fpcat: Functional PCA of probability densities among time

In dad: Three-Way / Multigroup Data Analysis Through Densities

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

xf	object of class "foldert" or data.frame. An object of class "foldert" is a list of data frames with the same column names, each of them corresponding to a time of observation. Its elements are data frames with p numeric columns. If there are non numeric columns, there is an error. The t^{th} element (t = 1, \ldots, T) matches with the t^{th} time of observation. If it is a data frame: If method=1: the column with name given by the group.name argument is a factor giving the groups. The other columns are all numeric; otherwise, there is an error. If method=2: the column named after the ind argument contains the identifiers of the measured objects, and the observations are organized as follows: Given timecol the number of the column named by the group.name argument, the observations corresponding to the 1st time are on columns timecol : (timecol + nvar - 1) the observations corresponding to the 2nd time are on columns (timecol + nvar) : (timecol + 2 * nvar - 1) and so on.
group.name	string or numeric. If xf is an object of class "foldert", string. Name of the grouping variable, that is the observation times. The default is groupname = "time". If xf is a data frame, string or numeric, as the ind argument of as.foldert.data.frame. If method = 1, timecol is the name or the number of the column of x containing the times of observation, or the number of this column. x[, timecol] must be of class "numeric", "ordered", "Date", "POSIXlt" or "POSIXct", otherwise, there is an error. If method=2, timecol is the name or the number of the first column corresponding to the first observation. If there are duplicated column names and several columns are named by timecol, the first one is considered.
method	if xf is a data frame, 1 or 2. Omitted if xf is an object of class "foldert". If xf is a data frame, method indicates the layout of this data frame and, therefore, the method used to extract the data and build the foldert. If method = 1, there is a column containing the identifiers of the measured objects and a column containing the times. The other columns contain the observations. If method = 2, there is a column containing the identifiers of the measured objects, and the observations are organized as follows: the observations corresponding to the 1st time are on columns timecol : (timecol + nvar - 1) the observations corresponding to the 2nd time are on columns (timecol + nvar) : (timecol + 2 * nvar - 1) and so on.
ind	if xf is a data frame, string or numeric. Omitted if xf is an object of class "foldert". The name of the column of x containing the indentifiers of the measured objects, or the number of this column. See the ind argument of as.foldert.data.frame.
nvar	if xf is a data frame and mathod=2, string or numeric. Omitted if xf is an object of class "foldert" or if method=1. The number of variable measured at each observation time. See the ind argument of as.foldert.data.frame.

All other arguments are the same as for fpcad.

gaussiand	logical. If TRUE (default), the probability densities are supposed Gaussian. If FALSE, densities are estimated using the Gaussian kernel method (as fpcad).
windowh	either a list of T bandwidths (one per density associated to a group), or a strictly positive number. If windowh = NULL (default), the bandwidths are automatically computed (as fpcad). See Details.
normed	logical. If TRUE (default), the densities are normed before computing the distances (as fpcad).
centered	logical. If TRUE (default), the densities are centered (as fpcad).
data.centered	logical. If TRUE (default is FALSE), the data of each group are centered (as fpcad).
data.scaled	logical. If TRUE (default is FALSE), the data of each group are centered (even if data.centered = FALSE) and scaled (as fpcad).
common.variance	logical. If TRUE (default is FALSE), a common covariance matrix (or correlation matrix if data.scaled = TRUE), computed on the whole data, is used. If FALSE (default), a covariance (or correlation) matrix per group is used (as fpcad).
nb.factors	numeric. Number of returned principal scores (default nb.factors = 3) (as fpcad). Warning: The plot.fpcad and interpret.fpcad functions cannot take into account more than nb.factors principal factors (as fpcad).
nb.values	numerical. Number of returned eigenvalues (default nb.values = 10) (as fpcad).
sub.title	string. Subtitle for the graphs (default NULL) (as fpcad).
plot.eigen	logical. If TRUE (default), the barplot of the eigenvalues is plotted (as fpcad).
plot.score	logical. If TRUE, the graphs of principal scores are plotted. A new graphic device is opened for each pair of principal scores defined by nscore argument (as fpcad).
nscore	numeric vector. If plot.score = TRUE, the numbers of the principal scores which are plotted. By default it is equal to nscore = 1:3. Its components cannot be greater than nb.factors (as fpcad).
filename	string. Name of the file in which the results are saved. By default (filename = NULL) the results are not saved (as 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:

times	vector of the times of observation.
inertia	data frame of the eigenvalues and percentages of inertia.
contributions	data frame of the contributions to the first nb.factors principal components.
qualities	data frame of the qualities on the first nb.factors principal factors.
scores	data frame of the first nb.factors principal scores.
norm	vector of the L^2 norms of the densities.
means	list of the means.
variances	list of the covariance matrices.
correlations	list of the correlation matrices.
skewness	list of the skewness coefficients.
kurtosis	list of the kurtosis coefficients.

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)

dad documentation built on Aug. 30, 2023, 5:06 p.m.