| Title: | Fuzzy Measure Tools |
|---|---|
| Description: | Various tools for handling fuzzy measures, calculating Shapley value and interaction index, Choquet and Sugeno integrals, as well as fitting fuzzy measures to empirical data are provided. Construction of fuzzy measures from empirical data is done by solving a linear programming problem by using 'lpsolve' package, whose source in C adapted to the R environment is included. The description of the basic theory of fuzzy measures is in the manual in the Doc folder in this package. Please refer to the following: [1] <https://personal-sites.deakin.edu.au/~gleb/fmtools.html> [2] G. Beliakov, H. Bustince, T. Calvo, 'A Practical Guide to Averaging', Springer, (2016, ISBN: 978-3-319-24753-3). [3] G. Beliakov, S. James, J-Z. Wu, 'Discrete Fuzzy Measures', Springer, (2020, ISBN: 978-3-030-15305-2). |
| Authors: | Gleb Beliakov [aut, cre], Quan Vu [ctb], Andrei Kelarev [ctb], Michel Berkelaar [ctb], Kjell Eikland [ctb], Samuel E. Buttrey [ctb], Stefan I. Larimore [ctb], Timothy A. Davis [ctb], John Gilbert [ctb], Esmond Ng [ctb], Peter Notebaert [ctb], Richard Stallman [ctb], Jeroen Dirks [ctb], Daniela L. Calderon [ctb] |
| Maintainer: | Gleb Beliakov <[email protected]> |
| License: | LGPL-3 |
| Version: | 5.0.4 |
| Built: | 2024-10-31 06:22:46 UTC |
| Source: | CRAN |
This function shows a list of function included in this toolbox
fm()fm()
The following functions involve the parameters v (the array containing the fuzzy measure in standard representation) or Mob (in Mobius representation), n - the dimension and m = 2^n. The values of the fuzzy measure always obey the binary ordering.
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
fm()fm()
This is used for populating capacities which Add a pair v_ij to the structure, their Indices are 1-based.
fm.add_pair_sparse( i, j, v, envsp = NULL)fm.add_pair_sparse( i, j, v, envsp = NULL)
i |
One of the indices which are 1-based |
j |
One of the indices which are 1-based |
v |
The value to be added. |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n) |
output |
The output is an added pair v_ij to the structure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <-fm.add_pair_sparse(1,2, 0.4, envsp) envsp <-fm.add_pair_sparse(1,3, 0.3, envsp) envsp envsp <- fm.FreeSparseFM(envsp)n <- 3 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <-fm.add_pair_sparse(1,2, 0.4, envsp) envsp <-fm.add_pair_sparse(1,3, 0.3, envsp) envsp envsp <- fm.FreeSparseFM(envsp)
This is used for adding singletons to the structure.
fm.add_singletons_sparse(v, envsp=NULL)fm.add_singletons_sparse(v, envsp=NULL)
v |
The vector of singletons of size n. |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is added singletons to the structure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.add_singletons_sparse(c(0, 0.3, 0.5),envsp)n <- 3 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.add_singletons_sparse(c(0, 0.3, 0.5),envsp)
This is used for populating capacities which Add a tuple of size tupsize to the structure whose Indices are 1-based in tuple.
For populating capacities, adds a whose 1-based indices are in tuple
fm.add_tuple_sparse( tuple, v, envsp=NULL)fm.add_tuple_sparse( tuple, v, envsp=NULL)
tuple |
Collection of objects. It is a list of cardinalities of the nonzero tuples (cardinality, tuple composition) |
v |
The value of the tuple to be added |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is adding a tuple of size tupsize |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 4 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.add_tuple_sparse(c(1,2,3),0.2,envsp) envsp <- fm.add_tuple_sparse(c(1,3,4),0.3,envsp)n <- 4 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.add_tuple_sparse(c(1,2,3),0.2,envsp) envsp <- fm.add_tuple_sparse(c(1,3,4),0.3,envsp)
Calculates the Banzhaf indices of input criteria from general fuzzy measure.
fm.Banzhaf(v,env=NULL)fm.Banzhaf(v,env=NULL)
v |
Fuzzy measure in general representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size n, which contain Banzhaf indices of input criteria. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.Banzhaf(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env) env<-fm.Free(env)env<-fm.Init(3) fm.Banzhaf(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env) env<-fm.Free(env)
Calculate the Banzhaf values of a 2-additive fuzzy measure for n inputs given in Mobius representation. The results are in arrays.
fm.Banzhaf2addMob(n, Mob)fm.Banzhaf2addMob(n, Mob)
n |
Number of inputs |
Mob |
Fuzzy measure value in Mobius representation |
output |
The output is an array of size n, which contain Banzhaf indices of input criteria. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
Banzhaf <- fm.Banzhaf2addMob(3, c(0.2, 0.3, 0.5, -0.2, 0.4, 0.1))Banzhaf <- fm.Banzhaf2addMob(3, c(0.2, 0.3, 0.5, -0.2, 0.4, 0.1))
Calculates the Banzhaf indices of input criteria from general fuzzy measure in Mobius representation.
fm.BanzhafMob(Mob,env=NULL)fm.BanzhafMob(Mob,env=NULL)
Mob |
Fuzzy measure in Mobius representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size n, which contain Banzhaf indices of input criteria. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.BanzhafMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env) env<-fm.Free(env)env<-fm.Init(3) fm.BanzhafMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env) env<-fm.Free(env)
Calculates Banzhaf values vectors of size n of a sparse fuzzy measure
fm.BanzhafMob_sparse(n, envsp=NULL)fm.BanzhafMob_sparse(n, envsp=NULL)
n |
The size of values vectors |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is Banzhaf values vectors of size n of a sparse fuzzy measure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); fm.BanzhafMob_sparse(3, envsp) envsp <- fm.FreeSparseFM(envsp)n <- 3 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); fm.BanzhafMob_sparse(3, envsp) envsp <- fm.FreeSparseFM(envsp)
Calculates the Bipartition interaction indices of input criteria from general fuzzy measure.
fm.Bipartition(v,env=NULL)fm.Bipartition(v,env=NULL)
v |
Fuzzy measure in general representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size 2^n, which contain bipartition interaction indices of input criteria coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.Bipartition(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env) env<-fm.Free(env)env<-fm.Init(3) fm.Bipartition(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env) env<-fm.Free(env)
Calculates the Banzhaf Bipartition interaction indices of input criteria from general fuzzy measure.
fm.BipartitionBanzhaf(v,env=NULL)fm.BipartitionBanzhaf(v,env=NULL)
v |
Fuzzy measure in general representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size 2^n, which contain Banzhaf bipartition interaction indices of input criteria coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.BipartitionBanzhaf(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env) env<-fm.Free(env)env<-fm.Init(3) fm.BipartitionBanzhaf(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env) env<-fm.Free(env)
Checks supermodularity of the set function v in Mobius representation using stan- dard check.
fm.check_convexity_monotonicity_mob(v, len, env=NULL)fm.check_convexity_monotonicity_mob(v, len, env=NULL)
v |
matrix v stores fuzzy measurements consecutively in cardinal order v. |
len |
this is the length of array Mob (this array is usually smaller than 2^n), and is computed by fm.fm_arraysize_kadd(N, Kadd). |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 or 0 to check for monotonicity. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) step <- 0.001 Fn <- NULL Option<- 3 fuzzymeasures <- fm.generate_fm_randomwalk(1, 3, 2, 1000, Option, step, Fn, env) len <- fuzzymeasures$length check <- fm.check_convexity_monotonicity_mob(fuzzymeasures$V, len, env)env<-fm.Init(3) step <- 0.001 Fn <- NULL Option<- 3 fuzzymeasures <- fm.generate_fm_randomwalk(1, 3, 2, 1000, Option, step, Fn, env) len <- fuzzymeasures$length check <- fm.check_convexity_monotonicity_mob(fuzzymeasures$V, len, env)
Checks monotonicity of the set function v in standard representation using insert sort.
fm.check_monotonicity(v, env=NULL)fm.check_monotonicity(v, env=NULL)
v |
matrix v stores fuzzy measurements consecutively in cardinal order. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 or 0 to check for monotonicity. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) v <- fm.generate_fm_sorting(1, 1000, 0, env) monotonicity <- fm.check_monotonicity(v, env)env<-fm.Init(3) v <- fm.generate_fm_sorting(1, 1000, 0, env) monotonicity <- fm.check_monotonicity(v, env)
Checks monotonicity of the set function v in Mobius representation using standard check.
