Localized Linear Discriminant Analysis (LocLDA)
A localized version of Linear Discriminant Analysis.
loclda(x, ...)
## S3 method for class 'formula'
loclda(formula, data, ..., subset, na.action)
## Default S3 method:
loclda(x, grouping, weight.func = function(x) 1/exp(x),
k = nrow(x), weighted.apriori = TRUE, ...)
## S3 method for class 'data.frame'
loclda(x, ...)
## S3 method for class 'matrix'
loclda(x, grouping, ..., subset, na.action)formula |
Formula of the form ‘ |
data |
Data frame from which variables specified in |
x |
Matrix or data frame containing the explanatory variables
(required, if |
grouping |
(required if no |
weight.func |
Function used to compute local weights. Must be finite over the interval [0,1]. See Details below. |
k |
Number of nearest neighbours used to construct localized classification rules. See Details below. |
weighted.apriori |
Logical: if |
subset |
An index vector specifying the cases to be used in the training sample. |
na.action |
A function to specify the action to be taken if |
... |
Further arguments to be passed to |
This is an approach to apply the concept of localization described by Tutz and Binder (2005)
to Linear Discriminant Analysis. The function loclda generates an object of class loclda
(see Value below). As localization makes it necessary to build an
individual decision rule for each test observation,
this rule construction has to be handled by predict.loclda.
For convenience, the rule building procedure is still described here.
To classify a test observation x_s, only the k nearest neighbours of
x_s within the train data are used. Each of these k train observations
x_i, i=1,...,k, is assigned a weight w_i according to
w_i := K ( ||x_i - x_s|| / d_k ), i=1,...,k,
where K is the weighting function given by weight.func, ||x_i - x_s||
is the euclidian distance of x_i and x_s
and d_k is the euclidian distance of x_s
to its k-th nearest neighbour.
With these weights for each class A_g, g=1,...,G,
its weighted empirical mean mu_g_hat and weighted empirical
covariance matrix are computed. The estimated pooled (weighted) covariance matrix
Sigma_hat is then calculated from the individual weighted
empirical class covariance matrices. If weighted.apriori is TRUE (the default),
prior class probabilities are estimated according to:
prior_g := [ Sum_{i=1,..,k} ( w_i * I(x_i in A_g) ) ] / [ Sum_{i=1,...,k} ( w_i ) ], g = 1,...,G,
where I is the indicator function. If FALSE, equal priors for all classes are used.
In analogy to Linear Discriminant Analysis, the decision rule for x_s is
A_hat := argmax_{g in 1,...,G} (posterior_g),
where
posterior_g := prior_g * exp [ (-1/2) * t( x_s - mu_g_hat ) * Sigma_hat^(-1) * ( x_s - mu_g_hat ) ] .
If posterior_g < 1e-150 for all g in 1,...,G, posterior_g is set to 1/G for all g in 1,...,G and the test observation x_s is simply assigned to the class whose weighted mean has the lowest euclidian distance to x_s.
A list of class loclda containing the following components:
call |
The (matched) function call. |
learn |
Matrix containing the values of the explanatory variables for all train observations. |
grouping |
Factor specifying the class for each train observation. |
weight.func |
Value of the argument |
k |
Value of the argument |
weighted.apriori |
Value of the argument |
Marc Zentgraf (marc-zentgraf@gmx.de) and Karsten Luebke (karsten.luebke@fom.de)
Tutz, G. and Binder, H. (2005): Localized classification. Statistics and Computing 15, 155-166.
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