R : Copyright 2005, The R Foundation for Statistical Computing
Version 2.1.1  (2005-06-20), ISBN 3-900051-07-0

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> ### * <HEADER>
> ###
> attach(NULL, name = "CheckExEnv")
> assign(".CheckExEnv", as.environment(2), pos = length(search())) # base
> ## add some hooks to label plot pages for base and grid graphics
> setHook("plot.new", ".newplot.hook")
> setHook("persp", ".newplot.hook")
> setHook("grid.newpage", ".gridplot.hook")
> 
> assign("cleanEx",
+        function(env = .GlobalEnv) {
+ 	   rm(list = ls(envir = env, all.names = TRUE), envir = env)
+            RNGkind("default", "default")
+ 	   set.seed(1)
+    	   options(warn = 1)
+ 	   delayedAssign("T", stop("T used instead of TRUE"),
+ 		  assign.env = .CheckExEnv)
+ 	   delayedAssign("F", stop("F used instead of FALSE"),
+ 		  assign.env = .CheckExEnv)
+ 	   sch <- search()
+ 	   newitems <- sch[! sch %in% .oldSearch]
+ 	   for(item in rev(newitems))
+                eval(substitute(detach(item), list(item=item)))
+ 	   missitems <- .oldSearch[! .oldSearch %in% sch]
+ 	   if(length(missitems))
+ 	       warning("items ", paste(missitems, collapse=", "),
+ 		       " have been removed from the search path")
+        },
+        env = .CheckExEnv)
> assign("..nameEx", "__{must remake R-ex/*.R}__", env = .CheckExEnv) # for now
> assign("ptime", proc.time(), env = .CheckExEnv)
> grDevices::postscript("kappalab-Examples.ps")
> assign("par.postscript", graphics::par(no.readonly = TRUE), env = .CheckExEnv)
> options(contrasts = c(unordered = "contr.treatment", ordered = "contr.poly"))
> options(warn = 1)    
> library('kappalab')
Loading required package: quadprog
Loading required package: kernlab
> 
> assign(".oldSearch", search(), env = .CheckExEnv)
> assign(".oldNS", loadedNamespaces(), env = .CheckExEnv)
> cleanEx(); ..nameEx <- "Choquet.integral-methods"
> 
> ### * Choquet.integral-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Choquet.integral-methods
> ### Title: Choquet integral
> ### Aliases: Choquet.integral Choquet.integral-methods
> ###   Choquet.integral,Mobius.game,numeric-method
> ###   Choquet.integral,card.game,numeric-method
> ###   Choquet.integral,game,numeric-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a normalized capacity
> mu <- capacity(c(0:13/13,1,1))
> 
> ## and its Möbius transform
> a <- Mobius(mu)
> 
> ## a discrete positive function f
> f <- c(0.1,0.9,0.3,0.8)
> 
> ## the Choquet integral of f w.r.t mu
> Choquet.integral(mu,f)
[1] 0.6615385
> Choquet.integral(a,f)
[1] 0.6615385
> 
> ## a similar example with a cardinal capacity
> mu <- uniform.capacity(4)
> Choquet.integral(mu,f)
[1] 0.525
> 
> 
> 
> cleanEx(); ..nameEx <- "Mobius-methods"
> 
> ### * Mobius-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Mobius-methods
> ### Title: The Möbius transform
> ### Aliases: Mobius Mobius-methods Mobius,capacity-method
> ###   Mobius,card.set.func-method Mobius,game-method Mobius,set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(0:15)
> mu
		capacity
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		5.000000
{1,3}		6.000000
{1,4}		7.000000
{2,3}		8.000000
{2,4}		9.000000
{3,4}		10.000000
{1,2,3}		11.000000
{1,2,4}		12.000000
{1,3,4}		13.000000
{2,3,4}		14.000000
{1,2,3,4}		15.000000
> 
> ## its Möbius transform
> a <- Mobius(mu)
> a
		Mobius.capacity
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		2.000000
{1,3}		2.000000
{1,4}		2.000000
{2,3}		3.000000
{2,4}		3.000000
{3,4}		3.000000
{1,2,3}		-2.000000
{1,2,4}		-2.000000
{1,3,4}		-2.000000
{2,3,4}		-4.000000
{1,2,3,4}		0.000000
> 
> ## its zeta transform
> zeta(a)
		capacity
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		5.000000
{1,3}		6.000000
{1,4}		7.000000
{2,3}		8.000000
{2,4}		9.000000
{3,4}		10.000000
{1,2,3}		11.000000
{1,2,4}		12.000000
{1,3,4}		13.000000
{2,3,4}		14.000000
{1,2,3,4}		15.000000
> 
> ## a similar example with a game object
> mu <- game(c(0,-2:12))
> mu
		game
{}		0.000000
{1}		-2.000000
{2}		-1.000000
{3}		0.000000
{4}		1.000000
{1,2}		2.000000
{1,3}		3.000000
{1,4}		4.000000
{2,3}		5.000000
{2,4}		6.000000
{3,4}		7.000000
{1,2,3}		8.000000
{1,2,4}		9.000000
{1,3,4}		10.000000
{2,3,4}		11.000000
{1,2,3,4}		12.000000
> Mobius(mu)
		Mobius.game
{}		0.000000
{1}		-2.000000
{2}		-1.000000
{3}		0.000000
{4}		1.000000
{1,2}		5.000000
{1,3}		5.000000
{1,4}		5.000000
{2,3}		6.000000
{2,4}		6.000000
{3,4}		6.000000
{1,2,3}		-5.000000
{1,2,4}		-5.000000
{1,3,4}		-5.000000
{2,3,4}		-7.000000
{1,2,3,4}		3.000000
> zeta(Mobius(mu))
		game
{}		0.000000
{1}		-2.000000
{2}		-1.000000
{3}		0.000000
{4}		1.000000
{1,2}		2.000000
{1,3}		3.000000
{1,4}		4.000000
{2,3}		5.000000
{2,4}		6.000000
{3,4}		7.000000
{1,2,3}		8.000000
{1,2,4}		9.000000
{1,3,4}		10.000000
{2,3,4}		11.000000
{1,2,3,4}		12.000000
> 
> ## a similar example with a set.func object
> mu <- set.func(-7:8)
> mu
		set.func
{}		-7.000000
{1}		-6.000000
{2}		-5.000000
{3}		-4.000000
{4}		-3.000000
{1,2}		-2.000000
{1,3}		-1.000000
{1,4}		0.000000
{2,3}		1.000000
{2,4}		2.000000
{3,4}		3.000000
{1,2,3}		4.000000
{1,2,4}		5.000000
{1,3,4}		6.000000
{2,3,4}		7.000000
{1,2,3,4}		8.000000
> Mobius(mu)
		Mobius.set.func
{}		-7.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		2.000000
{1,3}		2.000000
{1,4}		2.000000
{2,3}		3.000000
{2,4}		3.000000
{3,4}		3.000000
{1,2,3}		-2.000000
{1,2,4}		-2.000000
{1,3,4}		-2.000000
{2,3,4}		-4.000000
{1,2,3,4}		0.000000
> zeta(Mobius(mu))
		set.func
{}		-7.000000
{1}		-6.000000
{2}		-5.000000
{3}		-4.000000
{4}		-3.000000
{1,2}		-2.000000
{1,3}		-1.000000
{1,4}		0.000000
{2,3}		1.000000
{2,4}		2.000000
{3,4}		3.000000
{1,2,3}		4.000000
{1,2,4}		5.000000
{1,3,4}		6.000000
{2,3,4}		7.000000
{1,2,3,4}		8.000000
> 
> ## a similar example with a card.set.func object
> mu <- card.set.func(-3:4)
> mu
  card.set.func
0            -3
1            -2
2            -1
3             0
4             1
5             2
6             3
7             4
> Mobius(mu)
  Mobius.card.set.func
0                   -3
1                    1
2                    0
3                    0
4                    0
5                    0
6                    0
7                    0
> zeta(Mobius(mu))
  card.set.func
0            -3
1            -2
2            -1
3             0
4             1
5             2
6             3
7             4
> 
> 
> 
> cleanEx(); ..nameEx <- "Mobius.capacity-class"
> 
> ### * Mobius.capacity-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Mobius.capacity-class
> ### Title: Class "Mobius.capacity"
> ### Aliases: Mobius.capacity-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(c(0,0,0:13))
> ## and its Möbius representation
> a <- Mobius(mu)
> a
		Mobius.capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		1.000000
{4}		2.000000
{1,2}		3.000000
{1,3}		3.000000
{1,4}		3.000000
{2,3}		5.000000
{2,4}		5.000000
{3,4}		5.000000
{1,2,3}		-3.000000
{1,2,4}		-3.000000
{1,3,4}		-3.000000
{2,3,4}		-6.000000
{1,2,3,4}		1.000000
> 
> # the attributes of object a
> a@n
[1] 4
> a@k
[1] 4
> a@data
 [1]  0  0  0  1  2  3  3  3  5  5  5 -3 -3 -3 -6  1
> a@subsets
 [1]  0  1  2  4  8  3  5  9  6 10 12  7 11 13 14 15
> 
> ## a test
> is.normalized(a)
[1] FALSE
> ## normalize it
> normalize(a)
		Mobius.capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		0.076923
{4}		0.153846
{1,2}		0.230769
{1,3}		0.230769
{1,4}		0.230769
{2,3}		0.384615
{2,4}		0.384615
{3,4}		0.384615
{1,2,3}		-0.230769
{1,2,4}		-0.230769
{1,3,4}		-0.230769
{2,3,4}		-0.461538
{1,2,3,4}		0.076923
> 
> ## a transformation
> zeta(a)
		capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		1.000000
{4}		2.000000
{1,2}		3.000000
{1,3}		4.000000
{1,4}		5.000000
{2,3}		6.000000
{2,4}		7.000000
{3,4}		8.000000
{1,2,3}		9.000000
{1,2,4}		10.000000
{1,3,4}		11.000000
{2,3,4}		12.000000
{1,2,3,4}		13.000000
> ## Let us check ...
> Mobius(zeta(a))
		Mobius.capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		1.000000
{4}		2.000000
{1,2}		3.000000
{1,3}		3.000000
{1,4}		3.000000
{2,3}		5.000000
{2,4}		5.000000
{3,4}		5.000000
{1,2,3}		-3.000000
{1,2,4}		-3.000000
{1,3,4}		-3.000000
{2,3,4}		-6.000000
{1,2,3,4}		1.000000
> 
> ## some summary indices
> orness(a)
[1] 0.4294872
> veto(a)
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 
> favor(a)
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 
> variance(a)
[1] 0.1439842
> entropy(a)
[1] 0.8073944
> ## the same
> summary(a)
Shapley value :
   1    2    3    4 
1.75 2.75 3.75 4.75 

Shapley interaction indices :
          1         2         3         4
1        NA 0.3333333 0.3333333 0.3333333
2 0.3333333        NA 0.8333333 0.8333333
3 0.3333333 0.8333333        NA 0.8333333
4 0.3333333 0.8333333 0.8333333        NA

