: 0 for the mean and the skweness parameter alpha, diag(p) for sigma, 1 for omega, and 0.5 for lambda.

Value

A n dimensional vector with the density from a coalesced generilzed hyperbolic distribution

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

C. Tortora, B.C. Franczak, R.P. Browne, and P.D. McNicholas (2019). A Mixture of Coalesced Generalized Hyperbolic Distributions. Journal of Classification (to appear).

Examples




x = seq(-3,3,length.out=30)
y = seq(-3,3,length.out=30)
xyS1 = matrix(0,nrow=length(x),ncol=length(y))
for(i in 1:length(x)){
  for(j in 1:length(y)){
      xy <- matrix(cbind(x[i],y[j]),1,2)	
      xyS1[i,j] =  dCGHD(xy,2) 
      
    }
  }
contour(x=x,y=y,z=xyS1, levels=c(.005,.01,.025,.05, .1,.25), main="CGHD",ylim=c(-3,3), xlim=c(-3,3))


x = seq(-3,3,length.out=30)
y = seq(-3,3,length.out=30)
xyS1 = matrix(0,nrow=length(x),ncol=length(y))
for(i in 1:length(x)){
  for(j in 1:length(y)){
      xy <- matrix(cbind(x[i],y[j]),1,2)	
      xyS1[i,j] =  dCGHD(xy,2) 
      
    }
  }
contour(x=x,y=y,z=xyS1, levels=c(.005,.01,.025,.05, .1,.25), main="CGHD",ylim=c(-3,3), xlim=c(-3,3))

Density of a generalized hyperbolic distribution (GHD).

Description

Compute the density of a p dimensional generalized hyperbolic distribution.

Usage

dGHD(data,p, mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omega=1,lambda=0.5, log=FALSE)dGHD(data,p, mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omega=1,lambda=0.5, log=FALSE)

Arguments

`data`	n x p data set
`p`	number of variables.
`mu`	(optional) the p dimensional mean
`alpha`	(optional) the p dimensional skewness parameter alpha
`sigma`	(optional) the p x p dimensional scale matrix
`omega`	(optional) the unidimensional concentration parameter omega
`lambda`	(optional) the unidimensional index parameter lambda
`log`	(optional) if TRUE returns the log of the density

Details

The default values are: 0 for the mean and the skweness parameter alpha, diag(p) for sigma, 1 for omega, and 0.5 for lambda.

Value

A n dimensional vector with the density from a generilzed hyperbolic distribution

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

R.P. Browne, and P.D. McNicholas (2015). A Mixture of Generalized Hyperbolic Distributions. Canadian Journal of Statistics, 43.2 176-198

Examples




x = seq(-3,3,length.out=50)
y = seq(-3,3,length.out=50)
xyS1 = matrix(0,nrow=length(x),ncol=length(y))
for(i in 1:length(x)){
  for(j in 1:length(y)){
      xy <- matrix(cbind(x[i],y[j]),1,2)	
      xyS1[i,j] =  dGHD(xy,2) 
      
    }
  }
contour(x=x,y=y,z=xyS1, levels=c(.005,.01,.025,.05, .1,.25), main="MGHD",ylim=c(-3,3), xlim=c(-3,3))




x = seq(-3,3,length.out=50)
y = seq(-3,3,length.out=50)
xyS1 = matrix(0,nrow=length(x),ncol=length(y))
for(i in 1:length(x)){
  for(j in 1:length(y)){
      xy <- matrix(cbind(x[i],y[j]),1,2)	
      xyS1[i,j] =  dGHD(xy,2) 
      
    }
  }
contour(x=x,y=y,z=xyS1, levels=c(.005,.01,.025,.05, .1,.25), main="MGHD",ylim=c(-3,3), xlim=c(-3,3))

Density of a mulitple-scaled generalized hyperbolic distribution (MSGHD).

Description

Compute the density of a p dimensional mulitple-scaled generalized hyperbolic distribution.

