New definitions (measures) of skewness, mean and dispersion of fuzzy numbers -- by way of a new representation as parameterized curves
NNew definitions (measures) of skewness, mean anddispersion of fuzzy numbers.- by way of a new representation as parameterized curves. (cid:63)
Jan Schneider
Faculty of Computer Science and Management, Wrocław University of Science and Technology, ul.Ignacego Łukasiewicza 5, 50-371 Wrocław, Poland
Abstract
We give a geometrically motivated measure of skewness, define a mean value trianglenumber, and dispersion (in that order) of a fuzzy number without reference or seekinganalogy to the namesake but parallel concepts in probability theory. These measurescome about by way of a new representation of fuzzy numbers as parameterized curvesrespectively their associated tangent bundle. Importantly skewness and dispersion aregiven as functions of α (the degree of membership) and such may be given separatelyand pointwise at each α -level, as well as overall. This allows for e.g., when a mathe-matical model is formulated in fuzzy numbers, to run optimization programs level-wisethereby encapsuling with deliberate accuracy the involved membership functions’ char-acteristics while increasing the computational complexity by only a multiplicative factorcompared to the same program formulated in real variables and parameters.As an example the work o ff ers a contribution to the recently very popular fuzzymean-variance-skewness portfolio optimization. Keywords: fuzzy numbers, parametric curves, polar coordinates, skewness of a fuzzynumber, dispersion of a fuzzy number, portfolio optimization
Contents1 Introduction 2 (cid:63)
This work was supported by the National Science Center (Poland), under Grant 394311,2017 / / B / HS4 / Email address: [email protected] (Jan Schneider)
Preprint submitted to Elsevier November 3, 2020 a r X i v : . [ s t a t . O T ] O c t .2.1 Transferring concepts of center variability and shape from prob-ability into the theory of fuzzy sets . . . . . . . . . . . . . . . . 4 ff distance from mean value triangle 267 Fuzzy skewness in Portfolio Optimization Theory 28 In a natural language the term “skewness” is in itself a vague concept, but when itoccurs in conversation is well understood intuitively by speakers of the English languageas “deviating from symmetry”, “ deviating from a straight line”, “slant to the left or theright”, obviously without mathematically specifying its meaning.In scientific context the term is most often associated on the grounds of probabilitytheory and statistics where many mathematical concepts have been introduced to cap-ture and fit the intuitively understood meaning of the term with respect to probabilitydistributions. 2his evolution of the concept started with 1985 Pearson [1] who gives his mode andmoment coe ffi cients of skewness of a probability distribution, and today quite a numberof measures of probabilistic skewness exist in parallel, many more are being developed,each useful for di ff erent given contexts.In search of universal properties which a skewness measure (coe ffi cient) γ shouldsatisfy to be considered useful in the conceptual structure of random variables van Zwetin 1964 [2] named four desiderata, namely P.1
A scale or location change for a random variable does not alter γ. Thus, if Y = cX + d for c > O and −∞ < d < ∞ then γ ( X ) = γ ( Y ) , P.2
For a symmetric distribution γ = , P.3 If Y = − X then γ ( Y ) = − γ ( X ) , P.4 If F and G CDFs for X and Y as above and F < G then γ ( X ) < γ ( Y ) . Although in this article much attention is devoted to underline the fact that the the-ory of probability and that of fuzzy sets provide for two completely di ff erent ways ofevaluating and expressing uncertainty, we see that the fuzzy analogues of P.1. to P.3.should hold for a fuzzy measure of skewness, whereas it is debatable, whether P.4 canbe coherently transferred. This depends e.g. on whether the linguistic value being mod-eled by a fuzzy number has an underlying base variable or not. This matter should bediscussed separately, and is not part of the discussion in this paper.This article attempts to give a definition of skewness in the context of membershipfunctions of fuzzy numbers, a definition which best matches the intuition naturally part-nered with the term, when practitioners use it in the context of uncertainty best describedand adequately quantifiable as fuzzy information by fuzzy numbers.Although we develop our own definition of skewness independently, and it arisesnaturally from the perspective of looking at fuzzy numbers as parameterized curveswhich own their specific di ff erential geometry, it is practically impossible to discuss thematter without referencing to the various measures grounded and employed in proba-bility and statistics, as the hitherto existing measures of fuzzy skewness are built withreference to the latter, that is as quite literal translations. This fact in itself does notnecessarily imply a lack of functional capacity of those measures, but as much as onemay track the roots of fuzzy set theory as ensembles flous to Karl Menger’s probabilisticmetric spaces [3] the two concepts of quantified uncertainty, probability and fuzziness,have grown apart very much apart into separate domains and the need for an indepen-dent definition of skewness that would be rooted in the very own characteristics of fuzzysets seems to be there.This next section gives a short recap of concepts of center, variability and shapein the theory of probability and discusses the various analogues which function in thefuzzy literature, but also suggests some hitherto uncovered directions.3 .2. Concepts of center, variability and shape in the theory of probability In probability theory one may distinguish between firstly • parameter free skewness coe ffi cients ,such as the original quartile (also called Bowley skewness, Galton’s measure ofskewness) coe ffi cient S = Q − Q + Q Q − Q (1)or the following skewness function introduced by R. A. Groeneveld and G. Mee-den in 1984 [4] γ ( u ) = F − ( u ) + F − (1 − u ) − F − (1 / F − ( u ) − F − (1 − u ) . (2)which incorporates (1) at u =
34 . • and secondly parameter measures of skewness the most prominent of which are Pearson’s mode and moment skewness coef-ficients, which assume as primary the parameters of mode and mean (variancefollows from mean) going Mean → Variance ( S D ) → S kewness , Kurtosis : Mode Skewness (Pearson’s first skewness coe ffi cient) S = µ − νσ , (3)but when speaking of skewness most people think of Pearson’s moment coe ffi cient of skewness around the mean S = E (cid:34)(cid:18) X − µσ (cid:19) (cid:35) . (4) • It should be noted that many more measures of skewness do exist and are inpermanent use, especially in the field of financial economics.
As above we divide between skewness measures which do or do not incorporateother descriptive parameters such as mean and variance:4 . Parameter free skewness measures.