Orness :
0.4294872

Veto indices :
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 

Favor indices :
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 

Normalized variance :
0.1439842

Normalized entropy :
0.8073944
> 
> 
> 
> cleanEx(); ..nameEx <- "Mobius.card.gen"
> 
> ### * Mobius.card.gen
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Mobius.card.set.func
> ### Title: Creates an object representing the Möbius transform of a
> ###   cardinal set function.
> ### Aliases: Mobius.card.set.func
> ### Keywords: math
> 
> ### ** Examples
> 
> Mobius.card.set.func(4:-2)
  Mobius.card.set.func
0                    4
1                    3
2                    2
3                    1
4                    0
5                   -1
6                   -2
> 
> 
> 
> cleanEx(); ..nameEx <- "Mobius.card.set.func-class"
> 
> ### * Mobius.card.set.func-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Mobius.card.set.func-class
> ### Title: Class "Mobius.card.set.func"
> ### Aliases: Mobius.card.set.func-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## the Möbius representation of a cardinal set function
> a <- Mobius.card.set.func(-3:2)
> 
> # the attributes of the object
> a@n
[1] 5
> a@data
[1] -3 -2 -1  0  1  2
> 
> ## some transformations
> as.set.func(a)
		set.func
{}		-3.000000
{1}		-2.000000
{2}		-2.000000
{3}		-2.000000
{4}		-2.000000
{5}		-2.000000
{1,2}		-1.000000
{1,3}		-1.000000
{1,4}		-1.000000
{1,5}		-1.000000
{2,3}		-1.000000
{2,4}		-1.000000
{2,5}		-1.000000
{3,4}		-1.000000
{3,5}		-1.000000
{4,5}		-1.000000
{1,2,3}		0.000000
{1,2,4}		0.000000
{1,2,5}		0.000000
{1,3,4}		0.000000
{1,3,5}		0.000000
{1,4,5}		0.000000
{2,3,4}		0.000000
{2,3,5}		0.000000
{2,4,5}		0.000000
{3,4,5}		0.000000
{1,2,3,4}		1.000000
{1,2,3,5}		1.000000
{1,2,4,5}		1.000000
{1,3,4,5}		1.000000
{2,3,4,5}		1.000000
{1,2,3,4,5}		2.000000
> zeta(a)
  card.set.func
0            -3
1            -5
2            -8
3           -12
4           -16
5           -16
> ## let us check ...
> Mobius(zeta(a))
  Mobius.card.set.func
0                   -3
1                   -2
2                   -1
3                    0
4                    1
5                    2
> 
> 
> 
> cleanEx(); ..nameEx <- "Mobius.game-class"
> 
> ### * Mobius.game-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Mobius.game-class
> ### Title: Class "Mobius.game"
> ### Aliases: Mobius.game-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a game (which is a capacity)
> mu <- game(c(0,rep(1,15)))
> ## and its Möbius representation
> a <- Mobius(mu)
> 
> # the attributes of object a
> a@n
[1] 4
> a@k
[1] 4
> a@data
 [1]  0  1  1  1  1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1
> a@subsets
 [1]  0  1  2  4  8  3  5  9  6 10 12  7 11 13 14 15
> 
> ## a transformation
> zeta(a)
		game
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{4}		1.000000
{1,2}		1.000000
{1,3}		1.000000
{1,4}		1.000000
{2,3}		1.000000
{2,4}		1.000000
{3,4}		1.000000
{1,2,3}		1.000000
{1,2,4}		1.000000
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		1.000000
> ## let us check ...
> Mobius(zeta(a))
		Mobius.game
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{4}		1.000000
{1,2}		-1.000000
{1,3}		-1.000000
{1,4}		-1.000000
{2,3}		-1.000000
{2,4}		-1.000000
{3,4}		-1.000000
{1,2,3}		1.000000
{1,2,4}		1.000000
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		-1.000000
> 
> ## integral calculations 
> f <- c(0.2,0.3,0.1,0.7)
> Choquet.integral(a,f)
[1] 0.7
> Sugeno.integral(a,f)
[1] 0.7
> f <- c(0.2,-0.3,0.1,-0.7)
> Sipos.integral(a,f)
[1] -0.5
> 
> 
> 
> cleanEx(); ..nameEx <- "Mobius.gen"
> 
> ### * Mobius.gen
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Mobius.set.func
> ### Title: Create objects representing the Möbius transform of a set
> ###   function.
> ### Aliases: Mobius.set.func Mobius.game Mobius.capacity additive.capacity
> ### Keywords: math
> 
> ### ** Examples
> 
> Mobius.set.func(c(0,1,1,2,1,2,1,2,2,3,2),4,2)
		Mobius.set.func
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		2.000000
{4}		1.000000
{1,2}		2.000000
{1,3}		1.000000
{1,4}		2.000000
{2,3}		2.000000
{2,4}		3.000000
{3,4}		2.000000
> Mobius.game(c(0,1,1,2,1,2,1,2,2,3,2),4,2)
		Mobius.game
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		2.000000
{4}		1.000000
{1,2}		2.000000
{1,3}		1.000000
{1,4}		2.000000
{2,3}		2.000000
{2,4}		3.000000
{3,4}		2.000000
> Mobius.capacity(c(0,1,1,2,1,2,1,2,2,3,2),4,2)
		Mobius.capacity
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		2.000000
{4}		1.000000
{1,2}		2.000000
{1,3}		1.000000
{1,4}		2.000000
{2,3}		2.000000
{2,4}		3.000000
{3,4}		2.000000
> additive.capacity(c(1,2,3,4))
		Mobius.capacity
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
> 
> 
> 
> cleanEx(); ..nameEx <- "Mobius.set.func-class"
> 
> ### * Mobius.set.func-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Mobius.set.func-class
> ### Title: Class "Mobius.set.func"
> ### Aliases: Mobius.set.func-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## the Möbius transform of a set function directly
> a <- Mobius.set.func(1:16,4,4)
> 
> ## the attributes of the object
> a@n
[1] 4
> a@k
[1] 4
> a@data
 [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
> a@subsets
 [1]  0  1  2  4  8  3  5  9  6 10 12  7 11 13 14 15
> 
> ## a set function
> mu <- set.func(7:-8)
> ## and its Möbius transform
> a <- Mobius(mu)
> 
> ## some conversions that cannot work
> ## as.game(a)
> ## as.capacity(a)
> ## as.card.set.func(a)
> 
> ## some tests
> is.cardinal(a)
[1] FALSE
> is.kadditive(a,2)
[1] FALSE
> is.monotone(a)
[1] FALSE
> 
> ## some transformations
> zeta(a)
		set.func
{}		7.000000
{1}		6.000000
{2}		5.000000
{3}		4.000000
{4}		3.000000
{1,2}		2.000000
{1,3}		1.000000
{1,4}		0.000000
{2,3}		-1.000000
{2,4}		-2.000000
{3,4}		-3.000000
{1,2,3}		-4.000000
{1,2,4}		-5.000000
{1,3,4}		-6.000000
{2,3,4}		-7.000000
{1,2,3,4}		-8.000000
> k.truncate.Mobius(a,2)
		Mobius.set.func
{}		7.000000
{1}		-1.000000
{2}		-2.000000
{3}		-3.000000
{4}		-4.000000
{1,2}		-2.000000
{1,3}		-2.000000
{1,4}		-2.000000
{2,3}		-3.000000
{2,4}		-3.000000
{3,4}		-3.000000
> 
> ## summary 
> Shapley.value(a)
        1         2         3         4 
-2.000000 -3.333333 -4.333333 -5.333333 
> interaction.indices(a)
   1  2  3  4
1 NA  0  0  0
2  0 NA  0  0
3  0  0 NA  0
4  0  0  0 NA
> # the same
> summary(a)
Shapley value :
        1         2         3         4 
-2.000000 -3.333333 -4.333333 -5.333333 

Shapley interaction indices :
   1  2  3  4
1 NA  0  0  0
2  0 NA  0  0
3  0  0 NA  0
4  0  0  0 NA
> 
> ## save the Möbius transform to a file
> d <- to.data.frame(a)
> write.table(d,"my.Mobius.set.func.csv",sep="\t")
> 
> # finally, some conversions that should work
> mu <- set.func(c(0,1,1,1,2,2,2,3))
> a <- Mobius(mu)
> as.Mobius.game(a)
		Mobius.game
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		0.000000
{1,3}		0.000000
{2,3}		0.000000
{1,2,3}		0.000000
> as.Mobius.capacity(a)
		Mobius.capacity
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		0.000000
{1,3}		0.000000
{2,3}		0.000000
{1,2,3}		0.000000
> as.Mobius.card.set.func(a)
  Mobius.card.set.func
0                    0
1                    1
2                    0
3                    0
> 
> 
> 
> cleanEx(); ..nameEx <- "Shapley.value-methods"
> 
> ### * Shapley.value-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Shapley.value-methods
> ### Title: The Shapley value
> ### Aliases: Shapley.value Shapley.value-methods
> ###   Shapley.value,Mobius.set.func-method
> ###   Shapley.value,card.set.func-method Shapley.value,set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a set function
> mu <- set.func(c(0:13/13,1,1))
> 
> ## the Shapley value
> Shapley.value(mu)
        1         2         3         4 
0.1346154 0.2115385 0.2884615 0.3653846 
> 
> ## the efficiency property should be satisfied
> sum(Shapley.value(mu))
[1] 1
> 
> ## a similar example using a Mobius.set.func object
> a <- Mobius(mu)
> Shapley.value(a)
        1         2         3         4 
0.1346154 0.2115385 0.2884615 0.3653846 
> 
> ## a similar example using a card.set.func object
> mu <- upper.capacity(6)
> Shapley.value(mu)
      All 
0.1666667 
> ## the efficiency property should be satisfied
> Shapley.value(mu)*6
All 
  1 
> 
> 
> 
> cleanEx(); ..nameEx <- "Sipos.integral-methods"
> 
> ### * Sipos.integral-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Sipos.integral-methods
> ### Title: Sipos integral
> ### Aliases: Sipos.integral Sipos.integral-methods
> ###   Sipos.integral,Mobius.game,numeric-method
> ###   Sipos.integral,card.game,numeric-method
> ###   Sipos.integral,game,numeric-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a normalized capacity
> mu <- capacity(c(0:13/13,1,1))
> 
> ## and its Möbius transform
> a <- Mobius(mu)
> 
> ## a discrete function f
> f <- c(0.1,-0.9,-0.3,0.8)
> 
> ## the Sugeno integral of f w.r.t mu
> Sipos.integral(mu,f)
[1] -0.007692308
> Sipos.integral(a,f)
[1] -0.007692308
> 
> ## a similar example with a cardinal capacity
> mu <- uniform.capacity(4)
> Sipos.integral(mu,f)
[1] -0.075
> 
> 
> 
> cleanEx(); ..nameEx <- "Sugeno.integral-methods"
> 
> ### * Sugeno.integral-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: Sugeno.integral-methods
> ### Title: Sugeno integral
> ### Aliases: Sugeno.integral Sugeno.integral-methods
> ###   Sugeno.integral,Mobius.game,numeric-method
> ###   Sugeno.integral,card.game,numeric-method
> ###   Sugeno.integral,game,numeric-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a normalized capacity
> mu <- capacity(c(0:13/13,1,1))
> 
> ## and its Möbius transform
> a <- Mobius(mu)
> 
> ## a discrete function f
> f <- c(0.1,0.9,0.3,0.8)
> 
> ## the Sugeno integral of f w.r.t mu
> Sugeno.integral(mu,f)
[1] 0.6923077
> Sugeno.integral(a,f) 
[1] 0.6923077
> 
> ## a similar example with a cardinal capacity
> mu <- uniform.capacity(4)
> Sugeno.integral(mu,f)
[1] 0.5
> 
> 
> 
> cleanEx(); ..nameEx <- "capacity-class"
> 
> ### * capacity-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: capacity-class
> ### Title: Class "capacity"
> ### Aliases: capacity-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(c(0:13,13,13)/13)
> 
> ## the attributes of the object
> mu@n
[1] 4
> mu@data
 [1] 0.00000000 0.07692308 0.15384615 0.38461538 0.23076923 0.46153846
 [7] 0.61538462 0.84615385 0.30769231 0.53846154 0.69230769 0.92307692
[13] 0.76923077 1.00000000 1.00000000 1.00000000
> mu@subsets
 [1]  0  1  2  4  8  3  5  9  6 10 12  7 11 13 14 15
> 
> ## a test
> is.normalized(mu)
[1] TRUE
> normalize(mu)
		capacity
{}		0.000000
{1}		0.076923
{2}		0.153846
{3}		0.230769
{4}		0.307692
{1,2}		0.384615
{1,3}		0.461538
{1,4}		0.538462
{2,3}		0.615385
{2,4}		0.692308
{3,4}		0.769231
{1,2,3}		0.846154
{1,2,4}		0.923077
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		1.000000
> 
> ## a conversion that should not work
> ## as.card.capacity(mu)
> 
> ## some transformations
> dual(mu)
		capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		0.076923
{4}		0.153846
{1,2}		0.230769
{1,3}		0.307692
{1,4}		0.384615
{2,3}		0.461538
{2,4}		0.538462
{3,4}		0.615385
{1,2,3}		0.692308
{1,2,4}		0.769231
{1,3,4}		0.846154
{2,3,4}		0.923077
{1,2,3,4}		1.000000
> Mobius(mu)
		Mobius.capacity
{}		0.000000
{1}		0.076923
{2}		0.153846
{3}		0.230769
{4}		0.307692
{1,2}		0.153846
{1,3}		0.153846
{1,4}		0.153846
{2,3}		0.230769
{2,4}		0.230769
{3,4}		0.230769
{1,2,3}		-0.153846
{1,2,4}		-0.153846
{1,3,4}		-0.153846
{2,3,4}		-0.384615
{1,2,3,4}		-0.076923
> ## let us check ...
> zeta(Mobius(mu))
		capacity
{}		0.000000
{1}		0.076923
{2}		0.153846
{3}		0.230769
{4}		0.307692
{1,2}		0.384615
{1,3}		0.461538
{1,4}		0.538462
{2,3}		0.615385
{2,4}		0.692308
{3,4}		0.769231
{1,2,3}		0.846154
{1,2,4}		0.923077
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		1.000000
> 
> ## some summary indices
> orness(mu)
[1] 0.5705128
> veto(mu)
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 
> favor(mu)
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 
> variance(mu)
[1] 0.1439842
> entropy(mu)
[1] 0.8073944
> ## the same
> summary(mu)
Shapley value :
        1         2         3         4 
0.1346154 0.2115385 0.2884615 0.3653846 

