Three-Way Data Analysis Through Densities

Description

The three-way data consists of a set of variables measured on several groups of individuals. To each group is associated an estimated probability density function. The package provides functional methods (principal component analysis, multidimensional scaling, cluster analysis, discriminant analysis...) for such probability densities.

Details

To cite dad, use citation("dad").

The main functions applying to the probability densities are:

• fpcad: functional principal component analysis,

• fpcat: functional principal component analysis applied to data indexed according to time,

• fmdsd: multidimensional scaling,

• fhclustd: hierarchical clustering,

• fdiscd.misclass: functional discriminant analysis in order to compute the misclassification ratio with the one-leave-out method,

• fdiscd.predict: discriminant analysis in order to predict the class (synonymous with cluster, not to be confused with the class attribute of an R object) of each probability density whose class is unknown,

• mdsdd: multidimensional scaling of discrete probability distributions,

• discdd.misclass: functional discriminant analysis of discrete probability distributions, in order to compute the misclassification ratio with the one-leave-out method,

• discdd.predict: discriminant analysis of discrete probability distributions, in order to predict the class of each probability distribution whose class is unknown,

The above functions are completed by:

• A print() method for objects of class fpcad, fmdsd, fdiscd.misclass, fdiscd.predict or mdsdd, in order to display the results of the corresponding function,

• A plot() method for objects of class fpcad, fmdsd, fhclustd or mdsdd, in order to display some useful graphics attached to the corresponding function,

• A generic function interpret that applies to objects of class fpcad fmdsd or mdsdd, helps the user to interpret the scores returned by the corresponding function, in terms of moments (fpcad or fmdsd) or in terms of marginal probability distributions (mdsdd).

We also introduce classes of objects and tools in order to handle collections of data frames:

• folder creates an object of class folder, that is a list of data frames which have in common the same columns.

  The following functions apply to a folder and compute some statistics on the columns of its elements: mean.folder, var.folder, cor.folder, skewness.folder or kurtosis.folder.

• folderh creates an object of class folderh, that is a list of data frames with a hierarchic relation between each pair of consecutive data frames.

• foldert creates an object of class foldert, that is a list of data frames indexed according to time, concerning the same individuals and variables or not.

• read.mtg creates an object of class foldermtg from an MTG (Multiscale Tree Graph) file containing plant architecture data.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard with the contributions from Gilles Hunault, Julie Bourbeillon and Besnik Pumo

References

Boumaza, R. (1998). Analyse en composantes principales de distributions gaussiennes multidimensionnelles. Revue de Statistique Appliqu?e, XLVI (2), 5-20.

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

Boumaza, R. (2004). Discriminant analysis with independently repeated multivariate measurements: an L^2 approach. Computational Statistics & Data Analysis, 47, 823-843.

Delicado, P. (2011). Dimensionality reduction when data are density functions. Computational Statistics & Data Analysis, 55, 401-420.

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

Pradal, C., Godin, C. and Cokelaer, T. (2023). MTG user guide

Rudrauf, J.M., Boumaza, R. (2001). Contribution à l'étude de l'architecture médiévale: les caractéristiques des pierres à bossage des châteaux forts alsaciens. Centre de Recherches Archéologiques Médiévales de Saverne, 5, 5-38.

Rachev, S.T., Klebanov, L.B., Stoyanov, S.V. and Fabozzi, F.J. (2013). The methods of distances in the theory of probability and statistics. Springer.

Yousfi, S., Boumaza, R., Aissani, D., Adjabi, S. (2014). Optimal bandwith matrices in functional principal component analysis of density functions. Journal of Statistical Computation and Simulation, 85 (11), 2315-2330.

Adds a data frame to a folderh.

Description

Creates an object of class folderh by appending a data frame to an object of class folderh. The appended data frame will be the first or last element of the returned folderh.

Usage

appendtofolderh(fh, df, key, after = FALSE)

Arguments

Value

Returns an object of class folderh, that is a list of n+1 data frames where n is the number of data frames of fh. The value of the attribute attr(, "keys") is c(key, attr(fh, "keys")) if after = FALSE), c(attr(fh, "keys"), key) otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folderh.

Folder to data frame

Description

Builds a data frame from an object of class folder.

Usage

## S3 method for class 'folder'
as.data.frame(x, row.names = NULL, optional = FALSE, ..., group.name = "group")

Arguments

Details

The data frame is simply obtained by row binding the data frames of the folder and adding a factor (as last column). The name of this column is given by group.name argument. The levels of this factor are the names of the elements of the folder.

Value

as.data.frame.folder returns a data frame.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder: object of class folder. as.folder.data.frame: build an object of class folder from a data frame.

Examples

data(iris)

iris.fold <- as.folder(iris, "Species")
print(iris.fold)

iris.df <- as.data.frame(iris.fold)
print(iris.df)

Hierarchic folder to data frame

Description

Builds a data frame from a folderh.

Usage

## S3 method for class 'folderh'
as.data.frame(x, row.names = NULL, optional = FALSE, ...,
        elt = names(x)[2], key = attr(x, "keys")[1])

Arguments

Value

as.data.frame.folderh returns a data frame whose row names are those of x[[elt]] (that is x[[j]]). The data frame contains the values of x[[elt]] and the corresponding values of the data frames x[[k]], these correspondances being defined by the keys of the hierarchic folder.

The column names of the returned data frame are organized in three parts.

1. The first part consists in the key names keys[k],..., keys[j-1].

2. The second part consists in the values of x[[j]].

3. The third part consists in the values of x[[k]] except the key keys[k].

See the examples to view these details.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder, folderh, as.folder.folderh.

Examples

# First example: rose flowers               
data(roseflowers)
flg <- roseflowers$variety
flx <- roseflowers$flower

flfh <- folderh(flg, "rose", flx)
print(flfh)

fldf <- as.data.frame(flfh)
print(fldf)

# Second example: castles               
data(castles.dated)
cag <- castles.dated$periods
cax <- castles.dated$stones

cafh <- folderh(cag, "castle", cax)
print(cafh)

cadf <- as.data.frame(cafh)
print(summary(cadf))

# Third example: leaves (example of a folderh with more than two data frames)
data(roseleaves)
lvr <- roseleaves$rose
lvs <- roseleaves$stem
lvl <- roseleaves$leaf
lvll <- roseleaves$leaflet

lfh <- folderh(lvr, "rose", lvs, "stem", lvl, "leaf", lvll)

lf1 <- as.data.frame(lfh, elt = "lvs", key = "rose")
print(lf1)

lf2 <- as.data.frame(lfh, elt = "lvl", key = "rose")
print(lf2)

lf3 <- as.data.frame(lfh, elt = "lvll", key = "rose")
print(lf3)

lf4 <- as.data.frame(lfh, elt = "lvll", key = "stem")
print(lf4)

foldert to data frame

Description

Builds a data frame from an object of class foldert.

Usage

## S3 method for class 'foldert'
as.data.frame(x, row.names = NULL, optional = FALSE, ..., group.name = "time")

Arguments

Details

as.data.frame.foldert uses as.data.frame.folder.

Value

as.data.frame.foldert returns a data frame.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

foldert: object of class foldert. as.foldert.data.frame: build an object of class foldert from a data frame. as.foldert.array: build an object of class foldert from a 3d-array.

Examples

data(floribundity)
ftflor <- foldert(floribundity, cols.select = "union", rows.select = "union")
print(ftflor)
dfflor <- as.data.frame(ftflor)
summary(dfflor)

Coerce to a folder

Description

Coerces a data frame or an object of class "folderh" to an object of class "folder".

Usage

as.folder(x, ...)

Arguments

Value

an object of class folder.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder: objects of class folder. as.data.frame.folder: build a data frame from an object of class folder. as.folder.data.frame: build an object of class folder from a data frame. as.folder.folderh: build an object of class folder from an object of class folderh.

Data frame to folder

Description

Builds an object of class folder from a data frame.

Usage

## S3 method for class 'data.frame'
as.folder(x, groups = tail(colnames(x), 1), ...)

Arguments

Value

as.folder.data.frame returns an object of class folder that is a list of data frames with the same column names.

Each element of the folder contains the data corresponding to one level of x[, groups].

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder: objects of class folder. as.data.frame.folder: build a data frame from an object of class folder. as.folder.folderh: build an object of class folder from an object of class folderh.

Examples

# First example: iris (Fisher)               
data(iris)
iris.fold <- as.folder(iris, "Species")
print(iris.fold)

# Second example: roses
data(roses)
roses.fold <- as.folder(roses, "rose")
print(roses.fold)

Hierarchic folder to folder

Description

Creates an object of class folder, that is a list of data frames with the same column names, from a folderh.

Usage

## S3 method for class 'folderh'
as.folder(x, elt = names(x)[2], key = attr(x, "keys")[1], ...)

Arguments

Value

as.folder.folderh returns an object of class folder, a list of data frames with the same columns. These data frames contain the values of x[[elt]] (or x[[j]]) and the corresponding values of the data frames x[[j-1]], ... x[[k]], these correspondances being defined by the keys of the hierarchic folder. The names of these data frames are given by the levels of the key attr(x, "keys")[k]).

The rows of the data frame x[[elt]] (or x[[j]]) are distributed among the data frames of the returned folder accordingly to the levels of the key attr(x, "keys")[k]. So the row names of the l-th data frame of the returned folder consist in the rows of x[[j]] corresponding to the l-th level of the key attr(x, "keys")[k].

The column names of the data frames of the returned folder are the union of the column names of the data frames x[[k]],..., x[[j]] and are organized in two parts.

1. The first part consists in the columns of x[[k]] except the column corresponding to the key attr(x, "keys")[k].

2. For each i=k+1,...,j the column names of the data frame x[[i]] are reorganized so that the key attr(x, "keys")[i] is its first column. The columns of the reorganized data frames x[[k+1]],..., x[[j]] are concatenated. The result forms the second part.

Notice that if:

• the folderh has two data frames df1 and df2, where the factor corresponding to the key has T levels, and one column of df2, say df2[, "Fa"], is a factor with levels "a1", ..., "ap"

• and the folder returned by as.folder includes T data frames dat1, ..., datT,

then each of dat1, ..., datT has a column named "Fa" which is a factor with the same levels "a1", ..., "ap" as df2[, "Fa"].

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder, folderh. as.folder.folderh to build an object of class folder from an object of class folderh. as.data.frame.folder to build a data frame from an object of class folder. as.data.frame.folderh to build a data frame from an object of class folderh.

Examples

# First example: flowers               
data(roseflowers)
flg <- roseflowers$variety
flx <- roseflowers$flower

flfh <- folderh(flg, "rose", flx)
print(flfh)

flf <- as.folder(flfh)
print(flf)

# Second example: castles               
data(castles.dated)
cag <- castles.dated$periods
cax <- castles.dated$stones

cafh <- folderh(cag, "castle", cax)
print(cafh)

caf <- as.folder(cafh)
print(caf)

# Third example: leaves (example of a folderh of more than two data frames)
data(roseleaves)
lvr <- roseleaves$rose
lvs <- roseleaves$stem
lvl <- roseleaves$leaf
lvll <- roseleaves$leaflet

lfh <- folderh(lvr, "rose", lvs, "stem", lvl, "leaf", lvll)

lf1 <- as.folder(lfh, elt = "lvs", key = "rose")
print(lf1)

lf2 <- as.folder(lfh, elt = "lvl", key = "rose")
print(lf2)

lf3 <- as.folder(lfh, elt = "lvll", key = "rose")
print(lf3)

lf4 <- as.folder(lfh, elt = "lvll", key = "stem")
print(lf4)

Coerce to a folderh

Description

Coerces an object to an object of class folderh.

Usage

as.folderh(x, classes)

Arguments

Value

an object of class folderh.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

as.folderh.foldermtg: build an object of class folderh from an object of class foldermtg.

Build a hierarchic folder from an object of class foldermtg

Description

Creates an object of class folderh from an object of class foldermtg.

Usage

## S3 method for class 'foldermtg'
as.folderh(x, classes)

Arguments

Details

This function uses folderh.

Value

An object of class folderh. Its elements are the data frames of x containing the features on vertices. Hence, each data frame matches with a class of vertex, and a scale. These data frames are in increasing order of the scale.

A column (factor) is added to the first data frame, containing the identifier of the vertex. Two columns are added to the second data frame:

1. the first one is a factor which gives, for each vertex, the name of the vertex of the first data frame which is its "parent",

2. and the second one is also a factor and contains the vertex's identifier.

And so on for the third and following data frames, if relevant.

The column containing the vertex identifiers is redundant with the row names; anyway, it is necessary for folderh.

The key of the relationship between the two first data frame is given by the first column of each of these data frames. If there are more than two data frames, the key of the relationship between the n-th and (n+1)-th data frames (n > 1) is given by the second column of the n$th data frame and the first column of the (n+1)-th data frame.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

read.mtg: reads a MTG file and creates an object of class "foldermtg". folderh : object of class folderh.

Examples

mtgfile <- system.file("extdata/plant1.mtg", package = "dad")
x <- read.mtg(mtgfile)

# folderh containing the plant ("P") and the stems ("A")
as.folderh(x, classes = c("P", "A"))

# folderh containing the plant ("P"), axes ("A") and phytomers ("M")
as.folderh(x, classes = c("P", "A", "M"))

# folderh containing the plant ("P") and the phytomers ("M")
as.folderh(x, classes = c("P", "M"))

# folderh containing the axes and phytomers
fhPM <- as.folderh(x, classes = c("A", "M"))
# coerce this folderh into a folder, and compute statistics on this folder
fPM <- as.folder(fhPM)
mean(fPM)

Coerce to a foldert

Description

Coerces a data frame or array to an object of class foldert.

Usage

as.foldert(x, ...)

Arguments

Value

an object of class foldert.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

Data frame to foldert

Description

Builds an object of class foldert from a 3d-array.

Usage

## S3 method for class 'array'
as.foldert(x, ind = 1, var = 2, time = 3, ...)

Arguments

Value

an object ft of class foldert that is a list of data frames, each of them corresponding to a time of observation; these data frames have the same column names.

They necessarily have the same row names (attr(ft, "same.rows")=TRUE). The "times" attribute of ft: attr(ft, "times") is a numeric vector, an ordered factor or an object of class Date, and contains the values nf the dimension of x given by time argument.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

foldert: objects of class foldert.

as.foldert.data.frame: build an object of class foldert from a data frame.

Examples

x <- array(c(rep(0, 5), rep(0, 5), rep(0, 5),
             rnorm(5, 2, 1), rnorm(5, 3, 2), rnorm(5, -2, 0.5),
             rnorm(5, 4, 1), rnorm(5, 5, 3), rnorm(5, -3, 1)),
           dim = c(5, 3, 3),
           dimnames = list(1:5, c("z1", "z2", "z3"), c("t1", "t2", "t3")))
# The individuals which were observed are on the 1st dimension,
# the variables are on the 2nd dimension and the times are on the 3rd dimension.
ft <- as.foldert(x, ind = 1, var = 2, time = 3)

Data frame to foldert

Description

Builds an object of class foldert from a data frame.

Usage

## S3 method for class 'data.frame'
as.foldert(x, method = 1, ind = 1, timecol = 2, nvar = NULL, same.rows = TRUE, ...)

Arguments

Value

an object ft of class foldert, that is a list of data frames organised according to time; these data frames have the same column names.

If method = 1, they can have the same row names (attr(ft, "same.rows") = TRUE) or not (attr(ft, "same.rows") = FALSE). The time attribute attr(ft, "times") has the same class as x[, timecol] (numeric vector, ordered factor or object of class "Date", "POSIXlt" or "POSIXct") and contains the values of x[, timecol].