The skewness function (2) may appear particu-larly interesting and a possible subject of future research from the fuzzy number pointof view: - One may construct a fuzzy number ξ by using the CDFs F l and (reflected) F r of two random variables X l and X r as the left and right fuzzy sides l ( x ) and r ( x ) of ξ and transfer and apply (2) directly from probability distributions to fuzzy numbers gen-erated like that. One may also conversely redefine the left and right endpoints of a fuzzynumber as the CDFs of two distributions F l and (reflected) F r and set for instance: γ ∗ ( u ) = γ l ( u ) + γ r ( u )2 . (5)Interestingly, to the author’s knowledge, no e ff ort has been taken to transfer (2), directlyinto the realm of fuzzy membership functions.But, as in the next approach described in the next paragraph, while this approach isformally-mathematically viable and painless, the interpretational value is di ffi cult anddoubtful, because it demands for the linking of the value of a linguistic variable withthe CDF of a probability distribution, which are entirely di ff erent concepts, although thedensity function of a unimodal probability distribution looks similar to many people.
2. Analogues of Pearson’s mode and moment coe ffi cient in fuzzy set theory. The bulk offuzzy literature relates to Pearson’s moment coe ffi cient although (3) may be transferreddirectly e.g. by takingmode = most probable (cid:27) middle = C ( ξ ) , and any of the available ranking indices for mean value in combination with any metricfor absolute average deviation.In bringing over Pearson’s moment coe ffi cient two approaches suggest themselves: When one reads Pearson’s original 1895 argumentation for the use of the secondand third moments it is of physical nature (center of mass, moments of rotational inertia)and curve fitting and one sees that the reasoning may transfer directly, that is technicallywithout reference to probability distributions:Starting with the possibilistic mean value introduced by Goetschel and Voxman [5](later generalized in [6], [7]), which in our notation (see Notation (10)) is written µ P = E P ( ξ ) = (cid:90) d ( α ) + u ( α ) d α (6)higher possibilistic moments are introduced by µ nP ( ξ ) = (cid:90) ( d ( α ) − E P ) n d α + (cid:90) ( u ( α ) − E P ) n d α (7)5ith skewness being the third such defined moment S ∗ ( ξ ) = µ P ( ξ ) . (8)An approach along these lines, in the setting of possibility theory, was taken by E.Vercher, E. and J.D. Bermúdez [8, 9] in the context of fuzzy portfolio optimization.(The values this skewness coe ffi cient may take infinity as the probabilistic counter-part, which is counter-intuitive in many contexts, and its use may often not be appropri-ate for this or other reasons.) Other authors build a bridge to probabilistic moments by relating / associating agiven membership function ξ ( x ) with a probability density function f X ( x ) by either1. scaling f X ( x ) = ξ ( x ) / (cid:90) + ∞−∞ ξ ( t ) dt
2. or by f X ( x ) = ξ ( x ) ddx ξ ( x ) ( = ddx ξ ( x ))both implying the existence of a meaningful underlying random variable X directlyassociated with value Y of a linguistic variable which is represented by the membershipfunction ξ Y . (cid:27) (cid:45)(cid:27) (cid:45)(cid:63) (cid:63) .................................................... Xf X ξ Y Y Figure 1: A non-commutative diagram of linguistic (fuzzy) and probabilistic notions.
This is at the same time our foremost point of critique against this linking of fuzzymembership functions with probability densities, that this link should be accompaniedby an interpretation of the implied link between the represented linguistic value of alinguistic variable, and the random variable, which the density stands for.If such an interpretation cannot be given the identifies given above are mathematically-formally correct but without content and not fit for meaningful applications. To give anexample: for a trapezoidal fuzzy number (interval) the link f X ( x ) = ξ ( x ) ddx ξ ( x ) gives abimodal density function supported outside of C ( ξ ) . Remark 1.1.
Interestingly the mean value of the probabilistic distribution defined by coincides with the possibilistic mean value of ξ with weight α .
2. Skewness of triangular (linear) fuzzy numbers
It is assumed that the reader is generally familiar with the concept of fuzzy sets,fuzzy numbers and triangular fuzzy numbers in particular.(For a very clear and exhaustive account see the classic monograph of Buckley [11],for a little more recent text with special reference to fuzzy methods in statistics see R.Viertl [12])For an understanding of linguistic variables and their linguistic values, which aremodeled by fuzzy numbers the reader is best referred to the very founder and maincontributor of these concepts in science, i.e. Lotfi A. Zadeh in [13, 14, 15, 16].For easy reference regarding fuzzy numbers and to disambiguate, below are notationand terminology used throughout this particular paper.
Notation.
In this paper we denote fuzzy numbers by lowercase Greek letters ξ, ν, . . . .
By a fuzzy number we understand a fuzzy set on R , which attains x = x = m . We refer to m as the middle of ξ. We sometimes shorten “fuzzy number” to FN .A fuzzy number ξ may be given as an ordered function pair of a left and a right side : ξ : = ( l ( x ) , r ( x )) , (9)The ordered pair (9) may be comprised into a single function ξ ( x ), which is called thesingle membership function of the fuzzy number ξ. When given as either as a singlemembership function ξ ( x ) or as the ordered pair ( l ( x ) , r ( x )) we here refer to ξ such givenas in traditional representation .The support of l ( x ) is denoted by [ l , m ], and the support of r ( x ) by [ m , r ] and bothare onto [0 , . We assume invertibility and denote d ( α ) : = l − ( x ) and u ( α ) : = r − ( x ) and write ξ : = [ d ( α ) , u ( α )] (10)for the same fuzzy number ξ in parametric representation .7 a) ξ in traditional representation (b) ξ in parametric representation Figure 2: A fuzzy number ξ in both traditional and parametric representations For each α ∈ [0 ,
1] the closed interval [ d ( α ) , u ( α )] is a level set of the membershipfunction ξ ( x ) and is termed the α -level or α -cut of the fuzzy number ξ at α , denoted by C α ( ξ ) . For example C ( ξ ) = { m } for fuzzy numbers, as we use the name in this paper.We denote fuzzy triangular numbers, that is fuzzy numbers with linear sides l and r ,aka linear fuzzy numbers by tr ∗ ( l , m , r ) where ξ = tr ∗ ( l , m , r )( x ) : = l ( x ) = x − lm − l for x ∈ [ l , m ] , r ( x ) = r − xr − m for x ∈ [ m , r ] . (11)and l ( m ) = r ( m ) = , and l , r = l , r ] in traditional representation,respectively tr ∗ [ l , m , r ]( α ) = [ l + α ( m − l ) , r − α ( r − m )] , α ∈ [0 ,
1] (12)in parametric representation.In this section we will develop an angular skewness coe ffi cient γ ξ of a fuzzy trianglenumber ξ = tr ∗ ( l , m , r ) given by: γ ξ ( α ) = ± arccos (cid:16) √ · ( r − l ) | r | (cid:17) , for | r | (cid:44) , m − l = r − m (symmetry). (13)whereby the sign in (13) is determined by = + if m − l < r − m , − if m − l > r − m , (14)8y definition and design taking values γ ∈ (cid:20) − π , π (cid:21) . (15)The line of reasoning, which necessarily leads straight to this and no other measure γ originates from a new perspective of fuzzy numbers as parameterized curves, is laidout below. The line of reasoning is very geometric and graphical, so we show how to arrive at(13),(14),(15) by looking into a concrete example.