Shapley interaction indices :
            1           2           3           4
1          NA -0.02564103 -0.02564103 -0.02564103
2 -0.02564103          NA -0.06410256 -0.06410256
3 -0.02564103 -0.06410256          NA -0.06410256
4 -0.02564103 -0.06410256 -0.06410256          NA

Orness :
0.5705128

Veto indices :
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 

Favor indices :
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 

Normalized variance :
0.1439842

Normalized entropy :
0.8073944
> 
> 
> 
> cleanEx(); ..nameEx <- "card.capacity-class"
> 
> ### * card.capacity-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: card.capacity-class
> ### Title: Class "card.capacity"
> ### Aliases: card.capacity-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a capacity
> mu <- card.capacity(0:6/6)
> ## the same
> mu <- uniform.capacity(6)
> 
> # the attributes of the object
> mu@n
[1] 6
> mu@data
[1] 0.0000000 0.1666667 0.3333333 0.5000000 0.6666667 0.8333333 1.0000000
> 
> ## a test
> is.normalized(mu)
[1] TRUE
> normalize(mu)
  card.capacity
0     0.0000000
1     0.1666667
2     0.3333333
3     0.5000000
4     0.6666667
5     0.8333333
6     1.0000000
> 
> ## a transformation
> dual(mu)
  card.capacity
0     0.0000000
1     0.1666667
2     0.3333333
3     0.5000000
4     0.6666667
5     0.8333333
6     1.0000000
> 
> ## some summary indices
> orness(mu)
[1] 0.5
> veto(mu)
All 
0.5 
> favor(mu)
All 
0.5 
> variance(mu)
[1] 0
> entropy(mu)
[1] 1
> ## the same
> summary(mu)
Shapley value :
      All 
0.1666667 

Shapley interaction indices :
              All
All -5.551115e-18

Orness :
0.5

Veto indices :
All 
0.5 

Favor indices :
All 
0.5 

Normalized variance :
0

Normalized entropy :
1
> 
> 
> 
> cleanEx(); ..nameEx <- "card.game-class"
> 
> ### * card.game-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: card.game-class
> ### Title: Class "card.game"
> ### Aliases: card.game-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a cardinal game (which is a capacity)
> mu <- card.game(c(0,rep(1,4)))
> 
> # the attributes of the object
> mu@n
[1] 4
> mu@data
[1] 0 1 1 1 1
> 
> ## a conversion
> as.game(mu)
		game
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{4}		1.000000
{1,2}		1.000000
{1,3}		1.000000
{1,4}		1.000000
{2,3}		1.000000
{2,4}		1.000000
{3,4}		1.000000
{1,2,3}		1.000000
{1,2,4}		1.000000
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		1.000000
> 
> ## integral calculations 
> f <- c(0.2,0.3,0.1,0.7)
> Choquet.integral(mu,f)
[1] 0.7
> Sugeno.integral(mu,f)
[1] 0.7
> f <- c(0.2,-0.3,0.1,-0.7)
> Sipos.integral(mu,f)
[1] -0.5
> 
> 
> 
> cleanEx(); ..nameEx <- "card.set.func-class"
> 
> ### * card.set.func-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: card.set.func-class
> ### Title: Class "card.set.func"
> ### Aliases: card.set.func-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a cardinal set function
> mu <- card.set.func(-3:2)
> 
> # the attributes of the object
> mu@n
[1] 5
> mu@data
[1] -3 -2 -1  0  1  2
> 
> ## some conversions that cannot work
> ## Not run: as.card.game(mu)
> ## Not run: as.card.capacityfunc(mu)
> 
> ## the following should work
> as.set.func(mu)
		set.func
{}		-3.000000
{1}		-2.000000
{2}		-2.000000
{3}		-2.000000
{4}		-2.000000
{5}		-2.000000
{1,2}		-1.000000
{1,3}		-1.000000
{1,4}		-1.000000
{1,5}		-1.000000
{2,3}		-1.000000
{2,4}		-1.000000
{2,5}		-1.000000
{3,4}		-1.000000
{3,5}		-1.000000
{4,5}		-1.000000
{1,2,3}		0.000000
{1,2,4}		0.000000
{1,2,5}		0.000000
{1,3,4}		0.000000
{1,3,5}		0.000000
{1,4,5}		0.000000
{2,3,4}		0.000000
{2,3,5}		0.000000
{2,4,5}		0.000000
{3,4,5}		0.000000
{1,2,3,4}		1.000000
{1,2,3,5}		1.000000
{1,2,4,5}		1.000000
{1,3,4,5}		1.000000
{2,3,4,5}		1.000000
{1,2,3,4,5}		2.000000
> 
> ## some tests
> is.cardinal(mu)
[1] TRUE
> is.kadditive(mu,2)
[1] FALSE
> is.monotone(mu)
[1] TRUE
> 
> ## some transformations
> dual(mu)
  card.game
0         0
1         1
2         2
3         3
4         4
5         5
> Mobius(mu)
  Mobius.card.set.func
0                   -3
1                    1
2                    0
3                    0
4                    0
5                    0
> ## let us check ...
> zeta(Mobius(mu))
  card.set.func
0            -3
1            -2
2            -1
3             0
4             1
5             2
> 
> ## summary 
> Shapley.value(mu)
All 
  1 
> interaction.indices(mu)
    All
All   0
> # the same
> summary(mu)
Shapley value :
All 
  1 

