dad (version 4.1.6)

discdd.misclass: Misclassification ratio in functional discriminant analysis of discrete probability distributions.

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 probability distribution associated to the group to assign and the \(K\) probability distributions associated to the \(K\) classes.

Usage

discdd.misclass(xf, class.var, distance =  c("l1", "l2", "chisqsym", "hellinger",
           "jeffreys", "jensen", "lp"), crit = 1, p)

Value

Returns an object of class discdd.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 (\(q+1\)) 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 \(k\),

proximities

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

Arguments

xf

object of class folderh with two data frames or list of arrays (or tables).

• If it is a folderh:

  • The first data.frame 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 \((q+1)\) columns. The first \(q\) columns are factors (otherwise, they are coerced into factors). The last column is a factor with \(T\) levels defining \(T\) groups. Each group, say \(t\), consists of \(n_t\) individuals.

• If it is a list of arrays or tables, the \(t^{th}\) element (\(t = 1, \ldots, T\)) is the table of the joint distribution (absolute or relative frequencies) of the \(t^{th}\) group. These arrays have the same shape:

  Each array (or table) xf[[i]] has:

  • the same dimension(s). If \(q = 1\) (univariate), dim(xf[[i]]) is an integer. If \(q > 1\) (multivariate), dim(xf[[i]]) is an integer vector of length q.

  • the same dimension names dimnames(xf[[i]]) (is non NULL). These dimnames are the names of the variables.

class.var

string (if xf is an object of class "folderh") or data.frame with two columns (if xf is a list of arrays).

• If xf is of class "folder", class.var is the name of the class variable.

• If xf is a list of arrays or a list of tables, class.var is a data.frame with at least two columns named "group" and "class". The "group" column contains the names of the \(T\) groups (all the names must be different). The "class" column is a factor with \(K\) levels partitioning the \(T\) groups into \(K\) classes.

distance

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

• "l1" (default) the \(L^p\) distance with \(p = 1\)

• "l2" the \(L^p\) distance with \(p = 2\)

• "chisqsym" the symmetric Chi-squared distance

• "hellinger" the Hellinger metric (Matusita distance)

• "jeffreys" Jeffreys distance (symmetrised Kullback-Leibler divergence)

• "jensen" the Jensen-Shannon distance

• "lp" the \(L^p\) distance with \(p\) given by the argument p of the function.

crit

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

p

integer. Optional. When distance = "lp" (\(L^p\) distance with \(p>2\)), p is the parameter of the distance.

Author

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

Details

• If xf is an object of class "folderh" containing the data:

  The \(T\) probability distributions \(f_t\) corresponding to the \(T\) groups of individuals are estimated by frequency distributions within each group.

  To the class \(k\) consisting of \(T_k\) groups is associated the probability distribution \(g_k\), knowing that when using the one-leave-out method, we do not include the group to assign in its class \(k\). The crit argument selects the estimation method of the \(g_k\)'s.

  • crit=1 The probability distribution \(g_k\) 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 \(g_k\)'s uses the same method as the estimation of the \(f_t\)'s.

  • crit=2 The \(T_k\) probability distributions \(f_t\) are estimated using the corresponding data from xf. Then they are averaged to obtain an estimation of the density \(g_k\), that is \(g_k = \frac{1}{T_k} \, \sum{f_t}\).

• If xf is a list of arrays (or list of tables):

  The \(t^{th}\) array is the joint frequency distribution of the \(t^{th}\) group. The frequencies can be absolute or relative.

  To the class \(k\) consisting of \(T_k\) groups is associated the probability distribution \(g_k\), knowing that when using the one-leave-out method, we do not include the group to assign in its class \(k\). The crit argument selects the estimation method of the \(g_k\)'s.

  • crit=1 \(g_k = \frac{1}{\sum n_t} \sum n_t f_t\), where \(n_t\) is the total of xf[[t]].

    Notice that when xf[[t]] contains relative frequencies, its total is 1. That is equivalent to crit=2.

  • crit=2 \(g_k = \frac{1}{T_k} \, \sum f_t\).

References

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

# Example 1 with a folderh obtained by converting numeric variables
data("castles.dated")
stones <- castles.dated$stones
periods <- castles.dated$periods
stones$height <- cut(stones$height, breaks = c(19, 27, 40, 71), include.lowest = TRUE)
stones$width <- cut(stones$width, breaks = c(24, 45, 62, 144), include.lowest = TRUE)
stones$edging <- cut(stones$edging, breaks = c(0, 3, 4, 8), include.lowest = TRUE)
stones$boss <- cut(stones$boss, breaks = c(0, 6, 9, 20), include.lowest = TRUE )

castlefh <- folderh(periods, "castle", stones)

# Default: dist="l1", crit=1
discdd.misclass(castlefh, "period")

# Hellinger distance, crit=2
discdd.misclass(castlefh, "period", distance = "hellinger", crit = 2)


# Example 2 with a list of 96 arrays
data("dspgd2015")
data("departments")
classes <- departments[, c("coded", "namer")]
names(classes) <- c("group", "class")

# Default: dist="l1", crit=1
discdd.misclass(dspgd2015, classes)

# Hellinger distance, crit=2
discdd.misclass(dspgd2015, classes, distance = "hellinger", crit = 2)