The fuzzy triangle number tr ∗ (2 , ,
5) may be written in traditional representation as tr ∗ (2 , , x ) = ( x − / x ∈ [2 , , x = , (5 − x ) for x = ∈ (4 , , . (16)or given equivalently in by its nested sequence of α -cuts: C α ( tr ∗ (2 , , = [2 + α, − α ] , α ∈ [0 ,
1] (17)which corresponds to the parametric representation (see (10)) of tr ∗ (2 , , . Here is the key observation, key to newness, which is central to this paper:For each α : the α -cut given by [2 + α, − α ] may be naturally identified with thecorresponding point / row vector (2 + α, − α ) or equivalently column vector (cid:16) + α − α (cid:17) inthe closed half plane { x ≤ y } ⊆ R . This equivalence of is shown beneath in
Figure 3 for the arbitrarily chosen α -level α = . Thank You, Franck Barthe of
Institut de Mathématiques de Toulouse , for this one. igure 3: Equivalent objects: the level set ( α -cut) C (cid:0) tr ∗ (2 , , (cid:1) = [3 , . the interval value of tr ∗ [2 , , α ) at α = is [3 , . , and the point (cid:0) , . (cid:1) belonging to the half plane { x ≤ y } ⊆ R . The identification shown above for the single α -value α = . α ∈ [0 , Remark 2.1.
By identifying each closed interval [2 + α, − α ] ⊂ R with the cor-responding point (2 + α, − α ) ∈ R one may then define a parameterized curve σ ( α ) = ( σ ( α ) , σ ( α ) , σ ( α )) in R given by σ ( α ) = (2 + α, − α, α ) (18) as shown in Fig. 4 below: Figure 4: The parameterized curve σ ( α ) = (2 + α, − α, α ) corresponding to the fuzzy triangle number tr ∗ (2 , , t is clear, that in the case of fuzzy triangle numbers, because of the linearity of allthree coordinate functions of σ ( α ) and the resulting similarity of section triangles alongthe { x = y } - line the third dimension may be omitted (as the information, α , is given bythe first two coordinates), and one may substitute (18) with its projection onto the xy -plane: σ ( α ) = (2 + α, − α ) , α ∈ [0 , , where σ (0) = (2 , , σ (1) = (4 , and for all α ∈ [0 , the values of σ ( α ) are exactly the vector coordinates ∈ R corresponding tothe α -cuts C α (cid:0) tr ∗ (2 , , (cid:1) . This leads to the following way of looking at things, a representation of a fuzzytriangle number as a projected parameterized curve σ ( α ) in R (obviously the curvature κ of σ is constantly κ =
0, and the projected curve appears as a straight line):
Figure 5: The middle point x = tr ∗ (2 , ,
5) corresponds to (4 ,
4) lying on the { x = y } - line (with α = l = r = , α = Because σ ( α ) is linear in all coordinates it has a single unique tangent vector T ξ = σ (cid:48) ( α ) given by σ (cid:48) ( α ) = (cid:104) m − l , − r + m (cid:105) = (cid:104) , − (cid:105) = −(cid:104)− , (cid:105) (19)which together with the boundary condition ξ ( m ) = , that is knowledge of the locationof the middle point m , uniquely determines the fuzzy triangle, as shown in Fig. 6 :11 igure 6: The free tangent vector σ (cid:48) ( α ) = (cid:104) − ( m − l ) , r − m (cid:105) attached to the triangle’s middle point ( m , m )(for better visualization we take the vector’s negative against the run of the parametrisation). Then we note that • T ξ = σ (cid:48) ( α ) is of constant magnitude | σ (cid:48) ( α ) | = (cid:113)(cid:0) ( m − l ) + ( r − m ) (cid:1) , (20)which in the example case ξ = tr ∗ (2 , ,
5) gives = (cid:112) ( − + = √ . • The constant angle γ by which the curve’s tangent vector deviates from the line {− α = α } perpendicular to the { x = y } - line is γ ξ ( α ) = ± arccos (cid:104) σ (cid:48) ( α ) , (1 , − (cid:105)| σ (cid:48) ( α ) | · | (1 , − | = (21)which simplifies to = ± arccos (cid:32) √ · ( r − l ) | r | (cid:33) (22)with = r − m = l − m , (23)12nd + if m − l < r − m , − if m − l > r − m , (24)as anticipated in (13).Note that the for triangle numbers of the sort, ξ = tr ∗ ( l , r , r ) (25)or (26) ξ = tr ∗ ( l , l , r ) (27)that is of extreme skew, (22) produces an angle of γ = − ◦ or + ◦ , (28)respectively.In this particular example ξ = tr ∗ (2 , ,
5) the constant angle of deviation γ ( α ) is seento be given by cos ( γ c ) = / √ γ c + γ = ◦ that is to say γ = arccos(2 / √ − ◦ = − .
435 degrees.For the full picture the length of the projected curve is by definition C ( σ ) = (cid:90) | σ (cid:48) ( α ) | d α = √ + = √ , (30)which can, in this linear case, be seen by the Pythagoras theorem. Definition 2.2.