If method = 2, they necessarily have the same row names: attr(ft, "same.rows") = TRUE and attr(ft, "times") is 1:length(ft).

The rownames of each data frame are the identifiers of the individuals, as given by x[, ind].

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

foldert: objects of class foldert.

as.data.frame.foldert: build a data frame from an object of class foldert.

as.foldert.array: build an object of class foldert from a 3d-array.

Examples

# First example: method = 1

times <- as.Date(c("2017-03-01", "2017-04-01", "2017-05-01"))
x1 <- data.frame(t=times[1], ind=1:6,
                 f=c("a","a","a","b","b","b"), z1=rep(0,6), z2=rep(0,6),
                 stringsAsFactors = TRUE)
x2 <- data.frame(t=times[2], ind=c(1,4,6),
                 f=c("a","b","b"), z1=rnorm(3,1,1), z2=rnorm(3,3,2),
                 stringsAsFactors = TRUE)
x3 <- data.frame(t=times[3], ind=c(1,3:6),
                 f=c("a","a","a","b","b"), z1=rnorm(5,3,2), z2=rnorm(5,6,3),
                 stringsAsFactors = TRUE)
x <- rbind(x1, x2, x3)

ft1 <- as.foldert(x, method = 1, ind = "ind", timecol = "t", same.rows = TRUE)
print(ft1)

ft2 <- as.foldert(x, method = 1, ind = "ind", timecol = "t", same.rows = FALSE)
print(ft2)

data(castles.dated)
periods <- castles.dated$periods
stones <- castles.dated$stones
stones$stone <- rownames(stones)

castledf <- merge(periods, stones, by = "castle")
castledf$period <- as.numeric(castledf$period)
castledf$stone <- as.factor(paste(as.character(castledf$castle),
                            as.character(castledf$stone), sep = "_"))

castfoldt1 <- as.foldert(castledf, method = 1, ind = "stone", timecol = "period",
                         same.rows = FALSE)
summary(castfoldt1)


# Second example: method = 2

times <- as.Date(c("2017-03-01", "2017-04-01", "2017-05-01"))
y1 <- data.frame(z1=rep(0,6), z2=rep(0,6))
y2 <- data.frame(z1=rnorm(6,1,1), z2=rnorm(6,3,2))
y3 <- data.frame(z1=rnorm(6,3,2), z2=rnorm(6,6,3))
y <- cbind(ind = 1:6, y1, y2, y3)

ft3 <- as.foldert(y, method = 2, ind = "ind", timecol = 2, nvar = 2)
print(ft3)

Association measures between several categorical variables of a data frame

Description

Computes pairwise association measures (Cramer's V, Pearson's contingency coefficient, phi, Tschuprow's T) between the categorical variables of a data frame, using functions of the package DescTools (see Assocs).

Usage

cramer.data.frame(x, check = TRUE)
pearson.data.frame(x, check = TRUE)
phi.data.frame(x, check = TRUE)
tschuprow.data.frame(x, check = TRUE)

Arguments

Value

A square matrix whose elements are the pairwise association measures.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

Examples

data(roses)
xr = roses[,c("Sha", "Den", "Sym", "rose")]
xr$Sha = cut(xr$Sha, breaks = c(0, 5, 7, 10))
xr$Den = cut(xr$Den, breaks = c(0, 4, 6, 10))
xr$Sym = cut(xr$Sym, breaks = c(0, 6, 8, 10))
cramer.data.frame(xr)
pearson.data.frame(xr)
phi.data.frame(xr)
tschuprow.data.frame(xr)

Association measures between categorical variables of the data frames of a folder

Description

Computes the pairwise association measures (Cramer's V, Pearson's contingency coefficient, phi, Tschuprow's T) between the categorical variables of an object of class folder. The computation is carried out using the functions cramer.data.frame, tschuprow.data.frame, pearson.data.frame or phi.data.frame. These functions are built from corresponding functions of the package DescTools (see Assocs)

Usage

cramer.folder(xf)
tschuprow.folder(xf)
pearson.folder(xf)
phi.folder(xf)

Arguments

Value

A list the length of which is equal to the number of data frames of the folder. Each element of the list is a square matrice giving the pairwise association measures of the variables of the corresponding data frame.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

Examples

data(roses)
xr = roses[,c("Sha", "Den", "Sym", "rose")]
xr$Sha = cut(xr$Sha, breaks = c(0, 5, 7, 10))
xr$Den = cut(xr$Den, breaks = c(0, 4, 6, 10))
xr$Sym = cut(xr$Sym, breaks = c(0, 6, 8, 10))
xfolder = as.folder(xr, groups = "rose")
cramer.folder(xfolder)
pearson.folder(xfolder)
phi.folder(xfolder)
tschuprow.folder(xfolder)

Parameter of the normal reference rule

Description

Computation of the parameter of the normal reference rule in order to estimate the (matrix) bandwidth.

Usage

bandwidth.parameter(p, n)

Arguments

Details

The parameter is equal to:

h = (\frac{4}{n(p+2)})^{\frac{1}{p+4}}

It is based on the minimisation of the asymptotic mean integrated square error in density estimation when using the Gaussian kernel method (Wand and Jones, 1995).

Value

Returns the value required by the functions fpcad, fmdsd, fdiscd.misclass and fdiscd.predict when their argument windowh is set to NULL.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

Wand, M. P., Jones, M. C. (1995). Kernel Smoothing. Boca Raton, FL: Chapman and Hall.

Examples

    # Sample size :
    n <- 20
    # Number of variables :
    p <- 3
    bandwidth.parameter(p, n)

Alsacian castles by year of building

Description

The data were collected by J.M. Rudrauf on Alsacian castles whose building year is known (even approximatively). On each castle, he measured 4 structural parameters on a sample of building stones.

These data are about the same castles as in castles.dated data set.

Usage

data(castles)

Format

castles is a list of 46 data frames. Each of these data frames matches with one year (between 1136 and 1510) and contains measures on one or several castles which have been built since that year.

Each data frame has 5 to 101 rows (stones) and 5 columns: height, width, edging, boss (numeric) and castle (factor).

Source

Rudrauf, J.M., Boumaza, R. (2001). Contribution a l'etude de l'architecture medievale: les caracteristiques des pierres a bossage des chateaux forts alsaciens. Centre de Recherches Archeologiques Medievales de Saverne, 5, 5-38.

Examples

data(castles)
foldert(castles)

Dated Alsacian castles

Description

The data were collected by J.M. Rudrauf on Alsacian castles whose building period is known (even approximately). On each castle, he measured 4 structural parameters on a sample of building stones.

Usage

data(castles.dated)

Format

castles.dated is a list of two data frames:

castles.dated$stones:

this first data frame has 1262 cases (rows) and 5 variables (columns) that are named height, width, edging, boss (numeric) and castle (factor).

castles.dated$periods:

this second data frame has 68 cases and 2 variables named castle and period; the column castle corresponds to the levels of the factor castle of the first data frame; the column period is a factor with 6 levels indicating the approximative building period. Thus this factor defines 6 classes of castles.

Source

Rudrauf, J.M., Boumaza, R. (2001). Contribution a l'etude de l'architecture medievale: les caracteristiques des pierres a bossage des chateaux forts alsaciens. Centre de Recherches Archeologiques Medievales de Saverne, 5, 5-38.

Examples

data(castles.dated)
summary(castles.dated$stones)
summary(castles.dated$periods)

Non dated Alsacian castles

Description

The data were collected by J.M. Rudrauf on Alsacian castles whose building period is unknown. On each castle, he measured 4 structural parameters on a sample of building stones.

Usage

data(castles.nondated)

Format

castles.nondated is a list of two data frames:

castles.nondated$stones:

this first data frame has 1280 cases (rows) and 5 variables (columns) that are named height, width, edging, boss (numeric) and castle (factor).

castles.nondated$periods:

this second data frame has 67 cases and 2 variables named castle and period; the column castle corresponds to the levels of the factor castle of the first data frame; the column period is a factor indicating NA as the building period is unknown.

Notice that the data frames corresponding to the castles whose building period is known are those in castles.dated.

Source

Rudrauf, J.M., Boumaza, R. (2001). Contribution a l'etude de l'architecture medievale: les caracteristiques des pierres a bossage des chateaux forts alsaciens. Centre de Recherches Archeologiques Medievales de Saverne, 5, 5-38.

Examples

data(castles.nondated)
summary(castles.nondated$stones)
summary(castles.nondated$periods)

Correlation matrices of a folder of data sets

Description

Computes the correlation matrices of the elements of an object of class folder.

Usage

cor.folder(x, use = "everything", method = "pearson")

Arguments

Details

It uses cor to compute the variance matrix of the numeric columns of each element of the folder. If some columns of the data frames are not numeric, there is a warning, and the variances are computed on the numeric columns only.

Value

A list whose elements are the correlation matrices of the elements of the folder.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder to create an object is of class folder. mean.folder, var.folder, skewness.folder, kurtosis.folder for other statistics for folder objects.

Examples

# First example: iris (Fisher)               
data(iris)
iris.fold <- as.folder(iris, "Species")
iris.cor <- cor.folder(iris.fold)
print(iris.cor)

# Second example: roses
data(roses)
roses.fold <- as.folder(roses, "rose")
roses.cor <- cor.folder(roses.fold)
print(roses.cor)

Change numeric variables into factors

Description

This function changes numerical columns of a data frame x into factors. For each of these columns, its range is divided into intervals and the values of this column is recoded according to which interval they fall.

For that, cut is applied to each column of x.

Usage

## S3 method for class 'data.frame'
cut(x, breaks, labels = NULL, include.lowest = FALSE, right = TRUE, dig.lab = 3L,
    ordered_result = FALSE, cutcol = NULL, ...)

Arguments

Value

A data frame with the same column and row names as x.

If cutcol is given, each numeric column x[, j] whose number is contained in cutcol is replaced by a factor. The other columns are unmodified.

If any column x[, j] whose number is in cutcol is not numeric, it is unmodified.

If cutcol is omitted, every numerical columns are replaced by factors.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

Examples

data("roses")
x <- roses[roses$rose %in% c("A", "B"), c("Sha", "Sym", "Den", "rose")]

cut(x, breaks = 3)
cut(x, breaks = 5)
cut(x, breaks = c(0, 4, 6, 10))
cut(x, breaks = list(c(0, 6, 8, 10), c(0, 5, 7, 10), c(0, 6, 7, 10)))
cut(x, breaks = list(c(0, 6, 8, 10), c(0, 5, 7, 10)), cutcol = 1:2)

In a folder: change numeric variables into factors

Description

This function applies to a folder. For each elements (data frames) of this folder, it changes its numerical columns into factors, using cut.data.frame.

Usage

## S3 method for class 'folder'
cut(x, breaks, labels = NULL, include.lowest = FALSE, right = TRUE, dig.lab = 3L,
    ordered_result = FALSE, cutcol = NULL, ...)

Arguments

Value

An object of class folder with the same length and names as x. Its elements (data frames) have the same column and row names as the elements of x.

For more details, see cut.data.frame

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

Examples

data("roses")

x <- as.folder(roses[, c("Sha", "Den", "Sym", "rose")], groups = "rose")
summary(x)

x3 <- cut(x, breaks = 3)
summary(x3)

x7 <- cut(x, breaks = 7)
summary(x7)

Distance between probability distributions of discrete variables given samples

Description

Symmetrized chi-squared distance between two multivariate (q > 1) or univariate (q = 1) discrete probability distributions, estimated from samples.

Usage

ddchisqsym(x1, x2)

Arguments

Details

Let p_1 and p_2 denote the estimated probability distributions of the discrete samples x_1 and x_2. The symmetrized chi-squared distance between the discrete probability distributions of the samples are computed using the ddchisqsympar function.

Value

The distance between the two probability distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddchisqsympar: chi-squared distance between two discrete distributions, given the probabilities on their common support.

Other distances: ddhellinger, ddjeffreys, ddjensen, ddlp.

Examples

# Example 1
x1 <- c("A", "A", "B", "B")
x2 <- c("A", "A", "A", "B", "B")
ddchisqsym(x1, x2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
ddchisqsym(x1, x2)

Distance between discrete probability distributions given the probabilities on their common support

Description

Symmetrized chi-squared distance between two discrete probability distributions on the same support (which can be a Cartesian product of q sets) , given the probabilities of the states (which are q-tuples) of the support.

Usage

ddchisqsympar(p1, p2)

Arguments

Details

The chi-squared distance between two discrete distributions p_1 and p_2 is given by:

\sum_x{(p_1(x) - p_2(x))^2}/p_2(x)

Then the symmetrized chi-squared distance is given by the formula:

||p_1 - p_2|| = \sum_x{(p_1(x) - p_2(x))^2}/(p_1(x) + p_2(x))

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddchisqsym: chi-squared distance between two estimated discrete distributions, given samples.

Other distances: ddhellingerpar, ddjeffreyspar, ddjensenpar, ddlppar.

Examples

# Example 1
p1 <- array(c(1/2, 1/2), dimnames = list(c("a", "b"))) 
p2 <- array(c(1/4, 3/4), dimnames = list(c("a", "b"))) 
ddchisqsympar(p1, p2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
p1 <- table(x1)/nrow(x1)                 
p2 <- table(x2)/nrow(x2)
ddchisqsympar(p1, p2)

Distance between probability distributions of discrete variables given samples

Description

Hellinger (or Matusita) distance between two multivariate (q > 1) or univariate (q = 1) discrete probability distributions, estimated from samples.

Usage

ddhellinger(x1, x2)

Arguments

Details

Let p_1 and p_2 denote the estimated probability distributions of the discrete samples x_1 and x_2. The Matusita distance between the discrete probability distributions of the samples are computed using the ddhellingerpar function.

Value

The distance between the two probability distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddhellingerpar: Hellinger metric (Matusita distance) between two discrete distributions, given the on their common support probabilities.

Other distances: ddchisqsym, ddjeffreys, ddjensen, ddlp.

Examples

# Example 1
x1 <- c("A", "A", "B", "B")
x2 <- c("A", "A", "A", "B", "B")
ddhellinger(x1, x2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
ddhellinger(x1, x2)

Distance between discrete probability distributions given the probabilities on their common support

Description

Hellinger (or Matusita) distance between two discrete probability distributions on the same support (which can be a Cartesian product of q sets) , given the probabilities of the states (which are q-tuples) of the support.

Usage

ddhellingerpar(p1, p2)

Arguments

Details

The Hellinger distance between two discrete distributions p_1 and p_2 is given by: \sqrt{ \sum_x{(\sqrt{p_1(x)} - \sqrt{p_2(x)})^2}}

Notice that some authors divide this expression by \sqrt{2}.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddhellinger: Hellinger distance between two estimated discrete distributions, given samples.

Other distances: ddchisqsympar, ddjeffreyspar, ddjensenpar, ddlppar.

Examples

# Example 1
p1 <- array(c(1/2, 1/2), dimnames = list(c("a", "b"))) 
p2 <- array(c(1/4, 3/4), dimnames = list(c("a", "b"))) 
ddhellingerpar(p1, p2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
p1 <- table(x1)/nrow(x1)                 
p2 <- table(x2)/nrow(x2)
ddhellingerpar(p1, p2)

Divergence between probability distributions of discrete variables given samples

Description

jeffreys's divergence (symmetrized Kullback-Leibler divergence) between two multivariate (q > 1) or univariate (q = 1) discrete probability distributions, estimated from samples.

