# fm.FuzzyMeasureFitLPMob: Mobius Fuzzy Measure Fitting function, R wrapper for... In Rfmtool: Fuzzy Measure Tools for R

## Description

Esimate values of the Mobius fuzzy measures from empirical data. The result is an array containing the values of the fuzzy measure in Mobius, ordered according to set cardinalities. kadd defines the complexity of fuzzy measure. if kadd is not provided, its default value is equal to the number of inputs.

## Usage

 ```1 2 3``` ```fm.FuzzyMeasureFitLPMob(data, env=NULL, kadd="NA", options=0, indexlow=(NULL), indexhigh=(NULL) , option1=0, orness=(NULL) ) ```

## Arguments

 `data` is the 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` is the 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 Mobius 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, Deakin University

## References

[1] http://www.deakin.edu.au/~gleb/fmtools.html [2] H.Q. Vu, G. Beliakov, G. Li, 2014, A choquet integral toolbox and its application in customer preference analysis, in Data mining applications with R, Elsevier Inc., Waltham, MA., pp.247-272.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29``` ```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); env<-fm.Init(3) fm.FuzzyMeasureFitLPMob(d,env) indexlow=c(0.1,0.1,0.2); indexhigh=c(0.9,0.9,0.5); fm.FuzzyMeasureFitLPMob(d,env, kadd=2, indexlow, indexhigh, options=3, option1=1, orness=c(0.1,0.7)) ```

Rfmtool documentation built on May 29, 2017, 7:58 p.m.