Version: 1.0
Title: Tools to Work with the Flexible Dirichlet Distribution
Author: Sonia Migliorati [aut], Agnese Maria Di Brisco [aut, cre], Matteo Vestrucci [aut]
Maintainer: Agnese Maria Di Brisco <agnese.dibrisco@unimib.it>
Description: Provides tools to work with the Flexible Dirichlet distribution. The main features are an E-M algorithm for computing the maximum likelihood estimate of the parameter vector and a function based on conditional bootstrap to estimate its asymptotic variance-covariance matrix. It contains also functions to plot graphs, to generate random observations and to handle compositional data.
Depends: R (≥ 3.0.0)
Imports: stats, graphics, utils, grDevices
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
LazyData: true
NeedsCompilation: no
Packaged: 2015-08-07 08:47:02 UTC; everett
RoxygenNote: 6.0.0
Suggests: testthat
Repository: CRAN
Date/Publication: 2017-03-16 13:35:03

Information Criterions of a Flexible Dirichlet Model

Description

Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) of a fitted Flexible Dirichlet model. An Information Criterion for one fitted model object for which a log-likelihood value can be obtained is defined as -2*log-likelihood + k*npar, where npar represents the number of parameters in the fitted model, and k = 2 for AIC, or k = log(n) for BIC ( n being the number of observations).

Usage

FD.aicbic(x)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

See Also

FD.estimation, FD.stddev, FD.barycenters

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
FD.aicbic(results)

Amalgamation

Description

Given a matrix or a numeric dataframe, this function returns a composition where a set of specified columns is amalgamated together. The compositional operation of amalgamation provides sums of composition elements aimed at grouping homogeneous parts of the whole.

Usage

FD.amalgamation(data, columns, name = NULL)

Arguments

data

a matrix or a dataframe containing only variables to be transformed into compositional variables, after amalgamation.

columns

numeric vector containing the position of the columns to be amalgamated together.

name

string containing the name of the new column resulted from the amalgamation.

Details

Values must be positive. In case one row-entry (or more) is NA, the whole row will be returned as NA.

See Also

FD.generate, FD.subcomposition, FD.normalization

Examples

data(oliveoil)
dataoil <- oliveoil
head(dataoil)
data <- FD.normalization(dataoil[,3:10])
head(data)
data.sub <- FD.subcomposition(data,c(1,3,4,5))
head(data.sub)
data.amalg <- FD.amalgamation(data,c(2,6,7,8),name='others')
head(data.amalg)

Cluster Barycenters of a Flexible Dirichlet model

Description

Cluster barycenters of a fitted Flexible Dirichlet distribution.

Usage

FD.barycenters(x)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.clusterdistances, FD.moments

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
FD.barycenters(results)

Flexible Dirichlet Cluster Distances

Description

Returns a measure of symmetrized Kullback-Leibler distance between mixture component densities of a fitted Flexible Dirichlet distribution.

Usage

FD.clusterdistances(x)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.barycenters, FD.moments

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
FD.clusterdistances(results)

The Flexible Dirichlet Density Function

Description

Density function on the simplex for the Flexible Dirichlet distribution with parameters a, p and t.

Usage

FD.density(x, a, p, t)

Arguments

x

vector of a point on the simplex. It must sum to one.

a

vector of the non-negative alpha parameters.

p

vector of the clusters' probabilities. It must sum to one.

t

non-negative scalar tau parameter.

Details

Vectors x, a and p must be of the same length.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.theorcontours, FD.generate

Examples

x <- c(0.1,0.25,0.65)
alpha <- c(12,7,15)
prob <- c(0.3,0.4,0.3)
tau <- 8
FD.density(x,alpha,prob,tau)

Flexible Dirichlet Estimation

Description

Estimates the vector of parameters of a Flexible Dirichlet distribution through an EM-based maximum likelihood approach.

