EM Algorithm for Maximum Likelihood Estimation by Non-Precise Information

Description

The main goal of this package is easy estimation of the unknown parameter of a continues distribution by EM algorithm where the observed data are fuzzy rather than crisp. This package contains two major functions: (1) the function EM.Triangular works by Triangular Fuzzy Numbers (TFNs), and (2) the function EM.Trapezoidal works by Trapezoidal Fuzzy Numbers (TrFNs).

Author(s)

Abbas Parchami

References

Denoeux, T. (2011) Maximum likelihood estimation from fuzzy data using the EM algorithm, Fuzzy Sets and Systems 183, 72-91.

Gagolewski, M., Caha, J. (2015) FuzzyNumbers Package: Tools to deal with fuzzy numbers in R. R package version 0.4-1, https://cran.r-project.org/web/packages=FuzzyNumbers

Gagolewski, M., Caha, J. (2015) A guide to the FuzzyNumbers package for R (FuzzyNumbers version 0.4-1) http://FuzzyNumbers.rexamine.com

Examples

    library(FuzzyNumbers)
    library(DISTRIB, warn.conflicts = FALSE)

    # Let us we are going to estimation the unknown mean of Normal population with known variance 
    # (e.g, sd(X) = 0.5) on the basis of 11 trapezoidal fuzzy numbers (which we simulate them in 
    # bellow for simplification).
    n = 11
    set.seed(1000)
    c1 = rnorm(n, 10,.5)
    c2 = rnorm(n, 10,.5)
    for(i in 1:n) {if (c1[i] > c2[i]) { zarf <- c1[i]; c1[i] <- c2[i]; c2[i] <- zarf }}
    round(c1,3); round(c2,3)
    c1 <= c2
    l = runif(n, 0,1); round(l,3)
    u = runif(n, 0,1); round(u,3)

    
        EM.Trapezoidal(T.dist="norm", T.dist.par=c(NA,0.5), par.space=c(-5,30), c1, c2, l, u,
          start=4, ebs=.0001, fig=2)
    

MLE by EM algorithm based on Trapezoidal Fuzzy Data

Description

This function can easily obtain Maximum Likelihood Estimation (MLE) for the unknown one-dimensional parameter on the basis of Trapezoidal Fuzzy observation.

Usage

EM.Trapezoidal(T.dist, T.dist.par, par.space, c1, c2, l, u, start, ebs=0.001,
 fig = 2)

Arguments

Value

The parameter computed / estimated in each iteration separately and also the computation of the following values can be asked directly.

Note

In using this package it must be noted that:

(1) The sample size of TrFNs must be less than 16. This package is able to work with small sample sizes (n \leq 15) and can be extended by the user if needs.

(2) Considering a suitable interval for par.space is very important to obtain a true result for EM algorithm. It must be mentioned that this interval must be a sub-interval of the parameter space and the user must check the result of algorithm (MLE). It means that if the obtained MLE (by EM.Trapezoidal) overlay on the boundary of par.space, then the result is not acceptable and the EM algorithm must be repeated once again with a wider par.space.

(3) This package is able to work for continuous distributions with one or two parameter which only one of them is unknown and the user wants to estimate it based on TrFNs.

See Also

DISTRIB FuzzyNumbers

Examples

    library(FuzzyNumbers)
    library(DISTRIB, warn.conflicts = FALSE)

    # Example 1: Estimation the unknown mean of Normal population with known variance (e.g, 
    # var=0.5^2) based of Trapezoidal FNs.
    n = 2
    set.seed(1000)
    c1 = rnorm(n, 10,.5)
    c2 = rnorm(n, 10,.5)
    for(i in 1:n) {if (c1[i] > c2[i]) { zarf <- c1[i]; c1[i] <- c2[i]; c2[i] <- zarf }}
    round(c1,3); round(c2,3)
    c1 <= c2
    l = runif(n, 0,1); round(l,3)
    u = runif(n, 0,1); round(u,3)
    
    EM.Trapezoidal(T.dist="norm", T.dist.par=c(NA,0.5), par.space=c(-5,30), c1, c2, l, u, start=4,
    ebs=.1, fig=2)


    
        # Example 2: 
        n = 4	
        set.seed(10)
        c1 = rexp(n, 2)
        c2 = rexp(n, 2)
        for(i in 1:n) {if (c1[i] > c2[i]) { zarf <- c1[i]; c1[i] <- c2[i]; c2[i] <- zarf }}
        round(c1,3); round(c2,3)
        c1 <= c2
        l = runif(n, 0,1); round(l,3)
        u = runif(n, 0,2); round(u,3)

        EM.Trapezoidal(T.dist="exp", T.dist.par=NA, par.space=c(.1,20), c1, c2, l, u, start=7, 
          ebs=.001)


        # Example 3: Estimation the unknown standard deviation of Normal population with known 
        #  mean (e.g, mean=7) based of Trapezoidal FNs.
        n = 10	
        set.seed(123)
        c1 = rnorm(n, 4,1)
        c2 = rnorm(n, 4,1)
        for(i in 1:n) {if (c1[i] > c2[i]) { zarf <- c1[i]; c1[i] <- c2[i]; c2[i] <- zarf }}
        round(c1,3); round(c2,3)
        c1 <= c2
        l = runif(n, 0,.5); round(l,3)
        u = runif(n, 0,.75); round(u,3)