It is this angle γ ξ (21) by which the curve’s tangent vector deviates fromthe { α = α } -axis, that we take as skewness measure for fuzzy triangle numbers.To see why this choice makes sense as a measure of deviation from symmetry it isenough to realize that the projected parameterized curve as well the defining tangentvector representing a symmetric triangle number lies entirely on the {− α = α } line, thatis γ = tr ∗ (2 , , tr ∗ (3 , ,
6) and the symmetrictriangle tr ∗ (2 . , , . .1.2. Skewness as angle of deviation from symmetry (a) left skew (b) symmetry (c) right skew Figure 7: The tangent vector (cid:126) ( − ,
1) representing the fuzzy triangle number tr ∗ (2 , ,
5) 7a deviatesfrom symmetry by γ ∼ − , ◦ . The tangent vector (cid:126) ( − . , .
5) representing the symmetric triangle tr ∗ (2 . , , .
5) 7b has deviation γ = tr ∗ (3 , ,
6) 7c represented by thetangent vector (cid:126) ( − ,
2) deviates from symmetry by an angle of γ ∼ + . ◦ . The next observation is that the vector σ (cid:48) ( α ) , as any vector, may be given in polarrather than Cartesian coordinates, by setting r ξ = | T ξ | , (31) γ ξ = as in (13) . (32)The next logical step is to use this (31) as an alternative (eventually primary) repre-sentation of any fuzzy triangle number, whereby it is visually more convenient to take γ from verticality : Following the 1-1 correspondence between the ordered triples of( m , r , γ ) ⇔ ( l , m , r ) (33)for l ≤ m ≤ r ∈ R and γ ∈ (cid:20) − π , π (cid:21) , r ∈ [0 , ∞ ) (34)one may rewrite a linear fuzzy number ξ in polar representation as ξ = ( m , r , γ ) (35)14 a) Cartesian coordinates (b) Polar coordinates Figure 8: The tangent vector representing the fuzzy triangle number tr ∗ (2 , ,
5) in Cartesian and polarcoordinates
The visual change from cartesian to polar coordinates counting γ of the ordinateaxis and the characterizing vector’s tail anchored on the abscissa, in the real number m which is being fuzzified, may be e ff ected by a linear transformation, which amounts toa change of basis namely by F = · (cid:32) − , (cid:33) (36)which takes (cid:32) (cid:33) → (cid:32) (cid:33) and (cid:32) − (cid:33) → (cid:32) (cid:33) making the { x = y } axis the new abscissa and {− x = y } the new ordinate axis scalingboth by 1 √ . Note that F is an orthogonal transformation which leaves the angles γ unchanged.We use this transformation in the next section to better visualize our line of reasoning forthe one-to-one correspondence of non-linear fuzzy membership functions and a certainclass of parameterized curves.We finish this section with a comparative illustration of fuzzy triangle numbers intraditional versus vector representation, which also serves to show van Zwet’s P . − P . . F -coordinate system, that is vectorlengths | r | are scaled by 1 √ . Figure 9: Example fuzzy triangle numbers and their vector counterparts. Antisymmetry and translationinvariance.Figure 10: Special (border) cases: Triangle numbers of extreme skewness i.e. l = m , m = r , Symmetrictriangle numbers ( m − l = r − m ). and exact, real numbers i.e. l = m = r . tr ∗ ( m − δ, m , m + δ ) translate into vertical arrows oflength (height) r = √ δ, and angle γ = ,
2. real (exact) numbers translate into themselves on the real line ( r = , γ = tr ∗ ( l , m , m ) or tr ∗ ( m , r , r ) translate into vectors oflength ( m − l ), respectively ( r − m ) and angle γ = ± ◦ .
3. Non-linear fuzzy numbers in polar coordinates
For the purpose of this section we assume for a fuzzy number ξ = ( l ( x ) , r ( x )) to holdsome additional di ff erentiability properties: Properties (of a non-linear fuzzy number) . As had: l ( x ) , r ( x ) are supported on [ l , m ] , [ m , r ]respectively, and onto [0 , , invertible and such that d ( α ) = l − ( α ) , and now we also as-sume: d ( α ) , u ( α ) to be continuously di ff erentiable on [0 , . We also make the following assumptions whose sense becomes immediate below0 < d (cid:48) ( α ) < ∞ , ( d strictly increasing and d (cid:48) bounded) (37)0 < u (cid:48) ( α ) < ∞ . ( u strictly increasing and u (cid:48) bounded) (38) Remark 3.1.
Much less restrictive assumptions may be adopted and still achieve allresults of this paper, such as: non-strict monotonicity of l and r instead of strict, semi-continuity instead of continuous di ff erentiability, support unbounded but with lim x →∞ l ( x ) = and lim x →∞ r ( x ) = instead of bounded support. To overcome theappearing technical obstacles one would venture into the theory of distributions (in thesense of Laurent Schwartz, not probability distributions) but maximum generalization isnot a concern at this point. We now repeat step by step for non-linear numbers the line of reasoning taken insection 2 for fuzzy triangle (linear) numbers:
Given a fuzzy number as( l ( x ) , r ( x )) , x ∈ R (TRADITIONAL REPRESENTATION) (39)we may first switch to[ d ( α ) , u ( α )] , α ∈ [0 , (PARAMETRIC REPRESENTATION) (40)17hich one may choose to understand as a parameterized curve σ ( α ) : = ( d ( α ) , u ( α )) , α ∈ [0 , . (AS A PARAMETERIZED CURVE) (41)By assumptions (37), (38) one may equivalently give (41) as ξ = σ (cid:48) ( α ) = (cid:0) d (cid:48) ( α ) , u (cid:48) ( α ) (cid:1) with boundary values ( d (1) , u (1)) = ( m , m ) , (AS TANGENT BUNDLE) . (42)For convenient visualization one may choose to instate a change of coordinates by F (see (36)) as done below in figure 11. (a) The curve F ( σ )( α ) (b) (c) The tangent bundle Figure 11: A non-linear fuzzy number supported on F − (cid:16) . . (cid:17) = [2 ,
5] with middle point m = F -transformed parameterized curve 11a, the same curve with indicated tangent vectors, and as a tangentbundle. (cid:5) The key to all is again to represent ξ in polar coordinates:As done before in the linear case one may change coordinates to polar and rewrite eachtangent vector for each α as the triple ξ = ( m , r ( α ) , γ ( α )) (43)where as before r ( α ) = | σ (cid:48) ( α ) | = (cid:112) d (cid:48) ( α ) + u (cid:48) ( α ) . (44)and γ ( α ) : = ± arccos < ( d (cid:48) ( α ) , u (cid:48) ( α )) , (1 , − > | (cid:112) d (cid:48) ( α ) + u (cid:48) ( α ) | · | (1 , − | == ± arccos (cid:32) u (cid:48) ( α ) − d (cid:48) ( α ) √ r (cid:33) . (45)18ith the sign determined as in (14) by + f or | d (cid:48) | < | u (cid:48) |− f or | d (cid:48) | > | u (cid:48) | f or | d (cid:48) | = | u (cid:48) | (symmetry) (46)with sign changes taking place at α for which γ ( α ) = . Remark 3.2. Di ff erentiation (46) is redundant and artificial when the principle valuesof cosine to be taken as [ − π, π ] instead of [0 , π ] as is custom for the majority of textbooksand calculators / computer programs. The re-transformation from polar to cartesian runs: r (sin( γ ) ± π ) → m − l , r (cos( γ ) ± π ) → r − m , (47)With the sign determined by reversing (14). This transformation is clear from Fig. (8).