Usage

ddjeffreys(x1, x2)

Arguments

Details

Let p_1 and p_2 denote the estimated probability distributions of the discrete samples x_1 and x_2. The jeffreys's divergence between the discrete probability distributions of the samples are computed using the ddjeffreyspar function.

Value

The divergence between the two probability distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddjeffreyspar: Jeffrey's distances between two discrete distributions, given the probabilities on their common support.

Other distances: ddchisqsym, ddhellinger, ddjensen, ddlp.

Examples

# Example 1
x1 <- c("A", "A", "B", "B")
x2 <- c("A", "A", "A", "B", "B")
ddjeffreys(x1, x2)

# Example 2 (Its value can be infinity -Inf-)
x1 <- c("A", "A", "B", "C")
x2 <- c("A", "A", "A", "B", "B")
ddjeffreys(x1, x2)

# Example 3
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
ddjeffreys(x1, x2)

Distance between discrete probability distributions given the probabilities on their common support

Description

Jeffreys divergence (symmetrized Kullback-Leibler divergence) between two discrete probability distributions on the same support (which can be a Cartesian product of q sets) , given the probabilities of the states (which are q-tuples) of the support.

Usage

ddjeffreyspar(p1, p2)

Arguments

Details

Jeffreys divergence ||p_1 - p_2|| between two discrete distributions p_1 and p_2 is given by the formula:

||p_1 - p_2|| = \sum_x{(p_1(x) - p_2(x)) log(p_1(x)/p_2(x))}

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddjeffreys: Jeffreys distance between two estimated discrete distributions, given samples.

Other distances: ddchisqsympar, ddhellingerpar, ddjensenpar, ddlppar.

Examples

# Example 1
p1 <- array(c(1/2, 1/2), dimnames = list(c("a", "b"))) 
p2 <- array(c(1/4, 3/4), dimnames = list(c("a", "b"))) 
ddjeffreyspar(p1, p2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
p1 <- table(x1)/nrow(x1)                 
p2 <- table(x2)/nrow(x2)
ddjeffreyspar(p1, p2)

Divergence between probability distributions of discrete variables given samples

Description

Jensen-Shannon divergence between two multivariate (q > 1) or univariate (q = 1) discrete probability distributions, estimated from samples.

Usage

ddjensen(x1, x2)

Arguments

Details

Let p_1 and p_2 denote the estimated probability distributions of the discrete samples x_1 and x_2. The Jensen-Shannon divergence between the discrete probability distributions of the samples are computed using the ddjensenpar function.

Value

The distance between the two probability distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddjensenpar: Jensen-Shannon distance between two discrete distributions, given the probabilities on their common support.

Other distances: ddchisqsym, ddhellinger, ddjeffreys, ddlp.

Examples

# Example 1
x1 <- c("A", "A", "B", "B")
x2 <- c("A", "A", "A", "B", "B")
ddjensen(x1, x2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
ddjensen(x1, x2)

Divergence between discrete probability distributions given the probabilities on their common support

Description

Jensen-Shannon divergence between two discrete probability distributions on the same support (which can be a Cartesian product of q sets), given the probabilities of the states (which are q-tuples) of the support.

Usage

ddjensenpar(p1, p2)

Arguments

Details

The Jensen-Shannon divergence ||p_1 - p_2|| between two discrete distributions p_1 and p_2 is given by the formula:

||p_1 - p_2|| = \sum_x{(p_1(x) log(2 p_1(x) / (p_1(x)+p_2(x)))) + (p_2(x) log(2 p_2(x) / (p_1(x)+p_2(x))))}

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddjensen: Jensen-Shannon distance between two estimated discrete distributions, given samples.

Other distances: ddchisqsympar, ddhellingerpar, ddjeffreyspar, ddlppar.

Examples

# Example 1
p1 <- array(c(1/2, 1/2), dimnames = list(c("a", "b"))) 
p2 <- array(c(1/4, 3/4), dimnames = list(c("a", "b"))) 
ddjensenpar(p1, p2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
p1 <- table(x1)/nrow(x1)                 
p2 <- table(x2)/nrow(x2)
ddjensenpar(p1, p2)

Distance between probability distributions of discrete variables given samples

Description

L^p distance between two multivariate (q > 1) or univariate (q = 1) discrete probability distributions, estimated from samples.

Usage

ddlp(x1, x2, p = 1)

Arguments

Details

Let p_1 and p_2 denote the estimated probability distributions of the discrete samples x_1 and x_2. The L^p distance between the discrete probability distributions of the samples are computed using the ddlppar function.

Value

The distance between the two discrete probability distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddlppar: L^p distance between two discrete distributions, given the probabilities on their common support.

Other distances: ddchisqsym, ddhellinger, ddjeffreys, ddjensen.

Examples

# Example 1
x1 <- c("A", "A", "B", "B")
x2 <- c("A", "A", "A", "B", "B")
ddlp(x1, x2)
ddlp(x1, x2, p = 2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
ddlp(x1, x2)

Distance between discrete probability distributions given the probabilities on their common support

Description

L^p distance between two discrete probability distributions on the same support (which can be a Cartesian product of q sets) , given the probabilities of the states (which are q-tuples) of the support.

Usage

ddlppar(p1, p2, p = 1)

Arguments

Details

The L^p distance ||p_1 - p_2|| between two discrete distributions p_1 and p_2 is given by the formula:

||p_1 - p_2||^p = \sum_x{|p_1(x)-p_2(x)|^p}

If p=1, it is the variational distance.

If p=2, it is the Patrick-Fisher distance.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddlp: L^p distance between two estimated discrete distributions, given samples.

Other distances: ddchisqsympar, ddhellingerpar, ddjeffreyspar, ddjensenpar.

Examples

# Example 1
p1 <- array(c(1/2, 1/2), dimnames = list(c("a", "b"))) 
p2 <- array(c(1/4, 3/4), dimnames = list(c("a", "b"))) 
ddlppar(p1, p2)
ddlppar(p1, p2, p=2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
p1 <- table(x1)/nrow(x1)                 
p2 <- table(x2)/nrow(x2)
ddlppar(p1, p2)

French departments and regions

Description

Departments and regions of metropolitan France.

Usage

data(departments)

Format

departments is a data frame with 96 rows and 4 columns (factors):

coded:

departments: numbers

named:

departments: names

coder:

regions: ISO code

namer:

region: names

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

Source

INSEE. Code officiel g\'eographique au 1er janvier 2018.

Examples

data(departments)
print(departments)

Misclassification ratio in functional discriminant analysis of discrete probability distributions.

Description

Computes the one-leave-out misclassification ratio of the rule assigning T groups of individuals, one group after another, to the class of groups (among K classes of groups) which achieves the minimum of the distances or divergences between the probability distribution associated to the group to assign and the K probability distributions associated to the K classes.

Usage

discdd.misclass(xf, class.var, distance =  c("l1", "l2", "chisqsym", "hellinger",
           "jeffreys", "jensen", "lp"), crit = 1, p)

Arguments

Details

• If xf is an object of class "folderh" containing the data:

  The T probability distributions f_t corresponding to the T groups of individuals are estimated by frequency distributions within each group.

  To the class k consisting of T_k groups is associated the probability distribution g_k, knowing that when using the one-leave-out method, we do not include the group to assign in its class k. The crit argument selects the estimation method of the g_k's.

  • crit=1 The probability distribution g_k is estimated using the whole data of this class, that is the rows of x corresponding to the T_k groups of the class k.

    The estimation of the g_k's uses the same method as the estimation of the f_t's.

  • crit=2 The T_k probability distributions f_t are estimated using the corresponding data from xf. Then they are averaged to obtain an estimation of the density g_k, that is g_k = \frac{1}{T_k} \, \sum{f_t}.

• If xf is a list of arrays (or list of tables):

  The t^{th} array is the joint frequency distribution of the t^{th} group. The frequencies can be absolute or relative.

  To the class k consisting of T_k groups is associated the probability distribution g_k, knowing that when using the one-leave-out method, we do not include the group to assign in its class k. The crit argument selects the estimation method of the g_k's.

  • crit=1 g_k = \frac{1}{\sum n_t} \sum n_t f_t, where n_t is the total of xf[[t]].

    Notice that when xf[[t]] contains relative frequencies, its total is 1. That is equivalent to crit=2.

  • crit=2 g_k = \frac{1}{T_k} \, \sum f_t.

Value

Returns an object of class discdd.misclass, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Rudrauf, J.M., Boumaza, R. (2001). Contribution à l'étude de l'architecture médiévale: les caractéristiques des pierres à bossage des châteaux forts alsaciens, Centre de Recherches Archéologiques médiévales de Saverne, 5, 5-38.

Examples

# Example 1 with a folderh obtained by converting numeric variables
data("castles.dated")
stones <- castles.dated$stones
periods <- castles.dated$periods
stones$height <- cut(stones$height, breaks = c(19, 27, 40, 71), include.lowest = TRUE)
stones$width <- cut(stones$width, breaks = c(24, 45, 62, 144), include.lowest = TRUE)
stones$edging <- cut(stones$edging, breaks = c(0, 3, 4, 8), include.lowest = TRUE)
stones$boss <- cut(stones$boss, breaks = c(0, 6, 9, 20), include.lowest = TRUE )

castlefh <- folderh(periods, "castle", stones)

# Default: dist="l1", crit=1
discdd.misclass(castlefh, "period")

# Hellinger distance, crit=2
discdd.misclass(castlefh, "period", distance = "hellinger", crit = 2)


# Example 2 with a list of 96 arrays
data("dspgd2015")
data("departments")
classes <- departments[, c("coded", "namer")]
names(classes) <- c("group", "class")

# Default: dist="l1", crit=1
discdd.misclass(dspgd2015, classes)

# Hellinger distance, crit=2
discdd.misclass(dspgd2015, classes, distance = "hellinger", crit = 2)

Predicting the class of a group of individuals with discriminant analysis of probability distributions.

Description

Assigns several groups of individuals, one group after another, to the class of groups (among K classes of groups) which achieves the minimum of the distances or divergences between the probability distribution associated to the group to assign and the K probability distributions associated to the K classes.

Usage

discdd.predict(xf, class.var, distance =  c("l1", "l2", "chisqsym", "hellinger",
           "jeffreys", "jensen", "lp"), crit = 1, misclass.ratio = FALSE, p)

Arguments

Details

• If xf is an object of class "folderh" containing the data:

  The T probability distributions f_t corresponding to the T groups of individuals are estimated by frequency distributions within each group.

  To the class k consisting of T_k groups is associated the probability distribution g_k. The crit argument selects the estimation method of the g_k's.

  • crit=1 The probability distribution g_k is estimated using the whole data of this class, that is the rows of x corresponding to the T_k groups of the class k.

    The estimation of the g_k's uses the same method as the estimation of the f_t's.

  • crit=2 The T_k probability distributions f_t are estimated using the corresponding data from xf. Then they are averaged to obtain an estimation of the density g_k, that is g_k = \frac{1}{T_k} \, \sum{f_t}.

• If xf is a list of arrays (or list of tables):

  The t^{th} array is the joint frequency distribution of the t^{th} group. The frequencies can be absolute or relative.

  To the class k consisting of T_k groups is associated the probability distribution g_k. The crit argument selects the estimation method of the g_k's.

  • crit=1 g_k = \frac{1}{\sum n_t} \sum n_t f_t, where n_t is the total of xf[[t]].

    Notice that when xf[[t]] contains relative frequencies, its total is 1. That is equivalent to crit=2.

  • crit=2 g_k = \frac{1}{T_k} \, \sum f_t.

Value

Returns an object of class discdd.predict, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Rudrauf, J.M., Boumaza, R. (2001). Contribution à l'étude de l'architecture médiévale: les caractéristiques des pierres à bossage des châteaux forts alsaciens, Centre de Recherches Archéologiques médiévales de Saverne, 5, 5-38.

Examples

data(castles.dated)
data(castles.nondated)
stones <- rbind(castles.dated$stones, castles.nondated$stones)
periods <- rbind(castles.dated$periods, castles.nondated$periods)
stones$height <- cut(stones$height, breaks = c(19, 27, 40, 71), include.lowest = TRUE)
stones$width <- cut(stones$width, breaks = c(24, 45, 62, 144), include.lowest = TRUE)
stones$edging <- cut(stones$edging, breaks = c(0, 3, 4, 8), include.lowest = TRUE)
stones$boss <- cut(stones$boss, breaks = c(0, 6, 9, 20), include.lowest = TRUE )

castlesfh <- folderh(periods, "castle", stones)

# Default: dist="l1", crit=1
discdd.predict(castlesfh, "period")

# With the calculation of the confusion matrix and misclassification ratio
discdd.predict(castlesfh, "period", misclass.ratio = TRUE)

# Hellinger distance
discdd.predict(castlesfh, "period", distance = "hellinger")

# crit=2
discdd.predict(castlesfh, "period", crit = 2)

L^2 distance between probability densities

Description

L^2 distance between two multivariate (p > 1) or univariate (dimension: p = 1) probability densities, estimated from samples.

Usage

distl2d(x1, x2, method = "gaussiand", check = FALSE, varw1 = NULL, varw2 = NULL)

Arguments

Details

The function distl2d computes the distance between f_1 and f_2 from the formula

||f_1 - f_2||^2 = <f_1, f_1> + <f_2, f_2> - 2 <f_1, f_2>

For some information about the method used to compute the L^2 inner product or about the arguments, see l2d.

Value

The L^2 distance between the two densities.

Be careful! If check = FALSE and one smoothing bandwidth matrix is degenerate, the result returned can not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

matdistl2d in order to compute pairwise distances between several densities.

Examples

require(MASS)
m1 <- c(0,0)
v1 <- matrix(c(1,0,0,1),ncol = 2) 
m2 <- c(0,1)
v2 <- matrix(c(4,1,1,9),ncol = 2)
x1 <- mvrnorm(n = 3,mu = m1,Sigma = v1)
x2 <- mvrnorm(n = 5, mu = m2, Sigma = v2)
distl2d(x1, x2, method = "gaussiand")
distl2d(x1, x2, method = "kern")
distl2d(x1, x2, method = "kern", varw1 = v1, varw2 = v2)

L^2 distance between L^2-normed probability densities

Description

L^2 distance between two multivariate (p > 1) or univariate (dimension: p = 1) L^2-normed probability densities, estimated from samples, where a L^2-normed probability density is the original probability density function divided by its L^2-norm.

Usage

distl2dnorm(x1, x2, method = "gaussiand", check = FALSE, varw1 = NULL, varw2 = NULL)

Arguments

Details

Given densities f_1 and f_2, the function distl2dnormpar computes the distance between the L^2-normed densities f_1 / ||f_1|| and f_2 / ||f_2||:

2 - 2 <f_1, f_2> / (||f_1|| ||f_2||)

For some information about the method used to compute the L^2 inner product or about the arguments, see l2d.

Value

The L^2 distance between the two L^2-normed densities.

Be careful! If check = FALSE and one smoothing bandwidth matrix is degenerate, the result returned can not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

distl2d for the distance between two probability densities.

matdistl2dnorm in order to compute pairwise distances between several L^2-normed densities.

Examples

require(MASS)
m1 <- c(0,0)
v1 <- matrix(c(1,0,0,1),ncol = 2) 
m2 <- c(0,1)
v2 <- matrix(c(4,1,1,9),ncol = 2)
x1 <- mvrnorm(n = 3,mu = m1,Sigma = v1)
x2 <- mvrnorm(n = 5, mu = m2, Sigma = v2)
distl2dnorm(x1, x2, method = "gaussiand")
distl2dnorm(x1, x2, method = "kern")
distl2dnorm(x1, x2, method = "kern", varw1 = v1, varw2 = v2)

L^2 distance between L^2-normed Gaussian densities given their parameters

Description

L^2 distance between two multivariate (p > 1) or univariate (dimension: p = 1) L^2-normed Gaussian densities, given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate) where a L^2-normed probability density is the original probability density function divided by its L^2-norm.