        EM.Trapezoidal(T.dist="norm", T.dist.par=c(4,NA), par.space=c(0,40), c1, c2, l, u, start=1,
         ebs=.0001, fig=2)


        # Example 4: Estimation alpha parameter in Beta distribution.
        n = 4	
        set.seed(12)
        c1 = rbeta(n, 2,1)
        c2 = rbeta(n, 2,1)
        for(i in 1:n) {if (c1[i] > c2[i]) { zarf <- c1[i]; c1[i] <- c2[i]; c2[i] <- zarf }}
        round(c1,3); round(c2,3)
        c1 <= c2
        l = rbeta(n, 1,1); round(l,3)
        u = rbeta(n, 1,1); round(u,3)

        EM.Trapezoidal(T.dist="beta", T.dist.par=c(NA,1), par.space=c(0,10), c1, c2, l, u, start=1,
         ebs=.01, fig=2)
    

MLE by EM algorithm based on Triangular Fuzzy Data

Description

This function can easily obtain Maximum Likelihood Estimation (MLE) for the unknown one-dimensional parameter on the basis of Triangular Fuzzy observation.

Usage

EM.Triangular(T.dist, T.dist.par, par.space, c, l, u, start, ebs = 0.001,
 fig = 2)

Arguments

Value

The parameter computed / estimated in each iteration separately and also the computation of the following values can be asked directly.

Note

In using this package it must be noted that:

(1) The sample size of TFNs must be less than 16. This package is able to work with small sample sizes (n \leq 15) and can be extended by the user if needs.

(2) Considering a suitable interval for par.space is very important to obtain a true result for EM algorithm. It must be mentioned that this interval must be a sub-interval of the parameter space and the user must check the result of algorithm (MLE). It means that if the obtained MLE (by EM.Triangular) overlay on the boundary of par.space, then the result is not acceptable and the EM algorithm must be repeated once again with a wider par.space.

(3) This package is able to work for continuous distributions with one or two parameter which only one of them is unknown and the user wants to estimate it based on TFNs.

See Also

DISTRIB FuzzyNumbers

Examples

    library(FuzzyNumbers)
    library(DISTRIB, warn.conflicts = FALSE)

    # Example 1: 
    n = 2	
    set.seed(131)
    c = rexp(n, 2);    round(c,3)
    l = runif(n, 0,1); round(l,3)
    u = runif(n, 0,2); round(u,3)
 
    EM.Triangular(T.dist="exp", T.dist.par=NA, par.space=c(0,30), c, l, u, start=5, ebs=.1,
     fig=0)

    
        EM.Triangular(T.dist="exp", T.dist.par=NA, par.space=c(0,30), c, l, u, start=50, ebs=.001,
         fig=1) #Fast Convergence

        EM.Triangular(T.dist="exp", T.dist.par=NA, par.space=c(0,30), c, l, u, start=50, ebs=.1^6,
         fig=2)


        #Example 2:  Computing the mean and the standard deviation of 20 EM estimations:
        n = 15
        MLEs=c()
        for(j in 100:120){
                         print(j)
                         set.seed(j)
                         c = rexp(n, 2)
                         l = runif(n, 0,1)
                         u = runif(n, 0,2)
                         MLEs = c(MLEs, EM.Triangular(T.dist="exp", T.dist.par=NA, 
                                                      par.space=c(0,30), c, l, u, start=5, 
                                                      ebs=.01, fig=0)$MLE )
                         }

        MLEs       #  3.283703 2.475541 3.171026 ...
        mean(MLEs) # 2.263996
        sd(MLEs)   # 0.4952257
        hist(MLEs)


        # Example 3: Estimation the unknown mean of Normal population with known variance
        #  (e.g, var=1) based of TFNs.
        n = 5	
        set.seed(100)
        c = rnorm(n, 10,1);    round(c,3)
        l = runif(n, 0,1); round(l,3)
        u = runif(n, 0,1); round(u,3)

        EM.Triangular(T.dist="norm", T.dist.par=c(NA,1), par.space=c(-10,30), c, l, u, start=20,
          ebs=.001, fig=2)


        # Example 4: Estimation the unknown standard deviation of Normal population with known
        #  mean (e.g, mean=7) based of TFNs.
        n = 10	
        set.seed(123)
        c = rnorm(n, 7,2);    round(c,3)
        l = runif(n, 0,2.5); round(l,3)
        u = runif(n, 0,1); round(u,3)

        EM.Triangular(T.dist="norm", T.dist.par=c(7,NA), par.space=c(0,10), c, l, u, start=5,
          ebs=.0001, fig=2)


        # Example 5: Estimation the unknown parameter b where  X ~ U(a=0,b).
        n = 15	
        set.seed(101)
        c = runif(n, 0,5); round(c,3)
        l = runif(n, 0,1); round(l,3)
        u = runif(n, 0,1); round(u,3)

        b <- EM.Triangular(T.dist="unif", T.dist.par=c(0,NA), par.space=c(0,10), c, l, u,
                           start=5, ebs=.001, fig=2)$MLE
        print(b)