Package 'mixsmsn'

Title: Fitting Finite Mixture of Scale Mixture of Skew-Normal Distributions
Description: Functions to fit finite mixture of scale mixture of skew-normal (FM-SMSN) distributions, details in Prates, Lachos and Cabral (2013) <doi: 10.18637/jss.v054.i12>, Cabral, Lachos and Prates (2012) <doi:10.1016/j.csda.2011.06.026> and Basso, Lachos, Cabral and Ghosh (2010) <doi:10.1016/j.csda.2009.09.031>.
Authors: Marcos Prates [aut, cre, trl], Victor Lachos [aut], Celso Cabral [aut]
Maintainer: Marcos Prates <[email protected]>
License: GPL (>= 2.0)
Version: 1.1-10
Built: 2024-11-29 08:47:29 UTC
Source: CRAN

Help Index

Body Mass Index

Description

The data set has the measure of the Body Mass Index (bmi) for 2107 people.

Usage

data(bmi)

Format

A data frame with 2107 observations of bmi

Source

Rodrigo M. Basso, Victor H. Lachos, Celso R. B. Cabral, Pulak Ghosh (2009). "Robust mixture modeling based on scale mixtures of skew-normal distributions". Computational Statistics and Data Analysis (in press). doi: 10.1016/j.csda.2009.09.031

References

Marcos Oliveira Prates, Celso Romulo Barbosa Cabral, Victor Hugo Lachos (2013)."mixsmsn: Fitting Finite Mixture of Scale Mixture of Skew-Normal Distributions". Journal of Statistical Software, 54(12), 1-20., URL https://doi.org/10.18637/jss.v054.i12.

Examples

## Not run: 
data(bmi)
y <-bmi$bmi

hist(y,breaks=40)

## Maximum likelihood estimaton (MLE) with generated values
bmi.analysis <- smsn.mix(y, nu = 3, g = 2, get.init = TRUE, criteria = TRUE, 
                         group = TRUE, calc.im=TRUE)
mix.hist(y,bmi.analysis)

## Passing initial values to MLE
mu1 <- 20; mu2 <- 35
sigma2.1 <- 9; sigma2.2 <- 9;
lambda1 <- 0; lambda2 <- 0;
pii<- c(0.5,0.5)

mu <- c(mu1,mu2)
sigma2 <- c(sigma2.1,sigma2.2)
shape <- c(lambda1,lambda2)

bmi.analysis <- smsn.mix(y, nu = 3, mu, sigma2 , shape, pii, get.init = FALSE,
                         criteria = TRUE, group = TRUE, calc.im=FALSE)
mix.hist(y,bmi.analysis)

## Calculate the information matrix (when the calc.im option in smsn.mix is set FALSE)
bmi.im <-  im.smsn(y, bmi.analysis)

## Search for the best number of clusters from g=1 to g=5
bmi.analysis <- smsn.search(y, nu = 3, g.min = 1, g.max=5)
mix.hist(y,bmi.analysis$best.model)

## End(Not run)

Old Faithful Geyser Data

Description

Waiting time between eruptions and the duration of the eruption for the Old Faithful geyser in Yellowstone National Park, Wyoming, USA.

Usage

data(faithful)

Format

A data frame with 272 observations on 2 variables (p=2)

Source

H?rdle, W. (1991) "Smoothing Techniques with Implementation in S". New York: Springer.

Azzalini, A. and Bowman, A. W. (1990). "A look at some data on the Old Faithful geyser". Applied Statistics 39, 357–365.

References

Marcos Oliveira Prates, Celso Romulo Barbosa Cabral, Victor Hugo Lachos (2013)."mixsmsn: Fitting Finite Mixture of Scale Mixture of Skew-Normal Distributions". Journal of Statistical Software, 54(12), 1-20., URL https://doi.org/10.18637/jss.v054.i12.