Usage

distl2dnormpar(mean1, var1, mean2, var2, check = FALSE)

Arguments

Details

Given densities f_1 and f_2, the function distl2dnormpar computes the distance between the L^2-normed densities f_1 / ||f_1|| and f_2 / ||f_2||:

2 - 2 <f_1, f_2> / (||f_1|| ||f_2||)

.

For some information about the method used to compute the L^2 inner product or about the arguments, see l2dpar; the norm ||f|| of the multivariate Gaussian density f is equal to (4\pi)^{-p/4} det(var)^{-1/4}.

Value

The L^2 distance between the two L^2-normed Gaussian densities.

Be careful! If check = FALSE and one variance matrix is degenerated (or one variance is zero if the densities are univariate), the result returned must not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

distl2dpar for the distance between two probability densities.

matdistl2d in order to compute pairwise distances between several densities.

Examples

u1 <- c(1,1,1);
v1 <- matrix(c(4,0,0,0,16,0,0,0,25),ncol = 3);
u2 <- c(0,1,0);
v2 <- matrix(c(1,0,0,0,1,0,0,0,1),ncol = 3);
distl2dnormpar(u1,v1,u2,v2)

L^2 distance between Gaussian densities given their parameters

Description

L^2 distance between two multivariate (p > 1) or univariate (dimension: p = 1) Gaussian densities, given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate).

Usage

distl2dpar(mean1, var1, mean2, var2, check = FALSE)

Arguments

Details

The function distl2dpar computes the distance between two densities, say f_1 and f_2, from the formula:

||f_1 - f_2||^2 = <f_1, f_1> + <f_2, f_2> - 2 <f_1, f_2>

.

For some information about the method used to compute the L^2 inner product or about the arguments, see l2dpar.

Value

The L^2 distance between the two densities.

Be careful! If check = FALSE and one variance matrix is degenerated (or one variance is zero if the densities are univariate), the result returned must not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

matdistl2d in order to compute pairwise distances between several densities.

Examples

u1 <- c(1,1,1);
v1 <- matrix(c(4,0,0,0,16,0,0,0,25),ncol = 3);
u2 <- c(0,1,0);
v2 <- matrix(c(1,0,0,0,1,0,0,0,1),ncol = 3);
distl2dpar(u1,v1,u2,v2)

Diploma x Socio professional group

Description

Contingency tables of the counts of Diploma x Socio professional group of France

Usage

data(dspg)

Format

dspg is a list of 7 arrays (each one corresponding to a year: 1968, 1975, 1982, 1990, 1999, 2010, 2015) of 4 rows (each one corresponding to a level of diploma) and 6 columns (each one corresponding to a socio professional group).

csp:

Socio professional group

diplome:

Diploma

agri:

farmer (agriculteur)

arti:

craftsperson (artisan)

cadr:

senior manager (cadre sup\'erieur)

pint:

middle manager (profession interm\'ediaire)

empl:

employee (employ\'e)

ouvr:

worker (ouvrier)

bepc:

brevet

cap:

NVQ (cap)

bac:

baccalaureate

sup:

higher education (sup\'erieur)

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

Source

INSEE. Population active de 25 à 54 ans ayant un emploi et chômeurs par catégorie socioprofessionnelle et diplôme par commune et département (1968 à 2015).

Examples

data(dspg)
names(dspg)
print(dspg[[1]])

Diploma x Socio professional group by departement in 2015

Description

Contingency tables of the counts of Diploma x Socio professional group by metroplitan France departement in year 2015.

Usage

data(dspgd2015)

Format

dspgd2015 is a list of 96 arrays (each one corresponding to a department, designated by its official geographical code) of 4 rows (each one corresponding to a level of diploma) and 6 columns (each one corresponding to a socio professional group).

csp:

Socio professional group

diplome:

Diploma

agri:

farmer (agriculteur)

arti:

craftsperson (artisan)

cadr:

senior manager (cadre sup\'erieur)

pint:

middle manager (profession interm\'ediaire)

empl:

employee (employ\'e)

ouvr:

worker (ouvrier)

bepc:

brevet

cap:

NVQ (cap)

bac:

baccalaureate

sup:

higher education (sup\'erieur)

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

Source

INSEE. Population active de 25 à 54 ans ayant un emploi et chômeurs par catégorie socioprofessionnelle et diplôme par commune et département (1968 à 2015).

Examples

data(dspgd2015)
names(dspgd2015)
print(dspgd2015[[1]])

Dual STATIS method (interstructure stage)

Description

Performs the first stage (interstructure) of the dual STATIS method in order to describe a data folder, consisting of T groups of individuals on which are observed p variables. It returns an object of class dstatis.

Usage

dstatis.inter(xf, normed = TRUE, centered = TRUE, data.scaled = FALSE, nb.factors = 3,
      nb.values = 10, sub.title = "", plot.eigen = TRUE, plot.score = FALSE,
      nscore = 1:3, group.name = "group", filename = NULL)

Arguments

Details

The covariance matrices (if data.scale is FALSE) or correlation matrices (if TRUE) per group are computed. The matrix W of the scalar products between these covariance matrices is then computed.

To perform the STATIS method, see the function DSTATIS of the multigroup package.

Value

Returns an object of class dstatis, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Lavit, C., Escoufier, Y., Sabatier, R., Traissac, P. (1994). The ACT (STATIS method). Computational Statistics & Data Analysis, 18 (1994), 97-119.

See Also

print.dstatis, plot.dstatis, interpret.dstatis.

Examples

data(roses)
rosesf <- as.folder(roses[,c("Sha","Den","Sym","rose")])

# Dual STATIS on the covariance matrices
result1 <- dstatis.inter(rosesf, data.scaled = FALSE, group.name = "rose")
print(result1)
plot(result1)

# Dual STATIS on the correlation matrices
result2 <- dstatis.inter(rosesf, data.scaled = FALSE, group.name = "rose")
print(result2)
plot(result2)

Misclassification ratio in functional discriminant analysis of probability densities.

Description

Computes the one-leave-out misclassification ratio of the rule assigning T groups of individuals, one group after another, to the class of groups (among K classes of groups) which achieves the minimum of the distances or divergences between the density function associated to the group to assign and the K density functions associated to the K classes.

Usage

fdiscd.misclass(xf, class.var, gaussiand = TRUE,
           distance =  c("jeffreys", "hellinger", "wasserstein", "l2", "l2norm"),
           crit = 1, windowh = NULL)

Arguments

Details

The T probability densities f_t corresponding to the T groups of individuals are either parametrically estimated (gaussiand = TRUE) or estimated using the Gaussian kernel method (gaussiand = FALSE). In the latter case, the windowh argument provides the list of the bandwidths to be used. Notice that in the multivariate case (p>1), the bandwidths are positive-definite matrices.

The argument windowh is a numerical value, the matrix bandwidth is of the form h S, where S is either the square root of the covariance matrix (p>1) or the standard deviation of the estimated density.

If windowh = NULL (default), h in the above formula is computed using the bandwidth.parameter function.

To the class k consisting of T_k groups is associated the density denoted g_k. The crit argument selects the estimation method of the K densities g_k.

1. The density g_k is estimated using the whole data of this class, that is the rows of x corresponding to the T_k groups of the class k.

   The estimation of the densities g_k uses the same method as the estimation of the f_t.

2. The T_k densities f_t are estimated using the corresponding data from x. Then they are averaged to obtain an estimation of the density g_k, that is g_k = \frac{1}{T_k} \, \sum{f_t}.

3. Each previous density f_t is weighted by n_t (the number of rows of x corresponding to f_t). Then they are averaged, that is g_k = \frac{1}{\sum n_t} \sum n_t f_t.

The last two methods are only available for the L^2-distance. If the divergences between densities are computed using the Hellinger or Wasserstein distance or Jeffreys measure, only the first of these methods is available.

The distance or dissimilarity between the estimated densities is either the L^2 distance, the Hellinger distance, Jeffreys measure (symmetrised Kullback-Leibler divergence) or the Wasserstein distance.

• If it is the L^2 distance (distance="l2" or distance="l2norm"), the densities can be either parametrically estimated or estimated using the Gaussian kernel.

• If it is the Hellinger distance (distance="hellinger"), Jeffreys measure (distance="jeffreys") or the Wasserstein distance (distance="wasserstein"), the densities are considered Gaussian and necessarily parametrically estimated.

Value

Returns an object of class fdiscd.misclass, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R. (2004). Discriminant analysis with independently repeated multivariate measurements: an L^2 approach. Computational Statistics & Data Analysis, 47, 823-843.

Rudrauf, J.M., Boumaza, R. (2001). Contribution à l'étude de l'architecture médiévale: les caractéristiques des pierres à bossage des châteaux forts alsaciens. Centre de Recherches Archéologiques Médiévales de Saverne, 5, 5-38.

Examples

data(castles.dated)
castles.stones <- castles.dated$stones
castles.periods <- castles.dated$periods
castlesfh <- folderh(castles.periods, "castle", castles.stones)
result <- fdiscd.misclass(castlesfh, "period")
print(result)

Predicting the class of a group of individuals with discriminant analysis of probability densities.

Description

Assigns several groups of individuals, one group after another, to the class of groups (among K classes of groups) which achieves the minimum of the distances or divergences between the density function associated to the group to assign and the K density functions associated to the K classes.

Usage

fdiscd.predict(xf, class.var, gaussiand = TRUE,
           distance =  c("jeffreys", "hellinger", "wasserstein", "l2", "l2norm"),
           crit = 1, windowh = NULL, misclass.ratio = FALSE)

Arguments

Details

To the group t is associated the density denoted f_t. To the class k consisting of T_k groups is associated the density denoted g_k. The crit argument selects the estimation method of the K densities g_k.

1. The density g_k is estimated using the whole data of this class, that is the rows of x corresponding to the T_k groups of the class k.

2. The T_k densities f_t are estimated using the corresponding data from x. Then they are averaged to obtain an estimation of the density g_k, that is g_k = (1/T_k)\sum{f_t}.

3. Each previous density f_t is weighted by n_t (the number of rows of x corresponding to f_t). Then they are averaged, that is g_k = (1/\sum n_t) \sum n_t f_t.

The last two methods are available only for the L^2-distance. If the divergences between densities are computed using the Hellinger or Wasserstein distance or Jeffreys measure, only the first of these methods is available.

Value

Returns an object of class fdiscd.predict, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R. (2004). Discriminant analysis with independently repeated multivariate measurements: an L^2 approach. Computational Statistics & Data Analysis, 47, 823-843.

Rudrauf, J.M., Boumaza, R. (2001). Contribution à l'étude de l'architecture médiévale: les caractéristiques des pierres à bossage des châteaux forts alsaciens. Centre de Recherches Archéologiques Médiévales de Saverne, 5, 5-38.

Examples

data(castles.dated)
data(castles.nondated)
castles.stones <- rbind(castles.dated$stones, castles.nondated$stones)
castles.periods <- rbind(castles.dated$periods, castles.nondated$periods)
castlesfh <- folderh(castles.periods, "castle", castles.stones)

# With the L^2-distance

# - crit=1
resultl2.1 <- fdiscd.predict(castlesfh, "period", distance="l2", crit=1)
print(resultl2.1)

# - crit=2
## Not run: 
resultl2.2 <- fdiscd.predict(castlesfh, "period", distance="l2", crit=2)
print(resultl2.2)

## End(Not run)

# - crit=3
resultl2.3 <- fdiscd.predict(castlesfh, "period", distance="l2", crit=3)
print(resultl2.3)

# With the Hellinger distance
resulthelling <- fdiscd.predict(castlesfh, "period", distance="hellinger")
print(resulthelling)

# With jeffreys measure
resultjeff <- fdiscd.predict(castlesfh, "period", distance="jeffreys")
print(resultjeff)

Hierarchic cluster analysis of probability densities

Description

Performs functional hierarchic cluster analysis of probability densities. It returns an object of class fhclustd. It applies hclust to the distance matrix between the T densities.

Usage

fhclustd(xf, group.name  = "group", gaussiand = TRUE, distance = c("jeffreys",
             "hellinger", "wasserstein", "l2", "l2norm"), windowh=NULL,
             data.centered = FALSE, data.scaled = FALSE, common.variance = FALSE,
             sub.title = "", filename = NULL, method.hclust = "complete")

Arguments

Details

In order to compute the distances/dissimilarities between the groups, the T probability densities f_t corresponding to the T groups of individuals are either parametrically estimated (gaussiand = TRUE) or estimated using the Gaussian kernel method (gaussiand = FALSE). In the latter case, the windowh argument provides the list of the bandwidths to be used. Notice that in the multivariate case (p>1), the bandwidths are positive-definite matrices. The distances between the T groups of individuals are given by the L^2-distances between the T probability densities f_t corresponding to these groups. The hclust function is then applied to the distance matrix to perform the hierarchical clustering on the T groups.

If windowh is a numerical value, the matrix bandwidth is of the form h S, where S is either the square root of the covariance matrix (p>1) or the standard deviation of the estimated density.

If windowh = NULL (default), h in the above formula is computed using the bandwidth.parameter function.

The distance or dissimilarity between the estimated densities is either the L^2 distance, the Hellinger distance, Jeffreys measure (symmetrised Kullback-Leibler divergence) or the Wasserstein distance.

• If it is the L^2 distance (distance="l2" or distance="l2norm"), the densities can be either parametrically estimated or estimated using the Gaussian kernel.

• If it is the Hellinger distance (distance="hellinger"), Jeffreys measure (distance="jeffreys") or the Wasserstein distance (distance="wasserstein"), the densities are considered Gaussian and necessarily parametrically estimated.

Value

Returns an object of class fhclustd, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

fdiscd.predict, fdiscd.misclass

Examples

data(castles.dated)
stones <- castles.dated$stones
periods <- castles.dated$periods

periods123 <- periods[periods$period %in% 1:3, "castle"]
stones123 <- stones[stones$castle %in% periods123, ]
stones123$castle <- as.factor(as.character(stones123$castle))
yf <- as.folder(stones123)


# Jeffreys measure (default):

resultjef <- fhclustd(yf)
print(resultjef)
print(resultjef, dist.print = TRUE)
plot(resultjef)
plot(resultjef, hang = -1)

# Use cutree (stats package) to get the partition
cutree(resultjef$clust, k = 1:4)
cutree(resultjef$clust, k = 5)
cutree(resultjef$clust, h = 0.041)


# Applied to a data frame (Jeffreys measure):

fhclustd(stones123, group.name = "castle")

# Use cutree (stats package) to get the partition
cutree(resultjef$clust, k = 1:4)
cutree(resultjef$clust, k = 5)
cutree(resultjef$clust, h = 0.041)


# Hellinger distance:

resulthel <- fhclustd(yf, distance = "hellinger")
print(resulthel)
print(resulthel, dist.print = TRUE)
plot(resulthel)
plot(resulthel, hang = -1)

# Use cutree (stats package) to get the partition
cutree(resulthel$clust, k = 1:4)
cutree(resulthel$clust, k = 5)
cutree(resulthel$clust, h = 0.041)


## Not run: 
# L2-distance:

xf <- as.folder(stones)
result <- fhclustd(xf, distance = "l2")
print(result)
print(result, dist.print = TRUE)
plot(result)
plot(result, hang = -1)

# Use cutree (stats package) to get the partition
cutree(result$clust, k = 1:5)
cutree(result$clust, k = 5)
cutree(result$clust, h = 0.18)

## End(Not run)

periods123 <- periods[periods$period %in% 1:3, "castle"]
stones123 <- stones[stones$castle %in% periods123, ]
stones123$castle <- as.factor(as.character(stones123$castle))
yf <- as.folder(stones123)
result123 <- fhclustd(yf, distance = "l2")
print(result123)
print(result123, dist.print = TRUE)
plot(result123)
plot(result123, hang = -1)

# Use cutree (stats package) to get the partition
cutree(result123$clust, k = 1:4)
cutree(result123$clust, k = 5)
cutree(result123$clust, h = 0.041)

Rose flowering

Description

These data are collected on eight rosebushes from four varieties, during summer 2010 in Angers, France. They give measures of the flowering.

