Aitchison-Aitken kernel

Share:

Description

Discrete kernel for categorical data with k unordered categories.

Usage

1
dkern(x, y, k, lambda)

Arguments

x

categorical data vector

y

postive integer defining a fixed category

k

positive integer giving the number of categories

lambda

smoothing parameter

Details

This kernel was introduced in Aitchison & Aitken (1976); see also Titterington (1980).

The setting lambda =1/k corresponds to the extreme case 'maximal smoothing', while lambda = 1 means ‘no smoothing’. Statistically sensible settings are only 1/k<= lambda <=1.

Author(s)

Jochen Einbeck (2006)

References

Aitchison, J. and Aitken, C.G.G. (1976). Multivariate binary discrimination by kernel method. Biometrika 63, 413-420.

Titterington, D. M. (1980). A comparative study of kernel-based density estimates for categorical data. Technometrics, 22, 259-268.

Examples

1
2
3
4
5
6
7
8
9
k<-6; 
dkern(1:k,4,k,0.99)   
# Kernel centered at the 4th component with a very small amount of smoothing.


## The function is currently defined as
function(x,y,k,lambda){
ifelse(y==x, lambda, (1-lambda)/(k-1))
  }