Usage

FD.estimation(data, normalize = F, iter.initial.SEM = 50,
  iter.final.EM = 100, verbose = T)

Arguments

data

a matrix or a dataframe containing only the variables in the model. Rows must sum to one, or normalize must be set TRUE.

normalize

if TRUE, each row of data will be divided by its own total to become a point of the simplex. Values in data must be positive.

iter.initial.SEM

number of iterations for the initial SEM step. Default to 50.

iter.final.EM

number of iterations for the final EM step. Default to 100.

verbose

if TRUE, the progression of the elaboration and the results will be printed on screen.

Details

The procedure is made up of four stages:

  1. Clustering: The algorithm applies many different clustering rules to the dataset, in order to exploit the specific cluster patterns that the parameter structure of the model involves.

  2. Labelling: Once the initial partitions are obtained, group labeling needs to be established because any clustering algorithm assigns the group labels randomly, but the FD cluster structure entails a precise labelling scheme.

  3. Initial SEM: A Stochastic E-M algorithm is applied at every initial partition and every possible label permutation identified.

  4. Final E-M: The previous step must be seen as a multiple initialization strategy. At this point only the best one is selected and a final E-M algorithm is used to find the point that maximizes the likelihood of the parameter vector.

Value

an object of class FDfitted. It's a list composed by:

alpha

Estimated values of the parameter vector Alpha

p

Estimated values of the parameter vector P

tau

Estimated value of the parameter Tau

logL

LogLikelihood

data

Normalized dataset

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.generate, FD.stddev, FD.aicbic, FD.barycenters, FD.ternaryplot, FD.rightplot, FD.marginalplot

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
summary(results)

The Flexible Dirichlet Random Generation

Description

Random generation from the Flexible Dirichlet distribution with parameters a, p and t.

Usage

FD.generate(n, a, p, t)

Arguments

n

number of points on the simplex to be generated.

a

vector of the non-negative alpha parameters.

p

vector of the clusters' probabilities. It must sum to one.

t

non-negative scalar tau parameter.

Details

Vectors a and p must be of the same length. The Flexible Dirichlet distribution derives from the normalization of a basis of positive dependent random variables obtained by starting from a basis of independent equally scaled gamma random variables, and randomly allocating to the i-th element a further independent gamma random variable.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, 1–21.

See Also

FD.estimation, FD.density, FD.theorcontours, FD.subcomposition, FD.amalgamation

Examples

n <- 100
alpha <- c(12,7,15)
prob <- c(0.3,0.4,0.3)
tau <- 8
data <- FD.generate(n,alpha,prob,tau)
data

Marginal Plot of a Flexible Dirichlet

Description

Histogram of the observed marginal variable and estimated density function of the marginal variable of a fitted Flexible Dirichlet distribution.

Usage

FD.marginalplot(x, var, zoomed = T, showgrid = T, showdata = T)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

var

position of the variable to be plotted.

zoomed

if TRUE, shows only the area where most of the density is concentrated. If FALSE, shows the whole range [0;1].

showgrid

if TRUE, shows the axis and the labels. If FALSE, only the graph is printed.

showdata

if TRUE, prints the histogram of the data. If FALSE, shows only the density function.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.ternaryplot, FD.rightplot

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
FD.marginalplot(results, var=2)
FD.marginalplot(results, var=2, zoomed=FALSE, showgrid=TRUE, showdata=FALSE)

Flexible Dirichlet Moments

Description

Moments of a fitted Flexible Dirichlet distribution. The function returns the mean and variance vectors and the covariance and correlation matrices.

Usage

FD.moments(x)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.barycenters, FD.clusterdistances

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
FD.moments(results)

Normalization

Description

Given a matrix or a numeric dataframe, this function returns a composition (i.e. data summing up to 1).

Usage

FD.normalization(data)

Arguments

data

a matrix or a dataframe containing only variables to be transformed into compositional variables.

Details

Values must be positive. In case one row-entry (or more) is NA, the whole row will be returned as NA.