Shapley interaction indices :
    All
All   0
> 
> ## save the set function to a file
> d <- to.data.frame(mu)
> write.table(d,"my.card.set.func.csv",sep="\t")
> 
> # finally, some other conversions that should work
> mu <- card.set.func(0:5)
> as.card.game(mu)
  card.game
0         0
1         1
2         2
3         3
4         4
5         5
> as.card.capacity(mu)
  card.capacity
0             0
1             1
2             2
3             3
4             4
5             5
> 
> 
> 
> cleanEx(); ..nameEx <- "card.set.func.gen"
> 
> ### * card.set.func.gen
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: card.set.func
> ### Title: Create objects representing cardinal set functions.
> ### Aliases: card.set.func card.game card.capacity lower.capacity
> ###   upper.capacity uniform.capacity
> ### Keywords: math
> 
> ### ** Examples
> 
> card.set.func(4:-2)
  card.set.func
0             4
1             3
2             2
3             1
4             0
5            -1
6            -2
> card.game(c(0,-2:2))
  card.game
0         0
1        -2
2        -1
3         0
4         1
5         2
> card.capacity(0:5)
  card.capacity
0             0
1             1
2             2
3             3
4             4
5             5
> lower.capacity(3)
  card.capacity
0             0
1             0
2             0
3             1
> upper.capacity(4)
  card.capacity
0             0
1             1
2             1
3             1
4             1
> uniform.capacity(5)
  card.capacity
0           0.0
1           0.2
2           0.4
3           0.6
4           0.8
5           1.0
> 
> 
> 
> cleanEx(); ..nameEx <- "dual-methods"
> 
> ### * dual-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: dual-methods
> ### Title: The dual (or conjugate) transform
> ### Aliases: dual dual-methods dual,capacity-method
> ###   dual,card.capacity-method dual,card.set.func-method
> ###   dual,set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a game
> mu <- game(c(0,-7:7))
> mu
		game
{}		0.000000
{1}		-7.000000
{2}		-6.000000
{3}		-5.000000
{4}		-4.000000
{1,2}		-3.000000
{1,3}		-2.000000
{1,4}		-1.000000
{2,3}		0.000000
{2,4}		1.000000
{3,4}		2.000000
{1,2,3}		3.000000
{1,2,4}		4.000000
{1,3,4}		5.000000
{2,3,4}		6.000000
{1,2,3,4}		7.000000
> 
> ## its dual 
> dual(mu)
		game
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		5.000000
{1,3}		6.000000
{1,4}		7.000000
{2,3}		8.000000
{2,4}		9.000000
{3,4}		10.000000
{1,2,3}		11.000000
{1,2,4}		12.000000
{1,3,4}		13.000000
{2,3,4}		14.000000
{1,2,3,4}		7.000000
> ## and mu again 
> dual(dual(mu))
		game
{}		0.000000
{1}		-7.000000
{2}		-6.000000
{3}		-5.000000
{4}		-4.000000
{1,2}		-3.000000
{1,3}		-2.000000
{1,4}		-1.000000
{2,3}		0.000000
{2,4}		1.000000
{3,4}		2.000000
{1,2,3}		3.000000
{1,2,4}		4.000000
{1,3,4}		5.000000
{2,3,4}		6.000000
{1,2,3,4}		7.000000
> 
> ## a similar example with the upper capacity
> mu <- capacity(c(0,rep(1,15)))
> mu
		capacity
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{4}		1.000000
{1,2}		1.000000
{1,3}		1.000000
{1,4}		1.000000
{2,3}		1.000000
{2,4}		1.000000
{3,4}		1.000000
{1,2,3}		1.000000
{1,2,4}		1.000000
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		1.000000
> dual(mu)
		capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		0.000000
{4}		0.000000
{1,2}		0.000000
{1,3}		0.000000
{1,4}		0.000000
{2,3}		0.000000
{2,4}		0.000000
{3,4}		0.000000
{1,2,3}		0.000000
{1,2,4}		0.000000
{1,3,4}		0.000000
{2,3,4}		0.000000
{1,2,3,4}		1.000000
> dual(dual(mu))
		capacity
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{4}		1.000000
{1,2}		1.000000
{1,3}		1.000000
{1,4}		1.000000
{2,3}		1.000000
{2,4}		1.000000
{3,4}		1.000000
{1,2,3}		1.000000
{1,2,4}		1.000000
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		1.000000
> 
> ## a similar example with an object of class card.capacity
> mu <- upper.capacity(6)
> mu
  card.capacity
0             0
1             1
2             1
3             1
4             1
5             1
6             1
> dual(mu)
  card.capacity
0             0
1             0
2             0
3             0
4             0
5             0
6             1
> dual(dual(mu))
  card.capacity
0             0
1             1
2             1
3             1
4             1
5             1
6             1
> 
> ## the dual of a set function is a game
> mu <- set.func(-7:8)
> mu
		set.func
{}		-7.000000
{1}		-6.000000
{2}		-5.000000
{3}		-4.000000
{4}		-3.000000
{1,2}		-2.000000
{1,3}		-1.000000
{1,4}		0.000000
{2,3}		1.000000
{2,4}		2.000000
{3,4}		3.000000
{1,2,3}		4.000000
{1,2,4}		5.000000
{1,3,4}		6.000000
{2,3,4}		7.000000
{1,2,3,4}		8.000000
> dual(mu)
		game
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		5.000000
{1,3}		6.000000
{1,4}		7.000000
{2,3}		8.000000
{2,4}		9.000000
{3,4}		10.000000
{1,2,3}		11.000000
{1,2,4}		12.000000
{1,3,4}		13.000000
{2,3,4}		14.000000
{1,2,3,4}		15.000000
> mu <- card.set.func(-2:5)
> dual(mu)
  card.game
0         0
1         1
2         2
3         3
4         4
5         5
6         6
7         7
> 
> 
> 
> cleanEx(); ..nameEx <- "entropy-methods"
> 
> ### * entropy-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: entropy-methods
> ### Title: Normalized entropy of a capacity
> ### Aliases: entropy entropy-methods entropy,Mobius.capacity-method
> ###   entropy,capacity-method entropy,card.capacity-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(c(0,0,0:13))
> 
> ## its Möbius transform
> a <- Mobius(mu)
> 
> ## their normalized entropy
> entropy(mu)
[1] 0.8073944
> entropy(a)
[1] 0.8073944
> 
> ## similar examples with card.capacity objects
> mu <- lower.capacity(4)
> entropy(mu)
[1] 0
> mu <- uniform.capacity(4)
> entropy(mu)
[1] 1
> 
> 
> 
> cleanEx(); ..nameEx <- "entropy.capa.ident"
> 
> ### * entropy.capa.ident
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: entropy.capa.ident
> ### Title: Unsupervised identification of a capacity from profiles
> ### Aliases: entropy.capa.ident
> ### Keywords: math
> 
> ### ** Examples
> 
> ## a set of randomly generated data
> ## for instance, marks on a [0,20] scale
> p <- data.frame(matrix(runif(500,0,20),100,5))
> names(p) <- c("Stat","Prob","Alg","Cal","Eng")
> 
> ## discretization
> p[p <= 5] <- 1  
> p[p > 5 & p <= 10] <- 2 
> p[p > 10 & p <= 15] <- 3 
> p[p > 15] <- 4
> 
> d <- data.frame(factor(p[[1]]),
+                 factor(p[[2]]),
+                 factor(p[[3]]),
+                 factor(p[[4]]),
+                 factor(p[[5]]))
> 
> ## associated unsupervised capacity
> mu <- entropy.capa.ident(d)
> mu
		capacity
{}		0.000000
{1}		0.302819
{2}		0.303766
{3}		0.300483
{4}		0.300016
{5}		0.305746
{1,2}		0.596686
{1,3}		0.580858
{1,4}		0.585843
{1,5}		0.599376
{2,3}		0.590558
{2,4}		0.592248
{2,5}		0.602706
{3,4}		0.591646
{3,5}		0.598377
{4,5}		0.598055
{1,2,3}		0.826752
{1,2,4}		0.835746
{1,2,5}		0.828410
{1,3,4}		0.823276
{1,3,5}		0.838023
{1,4,5}		0.832737
{2,3,4}		0.826346
{2,3,5}		0.846925
{2,4,5}		0.842544
{3,4,5}		0.847677
{1,2,3,4}		0.968149
{1,2,3,5}		0.958942
{1,2,4,5}		0.952051
{1,3,4,5}		0.963922
{2,3,4,5}		0.962763
{1,2,3,4,5}		1.000000
> 
> 
> 
> cleanEx(); ..nameEx <- "favor-methods"
> 
> ### * favor-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: favor-methods
> ### Title: Favor indices
> ### Aliases: favor favor-methods favor,Mobius.capacity-method
> ###   favor,capacity-method favor,card.capacity-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(c(0:13,13,13))
> 
> ## its Möbius transform
> a <- Mobius(mu)
> 
> ## their favor indices
> favor(mu)
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 
> favor(a)
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 
> 
> ## the same with a card.capacity object
> mu <- lower.capacity(4)
> favor(mu)
All 
  0 
> 
> 
> 
> cleanEx(); ..nameEx <- "game-class"
> 
> ### * game-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: game-class
> ### Title: Class "game"
> ### Aliases: game-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a game (which is a capacity)
> mu <- game(c(0,rep(1,15)))
> 
> ## the attributes of the object
> mu@n
[1] 4
> mu@data
 [1] 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
> mu@subsets
 [1]  0  1  2  4  8  3  5  9  6 10 12  7 11 13 14 15
> 
> ## a conversion
> as.card.game(mu)
  card.game
0         0
1         1
2         1
3         1
4         1
> 
> ## a transformation
> Mobius(mu)
		Mobius.game
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{4}		1.000000
{1,2}		-1.000000
{1,3}		-1.000000
{1,4}		-1.000000
{2,3}		-1.000000
{2,4}		-1.000000
{3,4}		-1.000000
{1,2,3}		1.000000
{1,2,4}		1.000000
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		-1.000000
> ## let us check ...
> zeta(Mobius(mu))
		game
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{4}		1.000000
{1,2}		1.000000
{1,3}		1.000000
{1,4}		1.000000
{2,3}		1.000000
{2,4}		1.000000
{3,4}		1.000000
{1,2,3}		1.000000
{1,2,4}		1.000000
{1,3,4}		1.000000
{2,3,4}		1.000000
{1,2,3,4}		1.000000
> 
> ## integral calculations 
> f <- c(0.2,0.3,0.1,0.7)
> Choquet.integral(mu,f)
[1] 0.7
> Sugeno.integral(mu,f)
[1] 0.7
> f <- c(0.2,-0.3,0.1,-0.7)
> Sipos.integral(mu,f)
[1] -0.5
> 
> 
> 
> cleanEx(); ..nameEx <- "interaction.indices-methods"
> 
> ### * interaction.indices-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: interaction.indices-methods
> ### Title: The Shapley interaction indices
> ### Aliases: interaction.indices interaction.indices-methods
> ###   interaction.indices,Mobius.set.func-method
> ###   interaction.indices,card.set.func-method
> ###   interaction.indices,set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a set function
> mu <- set.func(c(-7:6,6,6))
> 
> ## the associated interaction indices
> interaction.indices(mu)
           1          2          3          4
1         NA -0.3333333 -0.3333333 -0.3333333
2 -0.3333333         NA -0.8333333 -0.8333333
3 -0.3333333 -0.8333333         NA -0.8333333
4 -0.3333333 -0.8333333 -0.8333333         NA
> 
> ## a similar example using a Mobius.set.func object
> a <- Mobius(mu)
> interaction.indices(a)
           1          2          3          4
1         NA -0.3333333 -0.3333333 -0.3333333
2 -0.3333333         NA -0.8333333 -0.8333333
3 -0.3333333 -0.8333333         NA -0.8333333
4 -0.3333333 -0.8333333 -0.8333333         NA
> 
> ## a similar example using a card.set.func object
> mu <- upper.capacity(6)
> interaction.indices(mu)
     All
All -0.2
> 
> 
> 
> cleanEx(); ..nameEx <- "is.cardinal-methods"
> 
> ### * is.cardinal-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: is.cardinal-methods
> ### Title: Test method
> ### Aliases: is.cardinal is.cardinal-methods
> ###   is.cardinal,Mobius.set.func-method is.cardinal,card.set.func-method
> ###   is.cardinal,set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> is.cardinal(set.func(-7:8))
[1] FALSE
> is.cardinal(uniform.capacity(8))
[1] TRUE
> is.cardinal(Mobius.game(0:10,4,2))
[1] FALSE
> 
> 
> 
> cleanEx(); ..nameEx <- "is.kadditive-methods"
> 
> ### * is.kadditive-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: is.kadditive-methods
> ### Title: Test method
> ### Aliases: is.kadditive is.kadditive-methods
> ###   is.kadditive,Mobius.set.func,numeric-method
> ###   is.kadditive,card.set.func,numeric-method
> ###   is.kadditive,set.func,numeric-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a set function
> mu <- set.func(c(0,1,1,1,2,2,2,3))
> mu
		set.func
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		2.000000
{1,3}		2.000000
{2,3}		2.000000
{1,2,3}		3.000000
> is.kadditive(mu,2)
[1] FALSE
> is.kadditive(mu,1)
[1] TRUE
> 
> ## the Möbius representation of a set function, 2-additive by construction 
> a <- Mobius.set.func(c(0,1,2,1,3,1,2,1,2,3,1),4,2)
> is.kadditive(a,2)
[1] TRUE
> is.kadditive(a,1)
[1] FALSE
> 
> 
> 
> cleanEx(); ..nameEx <- "is.monotone-methods"
> 
> ### * is.monotone-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: is.monotone-methods
> ### Title: Test method
> ### Aliases: is.monotone is.monotone-methods
> ###   is.monotone,Mobius.set.func-method is.monotone,card.set.func-method
> ###   is.monotone,set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a monotone set function
> mu <- set.func(c(0,1,1,1,2,2,2,3))
> mu
		set.func
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		2.000000
{1,3}		2.000000
{2,3}		2.000000
{1,2,3}		3.000000
> is.monotone(mu)
[1] TRUE
> 
> ## the Möbius representation of a monotone set function
> a <- Mobius.set.func(c(0,1,2,1,3,1,2,1,2,3,1),4,2)
> is.monotone(a)
[1] TRUE
> 
> ## non-monotone examples
> mu <- set.func(c(0,-7:7))
> is.monotone(mu,verbose=TRUE)
Violation of monotonicity constraint between { } and { 1 }. 
Violation of monotonicity constraint between { } and { 2 }. 
Violation of monotonicity constraint between { } and { 3 }. 
Violation of monotonicity constraint between { } and { 4 }. 
[1] FALSE
> a <- Mobius(mu)
> is.monotone(a,verbose=TRUE)
Violation of monotonicity constraint between { } and { 1 }. 
Violation of monotonicity constraint between { } and { 2 }. 
Violation of monotonicity constraint between { } and { 3 }. 
Violation of monotonicity constraint between { } and { 4 }. 
[1] FALSE
> 
> 
> 
> cleanEx(); ..nameEx <- "is.normalized-methods"
> 
> ### * is.normalized-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: is.normalized-methods
> ### Title: Test method
> ### Aliases: is.normalized is.normalized-methods
> ###   is.normalized,Mobius.capacity-method is.normalized,capacity-method
> ###   is.normalized,card.capacity-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(0:15)
> is.normalized(mu)
[1] FALSE
> normalize(mu)
		capacity
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.333333
{1,3}		0.400000
{1,4}		0.466667
{2,3}		0.533333
{2,4}		0.600000
{3,4}		0.666667
{1,2,3}		0.733333
{1,2,4}		0.800000
{1,3,4}		0.866667
{2,3,4}		0.933333
{1,2,3,4}		1.000000
> 
> ## its Möbius transform
> a <- Mobius(mu)
> is.normalized(a)
[1] FALSE
> normalize(a)
		Mobius.capacity
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.133333
{1,3}		0.133333
{1,4}		0.133333
{2,3}		0.200000
{2,4}		0.200000
{3,4}		0.200000
{1,2,3}		-0.133333
{1,2,4}		-0.133333
{1,3,4}		-0.133333
{2,3,4}		-0.266667
{1,2,3,4}		0.000000
> 
> ## a cardinal capacity
> mu <- uniform.capacity(7)
> is.normalized(mu)
[1] TRUE
> 
> 
> 
> cleanEx(); ..nameEx <- "k.truncate.Mobius-methods"
> 
> ### * k.truncate.Mobius-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: k.truncate.Mobius-methods
> ### Title: k-order truncation of the Möbius representation of a set
> ###   function.
> ### Aliases: k.truncate.Mobius k.truncate.Mobius-methods
> ###   k.truncate.Mobius,set.func,numeric-method
> ###   k.truncate.Mobius,Mobius.set.func,numeric-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a set function
> mu <- set.func(c(0,1,1,1,2,2,2,3))
> mu
		set.func
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		2.000000
{1,3}		2.000000
{2,3}		2.000000
{1,2,3}		3.000000
> 
> ## 2-truncate it
> k.truncate.Mobius(mu,2)
		Mobius.set.func
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		0.000000
{1,3}		0.000000
{2,3}		0.000000
> ## 2-truncate it
> k.truncate.Mobius(Mobius(mu),2)
		Mobius.set.func
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		0.000000
{1,3}		0.000000
{2,3}		0.000000
> 
> 
> 
> cleanEx(); ..nameEx <- "least.squares.capa.ident"
> 
> ### * least.squares.capa.ident
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: least.squares.capa.ident
> ### Title: Least squares capacity identification
> ### Aliases: least.squares.capa.ident
> ### Keywords: math
> 
> ### ** Examples
> 
> 
> ## the number of data
> n.d <- 20
> 
> ## a randomly generated 5-criteria matrix
> C <- matrix(rnorm(5*n.d,10,2),n.d,5)
> 
> ## the corresponding global scores
> g <- numeric(n.d)
> mu <- capacity(c(0:29,29,29)/29)
> for (i in 1:n.d)
+   g[i] <- Choquet.integral(mu,C[i,])
> 
> ## Not run: 
> ##D ## the full solution 
> ##D lsc <- least.squares.capa.ident(5,5,C,g)
> ##D a <- lsc$solution
> ##D a
> ##D mu.sol <- zeta(a)
> ##D 
> ##D ## the difference between mu and mu.sol
> ##D mu@data - mu.sol@data
> ##D 
> ##D ## the residuals
> ##D lsc$residuals
> ##D 
> ##D ## the mean square error
> ##D mean(lsc$residuals^2)
> ##D 
> ##D ## a 3-additive solution 
> ##D lsc <- least.squares.capa.ident(5,3,C,g)
> ##D a <- lsc$solution
> ##D mu.sol <- zeta(a)
> ##D mu@data - mu.sol@data
> ##D lsc$residuals
> ## End(Not run)
> 
> 
> ## a similar example based on the Sipos integral
> 
> ## a randomly generated 5-criteria matrix
> C <- matrix(rnorm(5*n.d,0,2),n.d,5)
> 
> ## the corresponding global scores
> g <- numeric(n.d)
> mu <- capacity(c(0:29,29,29)/29)
> for (i in 1:n.d)
+   g[i] <- Sipos.integral(mu,C[i,])
> 
> ## Not run: 
> ##D ## the full solution 
> ##D lsc <- least.squares.capa.ident(5,5,C,g,Integral = "Sipos")
> ##D a <- lsc$solution
> ##D mu.sol <- zeta(a)
> ##D mu@data - mu.sol@data
> ##D lsc$residuals
> ##D 
> ##D ## a 3-additive solution 
> ##D lsc <- least.squares.capa.ident(5,3,C,g,Integral = "Sipos")
> ##D a <- lsc$solution
> ##D mu.sol <- zeta(a)
> ##D mu@data - mu.sol@data
> ##D lsc$residuals
> ## End(Not run)
> 
> 
> ## additional constraints
> 
> ## a Shapley preorder constraint matrix
> ## Sh(1) - Sh(2) >= -delta.S
> ## Sh(2) - Sh(1) >= -delta.S
> ## Sh(3) - Sh(4) >= -delta.S
> ## Sh(4) - Sh(3) >= -delta.S
> ## i.e. criteria 1,2 and criteria 3,4
> ## should have the same global importances
> delta.S <- 0.01    
> Asp <- rbind(c(1,2,-delta.S),
+              c(2,1,-delta.S),
+              c(3,4,-delta.S),
+              c(4,3,-delta.S)
+             )
> 
> ## a Shapley interval constraint matrix
> ## 0.3 <= Sh(1) <= 0.9 
> Asi <- rbind(c(1,0.3,0.9))
> 
> ## an interaction preorder constraint matrix
> ## such that I(12) = I(45)
> delta.I <- 0.01
> Aip <- rbind(c(1,2,4,5,-delta.I),
+              c(4,5,1,2,-delta.I))
> 
> ## an interaction interval constraint matrix
> ## i.e. -0.20 <= I(12) <= -0.15 
> delta.I <- 0.01
> Aii <- rbind(c(1,2,-0.2,-0.15))
> 
> ## an inter-additive partition constraint
> ## criteria 1,2,3 and criteria 4,5 are independent 
> Aiap <- c(1,1,1,2,2)
> 
> 
> 
> ## a more constrained solution
> 
> lsc <- least.squares.capa.ident(5,5,C,g,Integral = "Sipos",
+                                  A.Shapley.preorder = Asp,
+                                  A.Shapley.interval = Asi,
+                                  A.interaction.preorder = Aip,
+                                  A.interaction.interval = Aii,
+                                  A.inter.additive.partition = Aiap,
+                                  sigf = 5)
Warning in rbind(sigma, beta) : number of columns of result
	not a multiple of vector length (arg 2)
> 
> a <- lsc$solution
> mu.sol <- zeta(a)
> mu@data - mu.sol@data
 [1]  0.000000e+00 -1.934924e-01 -3.629596e-01 -4.794334e-01  2.840685e-02
 [6] -4.144969e-01 -8.709869e-02 -1.346071e-01 -2.840028e-02 -1.184444e-01
[11] -2.189461e-01 -2.664544e-01  2.069031e-01 -1.670351e-01  1.603631e-01
[16]  4.388912e-02 -1.349237e-01 -2.249678e-01 -3.254695e-01 -3.729778e-01
[21]  1.003798e-01 -2.735585e-01  5.383972e-02 -6.263425e-02  2.035727e-01
[26]  1.824940e-01  8.199237e-02 -3.448144e-02  4.733588e-01  3.045509e-02
[31]  2.544050e-01  2.240198e-10
> lsc$residuals
 [1] -0.05963137 -0.04470915  0.63410159  0.11569759  0.02709655 -0.72004795
 [7] -0.16408733 -0.62527005 -0.24347211 -0.81616139  0.33233613  0.48537466
[13] -0.61578720  0.39969224 -0.54639406  0.31388259  0.96522926 -0.02193253
[19]  0.17259077  0.68902569
> summary(a)
Shapley value :
         1          2          3          4          5 
0.30000001 0.29000001 0.09633127 0.08633129 0.22733742 

