funHDDC (version 2.0)

funHDDC: Clustering Univariate and Multivariate Functional Data in Group-specific Functional Subspaces

Description

The funHDDC algorithm allows to cluster functional univariate and multivariate data by modeling each cluster within a specific functional subspace. See Bouveyron and Jacques (2011) <doi:10.1007/s11634-011-0095-6> and Schmutz et al. (2017) <hal:01652467> for details on the model parameters described below.

Usage

funHDDC(data,K=1:10,model="AkjBkQkDk",threshold=0.2,itermax=200,eps=1e-6,init="kmeans",
criterion="bic",algo='EM', d_select="Cattell", init.vector=NULL, show=TRUE,
mini.nb=c(5, 10),min.individuals=2, mc.cores=1, nb.rep=1, keepAllRes=TRUE,
kmeans.control = list(), d_max=100)

Arguments

data

In the univariate case: a functional data object produced by the fda package, in the multivariate case: a list of functional data objects.

K

The number of clusters, you can test one partition or multiple partitions at the same time with ':'

model

The chosen model among 'AkjBkQkDk', 'AkjBQkDk', 'AkBkQkDk', 'ABkQkDk', 'AkBQkDk', 'ABQkDk'. 'AkjBkQkDk' is the default.

threshold

The threshold of the Cattell' scree-test used for selecting the group-specific intrinsic dimensions.

itermax

The maximum number of iterations.

eps

The threshold of the convergence criterion.

init

A character string. It is the way to initialize the E-M algorithm. There are five ways of initialization: <U+201C>kmeans<U+201D> (default), <U+201C>param<U+201D>, <U+201C>random<U+201D>, <U+201C>mini-em<U+201D> or <U+201C>vector<U+201D>. See details for more information. It can also be directly initialized with a vector containing the prior classes of the observations.

criterion

The criterion used for model selection: bic (default) or icl. But we recommand using slopeHeuristic function provided in this package.

algo

A character string indicating the algorithm to be used. The available algorithms are the Expectation-Maximisation ("EM"), the Classification E-M ("CEM") and the Stochastic E-M ("SEM"). The default algorithm is the "EM".

d_select

Either <U+201C>Cattell<U+201D> (default) or <U+201C>BIC<U+201D>. This parameter selects which method to use to select the intrinsic dimensions of subgroups.

init.vector

A vector of integers or factors. It is a user-given initialization. It should be of the same length as of the data. Only used when init="vector".

show

Use show = FALSE to settle off the informations that may be printed.

mini.nb

A vector of integers of length two. This parameter is used in the <U+201C>mini-em<U+201D> initialization. The first integer sets how many times the algorithm is repeated; the second sets the maximum number of iterations the algorithm will do each time. For example, if init=<U+201C>mini-em<U+201D> and mini.nb=c(5,10), the algorithm wil be launched 5 times, doing each time 10 iterations; finally the algorithm will begin with the initialization that maximizes the log-likelihood.

min.individuals

This parameter is used to control for the minimum population of a class. If the population of a class becomes stricly inferior to 'min.individuals' then the algorithm stops and gives the message: 'pop<min.indiv.'. Here the meaning of "population of a class" is the sum of its posterior probabilities. The value of 'min.individuals' cannot be lower than 2.

mc.cores

Positive integer, default is 1. If mc.cores>1, then parallel computing is used, using mc.cores cores. Warning for Windows users only: the parallel computing can sometimes be slower than using one single core (due to how parLapply works).

nb.rep

A positive integer (default is 1 for kmeans initialization and 20 for random initialization). Each estimation (i.e. combination of (model, K, threshold)) is repeated nb.rep times and only the estimation with the highest log-likelihood is kept.

keepAllRes

Logical. Should the results of all runs be kept? If so, an argument all_results is created in the results. Default isTRUE.

kmeans.control

A list. The elements of this list should match the parameters of the kmeans initialization (see kmeans help for details). The parameters are <U+201C>iter.max<U+201D>, <U+201C>nstart<U+201D> and <U+201C>algorithm<U+201D>.

d_max

A positive integer. The maximum number of dimensions to be computed. Default is 100. It means that the instrinsic dimension of any cluster cannot be larger than dmax. It quickens a lot the algorithm for datasets with a large number of variables (e.g. thousands).