Example 3.3.
Let ξ be the fuzzy number around m = π whose sides are given in tradi-tional representation by the ordered function pair ξ = ( l ( x ) , r ( x )) : l ( x ) = − / + arccos (cid:16) (cid:112) (cos(4 / + π − x ) (cid:17) on [2 . , π ] r ( x ) = (2 − (1 /π ) · x ) / on [ π, π ] (48) Figure 12: The membership function of example (3.3) in traditional representation equivalently in parametric representation α ∈ [0 , ξ = [ d ( α ) , u ( α )] , d ( α ) = π + (cos(1 + / − (cos( α + / , u ( α ) = − πα + · π, (49)as tangent vector bundle σ (cid:48) ( α ) = (cid:34) d (cid:48) ( α ) u (cid:48) ( α ) (cid:35) , where d (cid:48) ( α ) = α + /
3) sin( α + / , u (cid:48) ( α ) = πα (50)and finally in polar coordinates as ( r ( α ) , γ ( α )) with r ( α ) = (cid:112) cos( α + / · sin( α + / + π α (51)and γ ( α ) = ± / √ (cid:16) π α + cos ( α + /
3) sin ( α + / (cid:17)(cid:112) π α − (cos ( α + / + (cos ( α + / (52)with a (single) sign change taking place at α = . . The parameterized curve and its polar coordinate functions are displayed below in
Fig. 13 : Figure 13: The fuzzy number ξ represented as an F -transformed parameterized curve, and its pseudo-polar coordinate functions r ( α ) and γ ( α ) . Note how γ ( α ) is contained between ± π ∼ . . Also mindthe direction of flow from α = α = We are now equipped to give our definitions of skewness, mean value and dispersionof non-linear fuzzy numbers. 20 . Skewness of non-linear fuzzy numbers
When looking at (43) the following definition of skewness very strongly suggests itself:
Definition 4.1 (Skewness at a point, α -level) . S ∗ ( ξ ) α = γ ( α ) = ± arccos (cid:32) d (cid:48) ( α ) − u (cid:48) ( α ) √ r (cid:33) . (53)By design − π ≤ S ∗ ( ξ ) α ≤ π , whereby both extreme values may be attained multiplyin a single fuzzy number, though in most typical cases there will be one single sign-change, or none at all.It is very valuable for practical applications to have skewness at a given α -leveldefined. The overall skewness coe ffi cient as defined below in (55) is eigen to infinitelymany very di ff erent membership functions, i. e. does not reflect the characteristics of aconcrete FN which stands for the very concrete value of a concrete linguistic variable.For practical purposes any given FN may be approximated with deliberate accuracyby the appropriate (finite) amount of α -cuts. (This can be shown in di ff erent ways butis most easily seen when regarding a fuzzy number as a Lebesgue integral achieved bysimple functions defined by these α -cuts). The huge gain from having skewness coe ffi -cients defined on all α -levels, instead of having only one, overall measure, is apparentwhen an optimization problem P ∗ involving skewness is formulated in fuzzy variablesand / or parameters. The problem becomes tractable by choosing a partition of [0 , / or parameters.If that is, say n , cuts, then the computational complexity of a linear program in-creases by the multiplicative factor n , with the characteristics of the FN reflected towhatever level chosen.Having a skewness coe ffi cient at every point α along the curve also allows to treateach α -level [ d ( α ) , u ( α )] as an ordered pair of interval and skewness coe ffi cient[ d ( α ) , u ( α )] , γ ( α ) . (54)Anyhow the overall skewness coe ffi cient of a given fuzzy number must then bedefined as Definition 4.2 (Overall skewness of a fuzzy number) . S ∗ ( ξ ) = C ( σ ) (cid:90) γ ( α ) · | σ (cid:48) ( α ) | d α, (55)21here C ( σ ) : = (cid:90) | σ (cid:48) ( α ) | d α (56)is the length of the parameterized curve σ. The values of both skewness coe ffi cients, S ∗ ( ξ ) and S ∗ ( ξ ) α are contained in theinterval γ ∈ (cid:20) − π , π (cid:21) . Remark 4.3.