Examples

## Not run: 
data(faithful)

## Maximum likelihood estimaton (MLE) for the multivariate FM-SMSN distribution
## with generated values
## Normal
Norm.analysis <- smsn.mmix(faithful, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                           group = TRUE, family = "Normal")
mix.contour(faithful,Norm.analysis,x.min=1,x.max=1,y.min=15,y.max=10,
            levels = c(0.1, 0.015, 0.005, 0.0009, 0.00015))

## Calculate the information matrix (when the calc.im option in smsn.mmix is set FALSE)
Norm.im <-  imm.smsn(faithful, Norm.analysis)

## Skew-Normal
Snorm.analysis <- smsn.mmix(faithful, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                           group = TRUE, family = "Skew.normal")
mix.contour(faithful,Snorm.analysis,x.min=1,x.max=1,y.min=15,y.max=10,
            levels = c(0.1, 0.015, 0.005, 0.0009, 0.00015))

## Calculate the information matrix (when the calc.im option in smsn.mmix is set FALSE)
Snorm.im <-  imm.smsn(faithful, Snorm.analysis)

## Skew-t
St.analysis <- smsn.mmix(faithful, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                           group = TRUE, family = "Skew.t")
mix.contour(faithful,St.analysis,x.min=1,x.max=1,y.min=15,y.max=10,
            levels = c(0.1, 0.015, 0.005, 0.0009, 0.00015))

## Calculate the information matrix (when the calc.im option in smsn.mmix is set FALSE)
St.im <-  imm.smsn(faithful, St.analysis)

## Passing initial values to MLE and automaticaly calculate the information matrix
mu1 <- c(5,77)
Sigma1 <- matrix(c(0.18,0.60,0.60,41), 2,2)
shape1 <- c(0.69,0.64)

mu2 <- c(2,52)
Sigma2 <- matrix(c(0.15,1.15,1.15,40), 2,2)
shape2 <- c(4.3,2.7)

pii<-c(0.65,0.35)

mu <- list(mu1,mu2)
Sigma <- list(Sigma1,Sigma2)
shape <- list(shape1,shape2)

Snorm.analysis <- smsn.mmix(faithful, nu=3, mu=mu, Sigma=Sigma, shape=shape, pii=pii,
                            g=2, get.init = FALSE, group = TRUE,
                            family = "Skew.normal", calc.im=TRUE)
                            
mix.contour(faithful,Snorm.analysis,x.min=1,x.max=1,y.min=15,y.max=10,
            levels = c(0.1, 0.015, 0.005, 0.0009, 0.00015))

## Search for the best number of clusters from g=1 to g=3
faithful.analysis <- smsn.search(faithful, nu = 3, g.min = 1, g.max=3)
mix.contour(faithful,faithful.analysis$best.model,x.min=1,x.max=1,
            y.min=15,y.max=10,levels = c(0.1, 0.015, 0.005, 0.0009,
            0.00015)) 

## End(Not run)

Information matrix

Description

Calculate the information matrix of returned analysis based on the model family choice (univariate case, p=1).

Usage

im.smsn(y, model)

Arguments

y

the response vector

model

a variable returned by smsn.mix

Value

Estimate the Information Matrix of the parameters.

Author(s)

Marcos Prates [email protected], Victor Lachos [email protected] and Celso Cabral [email protected]

See Also

smsn.mix

Examples

## see \code{\link{bmi}}

Information matrix

Description

Calculate the information matrix of returned analysis based on the model family choice (multivariate case, p>=2).

Usage

imm.smsn(y, model)

Arguments

y

the response vector (p>2)
model

a variable returned by smsn.mmix

Value

Estimate the Information Matrix of the parameters. Note: In the Information Matrix the scale parameters estimates are relative to the entries of square root matrix of Sigma.

Author(s)

Marcos Prates [email protected], Victor Lachos [email protected] and Celso Cabral [email protected]

See Also

smsn.mmix

Examples

## see \code{\link{faithful}}

Print the selected groups with contours

Description

Plot the contour of the observations with the group selection.

Usage

mix.contour(y, model,
            slice=100, ncontour=10,
            x.min=1, x.max=1,
            y.min=1,y.max=1,
            ...)