Value

d

The number of dimensions for each cluster.

a

Values of parameter a for each cluster.

b

Values of parameter b for each cluster.

mu

The mean of each cluster in the original space.

prop

The proportion of individuals in each cluster.

loglik

The maximum of log-likelihood.

loglik_all

The log-likelihood at each iteration.

posterior

The posterior probability for each individual to belong to each cluster.

class

The clustering partition.

BIC

The BIC value.

ICL

The ICL value.

complexity

the number of parameters that are estimated.

Details

If you choose init="random", the algorithm is run 20 times with the same model options and the solution which maximises the log-likelihood is printed.This explains why sometimes with this initialization it runs a bit slower than with 'kmeans' initialization.

If the Warning message : "In funHDDC(...) : All models diverged" is printed, it means that the algorithm found less classes that the number you choose (parameter K). Because of the EM algorithm, it should be because of a bad initialization of the EM algorithm. So you have to restart the algorithm multiple times in order to check if, with a new initialization of the EM algorithm the model converges, or if there is no solution with the number K you choose.

The different initializations are:

<U+201C>param<U+201D>:

it is initialized with the parameters, the means being generated by a multivariate normal distribution and the covariance matrix being common to the whole sample.

<U+201C>mini-em<U+201D>:

it is an initialization strategy, the classes are randomly initialized and the E-M algorithm makes several iterations, this action is repetead a few times (the default is 5 iterations and 10 times), at the end, the initialization choosen is the one which maximise the log-likelihood (see mini.nb for more information about its parametrization).

<U+201C>random<U+201D>:

the classes are randomly given using a multinomial distribution

<U+201C>kmeans<U+201D>:

the classes are initialized using the kmeans function (with: algorithm="Hartigan-Wong"; nstart=4; iter.max=50); note that the user can use his own arguments for kmeans using the dot-dot-dot argument

A prior class vector:

It can also be directly initialized with a vector containing the prior classes of the observations. To do so use init="vector" and provide the vector in the argument init.vector.

Note that the BIC criterion used in this function is to be maximized and is defined as 2*LL-k*log(n) where LL is the log-likelihood, k is the number of parameters and n is the number of observations.

References

- C. Bouveyron and J. Jacques, Model-based Clustering of Time Series in Group-specific Functional Subspaces, Advances in Data Analysis and Classification, vol. 5 (4), pp. 281-300, 2011. - A. Schmutz, C. Bouveyron, J. Jacques, L. Cheze and P. Martin, Clustering multivariate functional data in group-specic functional subspaces, Preprint HAL 01652467, Universit<U+00E9> C<U+00F4>te d'Azur, 2017.

Examples

# NOT RUN {

##Univariate example
library(fda)
data("trigo")
basis<- create.bspline.basis(c(0,1), nbasis=25)
var1<-smooth.basis(argvals=seq(0,1,length.out = 100),y=t(trigo[,1:100]),fdParobj=basis)$fd

res.uni<-funHDDC(var1,K=2,model="AkBkQkDk",init="kmeans",threshold=0.2)
plot(var1,col=res.uni$class)


##Multivariate example
library(fda)
data("triangle")
basis<- create.bspline.basis(c(1,21), nbasis=25)
var1<-smooth.basis(argvals=seq(1,21,length.out = 101),y=t(triangle[,1:101]),fdParobj=basis)$fd
var2<-smooth.basis(argvals=seq(1,21,length.out = 101),y=t(triangle[,102:202]),fdParobj=basis)$fd

res.multi<-funHDDC(list(var1,var2),K=3,model="AkBkQkDk",init="kmeans",threshold=0.2)
par(mfrow=c(1,2))
plot(var1,col=res.multi$class)
plot(var2,col=res.multi$class)


