fpcad: Functional PCA of probability densities
In dad: Three-Way / Multigroup Data Analysis Through Densities

fpcad

R Documentation

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

`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.

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:

`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.

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)

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