See Also

FD.generate, FD.subcomposition, FD.amalgamation

Examples

data(oliveoil)
dataoil <- oliveoil
head(dataoil)
data <- FD.normalization(dataoil[,3:10])
head(data)
data.sub <- FD.subcomposition(data,c(1,3,4,5))
head(data.sub)
data.amalg <- FD.amalgamation(data,c(2,6,7,8),name='others')
head(data.amalg)

Right Triangle Plot of a Flexible Dirichlet

Description

Right triangle plot and contour lines of the density function of a fitted Flexible Dirichlet distribution.

Usage

FD.rightplot(x, var = c(1, 2), zoomed = T, showgrid = T, showdata = T,
  nlevels = 10)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

var

numeric vector containing the two variables to be plotted on the axis.

zoomed

if TRUE, shows only the area where most of the density is concentrated. If FALSE, shows the whole area of the right triangle.

showgrid

if TRUE, shows the axis and the labels. If FALSE, only the graph is printed.

showdata

if TRUE, prints the data points. If FALSE, shows only the contour lines.

nlevels

approximate number of contour lines to be drawn.

Details

The number of variables in the fitted model must be 3 to draw a plot on the right triangle.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.ternaryplot, FD.marginalplot

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
FD.rightplot(results)
FD.rightplot(results, var=c(3,2), zoomed=FALSE, showgrid=TRUE, showdata=FALSE, nlevels=3)

Standard Deviation of the ML estimators of a Flexible Dirichlet

Description

Conditional Bootstrap evaluation of the standard errors of the maximum likelihood parameter estimates of a Flexible Dirichlet distribution.

Usage

FD.stddev(x, iter.bootstrap = 500)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

iter.bootstrap

number of iterations of the Bootstrap.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.aicbic, FD.barycenters

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
FD.stddev(results)

Subcomposition

Description

Given a matrix or a numeric dataframe, this function returns a subcomposition made up of the specified columns.

Usage

FD.subcomposition(data, columns)

Arguments

data

a matrix or a dataframe containing only variables in the model.

columns

numeric vector containing the position of the columns to keep in the new composition.

Details

Values must be positive. In case one row-entry (or more) is NA, the whole row will be returned as NA.

See Also

FD.generate, FD.amalgamation, FD.normalization

Examples

data(oliveoil)
dataoil <- oliveoil
head(dataoil)
data <- FD.normalization(dataoil[,3:10])
head(data)
data.sub <- FD.subcomposition(data,c(1,3,4,5))
head(data.sub)
data.amalg <- FD.amalgamation(data,c(2,6,7,8),name='others')
head(data.amalg)

Ternary Plot of a Flexible Dirichlet

Description

Ternary plot and contour lines of the density function of a fitted Flexible Dirichlet distribution.

Usage

FD.ternaryplot(x, zoomed = T, showgrid = T, showdata = T, nlevels = 10)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

zoomed

if TRUE, shows only the area where most of the density is concentrated. If FALSE, shows the whole area of the ternary diagram.

showgrid

if TRUE, shows the axis and the labels. If FALSE, only the graph is printed.

showdata

if TRUE, prints the data points. If FALSE, shows only the contour lines.

nlevels

approximate number of contour lines to be drawn.

Details

The number of variables in the fitted model must be 3 to draw a ternary plot.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.rightplot, FD.marginalplot

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
FD.ternaryplot(results)
FD.ternaryplot(results, zoomed=FALSE, showgrid=TRUE, showdata=FALSE, nlevels=3)

Contour Lines of a Flexible Dirichlet

Description

Contour lines of a Flexible Dirichlet with given parameters on the ternary diagram or on the right triangle.