For some a range of values of (cid:104) − π , π (cid:105) may seem more desirable for in-tuition or computability. This may of course be achieved without distortion and loss ofinformation by scaling.4.2. Properties (van Zwet 1964) It is straightforward to see that van Zwet’s three (of four) conditions for a “useful”skewness measure are fulfilled:
P.1
A scale or location change for a fuzzy number does not alter γ. In the context of fuzzy numbers this reads:if ν = c ξ + d then γ ( ν ) = γ ( ξ ) . (57)This is clear since:The location parameter d is annulled by di ff erentiation( c · σ ( α ) + d ) (cid:48) = c · σ (cid:48) ( α ) (58)and the scaling parameter also does not change the angle as < ( cd (cid:48) ( α ) , cu (cid:48) ( α )) , (1 , − > | (cid:112) c d (cid:48) ( α ) + c u (cid:48) ( α ) | · | (1 , − | = < ( d (cid:48) ( α ) , u (cid:48) ( α )) , (1 , − > | (cid:112) d (cid:48) ( α ) + u (cid:48) ( α ) | · | (1 , − | (59) P.2
For a symmetric distribution γ = . Symmetricity of a fuzzy number and when writing in parametric representationimplies u (cid:48) ( α ) = − d (cid:48) ( α ) (60)so the property follows immediately. See Fig. 7 . P.3 If Y = − X then γ ( Y ) = − γ ( X ) . In the context of fuzzy numbers this is to be interpreted as a reflection aroundsome axis x = s , Set in parametric representation X = [ d , u ] then Y = [ − u , − d ] + s and the property follows directly from (46).22eturning to example 3.3 , that is the fuzzy number ξ shown in Fig. 12 given in para-metric representation by ξ = d ( α ) = π + (cos(1 + / − (cos( α + / , u ( α ) = − πα + · π, (61)its point skewness is given by γ ( α ) of its polar representation (52), i.e.: S ∗ ( ξ ) α = ± / √ (cid:16) π α + cos ( α + /
3) sin ( α + / (cid:17)(cid:112) π α − (cos ( α + / + (cos ( α + / (62)with a (single) sign change taking place at α = . . To compute the overall skewness coe ffi cient of ξ by (55) we need compute the length C ( σ ) of the curve σ ( α ) by (56): C ( σ ) = (cid:90) | σ (cid:48) ( α ) | d α = (cid:90) r ( α ) d α = . . (63)The overall skewness coe ffi cient of ξ is thus(64) S ∗ ( ξ ) = C ( σ ) (cid:90) γ ( α ) · | σ (cid:48) ( α ) | d α = . · . = . rad ] = . π.
5. Mean by mean value theorem for integrals
In this section we o ff er a contribution to the already rich literature on what may betermed a mean value of a fuzzy number.Although fuzzy arithmetic is very straightforward (once given in parametric repre-sentation) and under certain assumptions also functional calculus is theoretically wellfounded the main drawback remains computational complexity, and in a systematicsense that the lack of an inverse element [17] (multiplicative or additive) typically leavesfunctional equations involving fuzzy numbers without a solution if understood as equa-tions in the usual “ = ” way and not some other, weaker relation.Hence the quest for a set of descriptive parameters which, to a satisfactory degree,reflect the characteristic basic attributes contained in a given fuzzy number while alle-viating aforementioned di ffi culties. 23he first thing that comes to mind is naturally that of a “mean” or “expected” value,and the earliest days the precursors of fuzzy set theory have set out to define appropriatemeasures:One may categorize three di ff erent approaches (strategies): a single (one-dimensional) mean value. As an example: the possibilistic meanvalue of Goetschel and Voxman [5] has proven to be particularly useful over time, asis its generalization by Carlson and Fullér [6]. This mean value also appears appearsnaturally when setting a density ρ ( x ) = µ ( x ) µ (cid:48) ( x ) in an e ff ort to relate directly to themachinery of probability as in [10], (see 1.2.1).Really every single so called ranking function (index, method) may play the role ofa mean value. The reader is referred to [18] for a comprehensive comparative treatmentof the wide variety of ranking indices and methods. An interval valued mean valuewhich may show a relation to probability theory as by Didier Dubois and HenriPrade [19] (who refer to Dempster [20]), or without such relation as “On possibilisticmean value and variance of fuzzy numbers” by R. Fullér, P. Majlender [7], who definean lower and upper possibilistic mean building on [5]. A third approach is to find a canonical most representative, triangle or trapezoidalfuzzy number (i.e. sets of three resp. four real numbers which are closed with respectto addition) which shares the same characteristic parameters (such as any of the rankingindices listed in [18] or later, or “value” and “ambiguity” introduced in [21], or fuzzinessintroduced in [22]), or is closest to it with respect to a chosen metric (see e.g [23] as astaring point) or other criterion.Having already stated (54) as a variation of approach above, in this paper we takethe third approach:Having defined the overall skewness coe ffi cient γ to a given FN we may define a“ mean value triangle ” i.e. a linear (triangle) number “most representative” of the non-linear FN in the sense that it shares, as the characteristic parameter, the same overallskewness coe ffi cient and the same middle point m .We start with an interesting representation theorem which needs no separate proof,as it follows directly from the preceding deliberations and definitions: Theorem 5.1.
Following the 1-1 correspondence between ordered triples of ( m , r ( α ) , γ ( α )) ⇔ tr ∗ ( l ( α ) , m , r ( α )) (65) for l ( α ) ≤ m ≤ r ( α ) ∈ R and γ ∈ (cid:20) − π , π (cid:21) , r ∈ [0 , ∞ ) (66) any given fuzzy number satisfying at least (37) , (38) may be represented by an or-dered set of triangle fuzzy numbers given by (65) . α mean ∈ [0 ,
1] at which level the overall skewness coe ffi cient of the measuredfuzzy number is attained: ∃ α mean : γ ( α mean ) = C ( σ ) (cid:90) γ ( α ) · | σ (cid:48) ( α ) | d α. (67)Having found α mean gives us the pseudo-polar coordinates of the corresponding tan-gent vector to the curve at α mean :( r ( α mean ) , γ ( α mean )) , (68)and in consequence by (47) a fuzzy triangle number tr ∗ ( l , m , r ) in traditional represen-tation, which has an overall skewness coe ffi cient of γ ( α mean ): Definition 5.2 (Mean value triangle number) . (69) tr ∗ mean ( ξ ) = (cid:18) m − cos (cid:18) γ (cid:18) α mean ± π (cid:19)(cid:19) · r ( α mean ) , m , m − sin (cid:18) γ (cid:18) α mean ± π (cid:19)(cid:19) · r ( α mean ) (cid:19) . This triangle number is uniquely determined and may be referred to as the meanvalue triangle of the fuzzy number ξ. We give a numerical instance returning to example 3.3 : From the definition of skew-ness (55) by the mean value theorem (67) we receive α mean = . , (70)hence r ( α mean ) = . , γ ( α mean ) = . , (71)giving a right-skewed fuzzy triangle number tr ∗ ( l , m , r ) in traditional notation by m − l = r m cos γ (cid:16) α m + π (cid:17) = . , r − m = r m sin γ (cid:16) α m + π (cid:17) = . . (72)thus tr ∗ mean ( ξ )( l , m , r ) = tr ∗ ( π − . , π, π + . , (73)as depicted below in Fig. 14. 25 igure 14: The Mean Value Triangle Number (blue) of a general fuzzy number (dark red). Remark 5.3.
