fm.FuzzyMeasureFitLP {Rfmtool}

Fuzzy Measure Fitting function.

Description

Estimate values of the fuzzy measures from empirical data. The result is an array containing the values of a standard fuzzy measure in binary ordering. kadd defines the complexity of fuzzy measure. If kadd is not provided, its default value is equal to the number of inputs.

Usage

fm.FuzzyMeasureFitLP(data, env=NULL, kadd="NA", 
        options=0, indexlow=(NULL), indexhigh=(NULL) , option1=0, orness=(NULL))

Arguments

data

Empirical data set in pairs (x_1,y_1),(x_2,y_2),...,(x_d,y_d) where x_i in [0,1]^n is a vector contains utility values of n input criteria x_i1,x_i2,...,x_in, y_i in [0,1] is a single aggregated value given by decision makers. The data is stored as a matrix of M by n+1 elements, where M is the number of data instances, and n is the number of input criteria,the column n + 1 store the observed aggregating value y.

env

Environment variable obtained from fm.Init(n).

kadd

Value of k-additivity, which is used for reducing the complexity of fuzzy measures. kadd is defined as an optional argument, its default value is kadd = n. kadd is k in k-additive f-measure, 1 < kadd < n+1; if kdd=n - f.m. is unrestricted

options

Options default value is 0. 1 - lower bounds on Shapley values supplied in indexlow, 2 - upper bounds on Shapley values supplied in indexhigh, 3 - lower and upper bounds on Shapley values supplied in indexlow and indexhigh, 4 - lower bounds on all interaction indices supplied in indexlow, 5 - upper bounds on all interaction indices supplied in indexhigh, 6 - lower and upper bounds on all interaction indices supplied inindexlow and indexhigh. All these value will be treated as additional constraints in the LP.

indexlow

Array of size n (options =1,2,3) or m (options=4,5,6) containing the lower bounds on the Shapley values or interaction indices

indexhigh

Array of size n (options =1,2,3) or m (options=4,5,6) containing the upper bounds on the Shapley values or interaction indices

option1

If the value is 1, the interval of orness values will be fitted (and the desired low and high orness values should be provided). If 0, no additional orness constraints.

orness

Array of size 2, for example c(0.1,1)

Value

output

The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering.

Note

The fit might not be perfect, and not all the constraints can be fully met.

Author(s)

Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University

Examples

d <-  matrix( c( 0.00125122, 0.563568, 0.193298, 0.164338, 
            0.808716, 0.584991, 0.479858, 0.544309, 
            0.350281, 0.895935, 0.822815, 0.625868, 
            0.746582, 0.174103, 0.858917, 0.480347, 
            0.71048, 0.513519, 0.303986, 0.387631, 
            0.0149841, 0.0914001, 0.364441, 0.134229, 
            0.147308, 0.165894, 0.988495, 0.388044, 
            0.445679, 0.11908, 0.00466919, 0.0897714, 
            0.00891113, 0.377869, 0.531647, 0.258585, 
            0.571167, 0.601746, 0.607147, 0.589803, 
            0.166229, 0.663025, 0.450775, 0.357412, 
            0.352112, 0.0570374, 0.607666, 0.270228, 
            0.783295, 0.802582, 0.519867, 0.583348, 
            0.301941, 0.875946, 0.726654, 0.562174, 
            0.955872, 0.92569, 0.539337, 0.633631, 
            0.142334, 0.462067, 0.235321, 0.228419, 
            0.862213, 0.209595, 0.779633, 0.498077, 
            0.843628, 0.996765, 0.999664, 0.930197, 
            0.611481, 0.92426, 0.266205, 0.334666, 
            0.297272, 0.840118, 0.0237427, 0.168081), 
       nrow=20, 
       ncol=4,byrow=TRUE);
env<-fm.Init(3)
fm.FuzzyMeasureFitLP(d,env)
indexlow=c(0.1,0.1,0.2);
indexhigh=c(0.9,0.9,0.5);
fm.FuzzyMeasureFitLP(d,env, kadd=2, indexlow, indexhigh, 
  options=3, option1=1, orness=c(0.1,0.7))