Model-based clustering in the discriminative functional subspaces with the funFEM algorithm

Description

The package provides the funFEM algorithm (Bouveyron et al., 2014) which allows to cluster functional data by modeling the curves within a common and discriminative functional subspace.

Details

Author(s)

Charles Bouveyron

Maintainer: <charles.bouveyron@parisdescartes.fr>

References

C. Bouveyron, E. Côme and J. Jacques, The discriminative functional mixture model for the analysis of bike sharing systems, Preprint HAL n.01024186, University Paris Descartes, 2014.

Examples

# Clustering the well-known "Canadian temperature" data (Ramsay & Silverman)
basis <- create.bspline.basis(c(0, 365), nbasis=21, norder=4)
fdobj <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"],basis,
        fdnames=list("Day", "Station", "Deg C"))$fd
res = funFEM(fdobj,K=4)

# Visualization of the partition and the group means
par(mfrow=c(1,2))
plot(fdobj,col=res$cls,lwd=2,lty=1)
fdmeans = fdobj; fdmeans$coefs = t(res$prms$my)
plot(fdmeans,col=1:max(res$cls),lwd=2)

The funFEM algorithm for the clustering of functional data.

Description

The funFEM algorithm allows to cluster time series or, more generally, functional data. It is based on a discriminative functional mixture model which allows the clustering of the data in a unique and discriminative functional subspace. This model presents the advantage to be parsimonious and can therefore handle long time series.

Usage

funFEM(fd, K=2:6, model = "AkjBk", crit = "bic", init = "kmeans", Tinit = c(), maxit = 50,
  eps = 1e-06, disp = FALSE, lambda = 0, graph = FALSE)

Arguments

Value

A list is returned:

Author(s)

Charles Bouveyron

References

C. Bouveyron, E. Côme and J. Jacques, The discriminative functional mixture model for the analysis of bike sharing systems, Preprint HAL n.01024186, University Paris Descartes, 2014.

Examples

# Clustering the well-known "Canadian temperature" data (Ramsay & Silverman)
basis <- create.bspline.basis(c(0, 365), nbasis=21, norder=4)
fdobj <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"],basis,
        fdnames=list("Day", "Station", "Deg C"))$fd
res = funFEM(fdobj,K=4)

# Visualization of the partition and the group means
par(mfrow=c(1,2))
plot(fdobj); lines(fdobj,col=res$cls,lwd=2,lty=1)
fdmeans = fdobj; fdmeans$coefs = t(res$prms$my)
plot(fdmeans); lines(fdmeans,col=1:max(res$cls),lwd=2)

# Visualization in the discriminative subspace (projected scores)
par(mfrow=c(1,1))
plot(t(fdobj$coefs) %*% res$U,col=res$cls,pch=19,main="Discriminative space")


###############################################################################
# Analysis of the Velib data set

# Load the velib data and smoothing
data(velib)
basis<- create.fourier.basis(c(0, 181), nbasis=25)
fdobj <- smooth.basis(1:181,t(velib$data),basis)$fd

# Clustrering with FunFEM
res = funFEM(fdobj,K=6,model='AkjBk',init='kmeans',lambda=0,disp=TRUE)

# Visualization of group means
fdmeans = fdobj; fdmeans$coefs = t(res$prms$my)
plot(fdmeans); lines(fdmeans,col=1:res$K,lwd=2,lty=1)
axis(1,at=seq(5,181,6),labels=velib$dates[seq(5,181,6)],las=2)

# # Choice of K (may be long!)
# res = funFEM(fdobj,K=2:20,model='AkjBk',init='kmeans',lambda=0,disp=TRUE)
# plot(2:20,res$plot$bic,type='b',xlab='K',main='BIC')

