Sierpinski (<1K) Fuzzy Multivectors

Fuzzy Logic

    A general rule of thumb for understanding fuzzy math, carrying over from Boolean logic, is:
     "and is inrsection is Min ; or is union is  Max".
    Min is typicaly denoted by Ù and Max by Ú in the fuzzy literature, but we will not follow this convention here.

Fuzzy Sets
    In Classical Set Theory a subset A of a set X can be defined via a Boolean valued membership function A : X ® {0,1} with A(x) = 1 Û x Î A. Generalising to a real-valued memebership function A: X ® [0,1] Ì Â we obtain the notion of a fuzzy set A of X. If A(x)=0 then x) is not in A;if A(x)=1 then x) is in A;  if A(x) has some intermediary value then x is considered tobe partially a member of A with A(x) representing a fractional degree of membership. This is signifiacntly distinct from the probability of full membership. Probability distributions sum to unity, whereas membership functions attain without exceeding unity. We will henceforth avoid using the unqualified term set and refer always to either fuzzy set or crisp set. A crisp set is a classical set with a membership function range discretisation to {0,1}.
    An a-cut of a fuzzy set A of X is a crisp set aa(A) =  { x Î X : A(x) > a }   for some 0 £ a £ 1. The 0-cut of a fuuzzy set is known as it's support. While the 1-cut of any fizzy set is f, the 1--cut =  { x Î X : A(x) n= 1 } is known as the core

    A fuzzy set A of X is a fuzzy subset of fuzzy set B of X if A(x) £ B(x) " x Î X. We write A Ì B.

    A fuzzy relatiion between two crisp sets X and Y is a fuzzy set R of X×Y, ie. a membership function R: X×Y ® [0,1] . When Y=X we have a fuzzy equivalence relation on X if the fuzzy relation is reflexive (R(x,x)=1 " xÎX), symmetric (R(x1,x2)=R(x2,x1) " x1,x2 Î X), and transitive (x1x2) ³ Min{ x1x3), x1x3) } " x3ÎX.
    Given fuzzy sets A on X and B on Y we have an induced fuzzy relation (A×B)(x,y)   =   Min{ A(x), B(y) }. Given fuzzy relations R on X×Y and S on Y×Z we have Min-composition (R_fatdotS)(x,z) º MaxyÎY{ Min{ R(x,y), S(y,z) } } and product-composition (R P =S)(x,z) º Maxy  Î |Y{ R(x,y) S(y,z) }. If X,Y, and Z have finite cardinality and R and S are expressed as real valued matrices then we combine the matrices with the standard matrix product but replacing multiplication with Min (for Min-composition) and addition with Max.

    A fuzzy number is a fuzzy subset of  with bounded support and a convex continuous membership function that attains 1 for just one x Î Â (ie. a singleton core). A fuzzy interval is a fuzzy subset with a a bounded support and a convex continuous membership function that attains unity over a core interval. Convexity ensures that all a-cuts are single intervals, connectedTypical fuzzy number membership functions include triangular spikes and smooth bulges centered around a single value x0 for which A(x0) = 1, and zero far from x0. We might refer to such a fuzzy number as "about x0" or "close to var(x)0" if the support is inextensive. One can define an aritmetic of fuzzy numbers using _newerm(Zadeh's extesion principle). Given a crisp function ¦: Â× ®  such as real aritmetic addition or multiplication and two fuzzy numbers a and b we can define a fuzzy set of Â
    ¦(a,b)(z) º Maxx,y : ¦(x,y)=z[ Min{ a(x), b(y) ] .
    This will generally be a fuzzy number for continous variables x and y (with caveats such as an unbounded support when dividing by a fuzzy number containing zero) but discretisation of x and/or y can result in nonconvex ¦. The non-linear optimisartion problem can also be computationally demanding and it is sometimes prefgerable to work with a-cuts.

    If a1,a2,..,aN are fuzzy numbers and we want to know the l-cut of fuzzy number ¦(a1,a2,...,aN)(z)   =   Maxx1,x2,...,xN : ¦(x1,x2,...,xN)=z[ Min{ a1(x1),a2(x2),...,aN(xN) } ] .  then we compute the N intervals al(a1), al(a2), .. al(aN) . These a subvolume of the domain of ¦ over which continuous ¦ will take its extremal values at the boundary verticies or at one or two of K(l) local extrema. We compute ¦ at these K(l) extrema and at all the 2N corners. The spread of these 2N+K(l) evaluations provides the l-cut of  ¦(a1,a2,...,aN) becausee ¦ attains l at the verticies. We can then categorise ¦ with a "sweep" of alpha-cuts, there being no real requirement for accuracy in membership functions.