fm.check_monotonicity_mob(v, len, env=NULL)fm.check_monotonicity_mob(v, len, env=NULL)
v |
matrix v stores fuzzy measurements consecutively in cardinal order. |
len |
this is the length of array Mob (this array is usually smaller than 2^n), and is computed by fm_arraysize_kadd(N, Kadd) |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 or 0 to check for monotonicity |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) step <- 100 Fn <- NULL Option<- 3 fuzzymeasures <- fm.generate_fm_randomwalk(1, 3, 2, 1000, Option, step, Fn, env) len <- fuzzymeasures$length check <- fm.check_monotonicity_mob(fuzzymeasures$V, len, env) checkenv<-fm.Init(3) step <- 100 Fn <- NULL Option<- 3 fuzzymeasures <- fm.generate_fm_randomwalk(1, 3, 2, 1000, Option, step, Fn, env) len <- fuzzymeasures$length check <- fm.check_monotonicity_mob(fuzzymeasures$V, len, env) check
Check the monotonicity of the 2-additive set function v in Mobius representation using fast check.
fm.check_monotonicity_mob_2additive(v, n, temp=NULL)fm.check_monotonicity_mob_2additive(v, n, temp=NULL)
v |
Random 2-additive fuzzy measure in Mobius representation. |
n |
Number of inputs |
temp |
Auxiliary array of length n^2 (e.g: array(0.0,n*n)). It may or may not be specified (if speed matters, then preallocate it). |
output |
The output is 1 or 0 to check for monotonicity. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
v <- fm.generate_fm_2additive(1, 10) n <- 10 v$len v$V check <- fm.check_monotonicity_mob_2additive(v$V, n) check temp <- array(0.0,10*10); check <- fm.check_monotonicity_mob_2additive(v$V, n, temp) checkv <- fm.generate_fm_2additive(1, 10) n <- 10 v$len v$V check <- fm.check_monotonicity_mob_2additive(v$V, n) check temp <- array(0.0,10*10); check <- fm.check_monotonicity_mob_2additive(v$V, n, temp) check
Checks monotonicity of the set function v in standard representation using insert sort.
fm.check_monotonicity_sort_insert(v, indices, env=NULL)fm.check_monotonicity_sort_insert(v, indices, env=NULL)
v |
matrix v stores fuzzy measurements consecutively in cardinal order. |
indices |
The indices can be used at subsequent steps of monotonicity verification. This function is called after merge sort, so the indices are already precomputed. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is a list of components (True/False, indices, values). The indices and values can be used at subsequent steps of monotonicity verification (e.g., values slightly perturbed) |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) v <- fm.generate_fm_sorting(1, 1000, 0, env) out <- fm.check_monotonicity_sort_merge(v, NULL, env) out$V[1] = out$V[1] *1.1 out<- fm.check_monotonicity_sort_insert(out$V, out$index, env) out$outenv<-fm.Init(3) v <- fm.generate_fm_sorting(1, 1000, 0, env) out <- fm.check_monotonicity_sort_merge(v, NULL, env) out$V[1] = out$V[1] *1.1 out<- fm.check_monotonicity_sort_insert(out$V, out$index, env) out$out
Checks monotonicity of the set function v in standard representation using merge sort.
fm.check_monotonicity_sort_merge(v, indices=NULL, env=NULL)fm.check_monotonicity_sort_merge(v, indices=NULL, env=NULL)
v |
matrix v stores fuzzy measurements consecutively in cardinal order. |
indices |
The indices can be used at subsequent steps of monotonicity verification. Initially indices need not be specified |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is a list of components (True/False, indices, values). The indices and values can be used at subsequent steps of monotonicity verification (e.g., values slightly perturbed) |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) v <- fm.generate_fm_sorting(1, 1000, 0, env) v measure <- fm.check_monotonicity_sort_merge(v,NULL, env) print(measure$out) measure$V[1] = measure$V[1] *1.1 measure <- fm.check_monotonicity_sort_merge(measure$V, measure$index, env)env<-fm.Init(3) v <- fm.generate_fm_sorting(1, 1000, 0, env) v measure <- fm.check_monotonicity_sort_merge(v,NULL, env) print(measure$out) measure$V[1] = measure$V[1] *1.1 measure <- fm.check_monotonicity_sort_merge(measure$V, measure$index, env)
Calculates the value of a discrete Choquet integral of input x, with fuzzy measure in general representation.
fm.Choquet(x, v, env=NULL)fm.Choquet(x, v, env=NULL)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
v |
The general fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is a single value of the computed Choquet integral. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.Choquet(c(0.6, 0.3, 0.8), c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env) env<-fm.Free(env)env<-fm.Init(3) fm.Choquet(c(0.6, 0.3, 0.8), c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env) env<-fm.Free(env)
Calculates the Choquet integral value of a 2-additive fuzzy measure for n inputs given in Mobius representation.
fm.Choquet2addMob(n, x, Mob)fm.Choquet2addMob(n, x, Mob)
n |
Number of inputs |
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
Mob |
The Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
output |
The output is the Choquet integral value in Mobius representation. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
Choquet <- fm.Choquet2addMob(3, c(0.2,0.5,0.4), c(0.2, 0.3, 0.5, -0.2, 0.4, 0.1))Choquet <- fm.Choquet2addMob(3, c(0.2,0.5,0.4), c(0.2, 0.3, 0.5, -0.2, 0.4, 0.1))
Calculates the Choquet integral of x with respect to dual k-interactive fuzzy measure in Mobius representation.
fm.ChoquetCoMobKInter(x, Mob, kadd, env=NULL)fm.ChoquetCoMobKInter(x, Mob, kadd, env=NULL)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
Mob |
The Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data |
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. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is the Choquet integral value in Mobius representation. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env <-fm.Init(4) step <- 0.0001 Fn <- NULL fuzzymeasures <- fm.generate_fm_kinteractivedualconvex(1, 4, 2, 1000, step, Fn, env) fuzzymeasures env fm.ChoquetCoMobKInter(c(0.2,0.5,0.4,0.1), fuzzymeasures$V, 2, env) env<-fm.Free(env)env <-fm.Init(4) step <- 0.0001 Fn <- NULL fuzzymeasures <- fm.generate_fm_kinteractivedualconvex(1, 4, 2, 1000, step, Fn, env) fuzzymeasures env fm.ChoquetCoMobKInter(c(0.2,0.5,0.4,0.1), fuzzymeasures$V, 2, env) env<-fm.Free(env)
This is an alternative calculation of the Choquet integral from the fuzzy measure in Mobius representation.
fm.ChoquetKinter(x, v, kint, env)fm.ChoquetKinter(x, v, kint, env)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
v |
The fuzzy measure of size less than m=2^n. Its values can be provided by users, or by estimating from empirical data. |
kint |
the k-interactivity parameter, must be smaller than n. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is a single value of the computed Choquet integral. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.ChoquetKinter(c(0.6,0.3,0.8),c(0,0.3,0.5,0.6,0.4,0.8,0.7,1),2,env) env<-fm.Free(env)env<-fm.Init(3) fm.ChoquetKinter(c(0.6,0.3,0.8),c(0,0.3,0.5,0.6,0.4,0.8,0.7,1),2,env) env<-fm.Free(env)
This is an alternative calculation of the Choquet integral from the fuzzy measure in Mobius representation.
fm.ChoquetMob(x, Mob, env=NULL)fm.ChoquetMob(x, Mob, env=NULL)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
Mob |
The Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is a single value of the computed Choquet integral. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.ChoquetMob(c(0.2,0.5,0.4), c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.ChoquetMob(c(0.2,0.5,0.4), c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Calculates the Choquet integral in Mobius sparse representation.
fm.ChoquetMob_sparse(x, envsp=NULL)fm.ChoquetMob_sparse(x, envsp=NULL)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the Choquet integral in Mobius sparse representation. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); ChoquetMobsparse <- fm.ChoquetMob_sparse(c(0.1,0.05,0.2),envsp) ChoquetMobsparse envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); ChoquetMobsparse <- fm.ChoquetMob_sparse(c(0.1,0.05,0.2),envsp) ChoquetMobsparse envsp <- fm.FreeSparseFM(envsp)
Finds the value of lambda and calculates the rest of the values of the fuzzy measure, given its values at singletons; singletons is an array of size n. The outputs are lambda and v, v is in standard representation and binary ordering.
fm.ConstructLambdaMeasure(singletons,env)fm.ConstructLambdaMeasure(singletons,env)
singletons |
Singletons is an array of n. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is the list (lambda, measure), where measure is a fuzzy measure in standard representation. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) w <- fm.ConstructLambdaMeasure(c(0, 0.3, 0.5),env)env<-fm.Init(3) w <- fm.ConstructLambdaMeasure(c(0, 0.3, 0.5),env)
Finds the value of lambda and calculates the rest of the values of the fuzzy measure, given its values at singletons; singletons is an array of size n. The outputs are lambda and measure, measure is in Mobius representation.