Depending on context it may make more sense to build a triangle of skew-ness coe ffi cient γ and magnitude r anchored not in the original fuzzy numbers middlepoint, but rather in its possibilistic mean value, or in fact any of the single mean valuesmentioned above, listed for instance in the very readable, comprehensive and accuratestudy [18].
6. Dispersion as Hausdor ff distance from mean value triangle Having defined a mean value triangle it suggests itself to take a look at the distanceof the fuzzy number to its mean. That is take some form of absolute deviation from it.It lies in the nature of fuzzy numbers, being a generalization of exact, real intervalsthat the Hausdor ff distance has proven most adequate for many purposes.It should be noted, that many other metrics have been and constantly are being devel-oped, some of which generalize the Haudor ff metric, at many times based on L p metricson [0 , × [0 ,
1] and mixing both, seldom entirely di ff erent approaches. See [24] as astarting point for examples.Now we may use definition equation (69) to establish a measure of dispersion as theHausdor ff distance of the fuzzy number in question from its own mean value triangle:Denote the mean value triangle of a fuzzy number ξ in parametric representation by tr ∗ mean ( ξ )( α ) = [ mean ξ ( α ) , mean ξ ( α )] . (74)As with the skewness coe ffi cient one may alternatively define this dispersion mea-sure level-wise (as a function of α ) or overall by integrating over [0 , Definition 6.1.
Dispersion on a given α -level d H ,α ( ξ ) = max (cid:16) | u ( α ) − mean ( α ) | , | d ( α ) − mean ( α ) | (cid:17) . (75)26n the running example 3.3 we get:(76) d H ,α ( ξ ) = max( − . α + . + . α, | ( | − . + cos( α + / + . α ))which is nicely illustrated in Fig. 15 : Figure 15: Dispersion: Hausdor ff distance of the example fuzzy number ξ (3.3) from its mean valuetriangle. The distance is easily seen from Fig. 14 to determined by the right sides of the FN, down untilthe lowest (nearing 0), when the maximum turns the switch. α -levels The overall dispersion may be defined by integration:
Definition 6.2.
Overall dispersion of a fuzzy number. d H ( ξ ) = (cid:90) d H ,α d α. (77)In the running example 3.3 the overall dispersion of ξ is given by: d H ( ξ ) = . . (78) Remark 6.3.
The dispersion measure introduced here in (75) is the most obvious choicebased on the definition of mean value triangle which again follows naturally from theintroduction of our measure of skewness. There is no claim to superiority over any othermeasures of dispersion being stated in the literature and currently in use. Also countlessvariations of (75) are possible (by changing the mean value, by changing the distancefunction), depending on context of the problem at hand, which may suggest a di ff erentmeasure. he main contribution of this paper remains the new perspective of fuzzy numbersas parameterized curves, and the skewness measure which follows so naturally, withoute ff ort, from it. Remark 6.4.
We use the term dispersion to emphasize (set apart not only in essence butalso in nomenclature) the di ff erence of fuzzy set theory from probability and statistics,which speaks of most often of variance and standard deviation. Also is the preference ofvariance (standard deviation) over average absolute deviation dictated by the algebraichandlebility which it brings. In the case of fuzzy numbers this advantage is not neces-sary, as unlike is the case with the intricate algebra of random variables (see [25, 26])the arithmetic of fuzzy numbers is comparatively clear and simple.
7. Fuzzy skewness in Portfolio Optimization Theory
In this section we make a general case for the use of fuzzy numbers in certain mod-elling scenarios, when the model inherent uncertainty does not allow for concrete un-derlying probability distributions to be inferred.Not wanting to venture into technicalities we begin with a simplified definition out-line of the problem:
Given a number I of financial assets i , each of which has attributes such as: expectedreturn r i and volatility v i .A portfolio is a convex choice of weights w i ∈ [0 ,
1] to be ascribed to each of thoseassets, that is a non-negative vector w = ( w , w , ..., w i ) , (cid:88) i w i = { obj m } and subject to various constraints { c n } .The primary objective would typically be the portfolio’s expected overall return,with secondary objectives including minimization of risk and volatility.Constraints may relate to budget, sectors, geography, etc. and may be relative withrespect to each other c i = R n ( w , .. w N ) = w k ≤ w l , or of absolute nature, e. g. d i ≤ i ≤ d i . Objectives and constraints may be formulated without reference to any given modelof uncertainty: probability, fuzziness or other: “expected return”, “ riskiness” may be28ut to numbers by gut feeling, inside knowledge, regression based on historical perfor-mance.But in modern practice all the factors contributing to both the objective functionsand constraints are modeled as random variables whose probability distributions in turnare generated / estimated by various methods.Understanding each asset’s return r i as a random variable, and having determinedprobability distributions f r i for each of the assets’ return r i a decision maker has athis disposal the moments µ ji of the random variable, that is µ i - expected values, µ i -variance and co-variance, but also skewness, kurtosis and higher moments (See [27] forhigher moments in portfolio selection).Set z = ( r , . . . , r I ) , µ i = E ( r i ) , m = ( µ , µ , ..., µ I ) and cov ( z ) = (cid:88) . If w = ( w , w , ..., w I ) is a set of weights associated with a portfolio, then the rate ofreturn of this portfolio I (cid:88) i = r i · w i is also a random variable with mean (cid:104) m , w (cid:105) and variance w (cid:88) w T . Before Markowitz [28] a portfolio optimization program would look like justMaximize ob j : (cid:88) w i · µ i , (79) subject to { c i } , (80) { c i } being a set of constraints.But with the arrival of MPT this objective function would change, and an optimalset of weights became one in which the portfolio achieves an acceptable baseline ex-pected rate of return with minimal volatility. In this theory the variance of the rate ofreturn of an asset is taken as a surrogate for its volatility.If µ b is the aforementioned acceptable baseline expected rate of return, then in theMarkowitz theory an optimal portfolio is any portfolio solving the following quadraticprogram: Minimize ob j : 12 w (cid:88) w T (81) subject to contstr : (cid:104) m , w (cid:105) ≥ µ b , (82) and { c i } . (83)where { c i } , µ i = E ( r i ) , m = ( µ , µ , ..., µ n ) , and cov ( z ) = (cid:88) , as above. Modern Portfolio Theory .2. The mean-variance-skewness