dad (version 4.1.6)

fpcad: 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)

Value

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

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.

Arguments

xf

object of class "folder" or data.frame.

• If it is an object of class "folder", 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 \(t^{th}\) group.

• If it is a data frame, 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.

group.name

string.

• If xf is an object of class "folder", name of the grouping variable in the returned results. The default is groupname = "group".

• If xf is a data frame, group.name is the name of the column of xf containing the groups.

gaussiand

logical. If TRUE (default), the probability densities are supposed Gaussian. If FALSE, densities are estimated using the Gaussian kernel method.

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. See Details.

normed

logical. If TRUE (default), the densities are normed before computing the distances.

centered

logical. If TRUE (default), the densities are centered.

data.centered

logical. If TRUE (default is FALSE), the data of each group are centered.

data.scaled

logical. If TRUE (default is FALSE), the data of each group are centered (even if data.centered = FALSE) and scaled.

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.

nb.factors

numeric. Number of returned principal scores (default nb.factors = 3).

Warning: The plot.fpcad and interpret.fpcad functions cannot take into account more than nb.factors principal factors.

nb.values

numerical. Number of returned eigenvalues (default nb.values = 10).

sub.title

string. If provided, the subtitle for the graphs.

plot.eigen

logical. If TRUE (default), the barplot of the eigenvalues is plotted.

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.

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.

filename

string. Name of the file in which the results are saved. By default (filename = NULL) the results are not saved.

Author

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

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.

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)