fm.ConstructLambdaMeasureMob(singletons,env)fm.ConstructLambdaMeasureMob(singletons,env)
singletons |
Singletons is an array of n. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is the list (lambda, measure), where measure is a fuzzy measure in Mobius representation. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) w <- fm.ConstructLambdaMeasureMob(c(0, 0.3, 0.5),env) w$measure fm.Free(env)env<-fm.Init(3) w <- fm.ConstructLambdaMeasureMob(c(0, 0.3, 0.5),env) w$measure fm.Free(env)
Converts dual k-interactive fuzzy measure from Mobius to standard representation.
fm.ConvertCoMob2Kinter(Mob,kadd, fullmu, env=NULL)fm.ConvertCoMob2Kinter(Mob,kadd, fullmu, env=NULL)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from emperical data. |
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. |
fullmu |
Integer flag. is 1 then all 2n are allocated, otherwise a more compact representation fo rk-interactive fuzzy measures is used. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is k-interactive fuzzy measure standard representation |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env <-fm.Init(4) fullmu <- 0 step<-0.001 Fn <- NULL fuzzymeasures <- fm.generate_fm_kinteractivedualconvex(1, 4, 2, 1000, step, Fn, env) fm.ConvertCoMob2Kinter(fuzzymeasures$V, 2, fullmu, env )env <-fm.Init(4) fullmu <- 0 step<-0.001 Fn <- NULL fuzzymeasures <- fm.generate_fm_kinteractivedualconvex(1, 4, 2, 1000, step, Fn, env) fm.ConvertCoMob2Kinter(fuzzymeasures$V, 2, fullmu, env )
Calculates the dual of fuzzy measure v, returns it as value of the function (array of size m).
fm.dualm(v, env=NULL)fm.dualm(v, env=NULL)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from emperical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is an array of size m with the dual of fuzzy measure v. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) w <- fm.dualm(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) w <- fm.dualm(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Calculates the dual of fuzzy measure v, returns it as value of the function (array of size m).
fm.dualmMob(Mob,env=NULL)fm.dualmMob(Mob,env=NULL)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from emperical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is an array of size m with the dual of fuzzy measure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) w <- fm.dualmMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) w <- fm.dualmMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Calculates the dual of a k-additive fuzzy measures for n inputs.
fm.dualMobKadd(Mob, env = NULL, kadd = "NA")fm.dualMobKadd(Mob, env = NULL, kadd = "NA")
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
kadd |
Value of k-interactivity, which is used for reducing the complexity of fuzzy measures. It is defined as an optional argument |
output |
The output is the dual of a k-additive fuzzy measures for n inputs |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) dualMob_Kadd <- fm.dualMobKadd(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1), env,2)env<-fm.Init(3) dualMob_Kadd <- fm.dualMobKadd(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1), env,2)
Calculates entropy value of the Choquet integral for the fuzzy measure v in general representation
fm.EntropyChoquet(v,env)fm.EntropyChoquet(v,env)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is the entropy value of the Choquet integral for the fuzzy measure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.EntropyChoquet(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1))env<-fm.Init(3) fm.EntropyChoquet(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1))
Calculates entropy value of the Choquet integral for the fuzzy measure v in Mobius representation
fm.EntropyChoquetMob(Mob,env)fm.EntropyChoquetMob(Mob,env)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is entropy value of the Choquet integral for the fuzzy measure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.EntropyChoquetMob(c(0.0,0.3,0.5,-0.2,0.4,0.1,-0.2,0.1),env)env<-fm.Init(3) fm.EntropyChoquetMob(c(0.0,0.3,0.5,-0.2,0.4,0.1,-0.2,0.1),env)
This function checks that the enviromnemt variable is internally consistent.
fm.errorcheck(env)fm.errorcheck(env)
env |
Environment variable obtained from fm.Init(n). |
output |
The ouput is TRUE or FALSE. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.errorcheck(env)env<-fm.Init(3) fm.errorcheck(env)
From sparse to full representation of 2-additive capacity (singletons and pairs, augmented with 0s).
fm.expand_2add_full(n, envsp=NULL)fm.expand_2add_full(n, envsp=NULL)
n |
Number of inputs |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is a sparse to full representation of 2-additive capacity (singletons and pairs, augmented with 0s) |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); cap2add <- fm.expand_2add_full(n,envsp) cap2add envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); cap2add <- fm.expand_2add_full(n,envsp) cap2add envsp <- fm.FreeSparseFM(envsp)
Exports from sparse to full capacity.
fm.expand_sparse_full(n, envsp=NULL)fm.expand_sparse_full(n, envsp=NULL)
n |
Number of inputs. |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
Exports from sparse to full capacity. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n<-3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); cap <- fm.expand_sparse_full(n, envsp) cap envsp <- fm.FreeSparseFM(envsp)n<-3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); cap <- fm.expand_sparse_full(n, envsp) cap envsp <- fm.FreeSparseFM(envsp)
Returns in mc the arrays of maximal chains (there are n! such arrays) of a fuzzy measure v. Each maximal chain corresponds to the coefficients of a linea. function on the respective simplex
fm.export_maximal_chains(v, env = NULL)fm.export_maximal_chains(v, env = NULL)
v |
Fuzzy measure in general representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is mc the arrays of maximal chains |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) exportmaximalchains <- fm.export_maximal_chains( c(0, 0.00224, 0.0649, 0.510, 0.00965, 0.374,0.154, 1),env)env<-fm.Init(3) exportmaximalchains <- fm.export_maximal_chains( c(0, 0.00224, 0.0649, 0.510, 0.00965, 0.374,0.154, 1),env)
Estimate values of the fuzzy measures from empirical data.
fm.fitting(data, env=NULL, kadd="NA")fm.fitting(data, env=NULL, kadd="NA")
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 |
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. |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fitting(d,env)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.fitting(d,env)
Estimate values of the fuzzy measures from empirical data tailored 2-additive standard fuzzy measure.
fm.fitting2additive(data, options=0, indexlow, indexhigh , option1=0, orness)fm.fitting2additive(data, options=0, indexlow, indexhigh , option1=0, orness)
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. |
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 |
optional 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 |
optional 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 |
optional array of size 2, for example c(0.1,1) |
output |
The output is an array containing the values of a standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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); indexlow=c(0.1,0.1,0.2); indexhigh=c(0.9,0.9,0.5); fm.fitting2additive(d, options=3, indexlow, indexhigh, option1=0, orness=c(0.1,0.7))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); indexlow=c(0.1,0.1,0.2); indexhigh=c(0.9,0.9,0.5); fm.fitting2additive(d, options=3, indexlow, indexhigh, option1=0, orness=c(0.1,0.7))
Estimate values of the k-interacive fuzzy measures from empirical data.
fm.fittingKinteractive(data, env=NULL, kadd="NA", K="NA")fm.fittingKinteractive(data, env=NULL, kadd="NA", K="NA")
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 |
Value of FM value for sets of cardinality kadd+1, its default value is K = 0.5. |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fittingKinteractive(d,env,2,0.8)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.fittingKinteractive(d,env,2,0.8)
Estimate values of the k-interacive fuzzy measures from empirical data.
fm.fittingKinteractiveAuto(data, env=NULL, kadd="NA")fm.fittingKinteractiveAuto(data, env=NULL, kadd="NA")
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. The constant K the value of FM value for sets of cardinality kadd+1 is computed from data. |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fittingKinteractiveAuto(d,env,2)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.fittingKinteractiveAuto(d,env,2)
Estimate values of the k-interacive fuzzy measures from empirical data using marginal representation.
fm.fittingKinteractiveMarginal(data, env=NULL, kadd="NA", K="NA", submod ="NA")fm.fittingKinteractiveMarginal(data, env=NULL, kadd="NA", K="NA", submod ="NA")
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 |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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)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)
Estimate values of the k-interacive fuzzy measures from empirical data using marginal representation and maximal chains method.
fm.fittingKinteractiveMarginalMC(data, env=NULL, kadd="NA", K="NA", submod ="NA")fm.fittingKinteractiveMarginalMC(data, env=NULL, kadd="NA", K="NA", submod ="NA")
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 |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fittingKinteractiveMarginalMC(d,env,2,0.6,0)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.fittingKinteractiveMarginalMC(d,env,2,0.6,0)
Estimate values of the k-interacive fuzzy measures from empirical data using maximal chains method.
fm.fittingKinteractiveMC(data, env=NULL, kadd="NA", K="NA")fm.fittingKinteractiveMC(data, env=NULL, kadd="NA", K="NA")
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. |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fittingKinteractiveMC(d,env,2,0.6)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.fittingKinteractiveMC(d,env,2,0.6)
Estimate values of the k-maxitive fuzzy measures from empirical data.
fm.fittingKmaxitive(data, env=NULL, kadd="NA")fm.fittingKmaxitive(data, env=NULL, kadd="NA")
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-maxitivity, which is used for reducing the complexity of fuzzy measures. kadd is defined as an optional argument, its default value is kadd = n. |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fittingKmaxitive(d,env,2)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.fittingKmaxitive(d,env,2)
Estimate values of the k-tolerant fuzzy measures from empirical data.
