fm.fittingKinteractiveMarginal {Rfmtool}

Fuzzy Measure Fitting function of the k-interactive using marginal representation

Description

Estimate values of the k-interacive fuzzy measures from empirical data using marginal representation.

Usage

fm.fittingKinteractiveMarginal(data, env=NULL, kadd="NA", K="NA", submod ="NA")

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 containing 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 stores the observed aggregated value y.

env

Environment variable obtained from fm.Init(n).

kadd

Value of k-interactivity, which is used for reducing the complexity of fuzzy measures. kadd is defined as an optional argument, its default value is kadd = 2.

K

The constant K, the value of FM value for sets of cardinality kadd+1 is computed from data, default 0.5.

submod

-1 indicates supermodular FM is needed, +1 indicates submodular, 0 otherwise. Should be consistent with K and n, see manual

Value

output

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

Author(s)

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

Examples

env<-fm.Init(3)
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);
fm.fittingKinteractiveMarginal(d,env,2,0.6, 0)