Usage

dMSGHD(data,p,mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omegav=rep(1,p),
 lambdav=rep(0.5,p),gam=NULL,phi=NULL,log=FALSE)dMSGHD(data,p,mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omegav=rep(1,p),
 lambdav=rep(0.5,p),gam=NULL,phi=NULL,log=FALSE)

Arguments

`data`	n x p data set
`p`	number of variables.
`mu`	(optional) the p dimensional mean
`alpha`	(optional) the p dimensional skewness parameter alpha
`sigma`	(optional) the p x p dimensional scale matrix
`omegav`	(optional) the p dimensional concentration parameter omega
`lambdav`	(optional) the p dimensional index parameter lambda
`gam`	(optional) the pxp gamma matrix
`phi`	(optional) the p dimensional vector phi
`log`	(optional) if TRUE returns the log of the density

Details

The default values are: 0 for the mean and the skweness parameter alpha, diag(p) for sigma, 1 for omega, and 0.5 for lambda.

Value

A n dimensional vector with the density from a multiple-scaled generilzed hyperbolic distribution

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

C. Tortora, B.C. Franczak, R.P. Browne, and P.D. McNicholas (2019). A Mixture of Coalesced Generalized Hyperbolic Distributions. Journal of Classification (to appear).

Examples




x = seq(-3,3,length.out=50)
y = seq(-3,3,length.out=50)
xyS1 = matrix(0,nrow=length(x),ncol=length(y))
for(i in 1:length(x)){
  for(j in 1:length(y)){
      xy <- matrix(cbind(x[i],y[j]),1,2)	
      xyS1[i,j] =  dMSGHD(xy,2) 
      
    }
  }
contour(x=x,y=y,z=xyS1, levels=seq(.005,.25,by=.005), main="MSGHD")

x = seq(-3,3,length.out=50)
y = seq(-3,3,length.out=50)
xyS1 = matrix(0,nrow=length(x),ncol=length(y))
for(i in 1:length(x)){
  for(j in 1:length(y)){
      xy <- matrix(cbind(x[i],y[j]),1,2)	
      xyS1[i,j] =  dMSGHD(xy,2) 
      
    }
  }
contour(x=x,y=y,z=xyS1, levels=seq(.005,.25,by=.005), main="MSGHD")

Mixture of coalesced generalized hyperbolic distributions (MCGHD).

Description

Carries out model-based clustering using the mixture of coalesced generalized hyperbolic distributions.

Usage

MCGHD(data=NULL,gpar0=NULL,G=2,max.iter=100,eps=1e-2,label=NULL,
	method="km",scale=TRUE,nr=10, modelSel="AIC")
MCGHD(data=NULL,gpar0=NULL,G=2,max.iter=100,eps=1e-2,label=NULL,
	method="km",scale=TRUE,nr=10, modelSel="AIC")

Arguments

`data`	A n x p matrix or data frame such that rows correspond to observations and columns correspond to variables.
`gpar0`	(optional) A list containing the initial parameters of the mixture model. See the 'Details' section.
`G`	The range of values for the number of clusters.
`max.iter`	(optional) A numerical parameter giving the maximum number of iterations each EM algorithm is allowed to use.
`eps`	(optional) A number specifying the epsilon value for the convergence criteria used in the EM algorithms. For each algorithm, the criterion is based on the difference between the log-likelihood at an iteration and an asymptotic estimate of the log-likelihood at that iteration. This asymptotic estimate is based on the Aitken acceleration.
`label`	( optional) A n dimensional vector, if label[i]=k then observation i belongs to group k, If label[i]=0 then observation i has no known group, if NULL then the data has no known groups.
`method`	( optional) A string indicating the initialization criterion, if not specified kmeans clustering is used. Alternative methods are: hierarchical "hierarchical", random "random", and model based "modelBased"
`scale`	( optional) A logical value indicating whether or not the data should be scaled, true by default.
`nr`	( optional) A number indicating the number of starting value when random is used, 10 by default.
`modelSel`	( optional) A string indicating the model selection criterion, if not specified AIC is used. Alternative methods are: BIC,ICL, and AIC3

Details

The arguments gpar0, if specified, has to be a list structure containing as much element as the number of components G. Each element must include the following parameters: one p dimensional vector mu, alpha and phi, a pxp matrix gamma, a px2 vector cpl containing the vectors omega and lambda, and a 2-dimensional vector containing the omega0 and lambda0.