Shapley interaction indices :
              1             2             3             4             5
1            NA -1.500001e-01  1.764299e-01  1.494118e-09  1.653554e-09
2 -1.500001e-01            NA -2.514700e-01  1.243012e-09  1.348793e-09
3  1.764299e-01 -2.514700e-01            NA  1.071158e-09  1.006189e-09
4  1.494118e-09  1.243012e-09  1.071158e-09            NA -1.600001e-01
5  1.653554e-09  1.348793e-09  1.006189e-09 -1.600001e-01            NA

Orness :
0.59626

Veto indices :
        1         2         3         4         5 
0.4850364 0.4342135 0.3471737 0.3320737 0.4202025 

Favor indices :
        1         2         3         4         5 
0.6399636 0.6782865 0.5232404 0.5258404 0.6139693 

Normalized variance :
0.1612258

Normalized entropy :
0.763725
> 
> 
> 
> cleanEx(); ..nameEx <- "ls.sorting.capa.ident"
> 
> ### * ls.sorting.capa.ident
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: ls.sorting.capa.ident
> ### Title: Least squares capacity identification in the framework of a
> ###   sorting procedure
> ### Aliases: ls.sorting.capa.ident
> ### Keywords: math
> 
> ### ** Examples
> 
> 
> ## generate a random problem with 10 prototypes and 4 criteria
> n.proto <- 10 ## prototypes
> n <- 4  ## criteria
> k <- 4  
> d <- 0.1
>         
> ## generating random data for the prototypes
> C <- matrix(runif(n.proto*n,0,1),n.proto,n)
> cl <- numeric(n.proto)
> 
> ## the corresponding global evaluations
> glob.eval <- numeric(n.proto)
> a <- capacity(c(0:(2^n-3),(2^n-3),(2^n-3))/(2^n-3))
> for (i in 1:n.proto)
+   glob.eval[i] <- Choquet.integral(a,C[i,])
> 
> ## and the classes for the prototypes
> cl[glob.eval <= 0.33] <- 1
> cl[glob.eval > 0.33 & glob.eval <= 0.66] <-2
> cl[glob.eval > 0.66] <- 3
> 
> cl
 [1] 2 2 2 1 2 2 3 2 3 2
> 
> ## Not run: 
> ##D # starting the calculations
> ##D # search for a capacity which satisfies the constraints
> ##D lsc <- ls.sorting.capa.ident(n ,k, C, cl, d)
> ##D 
> ##D ## output of the quadratic program (ipop, package kernlab)
> ##D lsc$how
> ##D 
> ##D ## the capacity satisfying the constraints
> ##D lsc$solution
> ##D summary(lsc$solution)
> ##D ## the global evaluations satisfying the constraints
> ##D lsc$glob.eval
> ## End(Not run)
> 
> ## let us now add some constraints        
> 
> ## a Shapley preorder constraint matrix
> ## Sh(1) > Sh(2)
> ## Sh(3) > Sh(4)
> delta.S <-0.01
> Asp <- rbind(c(1,2,delta.S), c(3,4,delta.S))
> 
> ## a Shapley interval constraint matrix
> ## 0.1 <= Sh(1) <= 0.2 
> Asi <- rbind(c(1,0.1,0.2))
>         
> ## an interaction preorder constraint matrix
> ## such that I(12) > I(34)
> delta.I <- 0.01
> Aip <- rbind(c(1,2,3,4,delta.I))
>         
> ## an interaction interval constraint matrix
> ## i.e. 0.2 <= I(12) <= 0.4 
> ## and 0 < I(34) <= 1
> Aii <- rbind(c(1,2,0.2,0.4), c(3,4,delta.I,1))
>         
> ## an inter-additive partition constraint
> ## criteria 1,2 and criteria 3,4 are independent 
> Aiap <- c(1,1,2,2)
> 
> ## starting the calculations
> ## search for a capacity which satisfies the constraints
> lsc <- ls.sorting.capa.ident(n ,k, C, cl, d,
+                                   A.Shapley.preorder = Asp,
+                                   A.Shapley.interval = Asi,
+                                   A.interaction.preorder = Aip,
+                                   A.interaction.interval = Aii,
+                                   A.inter.additive.partition = Aiap)
Warning in rbind(sigma, beta) : number of columns of result
	not a multiple of vector length (arg 2)
> 
> ## output of ipop
> lsc$how
[1] "converged"
> ## the capacity satisfying the constraints
> lsc$solution
		Mobius.capacity
{}		0.000000
{1}		0.099999
{2}		0.089999
{3}		0.305000
{4}		0.295000
{1,2}		0.200002
{1,3}		0.000000
{1,4}		-0.000000
{2,3}		0.000000
{2,4}		-0.000000
{3,4}		0.010000
{1,2,3}		0.000000
{1,2,4}		0.000000
{1,3,4}		0.000000
{2,3,4}		0.000000
{1,2,3,4}		0.000000
> summary(lsc$solution)
Shapley value :
        1         2         3         4 
0.1999998 0.1899998 0.3100002 0.3000002 

Shapley interaction indices :
             1            2            3            4
1           NA 2.000015e-01 2.979399e-13 2.816738e-13
2 2.000015e-01           NA 4.244533e-13 3.864737e-13
3 2.979399e-13 4.244533e-13           NA 1.000003e-02
4 2.816738e-13 3.864737e-13 1.000003e-02           NA

Orness :
0.4416662

Veto indices :
        1         2         3         4 
0.5355559 0.5288892 0.5877783 0.5811116 

Favor indices :
        1         2         3         4 
0.3977771 0.3911104 0.4922220 0.4855553 

