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.
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)Returns an object of class fpcat, that is a list including:
vector of the times of observation.
data frame of the eigenvalues and percentages of inertia.
data frame of the contributions to the first nb.factors principal components.
data frame of the qualities on the first nb.factors principal factors.
data frame of the first nb.factors principal scores.
vector of the \(L^2\) norms of the densities.
list of the means.
list of the covariance matrices.
list of the correlation matrices.
list of the skewness coefficients.
list of the kurtosis coefficients.
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.
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.
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.
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.
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.
logical. If TRUE (default), the probability densities are supposed Gaussian. If FALSE, densities are estimated using the Gaussian kernel method (as fpcad).
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.
logical. If TRUE (default), the densities are normed before computing the distances (as fpcad).
logical. If TRUE (default), the densities are centered (as fpcad).
logical. If TRUE (default is FALSE), the data of each group are centered (as fpcad).
logical. If TRUE (default is FALSE), the data of each group are centered (even if data.centered = FALSE) and scaled (as fpcad).
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).
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).
numerical. Number of returned eigenvalues (default nb.values = 10) (as fpcad).
string. Subtitle for the graphs (default NULL) (as fpcad).
logical. If TRUE (default), the barplot of the eigenvalues is plotted (as fpcad).
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).
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).
string. Name of the file in which the results are saved. By default (filename = NULL) the results are not saved (as fpcad).
Rachid Boumaza, Pierre Santagostini, Smail Yousfi, Gilles Hunault, Sabine Demotes-Mainard
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).
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.
print.fpcat, plot.fpcat, bandwidth.parameter
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)
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