Value

A S4 object of class MixGHD with slots:

`index`	Value of the index used for model selection (AIC or ICL or BIC or AIC3) for each G,the index used is specified by the user, if not specified AIC is used.
`BIC`	Bayesian information criterion.
`ICL`	Integrated completed likelihood..
`AIC`	Akaike information criterion.
`AIC3`	Akaike information criterion 3.
`gpar`	A list of the model parameters in the rotated space.
`loglik`	The log-likelihood values.
`map`	A vector of integers indicating the maximum a posteriori classifications for the best model.
`par`	A list of the model parameters.
`z`	A matrix giving the raw values upon which map is based.

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

Examples

##loading banknote data
data(banknote)

##model estimation
model=MCGHD(banknote[,2:7],G=2,max.iter=20)

#result
#summary(model)
#plot(model)
table(banknote[,1],model@map)##loading banknote data
data(banknote)

##model estimation
model=MCGHD(banknote[,2:7],G=2,max.iter=20)

#result
#summary(model)
#plot(model)
table(banknote[,1],model@map)

Mixture of generalized hyperbolic distributions (MGHD).

Description

Carries out model-based clustering and classification using the mixture of generalized hyperbolic distributions.

Usage

MGHD(data=NULL,gpar0=NULL,G=2,max.iter=100,label=NULL,eps=1e-2,
method="kmeans",scale=TRUE,nr=10, modelSel="AIC")
MGHD(data=NULL,gpar0=NULL,G=2,max.iter=100,label=NULL,eps=1e-2,
method="kmeans",scale=TRUE,nr=10, modelSel="AIC")

Arguments

`data`	A n x p matrix or data frame such that rows correspond to observations and columns correspond to variables.
`gpar0`	(optional) A list containing the initial parameters of the mixture model. See the 'Details' section.
`G`	The range of values for the number of clusters.
`max.iter`	(optional) A numerical parameter giving the maximum number of iterations each EM algorithm is allowed to use.
`label`	( optional) A n dimensional vector, if label[i]=k then observation i belongs to group k, If label[i]=0 then observation i has no known group, if NULL then the data has no known groups.
`eps`	(optional) A number specifying the epsilon value for the convergence criteria used in the EM algorithms. For each algorithm, the criterion is based on the difference between the log-likelihood at an iteration and an asymptotic estimate of the log-likelihood at that iteration. This asymptotic estimate is based on the Aitken acceleration.
`method`	( optional) A string indicating the initialization criterion, if not specified kmeans clustering is used. Alternative methods are: hierarchical "hierarchical", random "random", and model based "modelBased" clustering
`scale`	( optional) A logical value indicating whether or not the data should be scaled, true by default.
`nr`	( optional) A number indicating the number of starting value when random is used, 10 by default.
`modelSel`	( optional) A string indicating the model selection criterion, if not specified AIC is used. Alternative methods are: BIC,ICL, and AIC3

Details

The arguments gpar0, if specified, is a list structure containing at least one p dimensional vector mu, and alpha, a pxp matrix sigma, and a 2 dimensional vector containing omega and lambda.

Value

A S4 object of class MixGHD with slots:

`index`	Value of the index used for model selection (AIC or ICL or BIC or AIC3) for each G,the index used is specified by the user, if not specified AIC is used.
`BIC`	Bayesian information criterion.
`ICL`	Integrated completed likelihood..
`AIC`	Akaike information criterion.
`AIC3`	Akaike information criterion 3.
`gpar`	A list of the model parameters.
`loglik`	The log-likelihood values.
`map`	A vector of integers indicating the maximum a posteriori classifications for the best model.
`z`	A matrix giving the raw values upon which map is based.