Arguments

y

the response matrix (dimension nx2)

model

a variable returned by smsn.mmix

slice

number of slices in the sequenceo the contour

ncontour

number of contours to be ploted

x.min

value to be subtracted of the smallest observation in the x-axis

x.max

value to be added of the biggest observation in the x-axis

y.min

value to be subtracted of the smallest observation in the y-axis

y.max

value to be added of the biggest observation in the y-axis

...

further arguments to contour

See Also

smsn.mmix

Examples

## see \code{\link{smsn.mmix}}

Estimated densities

Description

Plot the estimated density or log-density (univariate case, p=1).

Usage

mix.dens(y, model, log=FALSE, ylab=NULL, xlab = NULL, main = NULL, ...)

Arguments

y

the response vector

model

a variable returned by smsn.mix

log

Logical, plot log-density if TRUE (default = FALSE)

ylab

Title of the ylab, if NULL default is selected

xlab

Title of the xlab, if NULL default is selected

main

Main Title, if NULL default is selected

...

further arguments to plot

See Also

smsn.mix

Examples

## see \code{\link{bmi}} and \code{\link{smsn.mix}}

Estimated densities

Description

Plot the histogram along with the estimated density (univariate case, p=1).

Usage

mix.hist(y, model, breaks, main, col.hist, col.dens, ...)

Arguments

y

the response vector

model

a variable returned by smsn.mix

breaks

the same option in histogram

main

the same option in histogram

col.hist

change the color of the histogram bars

col.dens

change the color of the density curve

...

further arguments to hist

See Also

smsn.mix

Examples

## see \code{\link{bmi}} and \code{\link{smsn.mix}}

Plot lines of smsn densities

Description

Add lines of smsn estimated denisty or log-density in mix.dens plots (univariate case, p=1).

Usage

mix.lines(y, model, log=FALSE, ...)

Arguments

y

the response vector

model

a variable returned by smsn.mix

log

Logical, plot log-density if TRUE (default = FALSE)

...

further arguments to lines

See Also

smsn.mix

Examples

## see \code{\link{bmi}} and \code{\link{smsn.mix}}

Printing mix object

Description

Printing a smsn.mix object (univariate case, p=1)

Usage

mix.print(model, digits = 3, ...)

Arguments

model

an object of class snsm.mix, see smsn.mix for details
digits

rounding for tabular output on the console (default is to round to 3 decimal place)
...

further arguments to print

See Also

smsn.mix

Random univariate FM-SMSN generator

Description

Random generator of univariate FM-SMSN distributions.

Usage

rmix(n, pii, family, arg, cluster=FALSE)

Arguments

n

number of observations

pii

a vector of weights for the mixture (dimension of the number g of clusters). Must sum to one!
family

distribution family to be used in fitting ("t", "Skew.t", "Skew.cn", "Skew.slash", "Skew.normal", "Normal")

arg

a list with each entry containing a vector of size equal to the number of clusters of the necessary parameters from a family

cluster

TRUE or FALSE if the true observations clusters must be returned.

Author(s)

Marcos Prates [email protected], Victor Lachos [email protected] and Celso Cabral [email protected]

See Also

smsn.mix

Examples

## see \code{\link{smsn.mix}}

Random multivariate FM-SMSN generator

Description

Random generator of multivariate FM-SMSN distributions.

Usage

rmmix(n, pii, family, arg, cluster=FALSE)

Arguments

n

number of observations

pii

a vector of weights for the mixture (dimension of the number g of clusters). Must sum to one!
family

distribution family to be used in fitting ("t", "Skew.t", "Skew.cn", "Skew.slash", "Skew.normal", "Normal")

arg

a list of g lists with each list containing the necessary parameters of the selected family

cluster

TRUE or FALSE if the true observations clusters must be returned.

Author(s)

Marcos Prates [email protected], Victor Lachos [email protected] and Celso Cabral [email protected]

See Also

smsn.mmix

Examples

## see \code{\link{smsn.mmix}}

Fit univariate FM-SMSN distribution

Description

Return EM algorithm output for FM-SMSN distributions (univaritate case, p=1).

