A Fuzzy Directional Distance Measure
Josie C. McCullochy, Chris J. Hinde, Christian Wagner, Uwe Aickelin
AA Fuzzy Directional Distance Measure
J.C. McCulloch † , Student Member, IEEE , C.J. Hinde ‡ , Senior Member, IEEE ,C. Wagner † , Senior Member, IEEE and U. Aickelin † Abstract —The measure of distance between two fuzzysets is a fundamental tool within fuzzy set theory, however,distance measures currently within the literature use acrisp value to represent the distance between fuzzy sets.A real valued distance measure is developed into a fuzzydistance measure which better reflects the uncertaintyinherent in fuzzy sets and a fuzzy directional distancemeasure is presented, which accounts for the directionof change between fuzzy sets. A multiplicative version isexplored as a full maximal assignment is computationallyintractable so an intermediate solution is offered.
I. I
NTRODUCTION
Distance measures for fuzzy sets (FSs) are an impor-tant tool and have been applied to many fields. There aremany distance measures that are appropriate in differentsituations, for example Mahalanobis distance [1] wasproposed in 1936 and many more are in use today, suchas Chaudhur and Rosenfeld’s [2] distance measure forFSs and work by Dubois [3]. An interesting applicationfor many workers is case based reasoning, Segura et al.present a variety of case based distance measures [4].While distance measures traditionally use a singlereal value to express distance, representing the distanceas a FS would give a richer, more accurate comparison,reflecting the uncertainty inherent in FSs. This workfollows and draws on work by [5], which describes areal-valued directional distance measure, and presents adistance measure which describes distance as a FS. In[5], alpha-cuts ( α -cuts) are used to measure distance bycomparing each α -cut of one FS with the same α -cutof another FS. This, however, introduces difficulties fornon-normal FSs where an α -cut results in the empty set.Though the problem was addressed, the method takenis limited by using a substituted value of distance for α -cuts where one of the fuzzy sets is not present. Themethod introduced in this paper removes this problemby comparing every α -cut (or mass assignment) of one † Department of Computer Science, University of Nottingham,Nottingham, Nottinghamshire, UK (email: psxjm5; christian.wagner;[email protected]). ‡ Department of Computer Science, Loughborough University,Loughborough, Leicestershire, UK (email: [email protected]).This work was partially funded by the EPSRCs TowardsData-Driven Environmental Policy Design grant, EP/K012479/1and the RCUKs Horizon Digital Economy Research Hub grant,EP/G065802/1.
FS with every α -cut of the other FS. This also resultsin a more accurate description of distance. This fuzzydistance measure is achieved using a mass assignment(MA) framework [6], [7].Section II provides a background on MAs and se-mantic unification of FSs which form the basis of thedistance measure. Following this, Sections III and IVintroduce both a non-directional and directional distancemeasure, respectively. Demonstrations of the distancemeasure are then given for non-normal and non-convexfuzzy FSs in Sections V and VI, respectively. Finally,Section VII presents some conclusions.II. B ACKGROUND
A background on MA and semantic unification is pre-sented first. Mass assignment uses a measure of supportbased on semantic unification, [6] that is generalised in[8] and further in [9]. Distance is commonly calculatedusing α -cuts, which are related to MAs, such that theyboth break down the FS along the membership axis. Thecrucial difference is that, using α -cuts, the membershipof an element is ascertained by the maximum α -cut towhich it belongs, whereas using masses the membershipvalue is given by the sum of the masses. As thetwo methods are related, the MA techniques shouldbe applicable to a distance measure just as they areapplicable to a support measure. A. Mass Assignments
Mass is a precise amount of probability assigned toa set of events, rather than individual events. A MAdefined on the domain C is written as [10]: X = X : x , ...., X N : x n where (cid:80) ni =1 x i = 1 . and X i ∈ C (1)where C is the powerset of the domain C , X i isa subset of the domain C , and x i is the amount ofmass assigned to X i . For example, consider the FS F expressed as F = { x, µ F ( x ) | x ∈ X } a r X i v : . [ c s . A I] S e p here X is the discrete space X = { x , x , ....x n } . To calculate the mass of F , its elements are first orderedsuch that [10] µ F ( x i ) ≥ µ F ( x j ) if i < j The MA of the FS F is then calculated as follows [10],[11] m F = {{ x , ..., x i } : µ F ( x i ) − µ F ( x i +1 ) , ∅ : 1 − µ ( x ) } with µ F ( x n +1 ) = 0 (2)Note that if the FS is normalised then the massassigned to the empty set will be 0. For example, giventwo FSs A and GA = { . | a, . | b, . | c } G = { . | a, . | b, . | c } with set of support { a, b, c } , the masses assigned to A and G are m A { A i : a i } and m G { G i : g i } as follows m A = { a } : 0 . , { a, b } : 0 . , { a, b, c } : 0 . m G = { a } : 0 . , { a, b } : 0 . , { a, b, c } : 0 . , ∅ : 0 . Having briefly covered MAs of FSs, the next sectionintroduces semantic unification which will be the basisof the distance measure in this paper.