Triangular Fuzzy Number Arithmetic
    A fuzzy number with a triangular spike membership function of the form Tria,b,c º (b-a)-1(x-a) for  a£x£b; (c-b)-1(c-x) for b£x£c; 0 elsewhere is known as a triangular fuzzy number. Triangular fuzzy numbers combine arithmetrically as follows:
    Tria1,a2,a3 + Trib1,b2,b3 = Tria1+b1,a2+b2,a3+b3 ;
    Tria1,a2,a3 - Trib1,b2,b3 = Tria1-b1,a2-b2,a3-b3 ;
    The product Tria1,a2,a3 Trib1,b2,b3 is not itself a triangular fuzzy number, having a humped membership function
    ( Tria1,a2,a3 Trib1,b2,b3)(x)   =   ½(a2-a1)-1(b2-b1)-1 ( -((a2-a1)b1 + (b2-b1)a1) + (((a2-a1)b1+(b2-b1)a1)2 - 4(a2-a1)(b2-b1)(a1b1-x))½ for a1b1 £ x £ a2b2 ½(a3-a2)-1(b3-b2)-1 ( -((b3-b2)a3 + (a2-a1)b3) - (((b3-b2)a3+(a3-a2)b3)2 - 4(a3-a2)(b3-b2)(a3b3-x))½ for a2b2 £ x £ a3b3 ; 0 elsewhere . This only holds for positive a1,b1.
    Also only holding for postive a1,b1 we have
    ( Tria1,a2,a3 / Trib1,b2,b3)(x)   =   ((a2-a1)+(b2-b3)x)-1 (b3x-a1) for b3-1a1 £ x £ b2-1a2 ; ((a3-a2)+(b2-b1)x)-1 (a3 - b1x) for b2-1a2 £ x £ b1-1a3 ; 0 elsewhere

Fuzzy Predicates
    Consider a fuzzy rule of the form IF x is A Ì X THEN z is C Ì Z ( such as IF Temperature is VERY HOT THEN Fanspeed is HIGH where VERY HOT is a linguistic variable fuzzy set).
    IF x is A THEN conditions on a fuzzy predicate. Rather than Boolean True or False, the condition is fractionally true to degree A(x).
    z is C is a fuzzy consequnce. We assign to z the fuzzy value C with degree A(x)
    Viewed as a logical predicate and recalling from Classical Logic that (P Þ Q) = Q  |  ( !  P) we can regard (i) as a fuzzy relation R : set(X)×Z ® [0,1] defined by
    R(x,y)   =   Max{ Min{ A(x), C(z) }, 1 - A(x) } .
    Alternative implication predicate relations mentioned in the literature are
    Mandini Implication: R(x,y)   =   Min{ A(x), C(z) }  
    Lukasiewicz Implication: R(x,y)   =   Min{ 1, 1 - A(x) + C(z) }  
    Product Implication: R(x,y)   =   A(x) C(z) .  
    Brouwerian Implication: R(x,y)   =   C(z) if  A(x) £ C(z) ; 1 else. .  

    A compound antecedent such as
    IF (x is A1 AND x is A2) OR x is A3 THEN  can be replaced by IF x is ((A1 Ç A2) È A3) THEN where fuzzy set (A1 Ç A2) È A3 has membership function
     ((A1 Ç A2) È A3)(x)   =    Max{ Min{ A1(x), A2(x) }, A3(x) }
    More generally, we have ((A1 Ç A2) È A3)(x)   =    S( T( A1(x), A2(x)),  A3(x) ) where T: [0,1]×[0,1] ® [0,1] is a symmetric, associative t-norm function providing fuzzy intersection (fuzzy and), and S: [0,1]×[0,1] ® [0,1] is a symmetric, associative t-conorm function proving a fuzzy union (fuzzy or). b2 We require t-norms to be monotonic ( T(x,y) £ T(x',y') if x£x' and y£y') and satisfy T(x,1) = x   so as to provide an "alternative or"; and t-conorms to be similarly monontonuic and satisfy S(x,0) = x so as to provide an "alternative and"

    Min is the maximal t-norm in that T(x,y) £ Min{x,y}. Weak(x,y) = Min{x,y} if Max{x,y}=1, 0 else is the minimal t-norm. Intermediary t-norms include the product xy and the Hamacher t-norm Hg(x,y) = (g + (1-g)(x+y-xy))-1 xy for g³| 0.
    Max is the minimal t-conorm.  Strong(x,y) = Max{x,y} if Min{x,y}=0, 1 else is the maximimal t-conorm. Any given t-norm T has an associated derived t-conorm S(x,y) = 1 - T(1-x,1-y).