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

R.P. Browne, and P.D. McNicholas (2015). A Mixture of Generalized Hyperbolic Distributions. Canadian Journal of Statistics, 43.2 176-198\ C. Tortora, R. P. Browne, A. ElSherbiny, B. C. Franczak, and P. D. McNicholas (2021). Model-Based Clustering, Classification, and Discriminant Analysis using the Generalized Hyperbolic Distribution: MixGHD R package, Journal of Statistical Software 98(3) 1–24, <doi:10.18637/jss.v098.i03>.

Examples

##loading crabs data
data(crabs)

##model estimation
model=MGHD(data=crabs[,4:8],  G=2   )

#result
plot(model)
table(model@map, crabs[,2])

## Classification
##loading bankruptcy data
data(bankruptcy)
#70% belong to the training set
 label=bankruptcy[,1]
#for a Classification porpuse the label cannot be 0
 label[1:33]=2
 a=round(runif(20)*65+1)
 label[a]=0
 
 
##model estimation
model=MGHD(data=bankruptcy[,2:3],  G=2, label=label )

#result
table(model@map,bankruptcy[,1])
plot(model)##loading crabs data
data(crabs)

##model estimation
model=MGHD(data=crabs[,4:8],  G=2   )

#result
plot(model)
table(model@map, crabs[,2])

## Classification
##loading bankruptcy data
data(bankruptcy)
#70% belong to the training set
 label=bankruptcy[,1]
#for a Classification porpuse the label cannot be 0
 label[1:33]=2
 a=round(runif(20)*65+1)
 label[a]=0
 
 
##model estimation
model=MGHD(data=bankruptcy[,2:3],  G=2, label=label )

#result
table(model@map,bankruptcy[,1])
plot(model)

Mixture of generalized hyperbolic factor analyzers (MGHFA).

Description

Carries out model-based clustering and classification using the mixture of generalized hyperbolic factor analyzers.

Usage

MGHFA(data=NULL, gpar0=NULL, G=2, max.iter=100, 
label =NULL  ,q=2,eps=1e-2 , method="kmeans", scale=TRUE ,nr=10)
MGHFA(data=NULL, gpar0=NULL, G=2, max.iter=100, 
label =NULL  ,q=2,eps=1e-2 , method="kmeans", scale=TRUE ,nr=10)

Arguments

`data`	A matrix or data frame such that rows correspond to observations and columns correspond to variables.
`gpar0`	(optional) A list containing the initial parameters of the mixture model. See the 'Details' section.
`G`	The range of values for the number of clusters.
`max.iter`	(optional) A numerical parameter giving the maximum number of iterations each EM algorithm is allowed to use.
`label`	( optional) A n dimensional vector, if label[i]=k then observation i belongs to group k, If label[i]=0 then observation i has no known group, if NULL then the data has no known groups.
`q`	The range of values for the number of factors.
`eps`	(optional) A number specifying the epsilon value for the convergence criteria used in the EM algorithms. For each algorithm, the criterion is based on the difference between the log-likelihood at an iteration and an asymptotic estimate of the log-likelihood at that iteration. This asymptotic estimate is based on the Aitken acceleration.
`method`	( optional) A string indicating the initialization criterion, if not specified kmeans clustering is used. Alternative methods are: hierarchical "hierarchical" and model based "modelBased" clustering
`scale`	( optional) A logical value indicating whether or not the data should be scaled, true by default.
`nr`	( optional) A number indicating the number of starting value when random is used, 10 by default.

Details

The arguments gpar0, if specified, is a list structure containing at least one p dimensional vector mu, alpha and phi, a pxp matrix gamma, a 2 dimensional vector cpl containing omega and lambda.

Value

A S4 object of class MixGHD with slots:

`Index`	Bayesian information criterion value for each combination of G and q.
`BIC`	Bayesian information criterion.
`gpar`	A list of the model parameters.
`loglik`	The log-likelihood values.
`map`	A vector of integers indicating the maximum a posteriori classifications for the best model.
`z`	A matrix giving the raw values upon which map is based.