model With authors such as [29], [30] and [31], or [8, 9], [10] for a fuzzy extension, themodel has been further refined by bringing in an additional skewness constraint. - Posi-tive skewness is desirable, since increasing skewness decreases the probability of largenegative rates of return. So by bringing in s i as the skewness of r i defined by the thirdmoment, and setting a base level of skewness s b and minimum co-variance v b one thenconsiders three variations of portfolio optimization programs:Minimize 12 w T (cid:88) w (84) subject to (cid:104) m , w (cid:105) ≥ µ b , (85) (cid:104) s , w (cid:105) ≥ s b , { c i } . and alternatively Maximize (cid:104) m , w (cid:105) (86) subject to 12 w T (cid:88) w ≤ v b , (87) (cid:104) s , w (cid:105) ≥ s b , { c i } . or Maximize (cid:104) s , w (cid:105) (88) subject to (cid:104) m , w (cid:105) ≥ µ b , (89) 12 w T (cid:88) w ≤ v b , { c i } . All of above linear (quadratic) optimization programs may be stated analogically byputting fuzzy numbers in place of random variables on each occurrence.We give justification to model the uncertainty associated with the assets not bymeans of probability theory, but resorting to fuzzy set theory instead, and refer to exist-ing literature for existing solution algorithms.30he main, true, core problem in the probabilistic modeling process and its practicalimplementation is to find the “right” probability distributions.When a viable probability distribution can not be found modelers often turn to so-called no-knowledge distributions, such as the uniform, triangular or PERT, setting pes-simistic, optimal, optimistic values for each µ i . But it must be noted that the underlyingassumptions (of exactly equal or otherwise placed probabilities) made here are actuallyvery strong, not at all “no knowledge”, and a fuzzy model, taking interval numbers orfuzzy triangle numbers instead of probabilistic uniform or triangular distributions mustbe favored.If a probability distribution governing an investigated process can not be found, itstill may often be possible, by various methods, to give sharp lower and upper boundson the descriptive parameters µ i , cov i j , s i . In this case, that is with parameters given as interval estimates µ i = [ µ i , µ i ] , (90) cov i j = [ cov i j , cov i j ] , (91) s i = [ s i , s i ] , (92)programs (79), (84), (86) or (88) fall into the realm of interval linear (quadratic)programming and become tractable as such and their solutions consist of vectors ofintervals which may be given explicitly.For an overview of existing literature on interval programming and leading to morerecent results see Milan Hladik’s [32] and [33].Now in consequence of the use of di ff erent methods, by di ff erent sources, a numberof di ff erent interval estimates of the investigated parameters may be given.If a number of di ff erent interval estimates E n are given from a number of sources (ex-perts) these individual estimates may be aggregated into a single staircase fuzzy numberby the procedure given in [34] for type-2 fuzzy intervals: ξ ( x ) = n (cid:88) / n · χ E n ( x ) . (93)with χ E n being the indicator (characteristic) function of E n . χ E n ( x ) = x ∈ E n , x (cid:60) E n , This very intuitive procedure is shown graphically below in figure 16:31 a) E (b) E (c) E (d) ξ ( x ) Figure 16: Three aggregated individual interval estimates aggregated into a single staircase fuzzy interval.
Remark 7.1.
The method presupposes that there be an overlap of all experts’ intervalestimates. Because it may be presumed that knowledge and prior information of allexperts be su ffi ciently similar this assumption is quite natural.In case there is no common overlap of all experts’ estimates (the level set at height α = is the empty set ∅ ), various methods can be applied. To facilitate the implementation of the parametric methods discussed in this papertwo steps must be taken:1. Add another single value level C ( ξ ) on top of the α -levels generated by the pro-cedure described in (93). This single value can be chosen as the possibilistic meanvalue or any ranking index which places in the intersection of all constituting E i .
2. Achieve piecewise di ff erentiability by linearly joining the endpoints at each level(the adaptation of the methods of the preceding sections to piecewise di ff erentia-bility is routine), or use a mollifier to achieve C ∞ , or do anything between thesetwo extremes.The resulting parameterized curve is symbolically graphed below in Fig. 17:32 a) ξ and F ( ξ ) (b) σ ( α ) Figure 17: A fuzzy staircase number in parametric representation (blue), the sides transformed by F , (17a) and the resulting parameterized curve (on the right, (17b)). An much more technical and more sophisticated approach is the attempt to gener-ate parameterized interval estimates from the outset, i.e. upper and lower bounds as µ ( α ) = (cid:104) µ ( α ) , µ ( α ) (cid:105) , v ( α ) = (cid:104) v ( α ) , v ( α ) (cid:105) and s ( α ) = (cid:104) s ( α ) , s ( α ) (cid:105) , which allows for theconstruction of fuzzy intervals straightforwardly. This so done in [35].We consciously refrain from displaying a random generated numerical example ofabove techniques.
8. Conclusion
The main result of this paper is really, that it adds to the here so-called traditional (9)and parametric (10) representations of a fuzzy number ξ two other: as a parameterizedcurve σ ( α ) (41) and as a tangent bundle (42). Then the representation of tangent vectorsin polar coordinates ( r ( α ) , γ ( α )) = (43,45) directly implicates measures of skewness ata point (53) and overall (55).The definition of a mean value triangle tr ∗ ξ (69) and dispersion d H ( ξ ) (75) then followjust by going through the motions.The results achieved in this paper may be developed and diversified in various di-rections: • Comparative studies.To compare the descriptive parameters, mean value triangle, dispersion, and skew-ness, developed in this paper to measures which have been developed before.33
Go into Sobolev spaces,to include non-di ff erentiable membership functions. As stated in remark 3.1 :Much less restrictive assumptions may be adopted and still achieve all results ofthis paper. • Parametric curves as primary representation:The arrow / vector representation of fuzzy triangle numbers may appear more in-tuitive to non-mathematical decision makers and takers than the traditional andparametric ones. • Di ff erential geometry of curves:This paper hints at, but does not exploit curvature, torsion, generally the TNB frame of a fuzzy number understood as a parameterized curve. • Computer aided visualisation and eye-tracking:Visualizing the twists and turns of a linguistic variable.
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