Usage

data("floribundity")

Format

floribundity is a list of 16 data frames, each corresponding to an observation date. Each one of these data frames has 3 or 4 columns:

• rose: the number of the rosebush, that is an identifier.

• variety: factor. The variety of the rosebush.

• area (when available): numeric. The ratio of flowering area to the whole plant area, measured on the photograph of the rosebush.

• nflowers (when available): integer. The number of flowers on the rosebush.

The row names of these data frames are the rose identifiers.

Examples

data(floribundity)
foldt <- foldert(floribundity, times = as.Date(names(floribundity)), rows.select = "union")
summary(foldt)

Multidimensional scaling of probability densities

Description

Applies the multidimensional scaling (MDS) method to probability densities in order to describe a data folder, consisting of T groups of individuals on which are observed p variables. It returns an object of class fmdsd. It applies cmdscale to the distance matrix between the T densities.

Usage

fmdsd(xf, group.name = "group", gaussiand = TRUE, distance = c("jeffreys", "hellinger",
    "wasserstein", "l2", "l2norm"), windowh=NULL, data.centered = FALSE,
    data.scaled = FALSE, common.variance = FALSE, add = TRUE, nb.factors = 3,
    nb.values = 10, sub.title = "", plot.eigen = TRUE, plot.score = FALSE, nscore = 1:3,
    filename = NULL)

Arguments

Details

In order to compute the distances/dissimilarities between the groups, the T probability densities f_t corresponding to the T groups of individuals are either parametrically estimated (gaussiand = TRUE) or estimated using the Gaussian kernel method (gaussiand = FALSE). In the latter case, the windowh argument provides the list of the bandwidths to be used. Notice that in the multivariate case (p>1), the bandwidths are positive-definite matrices.

If windowh is a numerical value, the matrix bandwidth is of the form h S, where S is either the square root of the covariance matrix (p>1) or the standard deviation of the estimated density.

If windowh = NULL (default), h in the above formula is computed using the bandwidth.parameter function.

The distance or dissimilarity between the estimated densities is either the L^2 distance, the Hellinger distance, Jeffreys measure (symmetrised Kullback-Leibler divergence) or the Wasserstein distance.

• If it is the L^2 distance (distance="l2" or distance="l2norm"), the densities can be either parametrically estimated or estimated using the Gaussian kernel.

• If it is the Hellinger distance (distance="hellinger"), Jeffreys measure (distance="jeffreys") or the Wasserstein distance (distance="wasserstein"), the densities are considered Gaussian and necessarily parametrically estimated.

Value

Returns an object of class fmdsd, i.e. a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

Delicado, P. (2011). Dimensionality reduction when data are density functions. Computational Statistics & Data Analysis, 55, 401-420.

Yousfi, S., Boumaza, R., Aissani, D., Adjabi, S. (2014). Optimal bandwith matrices in functional principal component analysis of density function. Journal of Statistical Computation and Simulation, 85 (11), 2315-2330.

Cox, T.F., Cox, M.A.A. (2001). Multimensional Scaling, second ed. Chapman & Hall/CRC.

See Also

fpcad print.fmdsd, plot.fmdsd, interpret.fmdsd, bandwidth.parameter

Examples

data(roses)
rosesf <- as.folder(roses[,c("Sha","Den","Sym","rose")])

# MDS on Gaussian densities (on sensory data)

# using jeffreys measure (default):
resultjeff <- fmdsd(rosesf, distance = "jeffreys")
print(resultjeff)
plot(resultjeff)

## Not run: 
# Applied to a data frame:
resultjeffdf <- fmdsd(roses[,c("Sha","Den","Sym","rose")],
                      distance = "jeffreys", group.name = "rose")
print(resultjeffdf)
plot(resultjeffdf)

## End(Not run)

# using the Hellinger distance:
resulthellin <- fmdsd(rosesf, distance = "hellinger")
print(resulthellin)
plot(resulthellin)

# using the Wasserstein distance:
resultwass <- fmdsd(rosesf, distance = "wasserstein")
print(resultwass)
plot(resultwass)

# Gaussian case, using the L2-distance:
resultl2 <- fmdsd(rosesf, distance = "l2")
print(resultl2)
plot(resultl2)

# Gaussian case, using the L2-distance between normed densities:
resultl2norm <- fmdsd(rosesf, distance = "l2norm")
print(resultl2norm)
plot(resultl2norm)

## Not run: 
# Non Gaussian case, using the L2-distance,
# the densities are estimated using the Gaussian kernel method:
result <- fmdsd(rosesf, distance = "l2", gaussiand = FALSE, group.name = "rose")
print(result)       
plot(result)

## End(Not run)

Folder of data sets

Description

Creates an object of class "folder" (called folder below), that is a list of data frames with the same column names. Thus, these data sets are on the same variables. They can be on the same individuals or not.

Usage

folder(x1, x2 = NULL, ..., cols.select = "intersect", rows.select = "")

Arguments

Details

The class folder has a logical attributes attr(,"same.rows").

The data frames in the returned folder all have the same column names. That means that the same variables are observed in every data sets.

If the rows.select argument is "union" or "intersect", the elements of the returned folder have the same rows. That means that the same individuals are present in every data sets. This allows to consider the evolution of each individual among time.

If rows.select is "", every rows of this folder are different, and the row names are made unique by adding the name of the data frame to the row names. In this case, The individuals of the data sets are assumed to be all different. Or, at least, the user does not mind if they are the same or not.

Value

Returns an object of class "folder", that is a list of data frames.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

is.folder to test if an object is of class folder. folderh to build a folder of several data frames with a hierarchic relation between each pair of consecutive data frames.

Examples

# First example              
x1 <- data.frame(x = rnorm(10), y = 1:10)
x2 <- data.frame(x = rnorm(10), z = runif(10, 1, 10))
f1 <- folder(x1, x2)
print(f1)

f2 <- folder(x1, x2, cols.select = "union")
print(f2)

#Second example
data(iris)
iris.set <- iris[iris$Species == "setosa", 1:4]
iris.ver <- iris[iris$Species == "versicolor", 1:4]
iris.vir <- iris[iris$Species == "virginica", 1:4]
irisf1 <- folder(iris.set, iris.ver, iris.vir)
print(irisf1)

listofdf <- list(df1 = iris.set,df2 = iris.ver,df3 = iris.vir)
irisf2 <- folder(listofdf,x2 = NULL)
print(irisf2)

Hierarchic folder of n data frames related in pairs by (n-1) keys

Description

Creates an object of class folderh, that is a list of n>1 data frames whose rows are related by (n-1) keys, each key defining a relation "1 to N" between the two adjacent data frames passed as arguments of the function.

Usage

folderh(df1, key1, df2, ..., na.rm = TRUE)

Arguments

Details

The object of class folderh is a list of n \ge 2 data frames.

• If no optional arguments are given via ..., that is n = 2, the two data frames of the list have a column named by the attribute attr(, "keys") (argument key1), which is a factor with the same levels. Each one of these levels occur exactly once in the first data frame of the list.

• If some supplementary data frames and supplementary strings key2, df3, ... are given as optional arguments, n is the number of data frames given as arguments. Then, the attribute attr(, "keys") is a vector of n-1 character strings. For i = 1, \ldots, N-1, its i-th element is the name of a column of the i-th and (i+1)-th data frames of the folderh, which are factors with the same levels. Each one of these levels occur exactly once in the i-th data frame.

If there are more than two data frames, folderh computes a folderh with the two last data frames, and then uses the function appendtofolderh to append each one of the other data frames to the folderh.

Value

Returns an object of class folderh. Its elements are the data frames passed as arguments, and the attribute attr(, "keys") contains the character arguments.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

is.folderh to test if an object is of class folderh. folder for a folder of data frames with no hierarchic relation between them. as.folder.folderh (or as.data.frame.folderh) to build an object of class folder (or a data frame) from an object of class folderh,

Examples

# First example: rose flowers
data(roseflowers)
df1 <- roseflowers$variety
df2 <- roseflowers$flower
fh1 <- folderh(df1, "rose", df2)
print(fh1)

# Second example
data(roseleaves)
roses <- roseleaves$rose
stems <- roseleaves$stem
leaves <- roseleaves$leaf
leaflets <- roseleaves$leaflet
fh2 <- folderh(roses, "rose", stems, "stem", leaves, "leaf", leaflets)
print(fh2)

foldermtg

Description

An object of S3 class "foldermtg" is built and returned by the function read.mtg.

Value

An object of this S3 class is a list of at least 5 data frames (see the Value section in read.mtg): classes, description, features, topology, coordinates...

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Pradal, C., Godin, C. and Cokelaer, T. (2023). MTG user guide

See Also

read.mtg print.foldermtg mtgorder

Examples

mtgfile1 <- system.file("extdata/plant1.mtg", package = "dad")
x1 <- read.mtg(mtgfile1)
print(x1)

mtgfile2 <- system.file("extdata/plant2.mtg", package = "dad")
x2 <- read.mtg(mtgfile2)
print(x2)

Folder of data sets among time

Description

Creates an object of class "foldert" (called foldert below), that is a list of data frames, each of them corresponding to a time of observation. These data sets are on the same variables. They can be on the same individuals or not.

Usage

foldert(x1, x2 = NULL, ..., times = NULL, cols.select = "intersect", rows.select = "")

Arguments

Details

The class "foldert" has an attribute attr(,"times") (the times argument, when provided) and a logical attributes attr(,"same.rows").

The data frames in the returned foldert all have the same column names. That means that the same variables are observed in every data sets.

If the rows.select argument is "union" or "intersect", the elements of the returned foldert have the same rows. That means that the same individuals are present in every data sets. This allows to consider the evolution of each individual among time.

If rows.select is "", every rows of this foldert are different, and the row names are made unique by adding the name of the data frame to the row names. In this case, The individuals of the data sets are assumed to be all different. Or, at least, the user does not mind if they are the same or not.

Value

Returns an object of class "foldert", that is a list of data frames. The elements of this list are ordered according to time.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

is.foldert to test if an object is of class foldert. as.foldert.data.frame: build an object of class foldert from a data frame. as.foldert.array: build an object of class foldert from a 3d-array.

Examples

x <- data.frame(xyz = rep(c("A", "B", "C"), each = 2),
                xy = letters[1:6],
                x1 = rnorm(6),
                x2 = rnorm(6, 2, 1),
                row.names = paste0("i", 1:6),
                stringsAsFactors = TRUE)
y <- data.frame(xyz = c("A", "A", "B", "C"),
                xy = c("a", "b", "a", "c"),
                y1 = rnorm(4, 4, 2),
                row.names = c(paste0("i", c(1, 2, 4, 6))),
                stringsAsFactors = TRUE)
z <- data.frame(xyz = c("A", "B", "C"),
                z1 = rnorm(3),
                row.names = c("i1", "i2", "i5"),
                stringsAsFactors = TRUE)

# Columns selected by the user
ftc. <- foldert(x, y, z, cols.select = c("xyz", "x1", "y1", "z1"))
print(ftc.)

# cols.select = "union": all the variables (columns) of each data frame are kept
ftcun <- foldert(x, y, z, cols.select = "union")
print(ftcun)

# cols.select = "intersect": only variables common to all data frames
ftcint <- foldert(x, y, z, cols.select = "intersect")
print(ftcint)

# rows.select = "": the rows of the data frames are unchanged
# and the rownames are made unique
ftr. <- foldert(x, y, z, rows.select = "")
print(ftr.)

# rows.select = "union": all the individuals (rows) of each data frame are kept
ftrun <- foldert(x, y, z, rows.select = "union")
print(ftrun)

# rows.select = "intersect": only individuals common to all data frames
ftrint <- foldert(x, y, z, rows.select = "intersect")
print(ftrint)

# Define the times (times argument)
ftimes <- foldert(x, y, z, times = as.Date(c("2018-03-01", "2018-04-01", "2018-05-01")))
print(ftimes)

Functional PCA of probability densities

Description

Performs functional principal component analysis of probability densities in order to describe a data folder, consisting of T groups of individuals on which are observed p variables. It returns an object of class fpcad.

Usage

fpcad(xf, group.name = "group", gaussiand = TRUE, windowh = NULL, normed = TRUE,
    centered = TRUE, data.centered = FALSE, data.scaled = FALSE,
    common.variance = FALSE, nb.factors = 3, nb.values = 10, sub.title = "",
    plot.eigen = TRUE, plot.score = FALSE, nscore = 1:3,
    filename = NULL)

Arguments

Details

The T probability densities f_t corresponding to the T groups of individuals are either parametrically estimated (gaussiand = TRUE) or estimated using the Gaussian kernel method (gaussiand = FALSE). In the latter case, the windowh argument provides the list of the bandwidths to use. Notice that in the multivariate case (p>1) the bandwidths are positive-definite matrices.

If windowh is a numerical value, the matrix bandwidth is of the form h S, where S is either the square root of the covariance matrix (p>1) or the standard deviation of the estimated density.

If windowh = NULL (default), h in the above formula is computed using the bandwidth.parameter function.

Value

Returns an object of class fpcad, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R. (1998). Analyse en composantes principales de distributions gaussiennes multidimensionnelles. Revue de Statistique Appliqu?e, XLVI (2), 5-20.

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

Delicado, P. (2011). Dimensionality reduction when data are density functions. Computational Statistics & Data Analysis, 55, 401-420.

Yousfi, S., Boumaza, R., Aissani, D., Adjabi, S. (2014). Optimal bandwith matrices in functional principal component analysis of density functions. Journal of Statistical Computation and Simulation, 85 (11), 2315-2330.

See Also

print.fpcad, plot.fpcad, interpret.fpcad, bandwidth.parameter

Examples

data(roses)
# Case of a normed non-centred PCA of Gaussian densities (on 3 architectural 
# characteristics of roses: shape (Sha), foliage density (Den) and symmetry (Sym))
rosesf <- as.folder(roses[,c("Sha","Den","Sym","rose")])
result3 <- fpcad(rosesf, group.name = "rose")
print(result3)
plot(result3)

# Applied to a data frame:
result3df <- fpcad(roses[,c("Sha","Den","Sym","rose")], group.name = "rose")
print(result3df)
plot(result3df)

# Flower colors of the roses
scores <- result3$scores
scores <- data.frame(scores, color = scores$rose, stringsAsFactors = TRUE)
colours <- scores$rose
colours <- factor(c(A = "yellow", B = "yellow", C = "pink", D = "yellow", E = "red",
                  F = "yellow", G = "pink", H = "pink", I = "yellow", J = "yellow"))
levels(scores$color) <- c(A = "yellow", B = "yellow", C = "pink", D = "yellow", E = "red",
                         F = "yellow", G = "pink", H = "pink", I = "yellow", J = "yellow")
# Scores according to the first two principal components, per color
plot(result3, nscore = 1:2, color = colours)

Functional PCA of probability densities among time

Description

Performs functional principal component analysis of probability densities in order to describe a data “foldert”, consisting of individuals on which are observed p variables on T times. It returns an object of class fpcat.