fm.fittingKtolerant(data, env=NULL, kadd="NA")fm.fittingKtolerant(data, env=NULL, kadd="NA")
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-tolerance, which is used for reducing the complexity of fuzzy measures. kadd is defined as an optional argument, its default value is kadd = n. |
output |
The output is an array of size 2^n containing estimated standard fuzzy measure in binary ordering. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fittingKtolerant(d,env,2)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.fittingKtolerant(d,env,2)
Estimate values of the Mobius fuzzy measures from empirical data.
fm.fittingMob(data, env=NULL ,kadd="NA")fm.fittingMob(data, env=NULL ,kadd="NA")
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 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 defaultvalue is kadd = n. |
output |
The output is an array of size 2^n containing estimated Mobius 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,byrow=TRUE); env<-fm.Init(3) fm.fittingMob(d,env)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.fittingMob(d,env)
Estimate values of the symmetric fuzzy measures from empirical data. The resulting Choquet integral is the OWA function.
fm.fittingOWA(data, env=NULL)fm.fittingOWA(data, env=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 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). |
output |
The output is an array of size n containing estimated OWA coefficients. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fittingOWA(d,env)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.fittingOWA(d,env)
Estimate values of an additive fuzzy measure from empirical data. In this case the Choquet integral is the weighted arithmetic mean WAM.
fm.fittingWAM(data, env=NULL)fm.fittingWAM(data, env=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 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). |
output |
The output is an array of size n containing estimated weighting vector of WAM. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
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.fittingWAM(d,env)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.fittingWAM(d,env)
Returns the length of the array of values of k-interactive fuzzy measures. Useful for reserving memory.
fm.fm_arraysize(env = NULL, kint = "NA")fm.fm_arraysize(env = NULL, kint = "NA")
env |
Environment variable obtained from fm.Init(n). |
kint |
Interactive fuzzy measure. 0 < kint <= n |
output |
The outputs is the length of the array of values of k-interactive fuzzy measures |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) arraylength <- fm.fm_arraysize(env,1)env<-fm.Init(3) arraylength <- fm.fm_arraysize(env,1)
Frees the memory previously allocated in env.
fm.Free(env)fm.Free(env)
env |
Structure required for auxiliary data. It is obtained from fm.Init(n). |
output |
Frees the memory previously allocated in envsp. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n<-3 env <- fm.Init(n) env<-fm.Free(env) envn<-3 env <- fm.Init(n) env<-fm.Free(env) env
Frees the memory previously allocated in envsp.
fm.FreeSparseFM(envsp)fm.FreeSparseFM(envsp)
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
Frees the memory previously allocated in envsp. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n<-3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.FreeSparseFM(envsp) envsp <- fm.PrepareSparseFM(n, c(0.2,0.4,0.1), c(2,1,2,2,1,3,3,1,2,3)) envsp <- fm.FreeSparseFM(envsp) envspn<-3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.FreeSparseFM(envsp) envsp <- fm.PrepareSparseFM(n, c(0.2,0.4,0.1), c(2,1,2,2,1,3,3,1,2,3)) envsp <- fm.FreeSparseFM(envsp) envsp
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))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,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))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))
Estimate 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.
fm.FuzzyMeasureFitLPMob(data, env=NULL, kadd="NA", options=0, indexlow=(NULL), indexhigh=(NULL) , option1=0, orness=(NULL))fm.FuzzyMeasureFitLPMob(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 Mobius 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,byrow=TRUE); 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))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.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))
Generates one antibuoyant random fuzzy measure in standard representation.
fm.generate_antibuoyant(env = NULL)fm.generate_antibuoyant(env = NULL)
env |
Environment variable obtained from fm.Init(n). |
output |
The output is one antibuoyant random fuzzy measure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fuzzymeasures <- fm.generate_antibuoyant(env) fuzzymeasuresenv<-fm.Init(3) fuzzymeasures <- fm.generate_antibuoyant(env) fuzzymeasures
Generate several balanced random fuzzy measures in standard representation.
fm.generate_balanced(num, env=NULL)fm.generate_balanced(num, env=NULL)
num |
Generates num random fuzzy measures stored in an array v of length num * 2n. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output are several random fuzzy measures containing in an array v of length num * 2n |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fuzzymeasures <- fm.generate_balanced(2, env) fuzzymeasuresenv<-fm.Init(3) fuzzymeasures <- fm.generate_balanced(2, env) fuzzymeasures
Generate several random k-additive belief measures in Mobius representation.
fm.generate_belief(num, kadd, env=NULL)fm.generate_belief(num, kadd, env=NULL)
num |
Generates num random belief measures stored in an array Mob of length num * fm_arraysize_kadd(n, kadd). |
kadd |
k-additivity |
env |
Environment variable obtained from fm.Init(n). |
output |
The output are several random belief measures containing in an array v of length num * fm_a |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(5) belief <- fm.generate_belief(2, 3, env) # 2 3-additive measures with n=5 beliefenv<-fm.Init(5) belief <- fm.generate_belief(2, 3, env) # 2 3-additive measures with n=5 belief
Generates num random 2-additive fuzzy measures in Mobius representation.
fm.generate_fm_2additive(num, n)fm.generate_fm_2additive(num, n)
num |
Generates num random fuzzy measures stored consecutively in cardinality ordering in the array v. |
n |
Number of inputs |
output |
The output are random fuzzy measures, it contains singletons and pairs but no emptyset |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
num <- 2 n <- 5 fuzzymeasures <- fm.generate_fm_2additive(num,n) fuzzymeasures$V fuzzymeasures$lennum <- 2 n <- 5 fuzzymeasures <- fm.generate_fm_2additive(num,n) fuzzymeasures$V fuzzymeasures$len
Generates num 2-additive concave (supermodular) fuzzy measures for n inputs.
fm.generate_fm_2additive_concave(num, n)fm.generate_fm_2additive_concave(num, n)
num |
Generated num concave random fuzzy measures stored consecutively in cardinality ordering in the array v |
n |
Number of inputs |
output |
The output is the length of the part of the array v allocated for each fuzzy measure, and the array with singletons and pairs in Mobius representation |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
fuzzymeasures <- fm.generate_fm_2additive_concave(10,20)fuzzymeasures <- fm.generate_fm_2additive_concave(10,20)
Generates num 2-additive convex (supermodular) fuzzy measures for n inputs.
fm.generate_fm_2additive_convex(num, n)fm.generate_fm_2additive_convex(num, n)
num |
Generates num convex random fuzzy measures stored consecutively in cardinality ordering in the array v |
n |
Number of inputs |
output |
The output is the length of the part of the array v allocated for each fuzzy measure, and the array with singletons and pairs in Mobius representation |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
addconvex <- fm.generate_fm_2additive_convex(5,20)addconvex <- fm.generate_fm_2additive_convex(5,20)
Generates a random 2-additive supermodular fuzzy measure in sparse representation.
fm.generate_fm_2additive_convex_sparse(n, envsp = NULL)fm.generate_fm_2additive_convex_sparse(n, envsp = NULL)
n |
Number of inputs |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output are 2-additive supermodular fuzzy measure in sparse representation |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 5 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.generate_fm_2additive_convex_sparse(n, envsp) envsp envsp <- fm.FreeSparseFM(envsp)n <- 5 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.generate_fm_2additive_convex_sparse(n, envsp) envsp envsp <- fm.FreeSparseFM(envsp)
Generates num 2-additive convex (supermodular) fuzzy measures for n inputs. Some of the interaction indices are set to 0 (independence).
fm.generate_fm_2additive_convex_withsomeindependent(num, n)fm.generate_fm_2additive_convex_withsomeindependent(num, n)
num |
Generates num convex random fuzzy measures stored consecutively in cardinality ordering in the array |
n |
Number of inputs |
output |
The output is the length of the part of the array v allocated for each fuzzy measure, and the array with singletons and pairs in Mobius representation |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
addconvex <- fm.generate_fm_2additive_convex_withsomeindependent(5,20)addconvex <- fm.generate_fm_2additive_convex_withsomeindependent(5,20)
Generate a random 2-additive fuzzy measures in Mobius representation by using random walk.
