fdiscd.misclass {dad}

Misclassification ratio in functional discriminant analysis of probability densities.

Description

Computes the one-leave-out misclassification ratio of the rule assigning T groups of individuals, one group after another, to the class of groups (among K classes of groups) which achieves the minimum of the distances or divergences between the density function associated to the group to assign and the K density functions associated to the K classes.

Usage

fdiscd.misclass(xf, class.var, gaussiand = TRUE,
           distance =  c("jeffreys", "hellinger", "wasserstein", "l2", "l2norm"),
           crit = 1, windowh = NULL)

Arguments

xf

object of class folderh with two data frames:

• The first one has at least two columns. One column contains the names of the T groups (all the names must be different). An other column is a factor with K levels partitionning the T groups into K classes.

• The second one has (p+1) columns. The first p columns are numeric (otherwise, there is an error). The last column is a factor with T levels defining T groups. Each group, say t, consists of n_t individuals.

class.var

string. The name of the class variable.

distance

The distance or dissimilarity used to compute the distance matrix between the densities. It can be:

• "jeffreys" (default) the Jeffreys measure (symmetrised Kullback-Leibler divergence),

• "hellinger" the Hellinger (Matusita) distance,

• "wasserstein" the Wasserstein distance,

• "l2" the L^2 distance,

• "l2norm" (only available when crit = 1) the densities are normed and the L^2 distance between these normed densities is used;

If gaussiand = FALSE, the densities are estimated by the Gaussian kernel method and the distance is "l2" or "l2norm".

crit

1, 2 or 3. In order to select the densities associated to the classes. See Details.

If distance is "hellinger", "jeffreys" or "wasserstein", crit is necessarily 1 (see Details).

gaussiand

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

If distance is "hellinger", "jeffreys" or "wasserstein", gaussiand is necessarily TRUE.

windowh

strictly positive numeric value. If windowh = NULL (default), the bandwidths are computed using the bandwidth.parameter function.

Omitted when distance is "hellinger", "jeffreys" or "wasserstein" (see Details).

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.

The argument 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.

To the class k consisting of T_k groups is associated the density denoted g_k. The crit argument selects the estimation method of the K densities g_k.

1. The density g_k is estimated using the whole data of this class, that is the rows of x corresponding to the T_k groups of the class k.

   The estimation of the densities g_k uses the same method as the estimation of the f_t.

2. The T_k densities f_t are estimated using the corresponding data from x. Then they are averaged to obtain an estimation of the density g_k, that is g_k = \frac{1}{T_k} \, \sum{f_t}.

3. Each previous density f_t is weighted by n_t (the number of rows of x corresponding to f_t). Then they are averaged, that is g_k = \frac{1}{\sum n_t} \sum n_t f_t.

The last two methods are only available for the L^2-distance. If the divergences between densities are computed using the Hellinger or Wasserstein distance or Jeffreys measure, only the first of these methods is available.

The distance or dissimilarity between the estimated densities is either the L^2 distance, the Hellinger distance, Jeffreys measure (symmetrised Kullback-Leibler divergence) or the Wasserstein distance.

• If it is the L^2 distance (distance="l2" or distance="l2norm"), the densities can be either parametrically estimated or estimated using the Gaussian kernel.

• If it is the Hellinger distance (distance="hellinger"), Jeffreys measure (distance="jeffreys") or the Wasserstein distance (distance="wasserstein"), the densities are considered Gaussian and necessarily parametrically estimated.

Value

Returns an object of class fdiscd.misclass, that is a list including:

classification

data frame with 4 columns:

• factor giving the group name. The column name is the same as that of the column (p+1) of x,

• the prior class of the group if it is available, or NA if not,

• alloc: the class allocation computed by the discriminant analysis method,

• misclassed: boolean. TRUE if the group is misclassed, FALSE if it is well-classed, NA if the prior class of the group is unknown.

confusion.mat

confusion matrix,

misalloc.per.class

the misclassification ratio per class,

misclassed

the misclassification ratio,

distances

matrix with T rows and K columns, of the distances (d_{tk}): d_{tk} is the distance between the group t and the class k, computed with the measure given by argument distance (L^2-distance, Hellinger distance or Jeffreys measure),

proximities

matrix of the proximity indices (in percents) between the groups and the classes. The proximity of the group t to the class k is computed as so: (1/d_{tk})/\sum_{l=1}^{l=K}(1/d_{tl}).

Author(s)

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

References

Boumaza, R. (2004). Discriminant analysis with independently repeated multivariate measurements: an L^2 approach. Computational Statistics & Data Analysis, 47, 823-843.

Rudrauf, J.M., Boumaza, R. (2001). Contribution à l'étude de l'architecture médiévale: les caractéristiques des pierres à bossage des châteaux forts alsaciens. Centre de Recherches Archéologiques Médiévales de Saverne, 5, 5-38.

Examples

data(castles.dated)
castles.stones <- castles.dated$stones
castles.periods <- castles.dated$periods
castlesfh <- folderh(castles.periods, "castle", castles.stones)
result <- fdiscd.misclass(castlesfh, "period")
print(result)