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

C.Tortora, P.D. McNicholas, and R.P. Browne (2016). Mixtures of Generalized Hyperbolic Factor Analyzers. Advanced in data analysis and classification 10(4) p.423-440.\ C. Tortora, R. P. Browne, A. ElSherbiny, B. C. Franczak, and P. D. McNicholas (2021). Model-Based Clustering, Classification, and Discriminant Analysis using the Generalized Hyperbolic Distribution: MixGHD R package, Journal of Statistical Software 98(3) 1–24, <doi:10.18637/jss.v098.i03>.

Examples

## Classification
#70% belong to the training set
data(sonar)
 label=sonar[,61]
 set.seed(4)
 a=round(runif(62)*207+1)
 label[a]=0
 
 
##model estimation
model=MGHFA(data=sonar[,1:60],  G=2, max.iter=25  ,q=2,label=label)

#result
table(model@map,sonar[,61])
summary(model)
## Classification
#70% belong to the training set
data(sonar)
 label=sonar[,61]
 set.seed(4)
 a=round(runif(62)*207+1)
 label[a]=0
 
 
##model estimation
model=MGHFA(data=sonar[,1:60],  G=2, max.iter=25  ,q=2,label=label)

#result
table(model@map,sonar[,61])
summary(model)

Class "MixGHD"

Description

This class pertains to results of the application of function MGHD, MSGHD, cMSGHD, MCGHD, and MGHFA.

Objects from the Class

Objects can be created as a result to a call to MGHD, MSGHD, cMSGHD, MCGHD, and MGHFA.

Slots

index: Value of the index used for model selection (AIC or ICL or BIC or AIC3) for each G,the index used is specified by the user, if not specified AIC is used.
BIC: Bayesian information criterion value.
ICL: ICL index.
AIC: AIC index.
AIC3: AIC3 index.
gpar: A list of the model parameters (in the rotated space for MCGHD).
loglik: The log-likelihood values.
map: A vector of integers indicating the maximum a posteriori classifications for the best model.
par: Only for MCGHD. A list of the model parameters.
z: A matrix giving the raw values upon which map is based.

Methods

plot

signature(x = "MixGHD") Provides plots of MixGHD-class by plotting the following elements:

the value of the log likelihood for each iteration.
Scatterplot of the data of all the possible couples of coordinates coloured according to the cluster. Only for less than 10 variables.
If the number of variables is two: scatterplot and contour plot of the data coloured according to the cluster

summary

summary(x = "MixGHD").

Provides a summary of MixGHD-class objects by printing the following elements:

The number components used for the model
BIC;
AIC;
AIC3;
ICL;
A table with the number of element in each cluster.

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

Examples

##loading bankruptcy data
data(bankruptcy)

##model estimation
#res=MCGHD(data=bankruptcy[,2:3],G=2,method="kmedoids",max.iter=30)
#result
#plot(res)
#summary(res)

##loading bankruptcy data
data(bankruptcy)

##model estimation
#res=MCGHD(data=bankruptcy[,2:3],G=2,method="kmedoids",max.iter=30)
#result
#plot(res)
#summary(res)

Class MixGHD.

Description

This class pertains to results of the application of function MGHD,MCGHD,MSGHD,cMSGHD.

Details

Plot the loglikhelyhood vale for each iteration of the EM algorithm. If p=2 it shows a contour plot. If 2<p<10 shows a splom of the data colored according to the cluster membership.

Slots

Index: Bayesian information criterion value for each combination of G and q.
BIC: Bayesian information criterion value.
gpar: A list of the model parameters.
loglik: The log-likelihood values.
map: A vector of integers indicating the maximum a posteriori classifications for the best model.
z: A matrix giving the raw values upon which map is based.
method: A string indicating the used method: MGHD, MGHFA, MSGHD, cMSGHD, MCGHD.
data: A matrix or data frame such that rows correspond to observations and columns correspond to variables.
par: (only for MCGHD)A list of the model parameters in the rotated space.