fm.generate_fm_2additive_randomwalk2(num, n, markov, option, step, Fn)fm.generate_fm_2additive_randomwalk2(num, n, markov, option, step, Fn)
num |
Generated num random fuzzy measures stored consecutively in cardinality ordering in the array v. |
n |
Number of inputs. |
markov |
Number of Markov steps to take, the randomness increases with that number. |
option |
Not used, reserved for future use. |
step |
The maximum size of random steps (with respect to each value). The actual step is a random value up to Step. |
Fn |
The callback function to verify any additional conditions on generated FM. Provided by the user or NULL. |
output |
The output are random 2-additive fuzzy measure, it contains singletons and pairs but no emptyset. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
num <- 10 n <- 5 fuzzymeasures <- fm.generate_fm_2additive_randomwalk2(num, n, 1000, 0, 0.001, NULL)num <- 10 n <- 5 fuzzymeasures <- fm.generate_fm_2additive_randomwalk2(num, n, 1000, 0, 0.001, NULL)
Generates a random k-additive Belief fuzzy measure in sparse representation
fm.generate_fm_kadditive_convex_sparse(n, kadd, nonzero, envsp = NULL)fm.generate_fm_kadditive_convex_sparse(n, kadd, nonzero, envsp = NULL)
n |
Inputs length. (n inputs) |
kadd |
kadd is the value of k-additivity, which is used for reducing the complexity of fuzzy measures. default value is kadd = n. 1 < kadd < n+1; if kdd=n - f.m. is unrestricted |
nonzero |
Values stored and indexed in the respective arrays which are part of the structure |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is k-additive Belief fuzzy measure in sparse representation |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 5 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.generate_fm_kadditive_convex_sparse(n,4,10, envsp) envsp envsp <- fm.FreeSparseFM(envsp)n <- 5 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.generate_fm_kadditive_convex_sparse(n,4,10, envsp) envsp envsp <- fm.FreeSparseFM(envsp)
Generates num k-interactive dual concave fuzzy measures in Mobius representation using random walk of length markov of stepsize step
fm.generate_fm_kinteractivedualconcave(num, n, kadd, markov, step, Fn, env)fm.generate_fm_kinteractivedualconcave(num, n, kadd, markov, step, Fn, env)
num |
Generated num random fuzzy measures stored consecutively in cardinality ordering in the array v. |
n |
Number of inputs. |
kadd |
kadd is the value of k-additivity, which is used for reducing the complexity of fuzzy measures. default value is kadd = n. 1 < kadd < n+1; if kdd=n - f.m. is unrestricted. |
markov |
Number of Markov steps to take, the randomness increases with that number. |
step |
The maximum size of random steps (with respect to each value). The actual step is a random value up to Step. |
Fn |
The callback function to verify any additional conditions on generated FM. Provided by the user. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output are k-interactive dual concave fuzzy measures in Mobius representation |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(4) step <- 0.001 Fn <- NULL fuzzymeasures <- fm.generate_fm_kinteractivedualconcave(10, 4, 2, 1000, step, Fn, env) fuzzymeasuresenv<-fm.Init(4) step <- 0.001 Fn <- NULL fuzzymeasures <- fm.generate_fm_kinteractivedualconcave(10, 4, 2, 1000, step, Fn, env) fuzzymeasures
Generates num k-interactive dual convex fuzzy measures in Mobius representation using random walk of length markov of stepsize step.
fm.generate_fm_kinteractivedualconvex(num, n, kadd, markov, step, Fn, env)fm.generate_fm_kinteractivedualconvex(num, n, kadd, markov, step, Fn, env)
num |
Generated num random fuzzy measures stored consecutively in cardinality ordering in the array v. |
n |
Number of inputs. |
kadd |
kadd is the value of k-additivity, which is used for reducing the complexity of fuzzy measures. default value is kadd = n. 1 < kadd < n+1; if kdd=n - f.m. is unrestricted. |
markov |
Number of Markov steps to take, the randomness increases with that number. |
step |
The maximum size of random steps (with respect to each value). The actual step is a random value up to Step. |
Fn |
The callback function to verify any additional conditions on generated FM. Provided by the user. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output are several k-interactive dual convex fuzzy measures in Mobius representation |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(4) step <- 0.0001 Fn <- NULL fuzzymeasures <- fm.generate_fm_kinteractivedualconvex(10, 4, 2, 1000, step, Fn, env) fuzzymeasures env<-fm.Free(env)env<-fm.Init(4) step <- 0.0001 Fn <- NULL fuzzymeasures <- fm.generate_fm_kinteractivedualconvex(10, 4, 2, 1000, step, Fn, env) fuzzymeasures env<-fm.Free(env)
Generate several random fuzzy measures (num is their number) stored in cardinality ordering in the array v using minimals_plus method.
fm.generate_fm_minplus(num, kint, markov, option, K, env = NULL)fm.generate_fm_minplus(num, kint, markov, option, K, env = NULL)
num |
Generated num random fuzzy measures stored consecutively in cardinality ordering in the array |
kint |
Interactive fuzzy measure. 0 < kint <= n |
markov |
Number of Markov steps to take, the randomness increases with that number |
option |
Option = 1 employs internal rejection method to improve uniformity, but for n > 5 is is not essential |
K |
K is the constant in k-interactive fuzzy measures |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is generate several random fuzzy measures |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fuzzymeasures <- fm.generate_fm_minplus(10,3,1000,0,0.7, env) fuzzymeasuresenv<-fm.Init(3) fuzzymeasures <- fm.generate_fm_minplus(10,3,1000,0,0.7, env) fuzzymeasures
Generates num k-additive fuzzy measures in the standard or Mobius representation using random walk of length markov of stepsize step.
fm.generate_fm_randomwalk(num, n, kadd, markov, option, step, Fn, env)fm.generate_fm_randomwalk(num, n, kadd, markov, option, step, Fn, env)
num |
Generated num random fuzzy measures stored consecutively in standard or cardinality ordering in the array v. |
n |
Number of inputs |
kadd |
kadd is the value of k-additivity, which is used for reducing the complexity of fuzzy measures. default value is kadd = n. 1 < kadd < n+1; if kdd=n - f.m. is unrestricted. The parameter kadd only matters for options 3 and 5 |
markov |
Number of Markov steps to take, the randomness increases with that number. |
option |
Option = 0 - normal, 1 convex (supermodular), 2 antibuoyant, 3 kadditive , 4 belief measure, 5 kadditive convex. The measure generated is in standard representation fo all options except 3,5. The parameter kadd only matters for options 3 and 5. In that case the measure is in more compact Mobius representation. |
step |
The maximum size of random steps (with respect to each value). The actual step is a random value up to Step. |
Fn |
The callback function to verify any additional conditions on generated FM. Provided by the user. if not NULL, is a callback function to perform additional check at every Markov step of the current set function, i.e., any extra conditions |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is named list with the first element v being the fuzzy measure and the second being the length of the array containing it |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
Fn <- function(n,v){ out <- 0.0 for(i in 1:n) out<- out+v[i]; if(out>1) { return(0) } else return(1) } env<-fm.Init(3) step <- 0.0010 Option <- 3 n <- 3 fuzzymeasures <- fm.generate_fm_randomwalk(2, 3, 2, 1000, Option, step, Fn, env) print(fuzzymeasures) print(fuzzymeasures$length)Fn <- function(n,v){ out <- 0.0 for(i in 1:n) out<- out+v[i]; if(out>1) { return(0) } else return(1) } env<-fm.Init(3) step <- 0.0010 Option <- 3 n <- 3 fuzzymeasures <- fm.generate_fm_randomwalk(2, 3, 2, 1000, Option, step, Fn, env) print(fuzzymeasures) print(fuzzymeasures$length)
Generate several random fuzzy measures in standard representation
fm.generate_fm_sorting(num, markov, option, env = NULL)fm.generate_fm_sorting(num, markov, option, env = NULL)
num |
Generates num random fuzzy measures stored in an array v of length num * 2n. |
markov |
Number of Markov steps to take, the randomness increases with that number. |
option |
Option = 1 employs internal rejection method to improve uniformity, but for n > 5 is not essential. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output are several random fuzzy measures containing in an array v of length num * 2n |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) markovsteps <- 100 fuzzymeasures <- fm.generate_fm_sorting(5, markovsteps, 0, env) fuzzymeasuresenv<-fm.Init(3) markovsteps <- 100 fuzzymeasures <- fm.generate_fm_sorting(5, markovsteps, 0, env) fuzzymeasures
Generate several random fuzzy measures (num is their number) stored in cardinality ordering in the array v using topological sort.
fm.generate_fm_tsort(num, kint, markov, option, K, env = NULL)fm.generate_fm_tsort(num, kint, markov, option, K, env = NULL)
num |
Generated num random fuzzy measures stored consecutively in cardinality ordering in the array |
kint |
Interactive fuzzy measure. 0 < kint <= n |
markov |
Number of Markov steps to take, the randomness increases with that number |
option |
Option = 1 employs internal rejection method to improve uniformity, but for n > 5 is is not essential |
K |
K is the constant in k-interactive fuzzy measures |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is generate several random fuzzy measures |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fuzzymeasures <- fm.generate_fm_tsort(10,3,1000,0,0.7, env) fuzzymeasuresenv<-fm.Init(3) fuzzymeasures <- fm.generate_fm_tsort(10,3,1000,0,0.7, env) fuzzymeasures
Generates num convex random fuzzy measures stored consecutively in cardinality ordering in the output array.