B. Semantic Unification
Semantic unification assesses the support of a claim A given a ground clause G . As defined in [7], itdoes not deal with claims or ground evidence that areinconsistent. However, an extended version describedin [8] deals with inconsistent FSs, or alternatively non-normalised FSs. This work starts with the extendedversion which is defined as follows for two MAs m A = { A i : a i } and m G = { G j : g j } : T ( A i | G j )= { t } if A i ⊇ G j ∨ ( A i = ∅ ∧ G j = ∅ ) { f } if A i ∩ G j = ∅ ∧ A i (cid:54) = ∅ ∧ G j (cid:54) = ∅∅ if A i (cid:54) = ∅ ∧ G j = ∅{ f, t } otherwise (3)This can be read as • The truth of A is true if G supports A • The truth of A is false if G denies A • The truth of A is unknown if G is unknown (noevidence exists) • The truth of A is inconsistent if G both supportsand denies AAn example of semantic unification using the MAsof A and G detailed above is given in Table I. The calculations multiply the masses of the contributingsets to calculate the mass of the resulting set. Byadding the final masses assigned to each set the result A | G = { t } : 0 . , { f, t } : 0 . , ∅ : 0 . is obtained. TABLE IS
EMANTIC UNIFICATION OPERATOR INCORPORATINGINCONSISTENCY FOR NON - NORMALISED FS S AND ASSIGNINGMASS MULTIPLICATIVELY .G A | G { a } : { a, b } : { a, b, c } : ∅ : { a } : { t } : { f, t } : { f, t } : ∅ : { a, b } : { t } : { t } : { f, t } : ∅ : { a, b, c } : { t } : { t } : { t } : ∅ : Semantic unification thus delivers a FS of truth valuesindicating the degree of support the fuzzy claim A receives from the fuzzy evidence G . Neither FS is nec-essarily normalised and so the FS representing the de-gree of support, similarly, is not necessarily normalised.Though the calculations above multiply the masses tocalculate the mass of the resulting set, this is not themost general answer possible. For example, Table IIshows a possible maximal assignments applied to theFSs A and G . Calculating the maximal assignmentinvolves maximising the value assigned to { f, t } , thenmaximising either to { f } or { t } , and then finally assign-ing mass to ∅ . Maximising first to { f, t } , then { t } , { f } and ∅ results in A | G = { t } : 0 . , { f, t } : 0 . , ∅ : 0 . ,more uncertain than the multiplicative result. TABLE IIS
EMANTIC UNIFICATION OPERATOR INCORPORATINGINCONSISTENCY FOR NON - NORMALISED FS S AND ASSIGNINGMASS MAXIMALLY .G A | G { a } : { a, b } : { a, b, c } : ∅ : { a } : { t } : { f, t } : { f, t } : ∅ : { a, b } : { t } : { t } : { f, t } : ∅ : { a, b, c } : { t } : { t } : { t } : ∅ : III. D
ISTANCE M EASURES
In [5] a distance measure is based on measuring dis-tance between individual α -cuts. As discussed earlier,MA is also based on α -cuts and so the generalisationis straightforward as presented next. For the distancemeasure proposed in this paper it is difficult to obtaina maximal MA in the general case, and even in thease analysed here a full maximal assignment is notavailable; however, a better approximation than themultiplicative case is presented. A. Mass based distance measure