Usage

FD.theorcontours(a, p, t, type = "ternary", var = c(1, 2), zoomed = T,
  showgrid = T, nlevels = 10)

Arguments

a

vector of the non-negative alpha parameters.

p

vector of the clusters' probabilities. It must sum to one.

t

non-negative scalar tau parameter.

type

string indicating whether to plot the contour lines on a ternary diagram 'ternary', or on a right triangle plot 'right'.

var

numeric vector containing the two variables to be plotted on the axis. Used only if type='right'.

zoomed

if TRUE, shows only the area where most of the density is concentrated. If FALSE, shows the whole area.

showgrid

if TRUE, shows the axis and the labels. If FALSE, only the graph is printed.

nlevels

approximate number of contour lines to be drawn.

Details

The number of variables in the Flexible Dirichlet must be 3 to draw a plot. Vectors a and p must be of the same length.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.generate, FD.density

Examples

alpha <- c(12,7,15)
prob <- c(0.3,0.4,0.3)
tau <- 8
FD.theorcontours(alpha,prob,tau)
FD.theorcontours(alpha,prob,tau, type='right', var=c(3,2), zoomed=FALSE, showgrid=TRUE, nlevels=3)

Olive oil data

Description

This data set represents eight chemical measurements on different specimen of olive oil produced in various regions in Italy (northern Apulia, southern Apulia, Calabria, Sicily, inland Sardinia and coast Sardinia, eastern and western Liguria, Umbria) and further classifiable into three macro-areas: Centre-North, South, Sardinia.

Format

This data frame contains 572 rows, each corresponding to a different specimen of olive oil, and 10 columns. The first and the second column correspond to the macro-area and the region of origin of the olive oils respectively; here, the term 'region' refers to a geographical area and only partially to administrative borders. Columns 3-10 represent the following eight chemical measurements on the acid components for the oil specimens: palmitic, palmitoleic, stearic, oleic, linoleic, linolenic, arachidic, eicosenoic.

Source

Originally included in the package pdfCluster.

Plot Method for FDfitted Objects

Description

This method plots the results of FD.estimation, using the functions FD.ternaryplot or FD.rightplot.

Usage

## S3 method for class 'FDfitted'
plot(x, type = "ternary", var = c(1, 2), zoomed = T,
  showgrid = T, showdata = T, nlevels = 10, ...)

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

type

string containing 'ternary' or 'right'.

var

numeric vector containing the two variables to be plotted on the axis. Used only if type='right'.

zoomed

if TRUE, shows only the area where most of the density is concentrated. If FALSE, shows the whole area.

showgrid

if TRUE, shows the axis and the labels. If FALSE, only the graph is printed.

showdata

if TRUE, prints the data points. If FALSE, shows only the contour lines.

nlevels

approximate number of contour lines to be drawn.

...

additional arguments

Details

The number of variables in the fitted model must be 3 to draw a plot.

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.ternaryplot, FD.rightplot, FD.marginalplot

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
plot(results)
plot(results, type='right', var=c(3,2), zoomed=FALSE, showgrid=TRUE, showdata=FALSE, nlevels=3)

Print Method for FDfitted Objects

Description

This method shows the results of FD.estimation.

Usage

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

Arguments

x

an object of class FDfitted, usually the result of FD.estimation.

...

additional arguments

Summary Method for FDfitted Objects

Description

This method summarizes the results of FD.estimation, adding also information from the functions FD.stddev and FD.aicbic.

Usage

## S3 method for class 'FDfitted'
summary(object, ...)

Arguments

object

an object of class FDfitted, usually the result of FD.estimation.

...

additional arguments

Value

A list composed by:

par

Estimated parameter vector

sd

Vector of the estimated standard deviations

goodness

Vector containing LogLikelihood, AIC and BIC

References

Ongaro, A. and Migliorati, S. (2013) A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati, S., Ongaro, A. and Monti, G. S. (2016) A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, doi:10.1007/s11222-016-9665-y.

See Also

FD.estimation, FD.stddev, FD.aicbic

Examples

data <- FD.generate(n=20,a=c(12,7,15),p=c(0.3,0.4,0.3),t=8)
data
results <- FD.estimation(data, normalize=TRUE,iter.initial.SEM = 5,iter.final.EM = 10)
results
summary(results)