Usage

smsn.mix(y, 
         nu, mu = NULL, sigma2 = NULL, shape = NULL, pii = NULL,
         g = NULL, get.init = TRUE,
         criteria = TRUE, group = FALSE, family = "Skew.normal",
         error = 0.00001, iter.max = 100, calc.im = TRUE, obs.prob = FALSE,
         kmeans.param = NULL)

Arguments

y

the response vector

nu

the parameter of the scale variable (vector or scalar) of the SMSN family (kurtosis parameter). It is necessary to all distributions. For the "Skew.cn" must be a vector of length 2 and values in (0,1)

mu

the vector of initial values (dimension g) for the location parameters

sigma2

the vector of initial values (dimension g) for the scale parameters

shape

the vector of initial values (dimension g) for the skewness parameters

pii

the vector of initial values (dimension g) for the weights for each cluster. Must sum one!

g

the number of cluster to be considered in fitting
get.init

if TRUE, the initial values are generated via k-means
criteria

if TRUE, AIC, DIC, EDC and ICL will be calculated

group

if TRUE, the vector with the classification of the response is returned

family

distribution family to be used in fitting ("Skew.t", "t", "Skew.cn", "Skew.slash", "slash", "Skew.normal", "Normal")

error

the covergence maximum error

iter.max

the maximum number of iterations of the EM algorithm. Default = 100

calc.im

if TRUE, the information matrix is calculated and the standard errors are reported
obs.prob

if TRUE, the posterior probability of each observation belonging to one of the g groups is reported
kmeans.param

a list with alternative parameters for the kmeans function when generating initial values, list(iter.max = 10, n.start = 1, algorithm = "Hartigan-Wong")

Value

Estimated values of the location, scale, skewness and kurtosis parameter.

Author(s)

Marcos Prates [email protected], Victor Lachos [email protected] and Celso Cabral [email protected]

References

Rodrigo M. Basso, Victor H. Lachos, Celso R. B. Cabral, Pulak Ghosh (2010). "Robust mixture modeling based on scale mixtures of skew-normal distributions". Computational Statistics and Data Analysis, 54, 2926-2941. doi: 10.1016/j.csda.2009.09.031

Marcos Oliveira Prates, Celso Romulo Barbosa Cabral, Victor Hugo Lachos (2013)."mixsmsn: Fitting Finite Mixture of Scale Mixture of Skew-Normal Distributions". Journal of Statistical Software, 54(12), 1-20., URL https://doi.org/10.18637/jss.v054.i12.

See Also

mix.hist, im.smsn and smsn.search

Examples

mu1 <- 5; mu2 <- 20; mu3 <- 35
sigma2.1 <- 9; sigma2.2 <- 16; sigma2.3 <- 9
lambda1 <- 5; lambda2 <- -3; lambda3 <- -6
nu = 5

mu <- c(mu1,mu2,mu3)
sigma2 <- c(sigma2.1,sigma2.2,sigma2.3)
shape <- c(lambda1,lambda2,lambda3)
pii <- c(0.5,0.2,0.3)

arg1 = c(mu1, sigma2.1, lambda1, nu)
arg2 = c(mu2, sigma2.2, lambda2, nu)
arg3 = c(mu3, sigma2.3, lambda3, nu)
y <- rmix(n=1000, p=pii, family="Skew.t", arg=list(arg1,arg2,arg3))

## Not run: 
par(mfrow=c(1,2))
## Normal fit
Norm.analysis <- smsn.mix(y, nu = 3, g = 3, get.init = TRUE, criteria = TRUE, 
                          group = TRUE, family = "Normal", calc.im=FALSE)
mix.hist(y,Norm.analysis)
mix.print(Norm.analysis)
mix.dens(y,Norm.analysis)

## Skew Normal fit
Snorm.analysis <- smsn.mix(y, nu = 3, g = 3, get.init = TRUE, criteria = TRUE, 
                           group = TRUE, family = "Skew.normal", calc.im=FALSE)
mix.hist(y,Snorm.analysis)
mix.print(Snorm.analysis)
mix.dens(y,Snorm.analysis)

## t fit
t.analysis <- smsn.mix(y, nu = 3, g = 3, get.init = TRUE, criteria = TRUE, 
                        group = TRUE, family = "t", calc.im=FALSE)
mix.hist(y,t.analysis)
mix.print(t.analysis)
mix.dens(y,t.analysis)