The MA operator based on a non-directional Haus-dorff distance measure [2], [5] is given in (4). UsingMAs, the two intervals a i,l and b j,k are sets of pos-sibilities, such that a i,l represents all points in A i and b j,k represents all points in B j . To calculate the distancebetween two subsets A i and B j the following equationis used: D ( A i , B j ) = (cid:40) {| a i,l − b j,k |} if A i (cid:54) = ∅ ∧ B j (cid:54) = ∅∅ otherwise (4)The non-directional distance measure is altered intoa directional distance measure [5] as follows (5): D ( A i , B j ) = (cid:40) { b j,k − a i,l } if A i (cid:54) = ∅ ∧ B j (cid:54) = ∅∅ otherwise (5)Note that the operation has been reversed from a i,l − b j,k in (4) to b j,k − a i,l within (5), and theabsolute value of the distance is no longer used. Thisis to account for the directional nature of the distancemeasure, and results in MAs assigned to the positivedomain where the FS B is placed to the right of A within the universe of discourse, and MAs in thenegative domain otherwise.Table III shows the calculation of the non-directionaldistance (4) between the two sets A and B as shownin Fig. 1. For simplicity, A and B are two highlydiscretised fuzzy numbers. The MAs of A and B using(2), are m A and m B as follows: m A = [1 . , .
0] : 0 . , [2 . , .
0] : 0 . m B = [6 . , .
0] : 0 . , [7 . , .
0] : 0 . B A Fig. 1. Fuzzy sets A and B . To derive the distance between A and B , the dis-tance measure given in (4) is used and the masses aremultiplied as shown in Table III. TABLE IIID
ISTANCE MEASURE BETWEEN A AND B ASSIGNEDMULTIPLICATIVELY BD ( A, B ) [6.0,10.0]: [7.0,9.0]:0.5 0.5[1.0,5.0]: [1.0,9.0]: [2.0,8.0]: A From Table III, the following MAs and correspondingFS are obtained m D ( A,B ) = [1 . , .
0] : 0 . , [2 . , .
0] : 0 . , [3 . , .
0] : 0 . (6) D ( A, B ) = { . | [1 . , . , . | [2 . , . , . | [3 . , . , . | [7 . , . , . | [8 . , . } (7)Fig. 2 shows the FS representing the distance between A and B with the masses assigned multiplicatively.Note that the smallest distance between any two pointsof A and B is 1 and the largest is 9, both of which areconveyed in the end points of the FS in Fig. 2. Distance A to B
Fig. 2. Distance between FSs A and B shown as a FS. A distance measure between FSs has been introducedusing multiplicative MA, distance using maximal MAsis addressed next .
B. Maximal assignments
The definition of a maximal assignment is one thatcannot be reached by means of restrictions or linearcombination of any of the other possible assignments.The two types of restriction of concern are defined in(8), Type 1, and (9), Type 2. m (cid:48) = { L i : m i } ∪ { L j : m j + x } ∪ { L k : m k − x } L k ⊇ L j , x ≤ m k , i (cid:54) = j, i (cid:54) = k (8) m (cid:48) = { L i : m i } ∪ { L k : m k − x } ∪{ L n : m n − x } ∪ { L u : m u + x }∪ { L p : m p + x } | L u = L k ∪ L n , L p = L k ∩ L n ,L i (cid:54) = L k , L n , L u , L p (9) he distance measures are special cases of MAs. Ifboth numbers are triangular FSs then the final result isalso a triangular FS. Theorem 1:
If the two FSs are similar isosceles tri-angles then all entries in a distance measure assignmentmatrix are subsets, supersets or equal to one another.
Proof:
Let the two triangles A and B be definedby the parameters as below, then the intervals will beof the form: [( B l + B n δ ) − ( A u − A n δ ) , ( B u − B n δ ) − ( A l + A n δ )] which may be rewritten as [( B l − A u + ( B n δ + A n δ ) , ( B u − A l − ( B n δ + A n δ )] where A l and A u , B l and B u are the lower boundand upper bound points of the triangles A and B ,respectively; A n and B n are the heights of the slices measured innumber of slices; δ is the amount the side of the triangle increases witheach slice.The rates of change of each quantity in the intervalsare identical so the result follows immediately. Once δ has been chosen the lower and upper bounds of theintervals are fixed. Corollary 0.1:
Theorem 1 essentially means thereare no type 2 restrictions for similar isosceles triangles.