Normalized variance :
0.04300053

Normalized entropy :
0.9433422
> ## the global evaluations satisfying the constraints
> lsc$glob.eval
 [1] 0.5183923 0.3486155 0.5256927 0.2486147 0.4580393 0.5256926 0.6256929
 [8] 0.4370614 0.6591117 0.3486151
> 
> 
> 
> 
> cleanEx(); ..nameEx <- "ls.sorting.treatment"
> 
> ### * ls.sorting.treatment
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: ls.sorting.treatment
> ### Title: Least squares capacity identification in the framework of a
> ###   sorting procedure: evaluation of the determined capacity
> ### Aliases: ls.sorting.treatment
> ### Keywords: math
> 
> ### ** Examples
> 
> 
> ## generate a random problem with 10 prototypes and 4 criteria
> n.proto <- 10 ## prototypes
> n <- 4  ## criteria
> P <- matrix(runif(n.proto*n,0,1),n.proto,n)
> 
> ## the corresponding global scores, based on a randomly generated
> ## capacity a
> glob.eval <- numeric(n.proto)
> a <- capacity(c(0:(2^n-3),(2^n-3),(2^n-3))/(2^n-3))
> for (i in 1:n.proto)
+   glob.eval[i] <- Choquet.integral(a,P[i,])
> 
> ## based on these global scores, let us create a classification (3 classes)
> cl.proto<-numeric(n.proto)
> cl.proto[glob.eval <= 0.33] <- 1
> cl.proto[glob.eval > 0.33 & glob.eval<=0.66] <-2
> cl.proto[glob.eval > 0.66] <- 3
> 
> ## search for a capacity which satisfies the constraints
> lsc <- ls.sorting.capa.ident(n ,4, P, cl.proto, 0.1)
Warning in rbind(sigma, beta) : number of columns of result
	not a multiple of vector length (arg 2)
> 
> ## output of the QP
> lsc$how
[1] "converged"
> 
> ## analyse the quality of the model (classify the prototypes by the
> ## model and compare both assignments)
> lst <- ls.sorting.treatment(P,cl.proto,lsc$solution,P,cl.proto)
> 
> ## assignments of the prototypes
> lst$class.A
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    2    2    2    1    2    2    3    2    3     2
[2,]    2    2    2    1    2    2    3    2    3     2
> ## assignment types
> lst$correct.A
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    1    1    1    1    1    1    1    1    1     1
[2,]    0    0    0    0    0    0    0    0    0     0
[3,]    0    0    0    0    0    0    0    0    0     0
[4,]    0    0    0    0    0    0    0    0    0     0
> ## evaluation
> lst$eval.correct
[1] 1 0 0 0
> 
> ## generate a second set of random alternatives (A)
> ## their "correct" class is determined as beforehand with the
> ## randomly generated capacity a
> ## the goal is to see if we can reproduce this classification
> ## by the capacity learnt from the prototypes
> 
> ## a randomly generated criteria matrix of 10 alternatives
> A <- matrix(runif(10*n,0,1),10,n)
> cl.orig.A <-numeric(10)
>         
> ## the corresponding global scores
> glob.eval.A <- numeric(10)
> for (i in 1:10)
+   glob.eval.A[i] <- Choquet.integral(a,A[i,])
> 
> ## based on these global scores, let us determine a classification
> cl.orig.A[glob.eval.A <= 0.33] <- 1
> cl.orig.A[glob.eval.A>0.33 & glob.eval.A<=0.66] <-2
> cl.orig.A[glob.eval.A > 0.66] <- 3
> 
> ## let us now classify the alternatives of A according to the model
> ## built on P
> 
> lst <- ls.sorting.treatment(P,cl.proto,lsc$solution,A,cl.orig.A)
> 
> ## assignment of the alternatives of A
> lst$class.A
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    2    2    2    1    2    2    2    2    2     3
[2,]    3    3    2    2    2    2    3    2    3     3
> ## type of assignments
> lst$correct.A
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    0    0    1    0    1    1    0    1    0     1
[2,]    1    1    0    1    0    0    1    0    1     0
[3,]    0    0    0    0    0    0    0    0    0     0
[4,]    0    0    0    0    0    0    0    0    0     0
> ## evaluation
> lst$eval.correct
[1] 0.5 0.5 0.0 0.0
> 
> ## show the learnt capacity
> ## x11()
> ## barplot(Shapley.value(lsc$solution), main="Learnt capacity", sub="Shapley")
> ## summary of the learnt capacity
> lsc$solution
		Mobius.capacity
{}		0.000000
{1}		0.002993
{2}		0.042950
{3}		0.171595
{4}		0.347543
{1,2}		0.011565
{1,3}		0.290460
{1,4}		0.167683
{2,3}		-0.013151
{2,4}		-0.013469
{3,4}		0.422559
{1,2,3}		0.355518
{1,2,4}		0.439593
{1,3,4}		-0.408164
{2,3,4}		0.029024
{1,2,3,4}		-0.846697
> summary(lsc$solution)
Shapley value :
         1          2          3          4 
0.15515489 0.09845918 0.30198049 0.44440544 

Shapley interaction indices :
            1           2           3           4
1          NA  0.12688808 -0.01809540 -0.09883497
2  0.12688808          NA -0.10311271 -0.06139333
3 -0.01809540 -0.10311271          NA -0.04924372
4 -0.09883497 -0.06139333 -0.04924372          NA

Orness :
0.5095702

Veto indices :
        1         2         3         4 
0.4396280 0.3965446 0.5174662 0.6080806 

Favor indices :
        1         2         3         4 
0.4339119 0.4014009 0.5518411 0.6511267 

Normalized variance :
0.3824951

Normalized entropy :
0.572115
> 
> 
> 
> 
> cleanEx(); ..nameEx <- "mini.var.capa.ident"
> 
> ### * mini.var.capa.ident
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: mini.var.capa.ident
> ### Title: Minimum variance capacity identification
> ### Aliases: mini.var.capa.ident
> ### Keywords: math
> 
> ### ** Examples
> 
> ## some alternatives
> a <- c(18,11,18,11,11)
> b <- c(18,18,11,11,11)
> c <- c(11,11,18,18,11)
> d <- c(18,11,11,11,18)
> e <- c(11,11,18,11,18)
>     
> ## preference threshold relative
> ## to the preorder of the alternatives
> delta.C <- 1
> 
> ## corresponding Choquet preorder constraint matrix 
> Acp <- rbind(c(d,a,delta.C),
+              c(a,e,delta.C),
+              c(e,b,delta.C),
+              c(b,c,delta.C)
+             )
> 
> ## a Shapley preorder constraint matrix
> ## Sh(1) - Sh(2) >= -delta.S
> ## Sh(2) - Sh(1) >= -delta.S
> ## Sh(3) - Sh(4) >= -delta.S
> ## Sh(4) - Sh(3) >= -delta.S
> ## i.e. criteria 1,2 and criteria 3,4
> ## should have the same global importances
> delta.S <- 0.01    
> Asp <- rbind(c(1,2,-delta.S),
+              c(2,1,-delta.S),
+              c(3,4,-delta.S),
+              c(4,3,-delta.S)
+             )
> 
> ## a Shapley interval constraint matrix
> ## 0.3 <= Sh(1) <= 0.9 
> Asi <- rbind(c(1,0.3,0.9))
> 
> ## an interaction preorder constraint matrix
> ## such that I(12) = I(34)
> delta.I <- 0.01
> Aip <- rbind(c(1,2,3,4,-delta.I),
+              c(3,4,1,2,-delta.I))
> 
> ## an interaction interval constraint matrix
> ## i.e. -0.20 <= I(12) <= -0.15 
> delta.I <- 0.01
> Aii <- rbind(c(1,2,-0.2,-0.15))
> 
> ## a minimum variance 2-additive solution
> min.var <- mini.var.capa.ident(5,2,A.Choquet.preorder = Acp)              
> m <- min.var$solution
> m
		Mobius.capacity
{}		0.000000
{1}		0.291962
{2}		0.183746
{3}		0.149105
{4}		0.036238
{5}		0.398582
{1,2}		-0.183745
{1,3}		0.136610
{1,4}		-0.000000
{1,5}		0.029991
{2,3}		0.005046
{2,4}		0.101568
{2,5}		-0.000000
{3,4}		-0.036237
{3,5}		-0.112867
{4,5}		0.000000
> 
> ## the resulting global evaluations
> rbind(c(a,mean(a),Choquet.integral(m,a)),
+       c(b,mean(b),Choquet.integral(m,b)),
+       c(c,mean(c),Choquet.integral(m,c)),
+       c(d,mean(d),Choquet.integral(m,d)),
+       c(e,mean(e),Choquet.integral(m,e)))
     [,1] [,2] [,3] [,4] [,5] [,6]     [,7]
[1,]   18   11   18   11   11 13.8 15.04374
[2,]   18   18   11   11   11 13.8 13.04374
[3,]   11   11   18   18   11 13.8 12.04374
[4,]   18   11   11   11   18 13.8 16.04374
[5,]   11   11   18   11   18 13.8 14.04374
> 
> ## the Shapley value
> Shapley.value(m)
         1          2          3          4          5 
0.28339006 0.14518087 0.14538137 0.06890384 0.35714386 
> 
> ## a minimum variance 3-additive more constrained solution
> min.var2 <- mini.var.capa.ident(5,3,
+                                    A.Choquet.preorder = Acp,
+                                    A.Shapley.preorder = Asp)
> m <- min.var2$solution
> m
		Mobius.capacity
{}		0.000000
{1}		0.244167
{2}		0.240948
{3}		0.118006
{4}		0.118006
{5}		0.289053
{1,2}		-0.224250
{1,3}		0.184406
{1,4}		-0.009865
{1,5}		0.156216
{2,3}		0.055736
{2,4}		0.111378
{2,5}		-0.070164
{3,4}		-0.118005
{3,5}		-0.003337
{4,5}		0.001006
{1,2,3}		-0.005691
{1,2,4}		0.086434
{1,2,5}		0.064969
{1,3,4}		0.009865
{1,3,5}		-0.261833
{1,4,5}		-0.109146
{2,3,4}		-0.000039
{2,3,5}		0.021339
{2,4,5}		-0.052047
{3,4,5}		0.152851
> rbind(c(a,mean(a),Choquet.integral(m,a)),
+       c(b,mean(b),Choquet.integral(m,b)),
+       c(c,mean(c),Choquet.integral(m,c)),
+       c(d,mean(d),Choquet.integral(m,d)),
+       c(e,mean(e),Choquet.integral(m,e)))
     [,1] [,2] [,3] [,4] [,5] [,6]     [,7]
[1,]   18   11   18   11   11 13.8 14.82605
[2,]   18   18   11   11   11 13.8 12.82605
[3,]   11   11   18   18   11 13.8 11.82605
[4,]   18   11   11   11   18 13.8 15.82605
[5,]   11   11   18   11   18 13.8 13.82605
> Shapley.value(m)
        1         2         3         4         5 
0.2256191 0.2156191 0.1495690 0.1395690 0.2696237 
> 
> ## a minimum variance 5-additive more constrained solution
> min.var3 <- mini.var.capa.ident(5,5,
+                                    A.Choquet.preorder = Acp,
+                                    A.Shapley.preorder = Asp,
+                                    A.Shapley.interval = Asi,
+                                    A.interaction.preorder = Aip,
+                                    A.interaction.interval = Aii)
> 
> m <- min.var3$solution
> m
		Mobius.capacity
{}		0.000000
{1}		0.277968
{2}		0.277968
{3}		0.117798
{4}		0.116752
{5}		0.141094
{1,2}		-0.277967
{1,3}		0.167917
{1,4}		0.095170
{1,5}		0.287479
{2,3}		0.118908
{2,4}		0.139738
{2,5}		0.098626
{3,4}		-0.099439
{3,5}		0.161934
{4,5}		-0.000044
{1,2,3}		0.096023
{1,2,4}		0.164392
{1,2,5}		-0.024410
{1,3,4}		-0.112483
{1,3,5}		-0.447648
{1,4,5}		-0.211878
{2,3,4}		-0.061228
{2,3,5}		-0.237635
{2,4,5}		-0.094907
{3,4,5}		-0.017268
{1,2,3,4}		-0.113658
{1,2,3,5}		0.184804
{1,2,4,5}		-0.019047
{1,3,4,5}		0.229190
{2,3,4,5}		0.071460
{1,2,3,4,5}		-0.029610
> rbind(c(a,mean(a),Choquet.integral(m,a)),
+       c(b,mean(b),Choquet.integral(m,b)),
+       c(c,mean(c),Choquet.integral(m,c)),
+       c(d,mean(d),Choquet.integral(m,d)),
+       c(e,mean(e),Choquet.integral(m,e)))
     [,1] [,2] [,3] [,4] [,5] [,6]     [,7]
[1,]   18   11   18   11   11 13.8 14.94578
[2,]   18   18   11   11   11 13.8 12.94578
[3,]   11   11   18   18   11 13.8 11.94578
[4,]   18   11   11   11   18 13.8 15.94578
[5,]   11   11   18   11   18 13.8 13.94578
> summary(m)
Shapley value :
        1         2         3         4         5 
0.3000000 0.2900000 0.1194055 0.1094055 0.1811889 

Shapley interaction indices :
            1            2           3           4            5
1          NA -0.150000000  0.02857307  0.03994551  0.069757382
2 -0.15000000           NA  0.05762099  0.11604925 -0.008179734
3  0.02857307  0.057620989          NA -0.14000000 -0.034926277
4  0.03994551  0.116049246 -0.14000000          NA -0.075605322
5  0.06975738 -0.008179734 -0.03492628 -0.07560532           NA

Orness :
0.5381275

Veto indices :
        1         2         3         4         5 
0.5271926 0.5254204 0.4053566 0.4042635 0.4471292 

Favor indices :
        1         2         3         4         5 
0.5978074 0.5870796 0.4939004 0.4824934 0.5293569 