fm.generate_fmconvex_tsort(num, kint, markov, option, K, env = NULL)fm.generate_fmconvex_tsort(num, kint, markov, option, K, env = NULL)
num |
Several random fuzzy measures stored in cardinality ordering in the array v (num is their number) |
kint |
Interactive fuzzy measure. 0 < kint <= n |
markov |
Number of Markov steps to take, the randomness increases with that number |
option |
Option = 1 employs internal rejection method to improve uniformity, but for n > 5 is is not essential |
K |
K is the constant in k-interactive fuzzy measures |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is the generation of num convex random fuzzy measures stored consecutively in cardinality ordering in the array v |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fuzzymeasures <- fm.generate_fmconvex_tsort(1,3,1000,0,1, env)env<-fm.Init(3) fuzzymeasures <- fm.generate_fmconvex_tsort(1,3,1000,0,1, env)
Returns the number of tuples.
fm.get_num_tuples(envsp=NULL)fm.get_num_tuples(envsp=NULL)
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the number of tuples. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.get_num_tuples(envsp) envsp <-fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.get_num_tuples(envsp) envsp <-fm.FreeSparseFM(envsp)
Returns the length of the array of tuples.
fm.get_sizearray_tuples(envsp=NULL)fm.get_sizearray_tuples(envsp=NULL)
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the length of the array of tuples. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.get_sizearray_tuples(envsp) envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.get_sizearray_tuples(envsp) envsp <- fm.FreeSparseFM(envsp)
This function initialises the internal structures which makes computations faster. The structures are saved in the output environment variable, which should be subsequently passed to other functions. Several environment variables (for different dimensions) can be initialised at the same time.
fm.Init(n1)fm.Init(n1)
n1 |
The number of variables. |
output |
The ouput is the enviromnet variable containing the internal structures. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3)env<-fm.Init(3)
Calculates all the interaction indices of input criteria for standard fuzzy measure.
fm.Interaction(v,env)fm.Interaction(v,env)
v |
Fuzzy measure value in standard representation |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is a matrix, whose first column stores the interaction index values, and the second column stores the indices of criteria in coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.Interaction(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.Interaction(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Calculates all the Banzhaf Interaction indices of input criteria for a standard fuzzy measure.
fm.InteractionB(v,env)fm.InteractionB(v,env)
v |
Fuzzy measure value in standard representation |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is a matrix, whose first column stores the Banzhaf Interaction index values, and the second column stores the indices of criteria in coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.InteractionB(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.InteractionB(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Calculates all the Banzhaf InteractionB indices of input criteria for a Mobius fuzzy measure.
fm.InteractionBMob(Mob,env)fm.InteractionBMob(Mob,env)
Mob |
Fuzzy measure value in Mobius representation |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is a matrix, whose first column stores the Banzhaf Interaction index values, and the second column stores the indices of criteria in coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.InteractionBMob(c( 0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.InteractionBMob(c( 0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Calculates all the interaction indices of input criteria for a Mobius fuzzy measure.
fm.InteractionMob(Mob,env )fm.InteractionMob(Mob,env )
Mob |
Fuzzy measure value in Mobius representation |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is a matrix, whose first column stores the interaction index values, and the second column stores the indices of criteria in coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.InteractionMob(c( 0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.InteractionMob(c( 0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Checks if element i (1-based) belongs to the tuple indexed A (whose cardinality can be 1,2, other (automatically determined)).
fm.is_inset_sparse(A, card, i, envsp=NULL)fm.is_inset_sparse(A, card, i, envsp=NULL)
A |
Tuple indexed. |
card |
Whose cardinality can be 1,2, other (automatically determined) |
i |
Element (1-based) |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is a logical value. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.is_inset_sparse(0,3,1,envsp) fm.is_inset_sparse(0,3,4,envsp) envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.is_inset_sparse(0,3,1,envsp) fm.is_inset_sparse(0,3,4,envsp) envsp <- fm.FreeSparseFM(envsp)
Checks if tuple B is a subset of tuple A, The cardinalities of both tuples need to be supplied.
fm.is_subset_sparse(A, cardA, B, cardB, envsp = NULL)fm.is_subset_sparse(A, cardA, B, cardB, envsp = NULL)
A |
Tuple |
cardA |
Whose cardinality can be 1,2, other (automatically determined) |
B |
Tuple, tup=0 |
cardB |
Whose cardinality can be 1,2, other (automatically determined) |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is a logical value. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); envsp <- fm.add_pair_sparse(1,2,0.2,envsp); envsp <- fm.add_pair_sparse(1,3,0.3,envsp); fm.is_subset_sparse(0,3,0,2,envsp) #is 0th pair a subset of the 0th tuple? fm.is_subset_sparse(0,3,1,2,envsp) #is 1th pair a subset of the 0th tuple? envsp<-fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); envsp <- fm.add_pair_sparse(1,2,0.2,envsp); envsp <- fm.add_pair_sparse(1,3,0.3,envsp); fm.is_subset_sparse(0,3,0,2,envsp) #is 0th pair a subset of the 0th tuple? fm.is_subset_sparse(0,3,1,2,envsp) #is 1th pair a subset of the 0th tuple? envsp<-fm.FreeSparseFM(envsp)
Returns 1 if yes, 0 if no; v is a fuzzy measure in standard representation.
fm.IsMeasureAdditive(v,env)fm.IsMeasureAdditive(v,env)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureAdditive(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureAdditive(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in Mobius representation.
fm.IsMeasureAdditiveMob(Mob,env)fm.IsMeasureAdditiveMob(Mob,env)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureAdditiveMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureAdditiveMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in standard representation.
fm.IsMeasureBalanced(v,env)fm.IsMeasureBalanced(v,env)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureBalanced(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureBalanced(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns 1 if yes, 0 if no; Mob is a fuzzy measure in Mobius representation.
fm.IsMeasureBalancedMob(Mob,env)fm.IsMeasureBalancedMob(Mob,env)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from emperical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureBalancedMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureBalancedMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Returns k; v is a fuzzy measure in standard representation.
fm.IsMeasureKmaxitive(v,env=NULL)fm.IsMeasureKmaxitive(v,env=NULL)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is k. If k=n then not k-maxitive |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureKmaxitive(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureKmaxitive(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns k; mob is a fuzzy measure in Mobius representation.
fm.IsMeasureKmaxitiveMob(Mob,env=NULL)fm.IsMeasureKmaxitiveMob(Mob,env=NULL)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from emperical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is k. If k=n then not k-maxitive |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureKmaxitiveMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureKmaxitiveMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in standard representation.
fm.IsMeasureSelfdual(v,env)fm.IsMeasureSelfdual(v,env)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSelfdual(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureSelfdual(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns 1 if yes, 0 if no; Mob is a fuzzy measure in Mobius representation.
fm.IsMeasureSelfdualMob(Mob,env)fm.IsMeasureSelfdualMob(Mob,env)
Mob |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSelfdualMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureSelfdualMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in standard representation.
fm.IsMeasureSubadditive(v,env)fm.IsMeasureSubadditive(v,env)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSubadditive(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureSubadditive(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in Mobius representation.
fm.IsMeasureSubadditiveMob(Mob,env)fm.IsMeasureSubadditiveMob(Mob,env)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSubadditiveMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureSubadditiveMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in standard representation.
fm.IsMeasureSubmodular(v,env=NULL)fm.IsMeasureSubmodular(v,env=NULL)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSubmodular(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureSubmodular(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns 1 if yes, 0 if no; Mob is a fuzzy measure in Mobius representation.
fm.IsMeasureSubmodularMob(Mob,env=NULL)fm.IsMeasureSubmodularMob(Mob,env=NULL)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSubmodularMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureSubmodularMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in standard representation.
fm.IsMeasureSuperadditive(v,env=NULL)fm.IsMeasureSuperadditive(v,env=NULL)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSuperadditive(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureSuperadditive(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns 1 if yes, 0 if no; Mob is a fuzzy measure in Mobius representation.
fm.IsMeasureSuperadditiveMob(Mob,env=NULL)fm.IsMeasureSuperadditiveMob(Mob,env=NULL)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSuperadditiveMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureSuperadditiveMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in standard representation.
fm.IsMeasureSupermodular(v,env=NULL)fm.IsMeasureSupermodular(v,env=NULL)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSupermodular(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureSupermodular(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns 1 if yes, 0 if no; Mob is a fuzzy measure in Mobius representation.
fm.IsMeasureSupermodularMob(Mob,env=NULL)fm.IsMeasureSupermodularMob(Mob,env=NULL)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSupermodularMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureSupermodularMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in standard representation.
fm.IsMeasureSymmetric(v,env=NULL)fm.IsMeasureSymmetric(v,env=NULL)
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSymmetric(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.IsMeasureSymmetric(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Returns 1 if yes, 0 if no; v is a fuzzy measure in Mobius representation.
