BC.positive.reg.FRST: Positive region based on fuzzy rough set
In janusza/RoughSets: Data Analysis Using Rough Set and Fuzzy Rough Set Theories

Description Usage Arguments Details Value Author(s) References See Also Examples

This is a function that implements a fundamental concept of fuzzy rough set theory which is the positive region and the corresponding degree of dependency. The explanation about this concept can be seen in B.Introduction-FuzzyRoughSets.

1	BC.positive.reg.FRST(decision.table, fuzzyroughset)

`decision.table`	a `"DecisionTable"` class representing the decision table. See `SF.asDecisionTable`.
`fuzzyroughset`	a `"LowerUpperApproximation"` class representing a fuzzy rough set that is produced by `BC.LU.approximation.FRST`.

In order to compute the function, we need to calculate the indiscernibility relation by executing BC.IND.relation.FRST and the lower and upper approximations by calling BC.LU.approximation.FRST.

A class "PositiveRegion" containing the following components:

positive.freg: a vector representing membership degrees to the fuzzy positive region for each index of objects.
degree.dependency: a value expressing the degree of dependency.
type.model: a string representing type of models. In this case, it is "FRST" which means fuzzy rough set theory.

Lala Septem Riza

R. Jensen and Q. Shen, "New Approaches to Fuzzy-Rough Feature Selection", IEEE Trans. on Fuzzy Systems, vol. 19, no. 4, p. 824 - 838 (2009).

BC.LU.approximation.FRST, BC.IND.relation.FRST, BC.IND.relation.RST,

BC.LU.approximation.RST, and BC.positive.reg.FRST.

###########################################################
##### 1. Example: Using a simple decision table containing 
#####             nominal values for the decision attribute
###########################################################
dt.ex1 <- data.frame(c(-0.4, -0.4, -0.3, 0.3, 0.2, 0.2), 
                     c(-0.3, 0.2, -0.4, -0.3, -0.3, 0),
			        c(-0.5, -0.1, -0.3, 0, 0, 0),
			        c("no", "yes", "no", "yes", "yes", "no"))
colnames(dt.ex1) <- c("a", "b", "c", "d")
decision.table <- SF.asDecisionTable(dataset = dt.ex1, decision.attr = 4)

## let us consider the first and second attributes only as conditional attribute
condAttr <- c(1, 2)

## let us consider the fourth attribute as decision attribute
decAttr <- c(4)

#### Calculate fuzzy indiscernibility relation ####
control.ind <- list(type.aggregation = c("t.tnorm", "lukasiewicz"), 
                    type.relation = c("tolerance", "eq.1"))
control.dec <- list(type.aggregation = c("crisp"), type.relation = "crisp")

IND.condAttr <- BC.IND.relation.FRST(decision.table, attributes = condAttr, 
                                     control = control.ind) 
IND.decAttr <- BC.IND.relation.FRST(decision.table, attributes = decAttr, 
                                     control = control.dec) 

#### Calculate fuzzy lower and upper approximation using type.LU : 
#### "implicator.tnorm" 
control <- list(t.implicator = "lukasiewicz")
FRST.LU <- BC.LU.approximation.FRST(decision.table, IND.condAttr, IND.decAttr, 
              type.LU = "implicator.tnorm", control = control)

#### Determine positive regions ####
res.1 <- BC.positive.reg.FRST(decision.table, FRST.LU)

###########################################################
##### 2. Example: Using the housing decision table containing 
#####             continuous values for the decision attribute
###########################################################

## In this case, we are using the housing dataset containing 7 objects
data(RoughSetData)
decision.table <- RoughSetData$housing7.dt

conditional.attr <- c(1, 2)
decision.attr = c(14)
control.ind <- list(type.aggregation = c("t.tnorm", "lukasiewicz"), 
                     type.relation = c("tolerance", "eq.1"))

#### Calculate fuzzy indiscernibility relation ####
IND.condAttr <- BC.IND.relation.FRST(decision.table, attributes = conditional.attr, 
                                     control = control.ind) 
IND.decAttr <- BC.IND.relation.FRST(decision.table, attributes = decision.attr, 
                                     control = control.ind) 

#### Calculate fuzzy lower and upper approximation using type.LU : 
#### "implicator.tnorm" 
control <- list(t.implicator = "lukasiewicz", t.tnorm = "lukasiewicz")

FRST.LU <- BC.LU.approximation.FRST(decision.table, IND.condAttr, IND.decAttr, 
              type.LU = "implicator.tnorm", control = control)

#### Determine fuzzy regions ####
res.2 <- BC.positive.reg.FRST(decision.table, FRST.LU)