Normalized variance :
0.108694

Normalized entropy :
0.8465558
> 
> 
> 
> 
> cleanEx(); ..nameEx <- "normalize-methods"
> 
> ### * normalize-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: normalize-methods
> ### Title: Normalizes a capacity.
> ### Aliases: normalize normalize-methods normalize,Mobius.capacity-method
> ###   normalize,capacity-method normalize,card.capacity-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(0:15)
> 
> ## normalize it
> is.normalized(mu)
[1] FALSE
> normalize(mu)
		capacity
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.333333
{1,3}		0.400000
{1,4}		0.466667
{2,3}		0.533333
{2,4}		0.600000
{3,4}		0.666667
{1,2,3}		0.733333
{1,2,4}		0.800000
{1,3,4}		0.866667
{2,3,4}		0.933333
{1,2,3,4}		1.000000
> 
> ## a similar example with a Mobius.capacity object
> a <- Mobius(mu)
> is.normalized(a)
[1] FALSE
> normalize(a)
		Mobius.capacity
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.133333
{1,3}		0.133333
{1,4}		0.133333
{2,3}		0.200000
{2,4}		0.200000
{3,4}		0.200000
{1,2,3}		-0.133333
{1,2,4}		-0.133333
{1,3,4}		-0.133333
{2,3,4}		-0.266667
{1,2,3,4}		0.000000
> zeta(normalize(a))
		capacity
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.333333
{1,3}		0.400000
{1,4}		0.466667
{2,3}		0.533333
{2,4}		0.600000
{3,4}		0.666667
{1,2,3}		0.733333
{1,2,4}		0.800000
{1,3,4}		0.866667
{2,3,4}		0.933333
{1,2,3,4}		1.000000
> 
> ## a similar example with a card.capacity object
> mu <- card.capacity(0:6)
> is.normalized(mu)
[1] FALSE
> normalize(mu)
  card.capacity
0     0.0000000
1     0.1666667
2     0.3333333
3     0.5000000
4     0.6666667
5     0.8333333
6     1.0000000
> Mobius(normalize(mu))
  Mobius.card.set.func
0         0.000000e+00
1         1.666667e-01
2         0.000000e+00
3         0.000000e+00
4        -1.110223e-16
5         6.661338e-16
6         0.000000e+00
> 
> 
> 
> 
> cleanEx(); ..nameEx <- "orness-methods"
> 
> ### * orness-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: orness-methods
> ### Title: Orness degree
> ### Aliases: orness orness-methods orness,Mobius.capacity-method
> ###   orness,capacity-method orness,card.capacity-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## the upper capacity
> mu <- capacity(c(0,rep(1,15)))
> 
> ## the Choquet integral w.r.t mu behaves like the maximum operator
> f <- c(0.1,0.1,0,0.9)
> Choquet.integral(mu,f)
[1] 0.9
> 
> ## its orness is 1
> orness(mu)
[1] 1
> 
> ## the same example with a Mobius.capacity object
> a <- Mobius(mu)
> Choquet.integral(a,f)
[1] 0.9
> orness(a)
[1] 1
> 
> ## the same example with a card.capacity object
> mu <- upper.capacity(4)
> Choquet.integral(mu,f)
[1] 0.9
> orness(mu)
[1] 1
> 
> 
> 
> cleanEx(); ..nameEx <- "round-methods"
> 
> ### * round-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: round-methods
> ### Title: Rounding of set function coefficients
> ### Aliases: round-methods round,superclass.set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(0:15/15)
> mu
		capacity
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.333333
{1,3}		0.400000
{1,4}		0.466667
{2,3}		0.533333
{2,4}		0.600000
{3,4}		0.666667
{1,2,3}		0.733333
{1,2,4}		0.800000
{1,3,4}		0.866667
{2,3,4}		0.933333
{1,2,3,4}		1.000000
> round(mu,2)
		capacity
{}		0.000000
{1}		0.070000
{2}		0.130000
{3}		0.200000
{4}		0.270000
{1,2}		0.330000
{1,3}		0.400000
{1,4}		0.470000
{2,3}		0.530000
{2,4}		0.600000
{3,4}		0.670000
{1,2,3}		0.730000
{1,2,4}		0.800000
{1,3,4}		0.870000
{2,3,4}		0.930000
{1,2,3,4}		1.000000
> 
> ## a similar example with a Mobius.capacity object
> a <- Mobius(mu)
> a
		Mobius.capacity
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.133333
{1,3}		0.133333
{1,4}		0.133333
{2,3}		0.200000
{2,4}		0.200000
{3,4}		0.200000
{1,2,3}		-0.133333
{1,2,4}		-0.133333
{1,3,4}		-0.133333
{2,3,4}		-0.266667
{1,2,3,4}		-0.000000
> round(a,1)
		Mobius.capacity
{}		0.000000
{1}		0.100000
{2}		0.100000
{3}		0.200000
{4}		0.300000
{1,2}		0.100000
{1,3}		0.100000
{1,4}		0.100000
{2,3}		0.200000
{2,4}		0.200000
{3,4}		0.200000
{1,2,3}		-0.100000
{1,2,4}		-0.100000
{1,3,4}		-0.100000
{2,3,4}		-0.300000
{1,2,3,4}		-0.000000
> 
> ## a similar example with a card.capacity object
> mu <- uniform.capacity(6)
> mu
  card.capacity
0     0.0000000
1     0.1666667
2     0.3333333
3     0.5000000
4     0.6666667
5     0.8333333
6     1.0000000
> round(mu)
  card.capacity
0             0
1             0
2             0
3             0
4             1
5             1
6             1
> 
> 
> 
> cleanEx(); ..nameEx <- "set.func-class"
> 
> ### * set.func-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: set.func-class
> ### Title: Class "set.func"
> ### Aliases: set.func-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a set function
> mu <- set.func(c(1:8,8:1)/8)
> 
> ## the attributes of the object
> mu@n
[1] 4
> mu@data
 [1] 0.125 0.250 0.375 0.750 0.500 0.875 1.000 0.625 0.625 1.000 0.875 0.500
[13] 0.750 0.375 0.250 0.125
> mu@subsets
 [1]  0  1  2  4  8  3  5  9  6 10 12  7 11 13 14 15
> 
> ## some conversions that cannot work
> ## Not run: as.game(mu)
> ## Not run: as.capacity(mu)
> ## Not run: as.card.set.func(mu)
> 
> ## some tests
> is.cardinal(mu)
[1] FALSE
> is.kadditive(mu,2)
[1] FALSE
> is.monotone(mu)
[1] FALSE
> 
> ## some transformations
> dual(mu)
		game
{}		0.000000
{1}		-0.125000
{2}		-0.250000
{3}		-0.375000
{4}		-0.500000
{1,2}		-0.625000
{1,3}		-0.750000
{1,4}		-0.875000
{2,3}		-0.875000
{2,4}		-0.750000
{3,4}		-0.625000
{1,2,3}		-0.500000
{1,2,4}		-0.375000
{1,3,4}		-0.250000
{2,3,4}		-0.125000
{1,2,3,4}		0.000000
> Mobius(mu)
		Mobius.set.func
{}		0.125000
{1}		0.125000
{2}		0.250000
{3}		0.375000
{4}		0.500000
{1,2}		0.250000
{1,3}		0.250000
{1,4}		0.250000
{2,3}		0.250000
{2,4}		0.000000
{3,4}		-0.250000
{1,2,3}		-1.000000
{1,2,4}		-1.000000
{1,3,4}		-1.000000
{2,3,4}		-1.000000
{1,2,3,4}		2.000000
> k.truncate.Mobius(mu,2)
		Mobius.set.func
{}		0.125000
{1}		0.125000
{2}		0.250000
{3}		0.375000
{4}		0.500000
{1,2}		0.250000
{1,3}		0.250000
{1,4}		0.250000
{2,3}		0.250000
{2,4}		0.000000
{3,4}		-0.250000
> 
> ## summary 
> Shapley.value(mu)
            1             2             3             4 
 0.000000e+00 -6.938894e-18  1.387779e-17  0.000000e+00 
> interaction.indices(mu)
            1           2           3           4
1          NA -0.08333333 -0.08333333 -0.08333333
2 -0.08333333          NA -0.08333333 -0.33333333
3 -0.08333333 -0.08333333          NA -0.58333333
4 -0.08333333 -0.33333333 -0.58333333          NA
> # the same
> summary(mu)
Shapley value :
            1             2             3             4 
 0.000000e+00 -6.938894e-18  1.387779e-17  0.000000e+00 

Shapley interaction indices :
            1           2           3           4
1          NA -0.08333333 -0.08333333 -0.08333333
2 -0.08333333          NA -0.08333333 -0.33333333
3 -0.08333333 -0.08333333          NA -0.58333333
4 -0.08333333 -0.33333333 -0.58333333          NA
> 
> ## save the set function to a file
> d <- to.data.frame(mu)
> write.table(d,"my.set.func.csv",sep="\t")
> 
> # finally, some conversions that should work
> mu <- set.func(c(0,1,1,1,2,2,2,3))
> as.game(mu)
		game
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		2.000000
{1,3}		2.000000
{2,3}		2.000000
{1,2,3}		3.000000
> as.capacity(mu)
		capacity
{}		0.000000
{1}		1.000000
{2}		1.000000
{3}		1.000000
{1,2}		2.000000
{1,3}		2.000000
{2,3}		2.000000
{1,2,3}		3.000000
> as.card.set.func(mu)
  card.set.func
0             0
1             1
2             2
3             3
> 
> 
> 
> cleanEx(); ..nameEx <- "set.func.gen"
> 
> ### * set.func.gen
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: set.func
> ### Title: Create objects of class "set.func", "game", or "capacity".
> ### Aliases: set.func game capacity
> ### Keywords: math
> 
> ### ** Examples
> 
> set.func(-12:3)
		set.func
{}		-12.000000
{1}		-11.000000
{2}		-10.000000
{3}		-9.000000
{4}		-8.000000
{1,2}		-7.000000
{1,3}		-6.000000
{1,4}		-5.000000
{2,3}		-4.000000
{2,4}		-3.000000
{3,4}		-2.000000
{1,2,3}		-1.000000
{1,2,4}		0.000000
{1,3,4}		1.000000
{2,3,4}		2.000000
{1,2,3,4}		3.000000
> game(c(0,rep(-1,15)))
		game
{}		0.000000
{1}		-1.000000
{2}		-1.000000
{3}		-1.000000
{4}		-1.000000
{1,2}		-1.000000
{1,3}		-1.000000
{1,4}		-1.000000
{2,3}		-1.000000
{2,4}		-1.000000
{3,4}		-1.000000
{1,2,3}		-1.000000
{1,2,4}		-1.000000
{1,3,4}		-1.000000
{2,3,4}		-1.000000
{1,2,3,4}		-1.000000
> capacity(0:15)
		capacity
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		5.000000
{1,3}		6.000000
{1,4}		7.000000
{2,3}		8.000000
{2,4}		9.000000
{3,4}		10.000000
{1,2,3}		11.000000
{1,2,4}		12.000000
{1,3,4}		13.000000
{2,3,4}		14.000000
{1,2,3,4}		15.000000
> 
> 
> 
> cleanEx(); ..nameEx <- "show-methods"
> 
> ### * show-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: show-methods
> ### Title: Methods for Function show in Package `kappalab'
> ### Aliases: show-methods show,superclass.set.func-method
> ###   show,set.func-method show,Mobius.set.func-method
> ###   show,summary.superclass.set.func-method
> ###   show,summary.superclass.capacity-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a set function
> mu <- set.func(0:15/15)
> show(mu)
		set.func
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.333333
{1,3}		0.400000
{1,4}		0.466667
{2,3}		0.533333
{2,4}		0.600000
{3,4}		0.666667
{1,2,3}		0.733333
{1,2,4}		0.800000
{1,3,4}		0.866667
{2,3,4}		0.933333
{1,2,3,4}		1.000000
> ## the same
> mu
		set.func
{}		0.000000
{1}		0.066667
{2}		0.133333
{3}		0.200000
{4}		0.266667
{1,2}		0.333333
{1,3}		0.400000
{1,4}		0.466667
{2,3}		0.533333
{2,4}		0.600000
{3,4}		0.666667
{1,2,3}		0.733333
{1,2,4}		0.800000
{1,3,4}		0.866667
{2,3,4}		0.933333
{1,2,3,4}		1.000000
> 
> ## a Möbius transform
> a <- Mobius.set.func(0:10,4,2)
> show(a)
		Mobius.set.func
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		5.000000
{1,3}		6.000000
{1,4}		7.000000
{2,3}		8.000000
{2,4}		9.000000
{3,4}		10.000000
> a
		Mobius.set.func
{}		0.000000
{1}		1.000000
{2}		2.000000
{3}		3.000000
{4}		4.000000
{1,2}		5.000000
{1,3}		6.000000
{1,4}		7.000000
{2,3}		8.000000
{2,4}		9.000000
{3,4}		10.000000
> 
> ## a cardinal capacity
> mu <- lower.capacity(5)
> show(mu)
  card.capacity
0             0
1             0
2             0
3             0
4             0
5             1
> mu
  card.capacity
0             0
1             0
2             0
3             0
4             0
5             1
> 
> 
> 
> cleanEx(); ..nameEx <- "summary.superclass.capacity-class"
> 
> ### * summary.superclass.capacity-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: summary.superclass.capacity-class
> ### Title: Class "summary.superclass.capacity"
> ### Aliases: summary.superclass.capacity-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(c(0:13,13,13)/13)
> a <- Mobius(mu)
> 
> ## its summary
> summary(mu)
Shapley value :
        1         2         3         4 
0.1346154 0.2115385 0.2884615 0.3653846 