fm.IsMeasureSymmetricMob(Mob,env=NULL)fm.IsMeasureSymmetricMob(Mob,env=NULL)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is 1 if yes, 0 if no. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.IsMeasureSymmetricMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.IsMeasureSymmetricMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Calculates maximum of x with the indices belonging to tuple indexed as S
fm.max_subset_sparse(x, S, cardS, envsp=NULL)fm.max_subset_sparse(x, S, cardS, envsp=NULL)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
S |
Indices belonging to tuple indexed |
cardS |
Cardinality cardS |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the maximum of x with the indices belonging to tuple indexed as S |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.max_subset_sparse(c(0.1,0.05,0.2),0,3,envsp) envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.max_subset_sparse(c(0.1,0.05,0.2),0,3,envsp) envsp <- fm.FreeSparseFM(envsp)
Calculates minimum of x with the indices belonging to tuple indexed as S
fm.min_subset_sparse(x, S, cardS, envsp=NULL)fm.min_subset_sparse(x, S, cardS, envsp=NULL)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
S |
Indices belonging to tuple indexed |
cardS |
Cardinality cardS |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the minimum of x with the indices belonging to tuple indexed as S |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.min_subset_sparse(c(0.1,0.05,0.2),0,3,envsp) envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_tuple_sparse(c(1,2,3),0.4,envsp); fm.min_subset_sparse(c(0.1,0.05,0.2),0,3,envsp) envsp <- fm.FreeSparseFM(envsp)
Calculates Mobius representation of general fuzzy measure, the input and output is an array of size 2^n=m in binary ordering.
fm.Mobius(v,env=NULL)fm.Mobius(v,env=NULL)
v |
Fuzzy measure value in standard representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is the fuzzy measure in Mobius representation. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.Mobius(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.Mobius(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Calculate the nonadditivity indices of input criteria from general fuzzy measure.
fm.NonadditivityIndex(v,env=NULL)fm.NonadditivityIndex(v,env=NULL)
v |
Fuzzy measure in general representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size 2^n, which contain nonadditivity indices of input criteria coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.NonadditivityIndex(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.NonadditivityIndex(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Calculate the nonadditivity indices of input criteria from general fuzzy measure in Mobius representation.
fm.NonadditivityIndexMob(Mob,env=NULL)fm.NonadditivityIndexMob(Mob,env=NULL)
Mob |
Fuzzy measure in Mobius representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size 2^n, which contain nonadditivity indices of input criteria coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) NonadditivityIndex <- fm.NonadditivityIndexMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) NonadditivityIndex <- fm.NonadditivityIndexMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Calculate all the m = 2^n nonmodularity indices of fuzzy measure v given in standard representation
fm.NonmodularityIndex(v, env = NULL)fm.NonmodularityIndex(v, env = NULL)
v |
Fuzzy measure in general representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size m |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) Nonmodularityindex <- fm.NonmodularityIndex(c(0,0.3,0.5,0.6,0.4,0.8,0.7,1),env)env<-fm.Init(3) Nonmodularityindex <- fm.NonmodularityIndex(c(0,0.3,0.5,0.6,0.4,0.8,0.7,1),env)
Calculate all 2^n nonmodularity indices using Mobius transform of a fuzzy measure of lenght 2^n=m, using sparse representation
fm.NonmodularityIndex_sparse( n, envsp=NULL)fm.NonmodularityIndex_sparse( n, envsp=NULL)
n |
Number of inputs |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is all 2^n nonmodularity indice. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); fm.NonmodularityIndex_sparse(3,envsp) envsp <- fm.FreeSparseFM(envsp)n <- 3 tups<-vector() tupsidx<-vector() envsp <- fm.PrepareSparseFM(n, tups,tupsidx) envsp <- fm.add_singletons_sparse(c(0.2,0.1,0.2),envsp) envsp <- fm.add_pair_sparse(1,2,0.4,envsp); fm.NonmodularityIndex_sparse(3,envsp) envsp <- fm.FreeSparseFM(envsp)
Calculate all the m = 2^n nonmodularity indices of k-interactive fuzzy measure v given in standard representation (in cardinality ordering)
fm.NonmodularityIndexKinteractive(v, env = NULL, kadd = "NA")fm.NonmodularityIndexKinteractive(v, env = NULL, kadd = "NA")
v |
Fuzzy measure in general representation. |
env |
Environment variable obtained from fm.Init(n). |
kadd |
kadd is the value of k-additivity, which is used for reducing the complexity of fuzzy measures. default value is kadd = n. 1 < kadd < n+1; if kdd=n - f.m. is unrestricted |
output |
The output is an array of size m. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.NonmodularityIndexKinteractive(c(0,0.3,0.5,0.6,0.4,0.8,0.7,1),env,2)env<-fm.Init(3) fm.NonmodularityIndexKinteractive(c(0,0.3,0.5,0.6,0.4,0.8,0.7,1),env,2)
Calculates all the nonmodularity indices of fuzzy measure in Mobius representation representation
fm.NonmodularityIndexMob(Mob, env = NULL)fm.NonmodularityIndexMob(Mob, env = NULL)
Mob |
Fuzzy measure in Mobius representation of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size m |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.NonmodularityIndexMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.NonmodularityIndexMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Calculate all the m = 2^n nonmodularity indices of k-additive in Mobius representation(in cardinality ordering)
fm.NonmodularityIndexMobkadditive(Mob, env = NULL, kadd = "NA")fm.NonmodularityIndexMobkadditive(Mob, env = NULL, kadd = "NA")
Mob |
Fuzzy measure in Mobius representation of size m=2^n. Its values can be provided by users, or by estimating from empirical data |
env |
Environment variable obtained from fm.Init(n). |
kadd |
kadd is the value of k-additivity, which is used for reducing the complexity of fuzzy measures. default value is kadd = n. 1 < kadd < n+1; if kdd=n - f.m. is unrestricted. |
output |
The output is an array of size m. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.NonmodularityIndexMobkadditive(c(0.0,0.3,0.5,-0.2,0.4,0.1,-0.2,0.1),env,2)env<-fm.Init(3) fm.NonmodularityIndexMobkadditive(c(0.0,0.3,0.5,-0.2,0.4,0.1,-0.2,0.1),env,2)
Calculate Orness value of the Choquet integral of the fuzzy measure, where v is a standard representation.
fm.OrnessChoquet(v,env=NULL)fm.OrnessChoquet(v,env=NULL)
v |
Standard fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is the Orness the Choquet integral for the fuzzy measure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.OrnessChoquet(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.OrnessChoquet(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Calculate Orness value of the Choquet integral of the fuzzy measure, where Mob is the Mobius representation.
fm.OrnessChoquetMob(Mob,env=NULL)fm.OrnessChoquetMob(Mob,env=NULL)
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is the Orness the Choquet integral for the fuzzy measure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.OrnessChoquetMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.OrnessChoquetMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Populate 2-additive sparse capacity with nonzero values using the singletons and two arrays of indices (of size numpairs).
fm.populate_fm_2add_sparse(singletons, numpairs, pairs, indicesp1, indicesp2, envsp)fm.populate_fm_2add_sparse(singletons, numpairs, pairs, indicesp1, indicesp2, envsp)
singletons |
Singletons 0-based. |
numpairs |
Size numpairs. |
pairs |
Array 0-based. |
indicesp1 |
Array of indices of Size numpairs.need to be 1-based. |
indicesp2 |
Array of indices of Size numpairs.need to be 1-based. |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is Populate 2-additive sparse capacity with nonzero values using the singletons and two arrays of indices (of size numpairs) |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.populate_fm_2add_sparse(c(0.1,0.2,0.3), 3, c(0.4,0.5,0.6), c(1,1,2), c(2,3,3), envsp) envsp envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.populate_fm_2add_sparse(c(0.1,0.2,0.3), 3, c(0.4,0.5,0.6), c(1,1,2), c(2,3,3), envsp) envsp envsp <- fm.FreeSparseFM(envsp)
Given 2-additive capacity singletons+pairs in one array v , selects nonzero pairs and populates sparse capacity envsp
fm.populate_fm_2add_sparse_from2add(n, v, envsp=NULL)fm.populate_fm_2add_sparse_from2add(n, v, envsp=NULL)
n |
Number of inputs |
v |
Pairs in one array v |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is a nonzero pairs and populates sparse capacity envsp |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.populate_fm_2add_sparse_from2add(3,c(0.4,0.5,0.6, 0, 0, 0.1),envsp) envsp envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.populate_fm_2add_sparse_from2add(3,c(0.4,0.5,0.6, 0, 0, 0.1),envsp) envsp envsp <- fm.FreeSparseFM(envsp)
This function initialises Sparse representation structure. It is used to allocate storage and later populate these values
fm.PrepareSparseFM(n, tups, tupsidx)fm.PrepareSparseFM(n, tups, tupsidx)
n |
Number of inputs |
tups |
Tuples to be added (can be null vector) |
tupsidx |
Cardinalities and indices (1-based) of the elements of tuples (can be null vector) |
output |
The output allocate storage and later populate these values. envsp |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n<-3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.FreeSparseFM(envsp) envsp <- fm.PrepareSparseFM(n, c(0.2,0.4,0.1), c(2,1,2,2,1,3,3,1,2,3)) envsp envsp <- fm.FreeSparseFM(envsp)n<-3 envsp <- fm.PrepareSparseFM(n, vector(), vector()) envsp <- fm.FreeSparseFM(envsp) envsp <- fm.PrepareSparseFM(n, c(0.2,0.4,0.1), c(2,1,2,2,1,3,3,1,2,3)) envsp envsp <- fm.FreeSparseFM(envsp)
Calculates the Shapley values of input criteria from general fuzzy measure,
fm.Shapley(v,env=NULL)fm.Shapley(v,env=NULL)
v |
Fuzzy measure in general representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size n, which contain Shapley values of input criteria. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.Shapley(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.Shapley(c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Calculate the Shapley values of a 2-additive fuzzy measure for n inputs given in Mobius representation. The results are in arrays.