## Skew t fit
St.analysis <- smsn.mix(y, nu = 3, g = 3, get.init = TRUE, criteria = TRUE, 
                        group = TRUE, family = "Skew.t", calc.im=FALSE)
mix.hist(y,St.analysis)
mix.print(St.analysis)
mix.dens(y,St.analysis)

## Skew Contaminated Normal fit
Scn.analysis <- smsn.mix(y, nu = c(0.3,0.3), g = 3, get.init = TRUE, criteria = TRUE, 
                         group = TRUE, family = "Skew.cn", calc.im=FALSE)
mix.hist(y,Scn.analysis)
mix.print(Scn.analysis)
mix.dens(y,Scn.analysis)

par(mfrow=c(1,1))
mix.dens(y,Norm.analysis)
mix.lines(y,Snorm.analysis,col="green")
mix.lines(y,t.analysis,col="red")
mix.lines(y,St.analysis,col="blue")
mix.lines(y,Scn.analysis,col="grey")

## End(Not run)

Fit multivariate FM-SMSN distributions.

Description

Return EM algorithm output for multivariate FM-SMSN distributions.

Usage

smsn.mmix(y, nu=1,
          mu = NULL, Sigma = NULL, shape = NULL, pii = NULL,
          g = NULL, get.init = TRUE, criteria = TRUE,
          group = FALSE, family = "Skew.normal", 
          error = 0.0001, iter.max = 100, uni.Gama = FALSE,
          calc.im=FALSE, obs.prob = FALSE, kmeans.param = NULL)

Arguments

y

the response matrix (dimension nxp)

nu

the parameter of the scale variable (vector or scalar) of the SMSN family (kurtosis parameter). It is necessary to all distributions. For the "Skew.cn" must be a vector of length 2 and values in (0,1)

mu

a list of g arguments of vectors of initial values (dimension p) for the location parameters

Sigma

a list of g arguments of matrices of initial values (dimension pxp) for the scale parameters

shape

a list of g arguments of vectors of initial values (dimension p)for the skewness parameters

pii

the vector of initial values (dimension g) for the weights for each cluster. Must sum one!

g

the number of cluster to be considered in fitting
get.init

if TRUE, the initial values are generated via k-means
criteria

if TRUE, log-likelihood (logLik), AIC, DIC, EDC and ICL will be calculated

group

if TRUE, the vector with the classification of the response is returned

family

distribution famility to be used in fitting ("Skew.t", "t", "Skew.cn", "Skew.slash", "slash", "Skew.normal", "Normal")

error

the covergence maximum error

iter.max

the maximum number of iterations of the EM algorithm. Default = 100

uni.Gama

if TRUE, the Gamma parameters are restricted to be the same for all clusters
calc.im

if TRUE, the information matrix is calculated and the starndard erros are reported

obs.prob

if TRUE, the posterior probability of each observation belonging to one of the g groups is reported
kmeans.param

a list with alternative parameters for the kmeans function when generating initial values, list(iter.max = 10, n.start = 1, algorithm = "Hartigan-Wong")

Value

Estimated values of the location, scale, skewness and kurtosis parameter. Note: The scale parameters estimated are relative to the entries of the squae root matrix of Sigma.

Author(s)

Marcos Prates [email protected], Victor Lachos [email protected] and Celso Cabral [email protected]

References

Cabral, C. R. B., Lachos, V. H. and Prates, M. O. (2012). "Multivariate Mixture Modeling Using Skew-Normal Independent Distributions". Computational Statistics & Data Analysis, 56, 126-142, doi:10.1016/j.csda.2011.06.026.

Marcos Oliveira Prates, Celso Romulo Barbosa Cabral, Victor Hugo Lachos (2013)."mixsmsn: Fitting Finite Mixture of Scale Mixture of Skew-Normal Distributions". Journal of Statistical Software, 54(12), 1-20., URL https://doi.org/10.18637/jss.v054.i12.