Methods

signature(x = "MixGHD", y = "missing"): S4 method for plotting objects of MixGHD-class.

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

Examples

##loading banknote data
data(bankruptcy)


##model estimation
model=MSGHD(bankruptcy[,2:3],G=2,max.iter=30)

#result
summary(model)
plot(model)##loading banknote data
data(bankruptcy)


##model estimation
model=MSGHD(bankruptcy[,2:3],G=2,max.iter=30)

#result
summary(model)
plot(model)

Mixture of multiple scaled generalized hyperbolic distributions (MSGHD).

Description

Carries out model-based clustering using the mixture of multiple scaled generalized hyperbolic distributions.

Usage

MSGHD(data=NULL,gpar0=NULL,G=2,max.iter=100,label=NULL,eps=1e-2,
	method="km",scale=TRUE,nr=10, modelSel="AIC")
MSGHD(data=NULL,gpar0=NULL,G=2,max.iter=100,label=NULL,eps=1e-2,
	method="km",scale=TRUE,nr=10, modelSel="AIC")

Arguments

`data`	A n x p matrix or data frame such that rows correspond to observations and columns correspond to variables.
`gpar0`	(optional) A list containing the initial parameters of the mixture model. See the 'Details' section.
`G`	The range of values for the number of clusters.
`max.iter`	(optional) A numerical parameter giving the maximum number of iterations each EM algorithm is allowed to use.
`label`	( optional) A n dimensional vector, if label[i]=k then observation i belongs to group k, If label[i]=0 then observation i has no known group, if NULL then the data has no known groups.
`eps`	(optional) A number specifying the epsilon value for the convergence criteria used in the EM algorithms. For each algorithm, the criterion is based on the difference between the log-likelihood at an iteration and an asymptotic estimate of the log-likelihood at that iteration. This asymptotic estimate is based on the Aitken acceleration.
`method`	( optional) A string indicating the initialization criterion, if not specified kmeans clustering is used. Alternative methods are: hierarchical "hierarchical", random "random", and model based "modelBased" clustering
`scale`	( optional) A logical value indicating whether or not the data should be scaled, true by default.
`nr`	( optional) A number indicating the number of starting value when random is used, 10 by default.
`modelSel`	( optional) A string indicating the model selection criterion, if not specified AIC is used. Alternative methods are: BIC,ICL, and AIC3

Details

Value

A S4 object of class MixGHD with slots:

`index`	Value of the index used for model selection (AIC or ICL or BIC or AIC3) for each G,the index used is specified by the user, if not specified AIC is used.
`BIC`	Bayesian information criterion.
`ICL`	Integrated completed likelihood.
`AIC`	Akaike information criterion.
`AIC3`	Akaike information criterion 3.
`gpar`	A list of the model parameters
`loglik`	The log-likelihood values.
`map`	A vector of integers indicating the maximum a posteriori classifications for the best model.
`z`	A matrix giving the raw values upon which map is based.

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

Examples

##loading banknote data
data(banknote)


##model estimation
model=MSGHD(banknote[,2:7],G=2,max.iter=30)

#result
table(banknote[,1],model@map)
summary(model)
plot(model)##loading banknote data
data(banknote)


##model estimation
model=MSGHD(banknote[,2:7],G=2,max.iter=30)

#result
table(banknote[,1],model@map)
summary(model)
plot(model)

Plot objects of class MixGHD.

Description

Plots the loglikelyhood function and for p<10 shows the splom of the data.

Usage

	## S4 method for signature 'MixGHD'
plot(x,y)
## S4 method for signature 'MixGHD'
plot(x,y)

Arguments

`x`	A object of `MixGHD-class`

;

`y`	Not used; for compatibility with generic plot.

Details

Plot the loglikhelyhood vale for each iteration of the EM algorithm. If p=2 it shows a contour plot. If 2<p<10 shows a splom of the data colored according to the cluster membership.

Methods

signature(x = "MixGHD", y = "missing"): S4 method for plotting objects of MixGHD-class.

