Introduction to the evclass package

Evidential K-nearest neighbor classifier

The principle of the evidential K-nearest neighbor (EK-NN) classifier is explained in (Denœux 1995), and the optimization of the parameters of this model is presented in (Zouhal and Denœux 1998). The reader is referred to these references. Here, we focus on the practical application of this method using the functions implemented in evclass.

Consider, for instance, the ionosphere data. This dataset consists in 351 instances grouped in two classes and described by 34 numeric attributes. The first 175 instances are training data, the rest are test data. Let us first load the data, and split them into a training set and a test set.

data(ionosphere)
xtr<-ionosphere$x[1:176,]
ytr<-ionosphere$y[1:176]
xtst<-ionosphere$x[177:351,]
ytst<-ionosphere$y[177:351]

The EK-NN classifier is implemented as three functions: EkNNinit for initialization, EkNNfit for training, and EkNNval for testing. Let us initialize the classifier and train it on the ionosphere data, with \(K=5\) neighbors. (If the argument param is not passed to EkNNfit, the function EkNNinit is called inside EkNNfit; here, we make the call explicit for clarity).

param0<- EkNNinit(xtr,ytr)
options=list(maxiter=300,eta=0.1,gain_min=1e-5,disp=FALSE)
fit<-EkNNfit(xtr,ytr,param=param0,K=5,options=options)

The list fit contains the optimized parameters, the final value of the cost function, the leave-one-out (LOO) error rate, the LOO predicted class labels and the LOO predicted mass functions. Here the LOO error rate and confusion matrix are:

print(fit$err)

## [1] 0.1079545

table(fit$ypred,ytr)

##    ytr
##       1   2
##   1 108  14
##   2   5  49

We can then evaluate the classifier on the test data:

val<-EkNNval(xtrain=xtr,ytrain=ytr,xtst=xtst,K=5,ytst=ytst,param=fit$param)
print(val$err)

## [1] 0.1142857

table(val$ypred,ytst)

##    ytst
##       1   2
##   1 107  15
##   2   5  48

To determine the best value of \(K\), we may compute the LOO error for different candidate value. Here, we will all values between 1 and 15

err<-rep(0,15)
i<-0
for(K in 1:15){
  fit<-EkNNfit(xtr,ytr,K,options=list(maxiter=100,eta=0.1,gain_min=1e-5,disp=FALSE))
  err[K]<-fit$err
}
plot(1:15,err,type="b",xlab='K',ylab='LOO error rate')

The minimum LOO error rate is obtained for \(K=8\). The test error rate and confusion matrix for that value of \(K\) are obtained as follows:

fit<-EkNNfit(xtr,ytr,K=8,options=list(maxiter=100,eta=0.1,gain_min=1e-5,disp=FALSE))
val<-EkNNval(xtrain=xtr,ytrain=ytr,xtst=xtst,K=8,ytst=ytst,param=fit$param)
print(val$err)

## [1] 0.09142857

table(val$ypred,ytst)

##    ytst
##       1   2
##   1 106  10
##   2   6  53

Evidential neural network classifier

In the evidential neural network classifier, the output mass functions are based on distances to protypes, which allows for faster classification. The prototypes and their class-membership degrees are leanrnt by minimizing a cost function. This function is defined as the sum of an error term and, optionally, a regularization term. As for the EK-NN classifier, the evidential neural network classifier is implemented as three functions: proDSinit for initialization, proDSfit for training and proDSval for evaluation.

Let us demonstrate this method on the glass dataset. This data set contains 185 instances, which can be split into 89 training instances and 96 test instances.

data(glass)
xtr<-glass$x[1:89,]
ytr<-glass$y[1:89]
xtst<-glass$x[90:185,]
ytst<-glass$y[90:185]

We then initialize a network with 7 prototypes:

param0<-proDSinit(xtr,ytr,nproto=7,nprotoPerClass=FALSE,crisp=FALSE)

and train this network without regularization:

options<-list(maxiter=500,eta=0.1,gain_min=1e-5,disp=20)
fit<-proDSfit(x=xtr,y=ytr,param=param0,options=options)

## [1]  1.0000000  0.3582862 10.0000000
## [1] 21.0000000  0.2933023  1.2426672
## [1] 41.0000000  0.2263345  0.2205550
## [1] 61.00000000  0.19483767  0.04260374
## [1] 81.000000000  0.188954121  0.007109284
## [1] 1.010000e+02 1.884715e-01 1.285468e-03
## [1] 1.210000e+02 1.883760e-01 3.389789e-04
## [1] 1.410000e+02 1.881880e-01 1.417006e-04

Finally, we evaluate the performance of the network on the test set:

val<-proDSval(xtst,fit$param,ytst)
print(val$err)

## [1] 0.28125

table(ytst,val$ypred)

##     
## ytst  1  2  4
##    1 32  8  0
##    2  5 28  4
##    3  5  3  0
##    4  0  2  9

If the training is done with regularization, the hyperparameter mu needs to be determined by cross-validation.