fm.Shapley2addMob(n, Mob)fm.Shapley2addMob(n, Mob)
n |
Number of inputs |
Mob |
Fuzzy measure value in Mobius representation |
output |
The output is an array of size n, which contain Shapley indices of input criteria. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
Shapley <- fm.Shapley2addMob(3, c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1))Shapley <- fm.Shapley2addMob(3, c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1))
Calculate the Shapley indices of input criteria from general fuzzy measure in Mobius representation.
fm.ShapleyMob(Mob,env=NULL)fm.ShapleyMob(Mob,env=NULL)
Mob |
Fuzzy measure in Mobius representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is an array of size n, which contain Shapley values of input criteria. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.ShapleyMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.ShapleyMob(c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
Calculate Shapley values vectors of size n of a sparse fuzzy measure
fm.ShapleyMob_sparse(n, envsp=NULL)fm.ShapleyMob_sparse(n, envsp=NULL)
n |
Size of values vectors |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is Shapley values vectors of size n of a sparse fuzzy measure. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, c(0.2,0.4,0.1), c(2,1,2,2,1,3,3,1,2,3)) fm.ShapleyMob_sparse(3, envsp) envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, c(0.2,0.4,0.1), c(2,1,2,2,1,3,3,1,2,3)) fm.ShapleyMob_sparse(3, envsp) envsp <- fm.FreeSparseFM(envsp)
Return the decimal expression for the subsets A. In binary and in cardinality ordering respectively.
fm.ShowCoalitions(env = NULL)fm.ShowCoalitions(env = NULL)
env |
Environment variable obtained from fm.Init(n). |
output |
is the array of integers which show the decimal expressions for all 2^n coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) ShowCoalitions <- fm.ShowCoalitions(env) ShowCoalitionsenv<-fm.Init(3) ShowCoalitions <- fm.ShowCoalitions(env) ShowCoalitions
Return the decimal expression for the subsets A. In binary and in cardinality ordering respectively.
fm.ShowCoalitionsCard(env = NULL)fm.ShowCoalitionsCard(env = NULL)
env |
Environment variable obtained from fm.Init(n). |
output |
The output the decimal expression for the subsets A. It is the array of integers containing the decimal expressions for all 2^n coalitions. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) CoalitionsCard <- fm.ShowCoalitionsCard(env) CoalitionsCardenv<-fm.Init(3) CoalitionsCard <- fm.ShowCoalitionsCard(env) CoalitionsCard
Export the internal arrays of the sparse capacity as arrays of singletons, pairs and tuples.
fm.sparse_get_pairs( envsp=NULL)fm.sparse_get_pairs( envsp=NULL)
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the array of pairs and their number. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n) envsp <-fm.add_pair_sparse(1,2, 0.4, envsp) envsp <-fm.add_pair_sparse(1,3, 0.3, envsp) pairs <- fm.sparse_get_pairs(envsp) pairs envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n) envsp <-fm.add_pair_sparse(1,2, 0.4, envsp) envsp <-fm.add_pair_sparse(1,3, 0.3, envsp) pairs <- fm.sparse_get_pairs(envsp) pairs envsp <- fm.FreeSparseFM(envsp)
Export the internal arrays of the sparse capacity as arrays of singletons, pairs and tuples.
fm.sparse_get_singletons(envsp=NULL)fm.sparse_get_singletons(envsp=NULL)
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the numbers of pairs and tuples. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n) envsp <- fm.add_singletons_sparse(c(0, 0.3, 0.5),envsp) singletons <- fm.sparse_get_singletons(envsp) singletons envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n) envsp <- fm.add_singletons_sparse(c(0, 0.3, 0.5),envsp) singletons <- fm.sparse_get_singletons(envsp) singletons envsp <- fm.FreeSparseFM(envsp)
Export the internal arrays of the sparse capacity as arrays of singletons, pairs and tuples.
fm.sparse_get_tuples(envsp=NULL)fm.sparse_get_tuples(envsp=NULL)
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the numbers of pairs and tuples. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n) envsp <- fm.add_tuple_sparse(c(1,2,3),0.2,envsp) envsp <- fm.add_tuple_sparse(c(1,3,4),0.3,envsp) tuples <- fm.sparse_get_tuples(envsp) tuples envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n) envsp <- fm.add_tuple_sparse(c(1,2,3),0.2,envsp) envsp <- fm.add_tuple_sparse(c(1,3,4),0.3,envsp) tuples <- fm.sparse_get_tuples(envsp) tuples envsp <- fm.FreeSparseFM(envsp)
Calculate the value of a Sugeno integral of input x, with fuzzy measure in standard representation
fm.Sugeno(x, v,env=NULL)fm.Sugeno(x, v,env=NULL)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
v |
General fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is a single value of the computed Sugeno integral. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.Sugeno(c(0.6, 0.3, 0.8), c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)env<-fm.Init(3) fm.Sugeno(c(0.6, 0.3, 0.8), c(0, 0.3, 0.5, 0.6, 0.4, 0.8, 0.7, 1),env)
Calculate the value of a Sugeno integral of input x, with fuzzy measure in mobius representation
fm.SugenoMob(x, Mob,env=NULL)fm.SugenoMob(x, Mob,env=NULL)
x |
Input vector of size n, containing utility value of input criteria. x is in [0,1]. |
Mob |
Mobius fuzzy measure of size m=2^n. Its values can be provided by users, or by estimating from empirical data. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is a single value of the computed Sugeno integral. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.SugenoMob(c(0.6, 0.3, 0.8), c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)env<-fm.Init(3) fm.SugenoMob(c(0.6, 0.3, 0.8), c(0.0, 0.3, 0.5, -0.2, 0.4, 0.1, -0.2, 0.1),env)
This function provide some examples of how fuzzy measure operation in this toolbox are used. It can be used to test if the toolbox has been installed successfully or not.
fm.test()fm.test()
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
fm.test()fm.test()
Returns the cardinality of the tuple numbered i in the list of tuples.
fm.tuple_cardinality_sparse(i, envsp = NULL)fm.tuple_cardinality_sparse(i, envsp = NULL)
i |
In the list of tuples. |
envsp |
Structure required for sparse representation which stores the relevant values (k-tuples). It is obtained from fm.PrepareSparseFM(n). |
output |
The output is the cardinality of the tuple numbered i in the list of tuple. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
n <- 3 envsp <- fm.PrepareSparseFM(n, c(0.2,0.4,0.1), c(2,1,2,2,1,3,3,1,2,3)) fm.tuple_cardinality_sparse(0,envsp) envsp <- fm.FreeSparseFM(envsp)n <- 3 envsp <- fm.PrepareSparseFM(n, c(0.2,0.4,0.1), c(2,1,2,2,1,3,3,1,2,3)) fm.tuple_cardinality_sparse(0,envsp) envsp <- fm.FreeSparseFM(envsp)
Calculate the general fuzzy measure from Mobius representation. The input and output is an array of size 2^n=m in binary ordering. This is the inverse of the Mobius function.
fm.Zeta(Mob,env)fm.Zeta(Mob,env)
Mob |
Fuzzy measure value in Mobius representation. |
env |
Environment variable obtained from fm.Init(n). |
output |
The output is the fuzzy measure in general representation. |
Gleb Beliakov, Andrei Kelarev, Quan Vu, Daniela L. Calderon, Deakin University
env<-fm.Init(3) fm.Zeta(c(0.0,0.3,0.5,-0.2,0.4,0.1,-0.2,0.1),env)env<-fm.Init(3) fm.Zeta(c(0.0,0.3,0.5,-0.2,0.4,0.1,-0.2,0.1),env)