Usage

fpcat(xf, group.name="time", method = 1, ind = 1, nvar = NULL, gaussiand = TRUE,
    windowh = NULL, normed=TRUE, centered=TRUE, data.centered = FALSE,
    data.scaled = FALSE, common.variance = FALSE, nb.factors = 3, nb.values = 10,
    sub.title = "", plot.eigen = TRUE, plot.score = FALSE, nscore = 1:3,
    filename = NULL)

Arguments

All other arguments are the same as for fpcad.

Details

The T probability densities f_t corresponding to the T times of observation are either parametrically estimated or estimated using the Gaussian kernel method (see fpcad for the use of the arguments indicating the method used to estimate these densities).

Value

Returns an object of class fpcat, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R. (1998). Analyse en composantes principales de distributions gaussiennes multidimensionnelles. Revue de Statistique Appliqu?e, XLVI (2), 5-20.

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

Delicado, P. (2011). Dimensionality reduction when data are density functions. Computational Statistics & Data Analysis, 55, 401-420.

Yousfi, S., Boumaza, R., Aissani, D., Adjabi, S. (2014). Optimal bandwith matrices in functional principal component analysis of density functions. Journal of Statistical Computation and Simulation, 85 (11), 2315-2330.

See Also

print.fpcat, plot.fpcat, bandwidth.parameter

Examples

times <- as.Date(c("2017-03-01", "2017-04-01", "2017-05-01", "2017-06-01"))
x1 <- data.frame(z1=rnorm(6,1,5), z2=rnorm(6,3,3))
x2 <- data.frame(z1=rnorm(6,4,6), z2=rnorm(6,5,2))
x3 <- data.frame(z1=rnorm(6,7,2), z2=rnorm(6,8,4))
x4 <- data.frame(z1=rnorm(6,9,3), z2=rnorm(6,10,2))
ft <- foldert(x1, x2, x3, x4, times = times, rows.select="intersect")
print(ft)
result <- fpcat(ft)
print(result)
plot(result)

Select columns in all elements of a folder

Description

Select columns in all data frames of a folder.

Usage

getcol.folder(object, name)

Arguments

Value

A folder with the same number of elements as object. Its k^{th} element is a data frame, and its columns are the columns of object[[k]] given by name.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder: object of class folder.

rmcol.folder: remove columns in all elements of a folder.

getrow.folder: select rows in all elements of a folder.

rmrow.folder: remove rows in all elements of a folder.

Examples

data(iris)

iris.fold <- as.folder(iris, "Species")
getcol.folder(iris.fold, "Sepal.Length")
getcol.folder(iris.fold, c("Petal.Length", "Petal.Width"))

Select columns in all elements of a foldert

Description

Select columns in all data frames of a foldert.

Usage

getcol.foldert(object, name)

Arguments

Value

A foldert with the same number of elements as object. Its k^{th} element is a data frame, and its columns are the columns of object[[k]] given by name.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

foldert: object of class foldert.

rmcol.foldert: remove columns in all elements of a foldert.

getrow.foldert: select rows in all elements of a foldert.

rmrow.foldert: remove rows in all elements of a foldert.

Examples

data(floribundity)

ft0 <- foldert(floribundity, cols.select = "union")
getcol.foldert(ft0, c("rose", "variety"))

Select rows in all elements of a folder

Description

Select rows in all data frames of a folder.

Usage

getrow.folder(object, name)

Arguments

Value

A folder with the same number of elements as object. Its k^{th} element is a data frame, and its rows are the rows of object[[k]] given by name.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder: object of class folder.

rmrow.folder: remove rows in all elements of a folder.

getcol.folder: select rows in all elements of a folder.

rmcol.folder: remove rows in all elements of a folder.

Examples

data(iris)

iris.fold <- as.folder(iris, "Species")
getrow.folder(iris.fold, c(1:5, 51:55, 101:105))

Select rows in all elements of a foldert

Description

Select rows in all data frames of a foldert.

Usage

getrow.foldert(object, name)

Arguments

Value

A foldert with the same number of elements as object. Its k^{th} element is a data frame, and its rows are the rows of object[[k]] given by name.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

foldert: object of class foldert.

rmrow.foldert: remove rows in all elements of a foldert.

getcol.foldert: select columns in all elements of a foldert.

rmcol.foldert: remove columns in all elements of a foldert.

Examples

data(floribundity)

ft0 <- foldert(floribundity, cols.select = "union", rows.select = "union")
getrow.foldert(ft0, c("16", "51"))

Hierarchic cluster analysis of discrete probability distributions

Description

Performs functional hierarchic cluster analysis of discrete probability distributions. It returns an object of class hclustdd. It applies hclust to the distance matrix between the T distributions.

Usage

hclustdd(xf, group.name = "group", distance = c("l1", "l2", "chisqsym", "hellinger",
             "jeffreys", "jensen", "lp"), 
             sub.title = "", filename = NULL,
             method.hclust = "complete")

Arguments

Details

In order to compute the distances/dissimilarities between the groups, the T probability distributions f_t corresponding to the T groups of individuals are estimated from observations. Then the distances/dissimilarities between the estimated distributions are computed, using the distance or divergence defined by the distance argument:

If the distance is "l1", "l2" or "lp", the distances are computed by the function matddlppar. Otherwise, it can be computed by matddchisqsympar ("chisqsym"), matddhellingerpar ("hellinger"), matddjeffreyspar ("jeffreys") or matddjensenpar ("jensen").

Value

Returns an object of class hclustdd, that is a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

hclustdd

Examples

# Example 1 with a folder (10 groups) of 3 factors 
# obtained by converting numeric variables 
data(roses)
xr = roses[,c("Sha", "Den", "Sym", "rose")]
xr = cut(xr, breaks = list(c(0, 5, 7, 10), c(0, 4, 6, 10), c(0, 6, 8, 10)))
xf = as.folder(xr, groups = "rose")
af = hclustdd(xf)
print(af)
print(af, dist.print = TRUE)
plot(af)
plot(af, hang = -1)

# Example 2 with a data frame obtained by converting numeric variables
ar = hclustdd(xr, group.name = "rose")
print(ar)
print(ar, dist.print = TRUE)
plot(ar)
plot(ar, hang = -1)

# Example 3 with a list of 7 arrays
data(dspg)
xl = dspg
hclustdd(xl)

Hellinger distance between Gaussian densities

Description

Hellinger distance between two multivariate (p > 1) or univariate (p = 1) Gaussian densities (see Details).

Usage

hellinger(x1, x2, check = FALSE)

Arguments

Details

The Hellinger distance between the two Gaussian densities is computed by using the hellingerpar function and the density parameters estimated from samples.

Value

Returns the Hellinger distance between the two probability densities.

Be careful! If check = FALSE and one smoothing bandwidth matrix is degenerate, the result returned can not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

McLachlan, G.J. (1992). Discriminant analysis and statistical pattern recognition. John Wiley & Sons, New York .

See Also

hellingerpar: Hellinger distance between Gaussian densities, given their parameters.

Examples

require(MASS)
m1 <- c(0,0)
v1 <- matrix(c(1,0,0,1),ncol = 2) 
m2 <- c(0,1)
v2 <- matrix(c(4,1,1,9),ncol = 2)
x1 <- mvrnorm(n = 3,mu = m1,Sigma = v1)
x2 <- mvrnorm(n = 5, mu = m2, Sigma = v2)
hellinger(x1, x2)

Hellinger distance between Gaussian densities given their parameters

Description

Hellinger distance between two multivariate (p > 1) or univariate (p = 1) Gaussian densities given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate) (see Details).

Usage

hellingerpar(mean1, var1, mean2, var2, check = FALSE)

Arguments

Details

The mean vectors (m1 and m2) and variance matrices (v1 and v2) given as arguments (mean1, mean2, var1 and var2) are used to compute the Hellinger distance between the two Gaussian densities, equal to:

( 2 (1 - 2^{p/2} det(v1 v2)^{1/4} det(v1 + v2)^{-1/2} exp((-1/4) t(m1-m2) (v1+v2)^{-1} (m1-m2)) ))^{1/2}

If p = 1 the means and variances are numbers, the formula is the same ignoring the following operators: t (transpose of a matrix or vector) and det (determinant of a square matrix).

Value

The Hellinger distance between two Gaussian densities.

Be careful! If check = FALSE and one covariance matrix is degenerated (multivariate case) or one variance is zero (univariate case), the result returned must not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

McLachlan, G.J. (1992). Discriminant analysis and statistical pattern recognition. John Wiley & Sons, New York .

See Also

hellinger: Hellinger distance between Gaussian densities estimated from samples.

Examples

m1 <- c(1,1)
v1 <- matrix(c(4,1,1,9),ncol = 2)
m2 <- c(0,1)
v2 <- matrix(c(1,0,0,1),ncol = 2)
hellingerpar(m1,v1,m2,v2)

Scores of fmdsd, dstatis, fpcad, or fpcat vs. moments, or scores of mdsdd vs. marginal distributions or association measures

Description

This generic function provides a tool for the interpretation of the results of fmdsd, dstatis, fpcad, fpcat or mdsdd function.

Usage

interpret(x, nscore = 1:3, ...)

Arguments

Value

Returns a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

See Also

interpret.fmdsd; interpret.dstatis; interpret.fpcad; interpret.fpcat; interpret.mdsdd.

Scores of the dstatis function vs. moments of the densities

Description

Applies to an object of class "dstatis", plots the principal scores vs. the moments of the densities (means, standard deviations, variances, correlations, skewness and kurtosis coefficients), and computes the correlations between these scores and moments.

Usage

## S3 method for class 'dstatis'
interpret(x, nscore = 1, moment=c("mean", "sd", "var", "cov", "cor",
    "skewness", "kurtosis"), ...)

Arguments

Details

A graphics device can contain up to 9 graphs. If there are too many (more than 36) graphs for each score, one can display the graphs in a multipage PDF file.

The number of principal scores to be interpreted cannot be greater than nb.factors of the data frame x$scores returned by the function dstatis.inter.

Value

Returns a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Lavit, C., Escoufier, Y., Sabatier, R., Traissac, P. (1994). The ACT (STATIS method). Computational Statistics & Data Analysis, 18 (1994), 97-119.

See Also

dstatis.inter; plot.dstatis.

Examples

data(roses)
rosesf <- as.folder(roses[,c("Sha","Den","Sym","rose")])

# Dual STATIS on the covariance matrices
## Not run: 
result <- dstatis.inter(rosesf, group.name = "rose")
interpret(result)
interpret(result, moment = "var")
interpret(result, moment = "cor")
interpret(result, nscore = 2)

## End(Not run)

Scores of the fmdsd function vs. moments of the densities

Description

Applies to an object of class "fmdsd", plots the scores vs. the moments of the densities (means, standard deviations, variances, correlations, skewness and kurtosis coefficients), and computes the correlations between these scores and moments.

Usage

## S3 method for class 'fmdsd'
interpret(x, nscore = 1, moment=c("mean", "sd", "var", "cov", "cor",
    "skewness", "kurtosis"), ...)

Arguments

Details

A graphics device can contain up to 9 graphs. If there are too many (more than 36) graphs for each score, one can display the graphs in a multipage PDF file.

The number of principal scores to be interpreted cannot be greater than nb.factors of the data frame x$scores returned by the function fmdsd.

Value

Returns a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

Delicado, P. (2011). Dimensionality reduction when data are density functions. Computational Statistics & Data Analysis, 55, 401-420.

See Also

fmdsd; plot.fmdsd.

Examples

data(roses)
x <- roses[,c("Sha","Den","Sym","rose")]
rosesfold <- as.folder(x)
result1 <- fmdsd(rosesfold)
interpret(result1)
## Not run: 
interpret(result1, moment = "var")

## End(Not run)
interpret(result1, nscore = 2)

Scores of the fpcad function vs. moments of the densities

Description

Applies to an object of class "fpcad", plots the principal scores vs. the moments of the densities (means, standard deviations, variances, correlations, skewness and kurtosis coefficients), and computes the correlations between these scores and moments.

Usage

## S3 method for class 'fpcad'
interpret(x, nscore = 1, moment=c("mean", "sd", "var", "cov", "cor",
    "skewness", "kurtosis"), ...)

Arguments

Details

A graphics device can contain up to 9 graphs. If there are too many (more than 36) graphs for each score, one can display the graphs in a multipage PDF file.

The number of principal scores to be interpreted cannot be greater than nb.factors of the data frame x$scores returned by the function fpcad.

Value

Returns a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

See Also

fpcad; plot.fpcad.

Examples

data(roses)
rosefold <- as.folder(roses[,c("Sha","Den","Sym","rose")])
result1 <- fpcad(rosefold)
interpret(result1)
## Not run: 
interpret(result1, moment = "var")

## End(Not run)
interpret(result1, moment = "cor")
interpret(result1, nscore = 2)

Scores of the "fpcat" function vs. moments of the densities

Description

This function applies to an object of class "fpcat" and does the same as for an object of class "fpcad": it plots the principal scores vs. the moments of the densities (means, standard deviations, variances, correlations, skewness and kurtosis coefficients), and computes the correlations between these scores and moments.

Usage

## S3 method for class 'fpcat'
interpret(x, nscore = 1, moment=c("mean", "sd", "var", "cov", "cor",
    "skewness", "kurtosis"), ...)

Arguments

Details

A graphics device can contain up to 9 graphs. If there are too many (more than 36) graphs for each score, one can display the graphs in a multipage PDF file.

The number of principal scores to be interpreted cannot be greater than nb.factors of the data frame x$scores returned by the function fpcat.

Value

Returns a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

See Also

fpcat; plot.fpcat.

Examples

# Alsacian castles with their building year
data(castles)
castyear <- foldert(lapply(castles, "[", 1:4))
fpcayear <- fpcat(castyear, group.name = "year")
interpret(fpcayear)
## Not run: 
interpret(fpcayear, moment="var")

## End(Not run)

Scores of the mdsdd function vs. marginal probability distributions or association measures

Description

Applies to an object of class "mdsdd", plots the scores vs. the marginal probability distributions or pairwise association measures of the discrete variables, and computes the correlations between these scores and probabilities or association measures (see Details).

Usage

## S3 method for class 'mdsdd'
interpret(x, nscore = 1, mma = c("marg1", "marg2", "assoc"), ...)

Arguments

Details

A graphics device can contain up to 9 graphs. If there are too many (more than 36) graphs for each score, one can display the graphs in a multipage PDF file.

The number of principal scores to be interpreted cannot be greater than nb.factors of the data frame x$scores returned by the function mdsdd.

Value

Returns a list including:

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

See Also

mdsdd; plot.mdsdd.

Examples

# INSEE (France): Diploma x Socio professional group, seven years.
data(dspg)
xlista = dspg
a <- mdsdd(xlista)
interpret(a)

# Example 3 with a list of 96 arrays (departments)
## Not run: 
data(dspgd2015)
xd = dspgd2015
res = mdsdd(xd, group.name = "coded")
interpret(res)
plot(res, fontsize.points = 0.7)

# Each department is represented by its name
data(departments)
coor = merge(res$scores, departments, by = "coded")
dev.new()
plot(coor$PC.1, coor$PC.2, type ="n")
text(coor$PC.1, coor$PC.2, coor$named, cex = 0.5)

# Each department is represented by its region
dev.new()
plot(coor$PC.1, coor$PC.2, type ="n")
text(coor$PC.1, coor$PC.2, coor$coder, cex = 0.7)

## End(Not run)

Class discdd.misclass

Description

Tests if its argument is an object of class discdd.misclass (see Details of the function discdd.misclass).