# Computation of the closest stations from the group means
par(mfrow=c(3,2))
for (i in 1:res$K) {
  matplot(t(velib$data[which.max(res$P[,i]),]),type='l',lty=i,col=i,xaxt='n',
          lwd=2,ylim=c(0,1))
  axis(1,at=seq(5,181,6),labels=velib$dates[seq(5,181,6)],las=2)
  title(main=paste('Cluster',i,' - ',velib$names[which.max(res$P[,i])]))
}

# Visualization in the discriminative subspace (projected scores)
par(mfrow=c(1,1))
plot(t(fdobj$coefs) %*% res$U,col=res$cls,pch=19,main="Discriminative space")
text(t(fdobj$coefs) %*% res$U)

# # Spatial visualization of the clustering (with library ggmap)
# library(ggmap)
# Mymap = get_map(location = 'Paris', zoom = 12, maptype = 'terrain')
# ggmap(Mymap) + geom_point(data=velib$position,aes(longitude,latitude),
#                           colour = I(res$cl), size = I(3))

# FunFEM clustering with sparsity
res2 = funFEM(fdobj,K=res$K,model='AkjBk',init='user',Tinit=res$P,
              lambda=0.01,disp=TRUE)

# Visualization of group means and the selected functional bases
split.screen(c(2,1))
fdmeans = fdobj; fdmeans$coefs = t(res2$prms$my)
screen(1); plot(fdmeans,col=1:res2$K,xaxt='n',lwd=2) 
axis(1,at=seq(5,181,6),labels=velib$dates[seq(5,181,6)],las=2)
basis$dropind = which(rowSums(abs(res2$U))==0)
screen(2); plot(basis,col=1,lty=1,xaxt='n',xlab='Disc. basis functions')
axis(1,at=seq(5,181,6),labels=velib$dates[seq(5,181,6)],las=2)
close.screen(all=TRUE)

The Vélib data set

Description

This data set contains data from the bike sharing system of Paris, called Vélib. The data are loading profiles of the bike stations over one week. The data were collected every hour during the period Sunday 1st Sept. - Sunday 7th Sept., 2014.

Usage

data(velib)

Format

The format is:

- data: the loading profiles (nb of available bikes / nb of bike docks) of the 1189 stations at 181 time points.

- position: the longitude and latitude of the 1189 bike stations.

- dates: the download dates.

- bonus: indicates if the station is on a hill (bonus = 1).

- names: the names of the stations.

Source

The real time data are available at https://developer.jcdecaux.com/ (with an api key).

References

The data were first used in C. Bouveyron, E. Côme and J. Jacques, The discriminative functional mixture model for the analysis of bike sharing systems, Preprint HAL n.01024186, University Paris Descartes, 2014.

Examples

data(velib)
matplot(t(velib$data[1:5,]),type='l',lty=1,col=2:5,xaxt='n',lwd=2,ylim=c(0,1))
axis(1,at=seq(5,181,6),labels=velib$dates[seq(5,181,6)],las=2)

The Vélov data set

Description

This data set contains data from the bike sharing system of Lyon, called Vélo'v. The data are loading profiles of the bike stations over one week. The data were collected every hour during the period Sunday 9th March - Sunday 16th March, 2014.

Usage

data(velov)

Format

The format is:

- data: the loading profiles (nb of available bikes / nb of bike docks) of the 345 stations at 181 times.

- position: the longitude and latitude of the 345 bike stations.

- dates: the download dates.

- bonus: indicates if the station is on a hill (bonus = 1).

- names: the names of the stations.

Source

The real time data are available at https://developer.jcdecaux.com/ (with an api key).

References

The data were first used in C. Bouveyron, E. Côme and J. Jacques, The discriminative functional mixture model for the analysis of bike sharing systems, Preprint HAL n.01024186, University Paris Descartes, 2014.

Examples

data(velov)
matplot(t(velov$data[1:5,]),type='l',lty=1,col=2:5,xaxt='n',lwd=2,ylim=c(0,1))
axis(1,at=seq(5,181,6),labels=velov$dates[seq(5,181,6)],las=2)