Approximations to the Normal Probability Distribution Function using Operators of Continuous-valued Logic

Transcription

1 Acta Cybernetica 3 08) Approximations to the Normal Probability Distribution Function using Operators of Continuous-value Logic József Dombi a an Tamás Jónás b Abstract In this stuy, novel approximation methos to the stanar normal probability istribution function are introuce. The techniques presente are foune on applications of certain operators of continuous-value logic. It is emonstrate here that application of the averaging Dombi conjunction operator to two symmetric Sigmoi fuzzy membership functions results in a function that is ientical with Tocher s approximation to the stanar normal probability istribution function. Next, an approximation connecte with a unary fuzzy moifier operator is iscusse. Namely, the so-calle Kappa function is applie for constructing a novel probability istribution function. It is shown here that the asymptotic Kappa function is just the Sigmoi function an the propose Quasi Logistic probability istribution function can be utilize to approximate the stanar normal probability istribution function. It is also explaine how the new probability istribution function is connecte with the generator function of Dombi operators. The propose approximation formula is very simple as it has only one constant parameter. It oes not inclue any exponential term, but has a goo approximation accuracy an fulfills certain requirements that only a few of the known approximation formulas o. Keywors: continuous logic, Dombi operators, sigmoi function, normal probability istribution, approximation Introuction The normal probability istribution plays a significant role in probability theory an mathematical statistics. Owing to the central limit theorem, it has an extremely wie range of applications in many areas of sciences. The fact that the a Department of Computer Algorithms an Artificial Intelligence, University of Szege, Árpá tér, H-670 Szege, Hungary, ombi@inf.u-szege.hu b Institute of Business Economics, Eötvös Lorán University, Szép utca., H-053 Buapest, Hungary, jonas@gti.elte.hu DOI: 0.43/actacyb

2 830 József Dombi an Tamás Jónás cumulative istribution function of the stanar normal ranom variable cannot be expresse in a close form an the practical nees for computing its values provie the motivations for researchers an practitioners over the last seven ecaes to approximate the stanar normal probability istribution function. These research efforts resulte in an extremely wie range of approximations with many applications. In this paper, we will introuce approximations to the stanar normal probability istribution function that are connecte with the well-known Dombi operators in continuous-value logic. Firstly, we will utilize the averaging Dombi conjunction operator to construct a probability ensity function from two Sigmoi functions. We will show that this approximation metho results in a probability istribution function that is ientical with Tocher s approximation from 963 [9]. Seconly, we will introuce the Kappa function an base on this function, we will construct the Quasi Logistic probability istribution function. We will show that the asymptotic Kappa function is just the Sigmoi function an using this result, we will also show how the Quasi Logistic probability istribution function can be utilize for approximating the stanar normal probability istribution function. Here, we will point out how the propose probability istribution function is connecte with the generator function of Dombi operators an with the Kappa function-base uniary operator that can be utilize as a general fuzzy moifier operator. The novelty of our methos lies in the fact that some mathematical constructions of continuousvalue logic can be successfully utilize to construct approximations to probability istribution functions. Many known approximations to the stanar probability istribution function focus mainly on the approximation accuracy, an so these methos result in highly accurate functions, without taking some other aspects of the approximation into account. It shoul be mentione here that we require our approximations to meet expectations that are base on certain theoretical an practical consierations. These expectations are simplicity an accuracy, asymptotic equality of the approximator function to the stanar normal istribution function to first orer at zero, symmetry of probability ensity function an a irect connection between the ensity an istribution functions. Finally, we propose the use of the following probability istribution function, which is a special case of the Quasi Logistic istribution function, to approximate the stanar normal probability istribution function: 0, if x π Φ κ,π x) = ), if x π, +π) π π x + π+x, if x +π. We call the function Φ κ,π x) the Dombi-Jónás probability istribution function. It has only one constant parameter, which is the number π, while its maximum absolute approximation error over the set of real numbers is Note that )

