belplau {dst}

Calculation of the degrees of Belief and Plausibility of a basic chance assignment (bca).

Description

Degrees of Belief Bel and Plausibility Pl of the focal elements of a bca are computed. The ratio of the plausibility of a focal element against the plausibility of its contrary is also computed. Subsets with zero mass can be excluded from the calculations.

Usage

belplau(x, remove = FALSE, h = NULL)

Arguments

x

A basic chance assignment mass function (see bca).

remove

= TRUE: Exclude subsets with zero mass.

h

= NULL: Hypothesis to be tested. Description matrix in the same format than x$tt

Details

The degree of belief Bel is defined by:

bel(A) = Sum((m(B); B \subseteq A))

for every subset B of A.
The degree of plausibility pl is defined by:

pl(A) = Sum[(m(B); B \cap A \neq \emptyset]

for every subset B of the frame of discernment.
The plausibility ratio of a focal element A versus its contrary not A is defined by: Pl(A)/(1-Bel(A)).

Value

A matrix of M rows by 3 columns is returned, where M is the number of focal elements:

• Column 1: the degree of Belief bel;

• Column 2: the degree of Disbellief (belief in favor of the contrary hypothesis) disbel;

• Column 3: the degree of Epistemic uncertainty unc;

• Column 4: the degree of Plausibility plau;

• Column 5: the Plausibility ratio rplau.

Author(s)

Claude Boivin, Peiyuan Zhu

References

• Shafer, G., (1976). A Mathematical Theory of Evidence. Princeton University Press, Princeton, New Jersey, p. 39-43.

• Williams, P., (1990). An interpretation of Shenoy and Shafer's axioms for local computation. International Journal of Approximate Reasoning 4, pp. 225-232.

Examples

x <- bca(tt = matrix(c(0,1,1,1,1,0,1,1,1),nrow = 3, 
byrow = TRUE), m = c(0.2,0.5, 0.3), 
cnames = c("a", "b", "c"), varnames = "x", idvar = 1)
belplau(x)
y <- bca(tt = matrix(c(1,0,0,1,1,1),nrow = 2, 
byrow = TRUE), m = c(0.6, 0.4),  
cnames = c("a", "b", "c"),  varnames = "y", idvar = 1)
xy <- nzdsr(dsrwon(x,y))
belplau(xy)
print("compare all elementary events")
xy1 <- addTobca(x = xy, tt = matrix(c(0,1,0,0,0,1), nrow = 2, byrow = TRUE))
belplau(xy1) 
belplau(xy1, remove = TRUE) 
belplau(xy1, h = matrix(c(1,0,0,0,1,1), nrow = 2, byrow = TRUE))