Estimation of a Discrete Distribution for Fuzzy Data (Data in Belief Function Framework)
This function estimates a discrete distribution for uncertain data based on the belief function framework (Denoeux, 2013; see Details).
fuzdiscr(X, theta0=NULL, maxiter=200, conv=1e-04)
X |
Matrix with fuzzy data. Rows corresponds to subjects and columns to values of the membership function |
theta0 |
Initial vector of parameter estimates |
maxiter |
Maximum number of iterations |
conv |
Convergence criterion |
For n subjects, membership functions m_n(k) are observed which indicate the belief in data X_n=k. The membership function is interpreted as epistemic uncertainty (Denoeux, 2011). However, associated parameters in statistical models are crisp which means that models are formulated at the basis of precise (crisp) data if they would be observed.
In the present estimation problem of a discrete distribution, the parameters of interest are category probabilities θ_k=P( X=k).
The parameter estimation follows the evidential EM algorithm (Denoeux, 2013).
Vector of probabilities of the discrete distribution
Denoeux, T. (2011). Maximum likelihood estimation from fuzzy data using the EM algorithm. Fuzzy Sets and Systems, 183, 72-91.
Denoeux, T. (2013). Maximum likelihood estimation from uncertain data in the belief function framework. IEEE Transactions on Knowledge and Data Engineering, 25, 119-130.
#############################################################################
# EXAMPLE 1: Binomial distribution Denoeux Example 4.3 (2013)
#############################################################################
#*** define uncertain data
X_alpha <- function( alpha ){
Q <- matrix( 0, 6, 2 )
Q[5:6,2] <- Q[1:3,1] <- 1
Q[4,] <- c( alpha, 1 - alpha )
return(Q)
}
# define data for alpha=0.5
X <- X_alpha( alpha=.5 )
## > X
## [,1] [,2]
## [1,] 1.0 0.0
## [2,] 1.0 0.0
## [3,] 1.0 0.0
## [4,] 0.5 0.5
## [5,] 0.0 1.0
## [6,] 0.0 1.0
## The fourth observation has equal plausibility for the first and the
## second category.
# parameter estimate uncertain data
fuzdiscr( X )
## > sirt::fuzdiscr( X )
## [1] 0.5999871 0.4000129
# parameter estimate pseudo likelihood
colMeans( X )
## > colMeans( X )
## [1] 0.5833333 0.4166667
##-> Observations are weighted according to belief function values.
#*****
# plot parameter estimates as function of alpha
alpha <- seq( 0, 1, len=100 )
res <- sapply( alpha, FUN=function(aa){
X <- X_alpha( alpha=aa )
c( sirt::fuzdiscr( X )[1], colMeans( X )[1] )
} )
# plot
plot( alpha, res[1,], xlab=expression(alpha), ylab=expression( theta[alpha] ), type="l",
main="Comparison Belief Function and Pseudo-Likelihood (Example 1)")
lines( alpha, res[2,], lty=2, col=2)
legend( 0, .67, c("Belief Function", "Pseudo-Likelihood" ), col=c(1,2), lty=c(1,2) )
#############################################################################
# EXAMPLE 2: Binomial distribution (extends Example 1)
#############################################################################
X_alpha <- function( alpha ){
Q <- matrix( 0, 6, 2 )
Q[6,2] <- Q[1:2,1] <- 1
Q[3:5,] <- matrix( c( alpha, 1 - alpha ), 3, 2, byrow=TRUE)
return(Q)
}
X <- X_alpha( alpha=.5 )
alpha <- seq( 0, 1, len=100 )
res <- sapply( alpha, FUN=function(aa){
X <- X_alpha( alpha=aa )
c( sirt::fuzdiscr( X )[1], colMeans( X )[1] )
} )
# plot
plot( alpha, res[1,], xlab=expression(alpha), ylab=expression( theta[alpha] ), type="l",
main="Comparison Belief Function and Pseudo-Likelihood (Example 2)")
lines( alpha, res[2,], lty=2, col=2)
legend( 0, .67, c("Belief Function", "Pseudo-Likelihood" ), col=c(1,2), lty=c(1,2) )
#############################################################################
# EXAMPLE 3: Multinomial distribution with three categories
#############################################################################
# define uncertain data
X <- matrix( c( 1,0,0, 1,0,0, 0,1,0, 0,0,1, .7, .2, .1,
.4, .6, 0 ), 6, 3, byrow=TRUE )
## > X
## [,1] [,2] [,3]
## [1,] 1.0 0.0 0.0
## [2,] 1.0 0.0 0.0
## [3,] 0.0 1.0 0.0
## [4,] 0.0 0.0 1.0
## [5,] 0.7 0.2 0.1
## [6,] 0.4 0.6 0.0
##-> Only the first four observations are crisp.
#*** estimation for uncertain data
fuzdiscr( X )
## > sirt::fuzdiscr( X )
## [1] 0.5772305 0.2499931 0.1727764
#*** estimation pseudo-likelihood
colMeans(X)
## > colMeans(X)
## [1] 0.5166667 0.3000000 0.1833333
##-> Obviously, the treatment uncertainty is different in belief function
## and in pseudo-likelihood framework.Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.