3 Continuous-value Logic an Approximations to the Normal Distribution 83 there are just a few known approximations with a single constant parameter in this accuracy range e.g. [6], [0], [], [3]), an all these approximations inclue exponential terms, while ours oes not contain any an has a very simple form. It shoul also be ae that the probability ensity function φ κ,π x) can be irectly expresse in the terms of the probability istribution function Φ κ,π x) without ifferentiating it. In many practical applications, the value of the stanar normal probability istribution function for an argument being less than -3 or greater than +3 is consiere to be zero an one, respectively, although the probability istribution oes not take these values. The propose Φ κ,π x) approximation has the value of zero, if x π, an it has the value of, if x +π, so the function Φ κ,π x) may be viewe as an alternative, with boune omain, to the stanar normal probability istribution function. The remaining part of the paper is organize as follows. In Section, we will review some notable approximations to the stanar normal probability istribution function. Next, in Section 3, we will set our approximation criteria an introuce novel approximation methos that are connecte with the Dombi operators. Lastly, in Section 4, we will summarize our approximation results an raw some key conclusions about the propose Quasi Logistic probability istribution function. Approximations to the Stanar Normal Probability Distribution Function Now, we will give a short review of the techniques that are wiely use for approximating the stanar normal probability istribution function an enumerate some notable approximations that have been constructe in the last seven ecaes. We will use the common notations φx) an Φx) for the probability ensity function an probability istribution function of the stanar normal ranom variable, respectively. That is, φx) = x e x ; Φx) = π φt)t. ) The approximation methos available in the literature can be categorize into two main approach categories []. One category is the group of approximations that are base on numerical methos, while the other category contains methos that are foune on a-hoc approximations. The numerical methos are typically base on numerical integration techniques, various power series, expansions in Hermite or Chebyshev polynomials an continue fraction expansions e.g. [6], [8], [], [5], [7]). In general, these methos can yiel a high-level approximation accuracy, but require complex computations. The a-hoc approximation methos typically utilize an a priori selecte parametric function an apply various mathematical techniques to estimate the parameters in orer minimize the approximation error. Matic et al. [], Soranzo

4 83 József Dombi an Tamás Jónás an Epure [8] an Yerukala an Boiroju [3] gave comprehensive overviews of the approximation formulas in their papers. Here, without striving for completeness, we enumerate some notable approximation formulas an inicate their maximum absolute errors MAE).. Pólya 949) [6]: Φx) + e x /π ; MAE = Hart 957) [5]: Φx) π e x /π x+0.8e 0.4x ; MAE = Tocher 963) [9]: Φx) e /π +e /π ; MAE = Zelen & Severo 964) [34]: Φx) a t a t + a 3 t 3) e x π, where t = x), a = , a = , a 3 = ; MAE = Hart 966) [6]: Φx) x e πx P 0+ +bx +ax P 0x +e x +bx +ax where a = + π +6π π, b = πa an P 0 = π/; MAE = Page 977) [4]: Φx) +tanhy), where y = π x x ) ; MAE = Hamaker 978) [4]: Φx) ) e y, where y = 0.806x 0.08x); MAE Lin 989) [9]: Φx) ; MAE = e 0.77x 0.46x {, if 0 x.7 9. Norton 989) [3]: Φx) e 0.77x 0.46x πx e x, if x >.7; MAE = Lin 990) [0]: Φx), where 0 x < 9; MAE = e 4.π 9 x x 0 3. Bagby 995) []: Φx) + 30 MAE , 7e x / + 6e x ) π4 x) e x ) ;. Waissi & Rossin 996) [3]: Φx) +e π0.9x x x 5 ) ; MAE = x 3. Bryc 00) [4]: Φx) πx3 +a x+a e x +b x, +b x+a where a = , a = , b = , b = ; MAE =