Theorem 2:
If the two base FSs are not similarisosceles triangles then there may be entries in a dis-tance measure assignment matrix that are overlappingintervals and are not subsets.
Proof:
Let the two triangles be defined by theparameters as below, then the intervals will be of theform: [( B l + B n δ L ) − ( A u − A n δ R ) , ( B u − B n δ R ) − ( A l + A n δ L )] where A n and B n are the heights of the slices measuredin number of slices; δ L is the amount the left hand side of the triangleincreases with each slice; δ R is the amount the right hand side of the triangledecreases with each slice.The two quantities of interest from above are: ( B n δ L ) + A n δ R ) , ( B n δ R + A n δ L ) If δ L > δ R then if B n is reduced by 1 and A n increased by one, ( B n δ L + A n δ R ) will be reduced whilesimultaneously ( B n δ R + A n δ L ) will be raised. Thus, thetwo intervals (10) and (11) overlap. [( B l + B n δ L ) − ( A h − A n δ R ) , ( B h − B n δ R ) − ( A l + A n δ L )] (10) [( B l + ( B n − δ L ) − ( A h − ( A n + 1) δ R ) , ( B h − ( B n − δ R ) − ( A l + ( A n + 1) δ L )] (11) Corollary 0.2:
Theorem 2 essentially means theremay be type 2 restrictions if the two triangles are notsimilar and isosceles.The theorems above show that type 2 restrictionsare likely to occur in many situations. If only type 1restrictions are considered then it is easy to see that thedistance measure between two nested FSs lies down themain diagonal. This is computationally straightforwardand results in more general assignments than the mul-tiplicative assignment. An assumption of independencebetween the two sets is now not necessary.Taking this approach with unification, the mass ismaximally assigned along the diagonal, as shown inTable IV which measures the FSs A and B in Fig. 1. TABLE IVD
ISTANCE MEASURE BETWEEN A AND B ASSIGNED MAXIMALLY BD ( A, B ) [6.0,10.0]: [7.0,9.0]:0.5 0.5[1.0,5.0]: [1.0,9.0]: [2.0,8.0]: A Resulting in m D ( A,B ) = [1 . , .
0] : 0 . , [3 . , .
0] : 0 . (12) D ( A, B ) = { . | [1 . , . , . | [3 . , . , . | [7 . , . } (13)Fig. 3 shows the FS representing the distance between A and B with the masses assigned down the diagonal. Distance A to B
Fig. 3. Distance between FSs A and B obtained down the diagonalshown as a FS. The assignment in Fig. 3 should be restrictable tothe assignment shown in Fig. 2 using type 1 or type2 restrictions, however neither assignment is reachablefrom the other. Alternatively, taking the assignmentfrom the other diagonal gives Table V resulting in theassignment in (14) and (15).Resulting in m D ( A,B ) = [2 . , .
0] : 1 . (14) D ( A, B ) = { . | [2 . , . } (15)Given the two assignments (12) and (14), a linearcombination results in the multiplication assignment (6). ABLE VD
ISTANCE MEASURE BETWEEN A AND B ASSIGNED DOWN THEOTHER DIAGONAL , DOWNWARDS AND RIGHT TO LEFT BD ( A, B ) [6.0,10.0]: [7.0,9.0]:0.5 0.5[1.0,5.0]: [1.0,9.0]: [2.0,8.0]: A There are no type 2 restrictions and the two orthogonalassignments, when linearly combined, result in theproduct assignment. At this point it is unclear that this isa reasonable assumption. Yet, consider a more detailedview of A and B , as AD and BD , shown in Fig. 4 BD AD Fig. 4. Fuzzy sets AD and BD . The MAs of AD and BD , denoted m AD and m BD ,are m AD = [1 . , .
0] : 0 . , [1 . , .
5] : 0 . , [2 . , .
0] : 0 . , [2 . , .
5] : 0 . m BD = [6 . , .
0] : 0 . , [6 . , .
5] : 0 . , [7 . , .
0] : 0 . , [7 . , .