Shapley interaction indices :
            1           2           3           4
1          NA -0.02564103 -0.02564103 -0.02564103
2 -0.02564103          NA -0.06410256 -0.06410256
3 -0.02564103 -0.06410256          NA -0.06410256
4 -0.02564103 -0.06410256 -0.06410256          NA

Orness :
0.5705128

Veto indices :
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 

Favor indices :
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 

Normalized variance :
0.1439842

Normalized entropy :
0.8073944
> summary(a)
Shapley value :
        1         2         3         4 
0.1346154 0.2115385 0.2884615 0.3653846 

Shapley interaction indices :
            1           2           3           4
1          NA -0.02564103 -0.02564103 -0.02564103
2 -0.02564103          NA -0.06410256 -0.06410256
3 -0.02564103 -0.06410256          NA -0.06410256
4 -0.02564103 -0.06410256 -0.06410256          NA

Orness :
0.5705128

Veto indices :
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 

Favor indices :
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 

Normalized variance :
0.1439842

Normalized entropy :
0.8073944
> 
> 
> 
> cleanEx(); ..nameEx <- "summary.superclass.set.func-class"
> 
> ### * summary.superclass.set.func-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: summary.superclass.set.func-class
> ### Title: Class "summary.superclass.set.func"
> ### Aliases: summary.superclass.set.func-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## a capacity
> mu <- set.func(c(0:13,13,13)/13)
> a <- Mobius(mu)
> 
> ## its summary
> summary(mu)
Shapley value :
        1         2         3         4 
0.1346154 0.2115385 0.2884615 0.3653846 

Shapley interaction indices :
            1           2           3           4
1          NA -0.02564103 -0.02564103 -0.02564103
2 -0.02564103          NA -0.06410256 -0.06410256
3 -0.02564103 -0.06410256          NA -0.06410256
4 -0.02564103 -0.06410256 -0.06410256          NA
> summary(a)
Shapley value :
        1         2         3         4 
0.1346154 0.2115385 0.2884615 0.3653846 

Shapley interaction indices :
            1           2           3           4
1          NA -0.02564103 -0.02564103 -0.02564103
2 -0.02564103          NA -0.06410256 -0.06410256
3 -0.02564103 -0.06410256          NA -0.06410256
4 -0.02564103 -0.06410256 -0.06410256          NA
> 
> 
> 
> cleanEx(); ..nameEx <- "superclass.capacity-class"
> 
> ### * superclass.capacity-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: superclass.capacity-class
> ### Title: Class "superclass.capacity"
> ### Aliases: superclass.capacity-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## three capacities
> mu1 <- uniform.capacity(5)
> mu2 <- capacity(c(0,0,0:13))
> a <- Mobius(mu2)
> 
> ## compute indices to summarize them
> summary(mu1)
Shapley value :
All 
0.2 

Shapley interaction indices :
              All
All -1.387779e-17

Orness :
0.5

Veto indices :
All 
0.5 

Favor indices :
All 
0.5 

Normalized variance :
0

Normalized entropy :
1
> summary(mu2)
Shapley value :
   1    2    3    4 
1.75 2.75 3.75 4.75 

Shapley interaction indices :
          1         2         3         4
1        NA 0.3333333 0.3333333 0.3333333
2 0.3333333        NA 0.8333333 0.8333333
3 0.3333333 0.8333333        NA 0.8333333
4 0.3333333 0.8333333 0.8333333        NA

Orness :
0.4294872

Veto indices :
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 

Favor indices :
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 

Normalized variance :
0.1439842

Normalized entropy :
0.8073944
> summary(a)
Shapley value :
   1    2    3    4 
1.75 2.75 3.75 4.75 

Shapley interaction indices :
          1         2         3         4
1        NA 0.3333333 0.3333333 0.3333333
2 0.3333333        NA 0.8333333 0.8333333
3 0.3333333 0.8333333        NA 0.8333333
4 0.3333333 0.8333333 0.8333333        NA

Orness :
0.4294872

Veto indices :
        1         2         3         4 
0.4871795 0.5470085 0.5982906 0.6495726 

Favor indices :
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 

Normalized variance :
0.1439842

Normalized entropy :
0.8073944
> 
> 
> 
> cleanEx(); ..nameEx <- "superclass.set.func-class"
> 
> ### * superclass.set.func-class
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: superclass.set.func-class
> ### Title: Class "superclass.set.func"
> ### Aliases: superclass.set.func-class
> ### Keywords: classes
> 
> ### ** Examples
> 
> ## three set functions
> mu1 <- card.set.func(-2:4)
> mu2 <- set.func(c(-2,-2,-2:11/11))
> mu3 <- Mobius.set.func(c(-7:6,6,6),4,4)
> 
> ## print mu1
> show(mu1)
  card.set.func
0            -2
1            -1
2             0
3             1
4             2
5             3
6             4
> ## the same
> mu1
  card.set.func
0            -2
1            -1
2             0
3             1
4             2
5             3
6             4
> ## the others
> mu2
		set.func
{}		-2.000000
{1}		-2.000000
{2}		-0.181818
{3}		-0.090909
{4}		0.000000
{1,2}		0.090909
{1,3}		0.181818
{1,4}		0.272727
{2,3}		0.363636
{2,4}		0.454545
{3,4}		0.545455
{1,2,3}		0.636364
{1,2,4}		0.727273
{1,3,4}		0.818182
{2,3,4}		0.909091
{1,2,3,4}		1.000000
> mu3
		Mobius.set.func
{}		-7.000000
{1}		-6.000000
{2}		-5.000000
{3}		-4.000000
{4}		-3.000000
{1,2}		-2.000000
{1,3}		-1.000000
{1,4}		0.000000
{2,3}		1.000000
{2,4}		2.000000
{3,4}		3.000000
{1,2,3}		4.000000
{1,2,4}		5.000000
{1,3,4}		6.000000
{2,3,4}		6.000000
{1,2,3,4}		6.000000
> 
> ## round mu2
> round(mu2,2)
		set.func
{}		-2.000000
{1}		-2.000000
{2}		-0.180000
{3}		-0.090000
{4}		0.000000
{1,2}		0.090000
{1,3}		0.180000
{1,4}		0.270000
{2,3}		0.360000
{2,4}		0.450000
{3,4}		0.550000
{1,2,3}		0.640000
{1,2,4}		0.730000
{1,3,4}		0.820000
{2,3,4}		0.910000
{1,2,3,4}		1.000000
> 
> ## compute indices to summarize them
> summary(mu1)
Shapley value :
All 
  1 

Shapley interaction indices :
    All
All   0
> summary(mu2)
Shapley value :
        1         2         3         4 
0.1590909 0.8560606 0.9469697 1.0378788 

Shapley interaction indices :
           1           2           3           4
1         NA  0.03030303  0.03030303  0.03030303
2 0.03030303          NA -0.83333333 -0.83333333
3 0.03030303 -0.83333333          NA -0.83333333
4 0.03030303 -0.83333333 -0.83333333          NA
> summary(mu3)
Shapley value :
        1         2         3         4 
-1.000000  2.000000  4.333333  6.666667 

Shapley interaction indices :
    1   2  3    4
1  NA 4.5  6  7.5
2 4.5  NA  8  9.5
3 6.0 8.0 NA 11.0
4 7.5 9.5 11   NA
> 
> 
> 
> cleanEx(); ..nameEx <- "to.data.frame-methods"
> 
> ### * to.data.frame-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: to.data.frame-methods
> ### Title: Puts a set function under the form of a data.frame
> ### Aliases: to.data.frame to.data.frame-methods
> ###   to.data.frame,Mobius.card.set.func-method
> ###   to.data.frame,Mobius.set.func-method
> ###   to.data.frame,card.set.func-method to.data.frame,set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## the Möbius representation of set function
> a <- Mobius.set.func(-1:-16,4,4)
> 
> ## to data.frame
> d <- to.data.frame(a)
> write.table(d,"my.set.func.csv",sep="\t") 
> 
> 
> 
> cleanEx(); ..nameEx <- "variance-methods"
> 
> ### * variance-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: variance-methods
> ### Title: Normalized variance of a capacity
> ### Aliases: variance variance-methods variance,Mobius.capacity-method
> ###   variance,capacity-method variance,card.capacity-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(c(0,0,0,0:12)/12)
> 
> ## its Möbius transform
> a <- Mobius(mu)
> 
> ## their normalized variance
> variance(mu)
[1] 0.1790123
> variance(a)
[1] 0.1790123
> 
> ## similar examples with card.capacity objects
> mu <- lower.capacity(4)
> variance(mu)
[1] 1
> mu <- uniform.capacity(4)
> variance(mu)
[1] 0
> 
> 
> 
> cleanEx(); ..nameEx <- "veto-methods"
> 
> ### * veto-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: veto-methods
> ### Title: Veto indices
> ### Aliases: veto veto-methods veto,Mobius.capacity-method
> ###   veto,capacity-method veto,card.capacity-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## a capacity
> mu <- capacity(c(0:13,13,13)/13)
> 
> ## its Möbius transform
> a <- Mobius(mu)
> 
> ## their veto indices
> veto(mu)
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 
> veto(a)
        1         2         3         4 
0.3589744 0.4017094 0.4529915 0.5042735 
> 
> ## the same with a card.capacity object
> mu <- lower.capacity(4)
> veto(mu)
All 
  1 
> 
> 
> 
> cleanEx(); ..nameEx <- "zeta-methods"
> 
> ### * zeta-methods
> 
> flush(stderr()); flush(stdout())
> 
> ### Name: zeta-methods
> ### Title: The zeta transform
> ### Aliases: zeta zeta-methods zeta,Mobius.capacity-method
> ###   zeta,Mobius.card.set.func-method zeta,Mobius.game-method
> ###   zeta,Mobius.set.func-method
> ### Keywords: methods
> 
> ### ** Examples
> 
> ## the Möbius transform of a capacity
> a <- Mobius.capacity(c(rep(0,15),1),4,4)
> a
		Mobius.capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		0.000000
{4}		0.000000
{1,2}		0.000000
{1,3}		0.000000
{1,4}		0.000000
{2,3}		0.000000
{2,4}		0.000000
{3,4}		0.000000
{1,2,3}		0.000000
{1,2,4}		0.000000
{1,3,4}		0.000000
{2,3,4}		0.000000
{1,2,3,4}		1.000000
> 
> ## its zeta transform
> zeta(a)
		capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		0.000000
{4}		0.000000
{1,2}		0.000000
{1,3}		0.000000
{1,4}		0.000000
{2,3}		0.000000
{2,4}		0.000000
{3,4}		0.000000
{1,2,3}		0.000000
{1,2,4}		0.000000
{1,3,4}		0.000000
{2,3,4}		0.000000
{1,2,3,4}		1.000000
> 
> ## let us check
> Mobius(zeta(a))
		Mobius.capacity
{}		0.000000
{1}		0.000000
{2}		0.000000
{3}		0.000000
{4}		0.000000
{1,2}		0.000000
{1,3}		0.000000
{1,4}		0.000000
{2,3}		0.000000
{2,4}		0.000000
{3,4}		0.000000
{1,2,3}		0.000000
{1,2,4}		0.000000
{1,3,4}		0.000000
{2,3,4}		0.000000
{1,2,3,4}		1.000000
> 
> ## a similar example with a Mobius.card.set.func object
> mu <- card.set.func(-3:4)
> a <- Mobius(mu)
> zeta(a)
  card.set.func
0            -3
1            -2
2            -1
3             0
4             1
5             2
6             3
7             4
> 
> 
> 
> ### * <FOOTER>
> ###
> cat("Time elapsed: ", proc.time() - get("ptime", env = .CheckExEnv),"\n")
Time elapsed:  6.45 0.35 6.81 0 0 
> grDevices::dev.off()
null device 
          1 
> ###
> ### Local variables: ***
> ### mode: outline-minor ***
> ### outline-regexp: "\\(> \\)?### [*]+" ***
> ### End: ***
> quit('no')