# }
# NOT RUN {
#You can test multiple number of groups at the same time
data("trigo")
basis<- create.bspline.basis(c(0,1), nbasis=25)
var1<-smooth.basis(argvals=seq(0,1,length.out = 100),y=t(trigo[,1:100]),fdParobj=basis)$fd
res.uni<-funHDDC(var1,K=2:4,model="AkBkQkDk",init="kmeans",threshold=0.2)

#You can test multiple models at the same time
data("trigo")
basis<- create.bspline.basis(c(0,1), nbasis=25)
var1<-smooth.basis(argvals=seq(0,1,length.out = 100),y=t(trigo[,1:100]),fdParobj=basis)$fd
res.uni<-funHDDC(var1,K=3,model=c("AkjBkQkDk","AkBkQkDk"),init="kmeans",threshold=0.2)

#Another example on Canadian data

#Clustering the "Canadian temperature" data (Ramsey & Silverman): univariate case
daybasis65 <- create.fourier.basis(c(0, 365), nbasis=65, period=365)
daytempfd <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"], daybasis65,
    fdnames=list("Day", "Station", "Deg C"))$fd

res.uni<-funHDDC(daytempfd,K=3,model="AkjBkQkDk",init="random",threshold=0.2)

#Graphical representation of groups mean curves
select1<-fd(daytempfd$coefs[,which(res.uni$class==1)],daytempfd$basis)
select2<-fd(daytempfd$coefs[,which(res.uni$class==2)],daytempfd$basis)
select3<-fd(daytempfd$coefs[,which(res.uni$class==3)],daytempfd$basis)

plot(mean.fd(select1),col="lightblue",ylim=c(-20,20),lty=1,lwd=3)
lines(mean.fd(select2),col="palegreen2",lty=1,lwd=3)
lines(mean.fd(select3),col="navy",lty=1,lwd=3)


#Clustering the "Canadian temperature" data (Ramsey & Silverman): multivariate case
daybasis65 <- create.fourier.basis(c(0, 365), nbasis=65, period=365)
daytempfd <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"], daybasis65,
    fdnames=list("Day", "Station", "Deg C"))$fd
dayprecfd<-smooth.basis(day.5, CanadianWeather$dailyAv[,,"Precipitation.mm"], daybasis65,
    fdnames=list("Day", "Station", "Mm"))$fd

res.multi<-funHDDC(list(daytempfd,dayprecfd),K=3,model="AkjBkQkDk",init="random",threshold=0.2)

#Graphical representation of groups mean curves
#Temperature
select1<-fd(daytempfd$coefs[,which(res.multi$class==1)],daytempfd$basis)
select2<-fd(daytempfd$coefs[,which(res.multi$class==2)],daytempfd$basis)
select3<-fd(daytempfd$coefs[,which(res.multi$class==3)],daytempfd$basis)

plot(mean.fd(select1),col="lightblue",ylim=c(-20,20),lty=1,lwd=3)
lines(mean.fd(select2),col="palegreen2",lty=1,lwd=3)
lines(mean.fd(select3),col="navy",lty=1,lwd=3)

#Precipitation
select1b<-fd(dayprecfd$coefs[,which(res.multi$class==1)],dayprecfd$basis)
select2b<-fd(dayprecfd$coefs[,which(res.multi$class==2)],dayprecfd$basis)
select3b<-fd(dayprecfd$coefs[,which(res.multi$class==3)],dayprecfd$basis)

plot(mean.fd(select1b),col="lightblue",ylim=c(0,5),lty=1,lwd=3)
lines(mean.fd(select2b),col="palegreen2",lty=1,lwd=3)
lines(mean.fd(select3b),col="navy",lty=1,lwd=3)
# }