5] : 0 . To derive the distance between AD and BD , thedistance measure given in (4) is used and the massesare multiplied resulting in (16) and (17). m D ( AD,BD ) = [1 . , .
0] : 0 . , [1 . , .
5] : 0 . , [2 . , .
0] : 0 . , [2 . , .
5] : 0 . , [3 . , .
0] : 0 . , [3 . , .
5] : 0 . , [4 . , .
0] : 0 . (16) D ( AD, BD ) = { . | [1 . , . , . | [2 . , . , . | [3 . , . , . | [7 . , . , . | [8 . , . } (17)The assignment down the left to right diagonal isshown in Table VI.From Table VI the following MAs and correspondingFS are obtained m D ( AD,BD ) = [1 . , .
0] : 0 . , [2 . , .
0] : 0 . , [3 . , .
0] : 0 . , [4 . , .
0] : 0 . (18) TABLE VID
ISTANCE MEASURE BETWEEN AD AND BD ASSIGNED DOWNTHE LEFT TO RIGHT DIAGONAL BD D( AD , BD ) [6.0,10.0]: [6.5,9.5]: [7.0,9.0]: [7.5,8.5]:0.25 0.25[1.0,5.0]: [1.0,9.0]: [1.5,8.5]: [2.0,8.0]: [2.5,7.5]:0.25 0.25 0.0 0.0 0.0[1.5,4.5]: [1.5,8.5]: [2.0,8.0]: [2.5,7.5]: [3.0,7.0]:0.25 0.0 0.25 0.0 0.0 AD [2.0,4.0]: [2.0,8.0]: [2.5,7.5]: [3.0,7.0]: [3.5,6.5]:0.25 0.0 0.0 0.25 0.0[2.5,4.5]: [2.5,7.5]: [3.0,7.0]: [3.5,6.5]: [4.0,6.0]:0.25 0.0 0.0 0.0 0.25 D ( AD, BD ) = { . | [1 . , . , . | [2 . , . , . | [3 . , . , . | [7 . , . , . | [8 . , . } (19)Again this does not restrict to the product assignmentand other orthogonal assignments are needed to makeit possible to create a linear combination resulting inthe product assignment. The crucial point to see is thatfour orthogonal assignments are needed to complete theprocess. The number required rises with the numberof slices taken, and the computation quickly becomesintractable.The diagonal assignment is one of the maximalorthogonal set and the assumption now is that takingthis is a better solution than the product MA whenapplied to the distance measure. It should be noted thatthis assumption may not be justified for a general MAoperator. IV. D IRECTIONAL DISTANCE
This section describes the directional version of thedistance operator (5) with examples of the calculation.Referring to the two FSs A and B in Fig. 1, thecalculation matrix shown in Table VII is the distancebetween A and B using the directional distance measure(5). The masses of A and B are as follows m A = [1 . , .
0] : 0 . , [2 . , .
0] : 0 . m B = [6 . , .
0] : 0 . , [7 . , .
0] : 0 . Calculating the distance using (5), as shown in TableVII, results in the following MAs and FS. m D ( A,B ) = [2 . , .
0] : 0 . , [4 . , .
0] : 0 . D ( A, B ) = { . | [2 . , . , . | [4 . , . , . | [6 . , . } Using the same FSs, and thus the same MAs, if thecalculation is reversed to measure the distance from B to A , as shown in Table VIII, the quantities are now ABLE VIIT
HE DIRECTIONAL DISTANCE MEASURE BETWEEN FS S A AND B CALCULATED MAXIMALLY . BD ( A, B ) [6.0,9.0]: [7.0,8.0]:0.5 0.5[1.0,4.0]: [2.0,8.0]: [3.0,7.0]: A IRECTIONAL DISTANCE MEASURE BETWEEN FS S B AND A . AD ( B, A ) [1.0,4.0]: [2.0,3.0]:0.5 0.5[6.0,9.0]: [-2.0,-8.0]: [-3.0,-7.0]: B reversed and the numbers are negative in comparisonto Table VII.The resulting FS is as follows and shown in Fig. 5. m D ( B,A ) = [ − . , − .
0] : 0 . , [ − . , − .