Usage

is.discdd.misclass(x)

Arguments

Value

TRUE if its argument is of class discdd.misclass, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

discdd.misclass.

Class discdd.predict

Description

Tests if its argument is an object of class discdd.predict (see Details of the function discdd.predict).

Usage

is.discdd.predict(x)

Arguments

Value

TRUE if its argument is of class discdd.predict, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

discdd.predict.

Class dstatis

Description

Tests if its argument is an object of class dstatis (see Details of the function dstatis.inter).

Usage

is.dstatis(x)

Arguments

Value

TRUE if its argument is of class dstatis, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

dstatis.inter.

Class fdiscd.misclass

Description

Tests if its argument is an object of class fdiscd.misclass (see Details of the function fdiscd.misclass).

Usage

is.fdiscd.misclass(x)

Arguments

Value

TRUE if its argument is of class fdiscd.misclass, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

fdiscd.misclass.

Class fdiscd.predict

Description

Tests if its argument is an object of class fdiscd.predict (see Details of the function fdiscd.predict)..

Usage

is.fdiscd.predict(x)

Arguments

Value

TRUE if its argument is of class fdiscd.predict, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

fdiscd.predict.

Class fhclustd

Description

Tests if its argument is an object of class fhclustd (see Details of the function fhclustd).

Usage

is.fhclustd(x)

Arguments

Value

TRUE if its argument is of class fhclustd, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

fhclustd.

Class fmdsd

Description

Tests if its argument is an object of class fmdsd (see Details of the function fmdsd).

Usage

is.fmdsd(x)

Arguments

Value

TRUE if its argument is of class fmdsd, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

fmdsd.

Class folder

Description

Tests if its argument is an object of class folder (see folder).

Usage

is.folder(x)

Arguments

Value

TRUE if its argument is of class folder, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder to create an object of class folder.

Class folderh

Description

Tests if its argument is an object of class folderh (see folderh).

Usage

is.folderh(x)

Arguments

Value

TRUE if its argument is of class folderh, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folderh to create an object of class folderh.

Class foldermtg

Description

Tests if its argument is an object of class foldermtg (see read.mtg).

Usage

is.foldermtg(x)

Arguments

Value

TRUE if its argument is of class foldermtg, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

read.mtg to read a MTG file and create an object of class foldermtg.

Class foldert

Description

Tests if its argument is an object of class foldert (see foldert).

Usage

is.foldert(x)

Arguments

Value

TRUE if its argument is of class foldert, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

foldert to create an object of class foldert.

Class fpcad

Description

Tests if its argument is an object of class fpcad (see Details of the function fpcad).

Usage

is.fpcad(x)

Arguments

Value

TRUE if its argument is of class fpcad, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

fpcad.

Class mdsdd

Description

Tests if its argument is an object of class mdsdd (see Details of the function mdsdd).

Usage

is.mdsdd(x)

Arguments

Value

TRUE if its argument is of class mdsdd, and FALSE otherwise.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

mdsdd.

Jeffreys measure between Gaussian densities

Description

Jeffreys measure (or symmetrised Kullback-Leibler divergence) between two multivariate (p > 1) or univariate (p = 1) Gaussian densities given samples (see Details).

Usage

jeffreys(x1, x2, check = FALSE)

Arguments

Details

The Jeffreys measure between the two Gaussian densities is computed by using the jeffreyspar function and the density parameters estimated from samples.

Value

Returns the Jeffrey's measure between the two probability densities.

Be careful! If check = FALSE and one smoothing bandwidth matrix is degenerate, the result returned must not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Thabane, L., Safiul Haq, M. (1999). On Bayesian selection of the best population using the Kullback-Leibler divergence measure. Statistica Neerlandica, 53(3): 342-360.

See Also

jeffreyspar: Jeffreys measure between Gaussian densities, given their parameters.

Examples

require(MASS)
m1 <- c(0,0)
v1 <- matrix(c(1,0,0,1),ncol = 2) 
m2 <- c(0,1)
v2 <- matrix(c(4,1,1,9),ncol = 2)
x1 <- mvrnorm(n = 3,mu = m1,Sigma = v1)
x2 <- mvrnorm(n = 5, mu = m2, Sigma = v2)
jeffreys(x1, x2)

Jeffreys measure between Gaussian densities given their parameters

Description

Jeffreys measure (or symmetrised Kullback-Leibler divergence) between two multivariate (p > 1) or univariate (p = 1) Gaussian densities, given their parameters (mean vectors and covariance matrices if they are multivariate, means and variances if univariate) (see Details).

Usage

jeffreyspar(mean1, var1, mean2, var2, check = FALSE)

Arguments

Details

Let m1 and m2 the mean vectors, v1 and v2 the covariance matrices, Jeffreys measure of the two Gaussian densities is equal to:

(1/2) t(m1 - m2) (v1^{-1} + v2^{-1}) (m1 - m2) - (1/2) tr( (v1 - v2) (v1^{-1} - v2^{-1}) )

.

If p = 1 the means and variances are numbers, the formula is the same ignoring the following operators: t (transpose of a matrix or vector) and tr (trace of a square matrix).

Value

Jeffreys measure between two Gaussian densities.

Be careful! If check = FALSE and one covariance matrix is degenerated (multivariate case) or one variance is zero (univariate case), the result returned must not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

McLachlan, G.J. (1992). Discriminant analysis and statistical pattern recognition. John Wiley & Sons, New York .

Thabane, L., Safiul Haq, M. (1999). On Bayesian selection of the best population using the Kullback-Leibler divergence measure. Statistica Neerlandica, 53(3): 342-360.

See Also

jeffreys: Jeffreys measure of two parametrically estimated Gaussian densities, given samples.

Examples

m1 <- c(1,1)
v1 <- matrix(c(4,1,1,9),ncol = 2)
m2 <- c(0,1)
v2 <- matrix(c(1,0,0,1),ncol = 2)
jeffreyspar(m1,v1,m2,v2)

Kurtosis coefficients of a folder of data sets

Description

Computes the kurtosis coefficient by column of the elements of an object of class folder.

Usage

kurtosis.folder(x, na.rm = FALSE, type = 3)

Arguments

Details

It uses kurtosis to compute the mean by numeric column of each element of the folder. If some columns of the data frames are not numeric, there is a warning, and the means are computed on the numeric columns only.

Value

A list whose elements are the kurtosis coefficients by column of the elements of the folder.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

folder to create an object is of class folder. mean.folder, var.folder, cor.folder, skewness.folder for other statistics for folder objects.

Examples

# First example: iris (Fisher)               
data(iris)
iris.fold <- as.folder(iris, "Species")
iris.kurtosis <- kurtosis.folder(iris.fold)
print(iris.kurtosis)

# Second example: roses
data(roses)
roses.fold <- as.folder(roses, "rose")
roses.kurtosis <- kurtosis.folder(roses.fold)
print(roses.kurtosis)

L^2 inner product of probability densities

Description

L^2 inner product of two multivariate (p > 1) or univariate (p = 1) probability densities, estimated from samples.

Usage

l2d(x1, x2, method = "gaussiand", check = FALSE, varw1 = NULL, varw2 = NULL)

Arguments

Details

• If method = "gaussiand", the mean vectors and the variance matrices (v1 and v2) of the two samples are computed, and they are used to compute the inner product using the l2dpar function.

• If method = "kern", the densities of both samples are estimated using the Gaussian kernel method. These estimations are then used to compute the inner product. if varw1 and varw2 arguments are omitted, the smoothing bandwidths are computed using the normal reference rule matrix bandwidth:

  h_1 v_1^{1/2}

  where

  h_1 = (4 / ( n_1 (p+2) ) )^{1 / (p+4)}

  for the first density. Idem for the second density after making the necessary changes.

Value

The L^2 inner product of the two probability densities.

Be careful! If check = FALSE and one smoothing bandwidth matrix is degenerate, the result returned can not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

Boumaza, R., Yousfi, S., Demotes-Mainard, S. (2015). Interpreting the principal component analysis of multivariate density functions. Communications in Statistics - Theory and Methods, 44 (16), 3321-3339.

Wand, M., Jones, M. (1995). Kernel smoothing. Chapman and Hall/CRC, London.

Yousfi, S., Boumaza R., Aissani, D., Adjabi, S. (2014). Optimal bandwith matrices in functional principal component analysis of density functions. Journal of Statistical Computational and Simulation, 85 (11), 2315-2330.

See Also

l2dpar for Gaussian densities whose parameters are given.

Examples

require(MASS)
m1 <- c(0,0)
v1 <- matrix(c(1,0,0,1),ncol = 2) 
m2 <- c(0,1)
v2 <- matrix(c(4,1,1,9),ncol = 2)
x1 <- mvrnorm(n = 3,mu = m1,Sigma = v1)
x2 <- mvrnorm(n = 5, mu = m2, Sigma = v2)
l2d(x1, x2, method = "gaussiand")
l2d(x1, x2, method = "kern")
l2d(x1, x2, method = "kern", varw1 = v1, varw2 = v2)

L^2 inner product of Gaussian densities given their parameters

Description

L^2 inner product of multivariate (p > 1) or univariate (p = 1) Gaussian densities, given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate).

Usage

l2dpar(mean1, var1, mean2, var2, check = FALSE)

Arguments

Details

Computes the inner product of two Gaussian densities, equal to:

(2\pi)^{-p/2} det(var1 + var2)^{-1/2} exp(-(1/2) t(mean1 - mean2) (var1 + var2)^{-1} (mean1 - mean2))

If p = 1 the means and variances are numbers, the formula is the same ignoring the following operators: t (transpose of a matrix or vector) and det (determinant of a square matrix).

Value

The L^2 inner product between two Gaussian densities.

Be careful! If check = FALSE and one covariance matrix is degenerated (multivariate case) or one variance is zero (univariate case), the result returned must not be considered.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

References

M. Wand and M. Jones (1995). Kernel Smoothing. Chapman and Hall, London.

See Also

l2d for parametrically estimated Gaussian densities or nonparametrically estimated densities, given samples;

Examples

m1 <- c(1,1)
v1 <- matrix(c(4,1,1,9),ncol = 2)
m2 <- c(0,1)
v2 <- matrix(c(1,0,0,1),ncol = 2)
l2dpar(m1,v1,m2,v2)

Matrix of distances between discrete probability densities given samples

Description

Computes the matrix of the symmetric Chi-squared distances between several multivariate or univariate discrete probability distributions, estimated from samples.

Usage

matddchisqsym(x)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of data frames (or distributions), consisting of the pairwise symmetric chi-squared distances between the distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddchisqsym.

matddchisqsympar for discrete probability densities, given the probabilities on the same support.

Examples

# Example 1
x1 <- data.frame(x = factor(c("A", "A", "B", "B")))
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")))
xf <- folder(x1, x2, x3)
matddchisqsym(xf)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")),
                 y = factor(c("a", "b", "a", "b", "a", "b")))
xf <- folder(x1, x2, x3)
matddchisqsym(xf)

Matrix of distances between discrete probability densities given the probabilities on their common support

Description

Computes the matrix of the symmetric Chi-squared distances between several multivariate or univariate discrete probability distributions on the same support (which can be a Cartesian product of q sets), given the probabilities of the states (which are q-tuples) of the support.

Usage

matddchisqsympar(freq)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of distributions, consisting of the pairwise symmetric chi-squared distances between these distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddchisqsympar.

matddchisqsym for discrete probability densities which are estimated from the data.

Matrix of distances between discrete probability densities given samples

Description

Computes the matrix of the Hellinger (or Matusita) distances between several multivariate or univariate discrete probability distributions, estimated from samples.

Usage

matddhellinger(x)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of data frames (or distributions), consisting of the pairwise Hellinger distances between the distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddhellinger.

matddhellingerpar for discrete probability densities, given the probabilities on the same support.

Examples

# Example 1
x1 <- data.frame(x = factor(c("A", "A", "B", "B")))
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")))
xf <- folder(x1, x2, x3)
matddhellinger(xf)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")),
                 y = factor(c("a", "b", "a", "b", "a", "b")))
xf <- folder(x1, x2, x3)
matddhellinger(xf)

Matrix of distances between discrete probability densities given the probabilities on their common support

Description

Computes the matrix of the Hellinger (or Matusita) distances between several multivariate or univariate discrete probability distributions on the same support (which can be a Cartesian product of q sets), given the probabilities of the states (which are q-tuples) of the support.

Usage

matddhellingerpar(freq)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of distributions, consisting of the pairwise Hellinger distances between these distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddhellingerpar.

matddhellinger for discrete probability densities which are estimated from the data.

Matrix of distances between discrete probability densities given samples

Description

Computes the matrix of Jeffreys divergences between several multivariate or univariate discrete probability distributions, estimated from samples.

Usage

matddjeffreys(x)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of data frames (or distributions), consisting of the pairwise Jeffreys divergences between the distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Dezaz E. (2013). Encyclopedia of distances. Springer.

See Also

ddjeffreys.

matddjeffreyspar for discrete probability densities, given the probabilities on the same support.

Examples

# Example 1
x1 <- data.frame(x = factor(c("A", "A", "B", "B")))
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")))
xf <- folder(x1, x2, x3)
matddhellinger(xf)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")),
                 y = factor(c("a", "b", "a", "b", "a", "b")))
xf <- folder(x1, x2, x3)
matddhellinger(xf)

Matrix of divergences between discrete probability densities given the probabilities on their common support

Description

Computes the matrix of Jeffreys divergences between several multivariate or univariate discrete probability distributions on the same support (which can be a Cartesian product of q sets), given the probabilities of the states (which are q-tuples) of the support.

Usage

matddjeffreyspar(freq)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of distributions, consisting of the pairwise Jeffreys divergences between these distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddjeffreyspar.

matddjeffreys for discrete probability densities which are estimated from the data.

Matrix of divergences between discrete probability densities given samples

Description

Computes the matrix of the Jensen-Shannon divergences between several multivariate or univariate discrete probability distributions, estimated from samples.

Usage

matddjensen(x)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of data frames (or distributions), consisting of the pairwise Jensen-Shannon divergences between the distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddjensen.

matddjensenpar for discrete probability densities, given the probabilities on the same support.

Examples

# Example 1
x1 <- data.frame(x = factor(c("A", "A", "B", "B")))
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")))
xf <- folder(x1, x2, x3)
matddhellinger(xf)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")),
                 y = factor(c("a", "b", "a", "b", "a", "b")))
xf <- folder(x1, x2, x3)
matddhellinger(xf)

Matrix of divergences between discrete probability densities given the probabilities on their common support

Description

Computes the matrix of the Jensen-Shannon divergences between several multivariate or univariate discrete probability distributions on the same support (which can be a Cartesian product of q sets), given the probabilities of the states (which are q-tuples) of the support.

Usage

matddjensenpar(freq)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise Jensen-Shannon divergences between the discrete probability densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddjensenpar.

matddjensen for discrete probability densities which are estimated from the data.

Matrix of distances between discrete probability distributions given samples

Description

Computes the matrix of the L^p distances between several multivariate or univariate discrete probability distributions, estimated from samples.

Usage

matddlp(x, p = 1)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of data frames (or distributions), consisting of the pairwise L^p distances between the distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddlp.

matddlppar for discrete probability distributions, given the probabilities on the same support.