Author(s)

Cristina Tortora. Maintainer: Cristina Tortora <[email protected]>

Examples

##loading banknote data
data(bankruptcy)


##model estimation
model=MCGHD(bankruptcy[,2:3],G=2,max.iter=30)

#result

plot(model)##loading banknote data
data(bankruptcy)


##model estimation
model=MCGHD(bankruptcy[,2:3],G=2,max.iter=30)

#result

plot(model)

Membership prediction for objects of class MixGHD

Description

Cluster membership

Usage

	## S4 method for signature 'MixGHD'
predict(object)

## S4 method for signature 'MixGHD'
predict(object)

Arguments

object

An S4 object of class MixGHD.

Value

The cluster membership

Author(s)

Cristina Tortora Maintainer: Cristina Tortora <[email protected]>

Examples

##loading bankruptcy data
data(bankruptcy)

##model estimation
res=MCGHD(data=bankruptcy[,2:3],G=2,method="kmedoids",max.iter=30)
#rcoefficients of the model
predict(res)

##loading bankruptcy data
data(bankruptcy)

##model estimation
res=MCGHD(data=bankruptcy[,2:3],G=2,method="kmedoids",max.iter=30)
#rcoefficients of the model
predict(res)

Pseudo random number generation from a coalesced generalized hyperbolic distribution (MSGHD).

Description

Generate n pseudo random numbers from a p dimensional coalesced generalized hyperbolic distribution.

Usage

rCGHD(n,p,mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omega=1,lambda=0.5
,omegav=rep(1,p),lambdav=rep(0.5,p),wg=0.5)rCGHD(n,p,mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omega=1,lambda=0.5
,omegav=rep(1,p),lambdav=rep(0.5,p),wg=0.5)

Arguments

`n`	number of observations.
`p`	number of variables.
`mu`	(optional) the p dimensional mean
`alpha`	(optional) the p dimensional skewness parameter alpha
`sigma`	(optional) the p x p dimensional scale matrix
`lambda`	(optional) the 1 dimensional index parameter lambda
`omega`	(optional) the 1 dimensional concentration parameter omega
`omegav`	(optional) the p dimensional concentration parameter omega
`lambdav`	(optional) the p dimensional index parameter lambda
`wg`	(optional) the weight

Details

The default values are: 0 for the mean and the skweness parameter alpha, diag(p) for sigma, 1 for omega, and 0.5 for lambda.

Value

A n times p matrix of numbers psudo randomly generated from a coalesced generilzed hyperbolic distribution

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

C. Tortora, B.C. Franczak, R.P. Browne, and P.D. McNicholas (2019). A Mixture of Coalesced Generalized Hyperbolic Distributions. Journal of Classification (to appear).

Examples




data=rCGHD(300,2,alpha=c(2,-2),omegav=c(2,2),omega=3)

plot(data)

data=rCGHD(300,2,alpha=c(2,-2),omegav=c(2,2),omega=3)

plot(data)

Pseudo random number generation from a generalized hyperbolic distribution (GHD).

Description

Generate n pseudo random numbers from a p dimensional generalized hyperbolic distribution.

Usage

rGHD(n,p, mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omega=1,lambda=0.5)rGHD(n,p, mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omega=1,lambda=0.5)

Arguments

`n`	number of observations.
`p`	number of variables.
`mu`	(optional) the p dimensional mean
`alpha`	(optional) the p dimensional skewness parameter alpha
`sigma`	(optional) the p x p dimensional scale matrix
`omega`	(optional) the unidimensional concentration parameter omega
`lambda`	(optional) the unidimensional index parameter lambda

Details

The default values are: 0 for the mean and the skweness parameter alpha, diag(p) for sigma, 1 for omega, and 0.5 for lambda.

Value

A n times p matrix of numbers psudo randomly generated from a generilzed hyperbolic distribution

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

R.P. Browne, and P.D. McNicholas (2015). A Mixture of Generalized Hyperbolic Distributions. Canadian Journal of Statistics, 43.2 176-198

Examples


data=rGHD(300,2,alpha=c(2,-2))

plot(data)
data=rGHD(300,2,alpha=c(2,-2))

plot(data)

Pseudo random number generation from a mulitple-scaled generalized hyperbolic distribution (MSGHD).