Fuzzy Rules
    Consider N fuzzy rules: IF x1 is X1 THEN z is Z1 ALSO IF x2 is X2 THEN z is Z2 ... ALSO IF xN is XN THEN z is ZN
    These are to be considered simultaneously true rules of fuzzy logic rather than sequential assignments. If xi is observed to have crisp value x then we consider rule i to fire with crisp real strength bi = Xi(a0) and output Z1 . We then assign to z some form of strength wieghted fuzzy aggregation ZS of Z1 and Z2, which we may ultimately defuzzify to a crisp value z representative of ZS.
    The wide variety of possible output aggregation and defuzzification strategies combined with the myriad choices of antecedent aggregation t-(co)norms combine to allow myriad alternate fuzzy logic schemeta. Combinned with the infinite potential for variouos antecedent fuzzy set membership functions we have an immensely rich modelling space.
    Mamdami inference uses ZS(z)   =   Maxi{ Min{ bi, Zi(z) } }
    Larsen inference uses ZS(z)   =   Maxi{ bi Zi(z) }
    Tsukamoto inference requires monotonic Zi and outputs crisp zS(z)   =   (åi=1N bi)-1 (åi=1N bi Zi-1(bi)) Maxi{ bi, Zi(z) } .
    Sugeno inference uses crisp output rules of the form IF xi is Xi THEN z = ¦i(xi) where ¦i is a crisp function of the crisp input. The output aggregation is thus a crisply weighted aggregation of crisp ouputs, avoiding fuzzy aggregation and defuzzification, restricting fuzzy analysis to the antecdents.sss

    For example, suppose we wish to model a function ¦: Â × Â ® Â , perhaps given multiple (x,y,z=¦(x,y)) samples. We can "cover" the domain of x,y with a number R of crisp real constant output fuzzy rules of the form
    IF x is NEAR xi AND y is NEAR yi then z is zi Using Gaussian member functions for each NEAR premise means the ith rule fires with degree
    bi(x,y)   =   T((½(sx,i-1(x-xi))2) , (½(sy,i-1(y-yi))2) ) where T(,) is the t-norm used for AND  and  sx,i, sy,i define the "spread" of the ith rule.
    Using the product t-norm gives
    bi(x,y)   =   (½(sx,i-1(x-xi))2 + sy,i-1(y-yi))2) . with the a-cuts of the compound premises being x,y axis aligned ellipses. Letting Q denote a 5R dimensional parameter vector containing the xi,yi,zi,sx,i, sy,i parameters, we have using product implication and center average defuzzification the ¦ approximator
    ¦Q(x,y)   =   (åj=1R bj(x,y))-1 (åj=1R bj(x,y)zj).
    Note that ¦(xi,yi) ¹ zi in general.

Fuzzy Multivectors
    There are two aproaches to fuzzing multivectors:
    (i) Replacing some or all of the 2N real coordinates with fuzzy real numbers and using fuzzy aritmetic in the geometric product. This is easily implimented given fuzzy real aritmetic but frame dependent and triangular fuzzy number cooprdinates wuill not remain traingular for long
    (ii) Defining a fuzzy multivector via a membership function a : Âp,q,r ® [0,1] and hoping to extend a function ¦ : Âp,q,r × Âp,q,r ® Âp,q,r to fuzzy multivectors with something like
    ¦( a, b)(x) º Max{ x, y : ¦(x,y) = x } [ Min{ a(x), b(y } ] in order to compose fuzzy multivector relations. This is somewhat formidable, however.
    Easier perhaps to use Sugeno fuzzy rule schemata of the form IF x is X THEN z = ¦(x) generating crisp multivector outputs from crisp multivector inputs.

Multivector membership functions
    Consdider a fuzzy point having  membership function mc,s,d(p)   =   Ivl-(s+d)2, -s2,s2,(s+d)2( (pÙc)2) ) where Ivl(a,b,c,d) is a real fuzzy interval of some sort having support [a,d] and core [b,c] . This can be seen as a fuzzy point with as core a solid sphere centered at c with of radius s and support radius s+d.
    If L = cÙdÙe¥ for unit dirertion vector d,  then mL,0,e is a fuzzy line through c having core L and support the solid cylindrical volume of radius e and L as axis.
    T( mc,,s,d) , mL,,0,e) ) where T(x,y) is a t-norm is the membership function for pointset c ÇT L which has as core the line segment from c-sd to c+sd.
    For circle S, the a-cuts of "triangular" membership function mS,,0,s) are torus like isosurfaces enclosing S rather than real intervals, making a-cut aritmetic problematic.
    Alternatively, we might use Gaussian membership function mb(p)   =   (-|(bÙp)2|) where b is a non-unit blade incorporating the desired spread s in it's scaling.

    Suppose we wish to model a scalar field ¦(p) . We could use axis aligned Gaussian ellipsoid patches as in the above Â×® example but degree functions of the form
    (bj(p)   =   (åk=1Kj -½|(bj,kÙp)2|)
    where the bj,k k=1,2,..,Kj are Kj scaled blades associated with the jth fuzzy pointset with defuzzification
    ¦Q(p)   =   (åj=1R bj(p)-1 (åj=1R bj(p)zj). provide more general "patches" .
    Such approximations are chacterised by sets of scaled blades, which can be transformed to transform the field.

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Copyright (c) Ian C G Bell 2015, 2014
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