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Last data update: 2014.03.03

R: Multidimensional scaling of probability densities

fmdsd

R Documentation

Multidimensional scaling of probability densities

Description

Applies the multidimensional scaling (MDS) method to probability densities, in order to describe a three-way data table, 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(x, gaussiand = TRUE, windowh = NULL, kern = NULL, normed=FALSE,
    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

`x`	data frame with (p+1) columns. The first p columns are numeric. The last column is a factor with T levels defining T groups. Each group, say t, consists of n_t individuals.
`gaussiand`	logical. If `TRUE` (default), the probability densities are supposed Gaussian. If `FALSE`, densities are estimated using the Gaussian kernel method.
`kern`	string. If `gaussiand = FALSE`, this argument sets the kernel used in the estimation method. Currently, only the Gaussian kernel is available: the settings `kern = "gauss"` and `kern = NULL` are equivalent.
`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.
`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 coordinates (default `nb.factors=3`). Warning: The `plot.fmdsd` and `interpret.fmdsd` functions cannot take into account more than `nb.factors` principal factors.
`nb.values`	numeric. Number of returned eigenvalues (default `nb.values=10`).
`sub.title`	string. Subtitle for the graphs (default `NULL`).
`plot.eigen`	logical. If `TRUE` (default), the barplot of the eigenvalues is plotted.
`plot.score`	logical. If `TRUE`, the graphs of new coordinates are plotted. A new graphic device is opened for each pair of coordinates defined by `nscore` argument.
`nscore`	numeric vector. If `plot.score=TRUE`, the numbers of the principal coordinates 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`) they 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 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.

Value

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

`inertia`	data frame of the eigenvalues and percentages of inertia.
`scores`	data frame of the coordinates along the `nb.factors` first principal components.
`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, Sabine Demotes-Mainard.

References

Boumaza, R. (1998). Analyse en composantes principales de distributions gaussiennes multidimensionnelles. Revue de Statistique Appliquee, 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 function. Journal of Statistical Computation and Simulation, 85 (11), 2315-2330.

Examples

# MDS on Gaussian densities (on sensory data)
data(roses)
result1 = fmdsd(roses)
print(result1)
plot(result1)