0] : 0 . D ( B, A ) = { . | [ − . , − . , . | [ − . , − . , . | [ − . , − . } Distance B to A
Fig. 5. Distance between FSs B and A shown as a FS. V. N ON - NORMAL FS S AND DISTANCES
This section shows the effect of non-normal FSs inthe distance calculation.
A. General case of non-normal distance
The definition in (4) caters for non-normalised sets soredefining A as AN in Fig. 6 results in the calculationshown in Table IX. In this case the maximal assignmenthas to take care of assignment to the empty set. Themaximal assignment would be to assign the mass alongthe diagonal, in this case the product assignment ismore satisfactory but requires the assumption of inde-pendence to be justified. m AN = [1 . , .
0] : 0 . , [] : 0 . m B = [6 . , .
0] : 0 . , [7 . , .
0] : 0 . AN B Fig. 6. FSs AN and B .TABLE IXM ULTIPLICATIVE DISTANCE MEASURE BETWEEN FS S AN AND B . BD ( AN, B ) [6.0,9.0]: [7.0,8.0]:0.5 0.5[1.0,4.0]: [2.0,8.0]: [3.0,7.0]: AN Table IX results in m D ( AN,B ) = [2 . , .
0] : 0 . , [3 . , .
0] : 0 . , [] : 0 . D ( AN, B ) = { . | [2 . , . , . | [3 . , . , . | [7 . , . } The product fuzzy distance between AN and B isshown pictorially in Fig. 7. Distance AN to B
Fig. 7. Distance between FSs AN and B shown as a FS. The last example, in Fig. 7, shows that the distancebetween numbers where non-normalisation is involvedis itself a non-normalised FS. It makes sense that it isnot possible to measure the distance between sets thatdon’t exist, and so the distance measure itself does notexist either.
B. Maximum likelihood estimates
All the distributions used may be transformed to asingle interval by taking the maximum likelihood, leastprejudiced values [7] and performing the operations onthe transformed values.Transforming FSs A and B in Fig. 1 to maximumlikelihood values results in A = [2 . , . and B =[7 . , . , and the distance between A and B comes outat D ( A, B ) = [4 . , . , which accords with standardinterval arithmetic. It should be noted here that takingthe centres of gravity yields a slightly different answer,with A = 2 . , B = 7 . and D ( A, B ) = 5 . , whichis more precise and also inaccurate as the uncertaintyis not preserved. This will become much clearer whenmultimodal FSs are dealt with in Section VI.I. M ULTIMODAL DISTANCE OF NON - CONVEX SETS
This section describes the effect of calculating thedistance between a bimodal FS and a unimodal FS.The distance between these FSs should intuitively bebimodal, but that is not necessarily the case. For ex-ample, take the two FSs AM and B , shown in Fig.8: m AM = [1 . , .
0] : 0 . , [1 . , . , [3 . , .
0] : 0 . m B = [6 . , .
0] : 0 . , [7 . , .
0] : 0 . B0.0 1.0 2.0 4.03.0 5.0 6.0 7.0 8.0 9.00.01.0 AM
Fig. 8. The multimodal FS AM and B .TABLE XD ISTANCE MEASURE BETWEEN FS S AM AND B . BD ( AM, B ) [6.0,9.0]: [7.0,8.0],0.5 0.5[1.0,4.0]: [2.0,8.0]: [3.0,7.0]: AM The distance is calculated in Table X, resulting in m D ( AM,B ) = [2 . , .
0] : 0 . , [3 . , .
0] : 0 . D ( AM, B ) = { . | [2 . , . , . | [3 . , . , . | [7 . , . } Distance AM to B0.0 1.0 2.0 4.03.0 5.0 6.0 7.0 8.0 9.00.01.0
Fig. 9. The unimodal FS showing the distance between themultimodal FS AM and B . The measure between AM and B , shown in Fig. 9,results in a wider FS than the measure between A and B , shown in Fig. 5, because of the uncertainty about AM . However, it is not multimodal. Extending thewidth of AM to AE (see Fig. 10) results in a bimodaldistance measure. m AE = [1 . , .
0] : 0 . , [1 . , . , [4 . , .
0] : 0 . m B = [6 . , .
0] : 0 . , [7 . , .