See Also

mix.contour, rmmix and smsn.search

Examples

mu1 <- c(0,0)
Sigma1 <- matrix(c(3,1,1,3), 2,2)
shape1 <-c(4,4)
nu1 <- 4

mu2 <- c(5,5)
Sigma2 <- matrix(c(2,1,1,2), 2,2)
shape2 <-c(2,2)
nu2 <- 4

pii<-c(0.6,0.4)

arg1 = list(mu=mu1, Sigma=Sigma1, shape=shape1, nu=nu1)
arg2 = list(mu=mu2, Sigma=Sigma2, shape=shape2, nu=nu2)
y <- rmmix(n= 500, p = pii, "Skew.t", list(arg1,arg2))

## Not run: 

## Normal fit giving intial values
mu <- list(mu1,mu2)
Sigma <- list(Sigma1,Sigma2)
shape <- list(shape1,shape2)
pii <- c(0.6,0.4)

Norm.analysis <- smsn.mmix(y, nu=3, mu=mu, Sigma=Sigma, shape=shape, pii = pii,
                           criteria = TRUE, g=2, get.init = FALSE, group = TRUE,
                           family = "Normal")
mix.contour(y,Norm.analysis)

## Normal fit 
Norm.analysis <- smsn.mmix(y, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                           group = TRUE, family = "Normal")
mix.contour(y,Norm.analysis)

## Normal fit with a unique Gamma
Norm.analysis <- smsn.mmix(y, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                           group = TRUE, family = "Normal", uni.Gama = TRUE)
mix.contour(y,Norm.analysis)


## Skew Normal fit
Snorm.analysis <- smsn.mmix(y, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                            group = TRUE, family = "Skew.normal")
mix.contour(y,Snorm.analysis)

## t fit
t.analysis <- smsn.mmix(y, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                         group = TRUE, family = "t")
mix.contour(y,t.analysis)

## Skew t fit
St.analysis <- smsn.mmix(y, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                         group = TRUE, family = "Skew.t")
mix.contour(y,St.analysis)

## Skew Contaminated Normal fit
Scn.analysis <- smsn.mmix(y, nu=c(0.1,0.1), g=2, get.init = TRUE, criteria = TRUE, 
                          group = TRUE, family = "Skew.cn",error=0.01)
mix.contour(y,Scn.analysis)

## Skew Contaminated Normal fit
Sslash.analysis <- smsn.mmix(y, nu=3, g=2, get.init = TRUE, criteria = TRUE, 
                             group = TRUE, family = "Skew.slash", error=0.1)
mix.contour(y,Sslash.analysis)


## End(Not run)

Find the best number of cluster for a determined data set.

Description

Search for the best fitting for number of cluster from g.min to g.max for a selected family and criteria for both univariate and multivariate distributions.

Usage

smsn.search(y, nu,
            g.min = 1, g.max = 3,
            family = "Skew.normal", criteria = "bic",
            error = 0.0001, iter.max = 100, 
            calc.im = FALSE, uni.Gama = FALSE, kmeans.param = NULL, ...)

Arguments

y

the response vector(matrix)

nu

the parameter of the scale variable (vector or scalar) of the SMSN family (kurtosis parameter). It is necessary to all distributions. For the "Skew.cn" must be a vector of length 2 and values in (0,1)

g.min

the minimum number of cluster to be modeled

g.max

the maximum number of cluster to be modeled

family

distribution famility to be used in fitting ("t", "Skew.t", "Skew.nc", "Skew.slash", "Skew.normal", "Normal")

criteria

the selection criteria method to be used ("aic", "bic", "edc", "icl")

error

the covergence maximum error

iter.max

the maximum number of iterations of the EM algorithm

calc.im

if TRUE, the infomation matrix is calculated and the starndard erros are reported

uni.Gama

if TRUE, the Gamma parameters are restricted to be the same for all clusters (Only valid in the multivariate case, p>1)

kmeans.param

a list with alternative parameters for the kmeans function when generating initial values, list(iter.max = 10, n.start = 1, algorithm = "Hartigan-Wong")
...

other parameters for the hist function

Value

Estimated values of the location, scale, skewness and kurtosis parameter from the optimum number of clusters.

Author(s)

Marcos Prates [email protected], Victor Lachos [email protected] and Celso Cabral [email protected]

See Also

smsn.mix and smsn.mmix

Examples

## see \code{\link{bmi}} and \code{\link{faithful}}