Examples

# Example 1
x1 <- data.frame(x = factor(c("A", "A", "B", "B")))
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")))
xf <- folder(x1, x2, x3)
matddlp(xf)
matddlp(xf, p = 2)

# Example 2
x1 <- data.frame(x = factor(c("A", "A", "A", "B", "B", "B")),
                 y = factor(c("a", "a", "a", "b", "b", "b")))                 
x2 <- data.frame(x = factor(c("A", "A", "A", "B", "B")),
                 y = factor(c("a", "a", "b", "a", "b")))
x3 <- data.frame(x = factor(c("A", "A", "B", "B", "B", "B")),
                 y = factor(c("a", "b", "a", "b", "a", "b")))
xf <- folder(x1, x2, x3)
matddlp(xf, p = 1)

Matrix of distances between discrete probability densities given the probabilities on their common support

Description

Computes the matrix of the L^p distances between several multivariate or univariate discrete probability distributions on the same support (which can be a Cartesian product of q sets), given the probabilities of the states (which are q-tuples) of the support.

Usage

matddlppar(freq, p = 1)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of distributions, consisting of the pairwise L^p distances between these distributions.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Sabine Demotes-Mainard

References

Deza, M.M. and Deza E. (2013). Encyclopedia of distances. Springer.

See Also

ddlppar.

matddlp for discrete probability distributions which are estimated from samples.

Matrix of L^2 distances between probability densities

Description

Computes the matrix of the L^2 distances between several multivariate (p > 1) or univariate (p = 1) probability densities, estimated from samples.

Usage

matdistl2d(x, method = "gaussiand", varwL = NULL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise distances between the probability densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

distl2d.

matdistl2dpar when the probability densities are Gaussian, given the parameters (means and variances).

Examples

    data(roses)
    
    # Multivariate:
    X <- as.folder(roses[,c("Sha","Den","Sym","rose")], groups = "rose")
    summary(X)
    mean.X <- mean(X)
    var.X <- var.folder(X)
    
    # Parametrically estimated Gaussian densities:
    matdistl2d(X)
    
    ## Not run: 
    # Estimated densities using the Gaussian kernel method ()normal reference rule bandwidth):
    matdistl2d(X, method = "kern")   

    # Estimated densities using the Gaussian kernel method (bandwidth provided):
    matdistl2d(X, method = "kern", varwL = var.X)
    
## End(Not run)

    # Univariate :
    X1 <- as.folder(roses[,c("Sha","rose")], groups = "rose")
    summary(X1)
    mean.X1 <- mean(X1)
    var.X1 <- var.folder(X1)
    
    # Parametrically estimated Gaussian densities:
    matdistl2d(X1)
    
    # Estimated densities using the Gaussian kernel method (normal reference rule bandwidth):
    matdistl2d(X1, method = "kern")
    
    # Estimated densities using the Gaussian kernel method (normal reference rule bandwidth):
    matdistl2d(X1, method = "kern", varwL = var.X1)

Matrix of L^2 distances between L^2-normed probability densities

Description

Computes the matrix of the L^2 distances between several multivariate (p > 1) or univariate (p = 1) L^2-normed probability densities, estimated from samples, where a L^2-normed probability density is the original probability density function divided by its L^2-norm.

Usage

matdistl2dnorm(x, method = "gaussiand", varwL = NULL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise distances between the L^2-normed probability densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

distl2dnorm.

matdistl2d for the distance matrix between probability densities.

matdistl2dnormpar when the probability densities are Gaussian, given the parameters (means and variances).

Examples

    data(roses)
    
    # Multivariate:
    X <- as.folder(roses[,c("Sha","Den","Sym","rose")], groups = "rose")
    summary(X)
    mean.X <- mean(X)
    var.X <- var.folder(X)
    
    # Parametrically estimated Gaussian densities:
    matdistl2dnorm(X)
    
    ## Not run: 
    # Estimated densities using the Gaussian kernel method ()normal reference rule bandwidth):
    matdistl2dnorm(X, method = "kern")   

    # Estimated densities using the Gaussian kernel method (bandwidth provided):
    matdistl2dnorm(X, method = "kern", varwL = var.X)
    
## End(Not run)

    # Univariate :
    X1 <- as.folder(roses[,c("Sha","rose")], groups = "rose")
    summary(X1)
    mean.X1 <- mean(X1)
    var.X1 <- var.folder(X1)
    
    # Parametrically estimated Gaussian densities:
    matdistl2dnorm(X1)
    
    # Estimated densities using the Gaussian kernel method (normal reference rule bandwidth):
    matdistl2dnorm(X1, method = "kern")
    
    # Estimated densities using the Gaussian kernel method (normal reference rule bandwidth):
    matdistl2dnorm(X1, method = "kern", varwL = var.X1)

Matrix of L^2 distances between L^2-normed Gaussian densities given their parameters

Description

Computes the matrix of the L^2 distances between several multivariate (p > 1) or univariate (p = 1) L^2-normed Gaussian densities, given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate), where a L^2-normed Gaussian density is the original probability density function divided by its L^2-norm.

Usage

matdistl2dnormpar(meanL, varL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise distances between the L^2-normed probability densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

distl2dnormpar.

matdistl2dpar for the distance matrix between Gaussian densities, given their parameters.

matdistl2dnorm for the distance matrix between normed probability densities which are estimated from the data.

Examples

    data(roses)
    
    # Multivariate:
    X <- roses[,c("Sha","Den","Sym","rose")]
    summary(X)
    mean.X <- as.list(by(X[, 1:3], X$rose, colMeans))
    var.X <- as.list(by(X[, 1:3], X$rose, var))
    
    # Gaussian densities, given parameters
    matdistl2dnormpar(mean.X, var.X)

    # Univariate :
    X1 <- roses[,c("Sha","rose")]
    summary(X1)
    mean.X1 <- by(X1$Sha, X1$rose, mean)
    var.X1 <- by(X1$Sha, X1$rose, var)
    
    # Gaussian densities, given parameters
    matdistl2dnormpar(mean.X1, var.X1)

Matrix of L^2 distances between Gaussian densities given their parameters

Description

Computes the matrix of the L^2 distances between several multivariate (p > 1) or univariate (p = 1) Gaussian densities, given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate).

Usage

matdistl2dpar(meanL, varL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise distances between the probability densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

distl2dpar.

matdistl2d for the distance matrix between probability densities which are estimated from the data.

Examples

    data(roses)
    
    # Multivariate:
    X <- roses[,c("Sha","Den","Sym","rose")]
    summary(X)
    mean.X <- as.list(by(X[, 1:3], X$rose, colMeans))
    var.X <- as.list(by(X[, 1:3], X$rose, var))
    
    # Gaussian densities, given parameters
    matdistl2dpar(mean.X, var.X)

    # Univariate :
    X1 <- roses[,c("Sha","rose")]
    summary(X1)
    mean.X1 <- by(X1$Sha, X1$rose, mean)
    var.X1 <- by(X1$Sha, X1$rose, var)
    
    # Gaussian densities, given parameters
    matdistl2dpar(mean.X1, var.X1)

Matrix of Hellinger distances between Gaussian densities

Description

Computes the matrix of the Hellinger distances between several multivariate (p > 1) or univariate (p = 1) Gaussian densities given samples and using hellinger.

Usage

mathellinger(x)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise Hellinger distances between the probability densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

hellinger.

mathellingerpar when the probability densities are Gaussian, given the parameters (means and variances).

Examples

    data(roses)
    
    # Multivariate:
    X <- as.folder(roses[,c("Sha","Den","Sym","rose")], groups = "rose")
    summary(X)
    mathellinger(X)
    
    # Univariate :
    X1 <- as.folder(roses[,c("Sha","rose")], groups = "rose")
    summary(X1)
    mathellinger(X1)

Matrix of Hellinger distances between Gaussian densities given their parameters

Description

Computes the matrix of the Hellinger distances between several multivariate (p > 1) or univariate (p = 1) Gaussian densities, given their means and variances, using hellingerpar.

Usage

mathellingerpar(meanL, varL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise distances between the Gaussian densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

hellingerpar.

mathellinger for the distance matrix between probability densities which are estimated from the data.

Examples

    data(roses)
    
    # Multivariate:
    X <- roses[,c("Sha","Den","Sym","rose")]
    summary(X)
    mean.X <- as.list(by(X[, 1:3], X$rose, colMeans))
    var.X <- as.list(by(X[, 1:3], X$rose, var))
    mathellingerpar(mean.X, var.X)

    # Univariate :
    X1 <- roses[,c("Sha","rose")]
    summary(X1)
    mean.X1 <- by(X1$Sha, X1$rose, mean)
    var.X1 <- by(X1$Sha, X1$rose, var)
    mathellingerpar(mean.X1, var.X1)

Matrix of L^2 inner products of probability densities

Description

Computes the matrix of the L^2 inner products between several multivariate (p > 1) or univariate (p = 1) probability densities, estimated from samples, using l2d.

Usage

matipl2d(x, method = "gaussiand", varwL = NULL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise inner products between the probability densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

l2d.

matipl2dpar when the probability densities are Gaussian, given the parameters (means and variances).

Examples

    data(roses)
    
    # Multivariate:
    X <- as.folder(roses[,c("Sha","Den","Sym","rose")], groups = "rose")
    summary(X)
    mean.X <- mean(X)
    var.X <- var.folder(X)
    
    # Parametrically estimated Gaussian densities:
    matipl2d(X)
    
    # Estimated densities using the Gaussian kernel method (normal reference rule bandwidth):
    matipl2d(X, method  = "kern")
    
    # Estimated densities using the Gaussian kernel method (bandwidth provided):
    matipl2d(X, method  = "kern", varwL = var.X)
    
    # Univariate :
    X1 <- as.folder(roses[,c("Sha","rose")], groups = "rose")
    summary(X1)
    mean.X1 <- mean(X1)
    var.X1 <- var.folder(X1)
    
    # Parametrically estimated Gaussian densities:
    matipl2d(X1)
    
    # Estimated densities using the Gaussian kernel method (normal reference rule bandwidth):
    matipl2d(X1, method = "kern")
    
    # Estimated densities using the Gaussian kernel method (bandwidth provided):
    matipl2d(X1, method = "kern", varwL = var.X1)

Matrix of L^2 inner products of Gaussian densities

Description

Computes the matrix of the L^2 inner products between several multivariate (p > 1) or univariate (p = 1) Gaussian densities, given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate).

Usage

matipl2dpar(meanL, varL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise inner products between the Gaussian densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

l2dpar.

matipl2d for the distance matrix between probability densities which are estimated from the data.

Examples

    data(roses)
    
    # Multivariate:
    X <- roses[,c("Sha","Den","Sym","rose")]
    summary(X)
    mean.X <- as.list(by(X[, 1:3], X$rose, colMeans))
    var.X <- as.list(by(X[, 1:3], X$rose, var))
    
    # Gaussian densities, given parameters
    matipl2dpar(mean.X, var.X)

    # Univariate :
    X1 <- roses[,c("Sha","rose")]
    summary(X1)
    mean.X1 <- by(X1$Sha, X1$rose, mean)
    var.X1 <- by(X1$Sha, X1$rose, var)
    
    # Gaussian densities, given parameters
    matipl2dpar(mean.X1, var.X1)

Matrix of the Jeffreys measures (symmetrised Kullback-Leibler divergences) between Gaussian densities

Description

Computes the matrix of Jeffreys measures between several multivariate (p > 1) or univariate (p = 1) Gaussian densities, given samples.

Usage

matjeffreys(x)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of pairwise Jeffreys measures between the Gaussian densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

matjeffreyspar if the parameters of the Gaussian densities are known.

Examples

    data(roses)
    
    # Multivariate:
    X <- as.folder(roses[,c("Sha","Den","Sym","rose")], groups = "rose")
    summary(X)
    matjeffreys(X)
    
    # Univariate :
    X1 <- as.folder(roses[,c("Sha","rose")], groups = "rose")
    summary(X1)
    matjeffreys(X1)

Matrix of Jeffreys measures (symmetrised Kullback-Leibler divergences) between Gaussian densities

Description

Computes the matrix of Jeffreys measures between several multivariate (p > 1) or univariate (p = 1) Gaussian densities, given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate), using jeffreyspar.

Usage

matjeffreyspar(meanL, varL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of pairwise Jeffreys measures between the Gaussian densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

jeffreyspar.

matjeffreys for the matrix of Jeffreys divergences between probability densities which are estimated from the data.

Examples

    data(roses)
    
    # Multivariate:
    X <- roses[,c("Sha","Den","Sym","rose")]
    summary(X)
    mean.X <- as.list(by(X[, 1:3], X$rose, colMeans))
    var.X <- as.list(by(X[, 1:3], X$rose, var))
    matjeffreyspar(mean.X, var.X)

    # Univariate :
    X1 <- roses[,c("Sha","rose")]
    summary(X1)
    mean.X1 <- by(X1$Sha, X1$rose, mean)
    var.X1 <- by(X1$Sha, X1$rose, var)
    matjeffreyspar(mean.X1, var.X1)

Matrix of 2-Wassterstein distance between Gaussian densities

Description

Computes the matrix of the 2-Wassterstein distances between several multivariate (p > 1) or univariate (p = 1) Gaussian densities, given samples.

Usage

matwasserstein(x)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise 2-Wassterstein distance between the Gaussian densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

matwassersteinpar if the parameters of the Gaussian densities are known.

Examples

    data(roses)
    
    # Multivariate:
    X <- as.folder(roses[,c("Sha","Den","Sym","rose")], groups = "rose")
    summary(X)
    matwasserstein(X)
    
    # Univariate :
    X1 <- as.folder(roses[,c("Sha","rose")], groups = "rose")
    summary(X1)
    matwasserstein(X1)

Matrix of 2-Wasserstein distances between Gaussian densities

Description

Computes the matrix of the 2-Wasserstein distances between several multivariate (p > 1) or univariate (p = 1) Gaussian densities, given their parameters (mean vectors and covariance matrices if the densities are multivariate, or means and variances if univariate), using wassersteinpar.

Usage

matwassersteinpar(meanL, varL)

Arguments

Value

Positive symmetric matrix whose order is equal to the number of densities, consisting of the pairwise 2-Wasserstein distances between the Gaussian densities.

Author(s)

Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard

See Also

wasserstein.

matwasserstein for the matrix of 2-Wasserstein distances between probability densities which are estimated from the data.

Examples

    data(roses)
    
    # Multivariate:
    X <- roses[,c("Sha","Den","Sym","rose")]
    summary(X)
    mean.X <- as.list(by(X[, 1:3], X$rose, colMeans))
    var.X <- as.list(by(X[, 1:3], X$rose, var))
    matwassersteinpar(mean.X, var.X)

    # Univariate :
    X1 <- roses[,c("Sha","rose")]
    summary(X1)
    mean.X1 <- by(X1$Sha, X1$rose, mean)
    var.X1 <- by(X1$Sha, X1$rose, var)
    matwassersteinpar(mean.X1, var.X1)

Multidimensional scaling of discrete probability distributions

Description

Applies the multidimensional scaling (MDS) method to discrete probability distributions in order to describe T groups of individuals on which are observed q categorical variables. It returns an object of class mdsdd. It applies cmdscale to the distance matrix between the T distributions.

Usage

mdsdd(xf, group.name = "group", distance = c("l1", "l2", "chisqsym", "hellinger",
    "jeffreys", "jensen", "lp"), nb.factors = 3, nb.values = 10, association = c("cramer",
    "tschuprow", "pearson", "phi"), sub.title = "", plot.eigen = TRUE,
    plot.score = FALSE, nscore = 1:3, filename = NULL, add = TRUE, p)

Arguments