0] : 0 . AE0.0 1.0 2.0 4.03.0 5.0 6.0 7.0 8.0 9.00.01.0 B
Fig. 10. The multimodal FS AE ( AM extended) and B .TABLE XIE XAMPLE DISTANCE MEASURE BETWEEN FS S AE AND B . BD ( AE, B ) [6.0,9.0]: [7.0,8.0],0.5 0.5[1.0,5.0]: [1.0,8.0]: [2.0,7.0]: AE XAMPLE DISTANCE MEASURE BETWEEN FS S AE AND B , SIMPLIFIED . BD ( AE, B ) [6.0,9.0]: [7.0,8.0],0.5 0.5[1.0,5.0]: [1.0,8.0]: [2.0,7.0]: AE Simplifying the intervals in Table XI gives Table XII,resulting in m D ( AE,B ) = [1 . , .
0] : 0 . , , [2 . , . , [5 . , .
0] : 0 . ,D ( AE, B ) = { . | [1 . , . , . | [2 . , . , . | [4 . , . , . | [5 . , . , . | [7 . , . } The resulting distance measure FS between AE and B is shown in Fig. 11. Distance AE to B0.0 1.0 2.0 4.03.0 5.0 6.0 7.0 8.0 9.00.01.0
Fig. 11. The multimodal FS of the distance between AE and B . A. Distances between non-normal multimodal FSs
This section describes the effect of calculating thedistance between a bimodal FS and a unimodal FS,where one of the bimodal modes is not normal. Takehe two FSs
AEN and B , shown in Fig. 12. Thishas the slices at different levels and the diagonal rulecannot be directly applied to arrive at the assignmentrequired. Slices must be taken at the same level. TableXIII rectifies this and the assignment is now down thediagonal. AEN0.0 1.0 2.0 4.03.0 5.0 6.0 7.0 8.0 9.00.01.0 B
Fig. 12. The multimodal FSs
AEN and B . m AEN = [1 . , .
0] : 0 . , [1 . , . , [4 . , .
0] : 0 . , [1 . , .
0] : 0 . TABLE XIIIE
XAMPLE MAXIMAL DISTANCE MEASURE BETWEEN FS S AEN
AND B . B D( AEN , B ) [6.0,9.0]: [7.0,8.0]: [7.0,8.0]:0.5 0.25 0.25[1.0,5.0]: [1.0,8.0]: [2.0,7.0]: [2.0,7.0]:0.5 0.5 0.0 0.0[1.0,2.0], [4.0,8.0], [5.0,7.0], [5.0,7.0], AEN [4.0,5.0]: [1.0,5.0]: [2.0,4.0]: [2.0,4.0]:0.25 0.0 0.25 0.0[1.0,2.0], [4.0,8.0]: [5.0,7.0]: [5.0,7.0]:0.25 0.0 0.0 0.25
Resulting in m D ( AEN,B ) = [1 . , .
0] : 0 . , [2 . , . , [5 . , .
0] : 0 . , [5 . , .
0] : 0 . D ( AEN, B ) = { . | [1 . , . , . | [2 . , . , . | [4 . , . , . | [5 . , . , . | [7 . , . } The maximal distance measure between AEN and B,see Table XIII, results in a bimodal FS, see Fig. 13,with one mode lower than the other.VII. C
ONCLUSIONS
This paper has introduced MA based distance mea-sures extending the work reported in [5]. The distanceresults are FSs and calculating the maximum likelihood
Distance AEN to B0.0 1.0 2.0 4.03.0 5.0 6.0 7.0 8.0 9.00.01.0
Fig. 13. The multimodal FS showing the maximally calculateddistance between
AEN , AM extended and not normalised, and B . values from the sets indicates that the measures accordwith intuition, and is a better result than the centre ofgravity approach. Ignoring the type 2 restrictions is anassumption that is likely to be broken often, howeverthe result is computable directly and is more generaland easier than the multiplicative method. The num-ber of orthogonal assignments rises with the increasedprecision of the FS leading an assignment down thediagonal being a less restrictive assignment but is auseful compromise.Demonstrations have shown the effects with bothnormal and non-normal as well as convex and non-convex FSs, and though the paper has dealt with veryblocky FSs which simplifies the calculations, the workgeneralises to countably continuous FSs.R EFERENCES[1] P. Mahalanobis, “On the generalized distance in statistics,”
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