Description

Generate n pseudo random numbers from a p dimensional mulitple-scaled generalized hyperbolic distribution.

Usage

rMSGHD(n,p, mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omegav=rep(1,p),lambdav=rep(0.5,p))rMSGHD(n,p, mu=rep(0,p),alpha=rep(0,p),sigma=diag(p),omegav=rep(1,p),lambdav=rep(0.5,p))

Arguments

`n`	number of observations.
`p`	number of variables.
`mu`	(optional) the p dimensional mean
`alpha`	(optional) the p dimensional skewness parameter alpha
`sigma`	(optional) the p x p dimensional scale matrix
`omegav`	(optional) the p dimensional concentration parameter omega
`lambdav`	(optional) the p dimensional index parameter lambda

Details

The default values are: 0 for the mean and the skweness parameter alpha, diag(p) for sigma, 1 for omega, and 0.5 for lambda.

Value

A n times p matrix of numbers psudo randomly generated from a generilzed hyperbolic distribution

Author(s)

Cristina Tortora, Aisha ElSherbiny, Ryan P. Browne, Brian C. Franczak, and Paul D. McNicholas. Maintainer: Cristina Tortora <[email protected]>

References

C. Tortora, B.C. Franczak, R.P. Browne, and P.D. McNicholas (2019). A Mixture of Coalesced Generalized Hyperbolic Distributions. Journal of Classification (to appear).

Examples


data=rMSGHD(300,2,alpha=c(2,-2),omegav=c(2,2))

plot(data)
data=rMSGHD(300,2,alpha=c(2,-2),omegav=c(2,2))

plot(data)

Sonar data

Description

The data report the patterns obtained by bouncing sonar signals at various angles and under various conditions. There are 208 patterns in all, 111 obtained by bouncing sonar signals off a metal cylinder and 97 obtained by bouncing signals off rocks. Each pattern is a set of 60 numbers (variables) taking values between 0 and 1.

Usage

data(sonar)data(sonar)

Format

A data frame with 208 observations and 61 columns. The first 60 columns contain the variables. The 61st column gives the material: 1 rock, 2 metal.

Source

UCI machine learning repository

References

R.P. Gorman and T. J. Sejnowski (1988) Analysis of hidden units in a layered network trained to classify sonar targets. Neural Networks 1: 75-89

Plot objects of class MixGHD.

Description

Methods for function summary aimed at summarizing the S4 classes included in the MixGHD-package

Arguments

object

A object of MixGHD-class.

Methods

signature(object = "MixGHD"): S4 method for summaryzing objects of MixGHD-class.

Author(s)

Cristina Tortora. Maintainer: Cristina Tortora <[email protected]>

Examples

##loading banknote data
data(bankruptcy)


##model estimation
model=MSGHD(bankruptcy[,2:3],G=2,max.iter=30)

#result

summary(model)##loading banknote data
data(bankruptcy)


##model estimation
model=MSGHD(bankruptcy[,2:3],G=2,max.iter=30)

#result

summary(model)

Package 'MixGHD'

Help Index

Adjusted Rand Index.

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Swiss Banknote data

Description

Usage

Format

References

Bankruptcy data

Description

Usage

Format

References

Convex mixture of multiple scaled generalized hyperbolic distributions (cMSGHD).

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Coefficients for objects of class MixGHD

Description

Usage

Arguments

Value

Author(s)

Examples

Contour plot

Description

Usage

Arguments

Value

Author(s)

Examples

Discriminant analysis using the mixture of generalized hyperbolic distributions.

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Density of a coalesced generalized hyperbolic distribution (MSGHD).

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Density of a generalized hyperbolic distribution (GHD).

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Density of a mulitple-scaled generalized hyperbolic distribution (MSGHD).

Description

Usage

Arguments

Details

Value

Author(s)

References