fm.FuzzyMeasureFitLP | R Documentation |
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.
fm.FuzzyMeasureFitLP(data, env=NULL, kadd="NA", options=0, indexlow=(NULL), indexhigh=(NULL) , option1=0, orness=(NULL))
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) |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
The fit might not be perfect, and not all the constraints can be fully met.
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.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))
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