5 Continuous-value Logic an Approximations to the Normal Distribution Shore 005) [7]: Φx) +g x)+gx), where gx) = e log eα/λ/s )+S x)λ/s ) )+S x λ = , S = , S = , α = ; MAE Aluaat an Aloat 008) []: Φx) + e π/8x ; MAE = Bowling et al. 009) [3]: Φx) +e x x) ; MAE = Yerukala{ et al. 0) [33]: H +.47H 3.03H 3, if 0 x 3.36 Φx), if x > 3.36, where H = tanh 0.695x), H = tanh0.546x) an H 3 = tanh0.434x); MAE = Vazquez-Leal et al. 0) [30]: Φx) tanh 79x MAE Chouhury 04) [5]: Φx) π MAE = e x x+0.33 ; x +3 e 0. Yerukala & Boiroju 05) [3]: Φx) x MAE = x+ 5 6 arctan )) 37x 94 + ; x +3 ;. Yerukala & Boiroju 05) [3]: Φx) wφ x) + w)φ x), where x > 0, w = 0.68, Φ x) is the approximation by Hart 966) an Φ x) is the approximation by Bryc 00); MAE = Matic et al. 06) []: Φx) + sgnx) e x where γ = 3 + π ; γ 4 = 7 90 γ 8 = π π 8 γ 0 = π 78 MAE = Eious an Al-Salman 06) [3]: Φx) MAE = π +γx +γ 4x 4 +γ 6x 6 +γ 8x 8 +γ 0x 0), 3π + 4 3π ; γ 6 = π 4 3π + π ; 3 3π π ; 4 567π π 6 3 3π π ; 5 + e 5/8x ) ; Base on the above approximation formulas, we may state that the accuracy of approximations increases with the complexity of formulas an with the number of parameters they possess.

6 834 József Dombi an Tamás Jónás 3 Novel Methos base on Operators of Continuous-value Logic Fist of all, we will lay own some expectations that we require from approximations an use these criteria to evaluate our results an compare them with some well-known ones. Next, we will introuce the Dombi operators that are familiar in continuous-value logic an construct novel approximation methos that are connecte with these operators. 3. Expectations towars Our Approximations The most basic expectation towars an approximation is that it is sufficiently accurate. In the literature, there are many approximations to the stanar probability istribution function that focus mainly on the approximation accuracy. These efforts have resulte in highly accurate functions, without taking some other features of the approximation into account. Here we set some criteria riven by theoretical an practical consierations that we require our approximations to meet. Simplicity an accuracy. The approximation functions shoul have a simple, easily computable formula, an the approximation accuracy shoul meet the requirements of practical applications. Ientity to first orer at zero. Let F x) be an approximating function to the stanar normal probability istribution function. We require F x) to be a probability istribution function an meet the following criteria: F 0) = Φ0) = 0.5 F x) x = Φx) x=0 x = φx) =. x=0 x=0 π 3) Symmetry. Since the probability ensity function φx) is an even function, Φ x) = Φx) hols for any x R. We require the approximation F x) to have the same feature; that is, F x) = F x) for any x R. Note that if F x) satisfies the F x) = F x) requirement, then the approximation error function δx) = Φx) F x) is an o function, an so the curve of Φx) F x) is symmetric with respect to the vertical axis. Direct connection between the ensity an istribution functions. In practice, it may be useful, if the probability istribution function can be expresse by the probability ensity function without integration, an vice versa, if the probability ensity function can be expresse by the probability istribution function without ifferentiation. Hence, we prefer the approximations that result in probability ensity an istribution functions with a irect connection between them; that is, one can be expresse by the other one in a close form.

7 Continuous-value Logic an Approximations to the Normal Distribution 835 It is worth emphasizing that only a few of the known approximations liste in Section meet all the requirements we emane. In general, the more complex an approximation formula is, the less of our criteria it meets. However, the approximations with more complex formulas an higher number of constant parameters result in a higher approximation precision. Note that many of the known approximations work just with positive values of variable x an let the user compute the approximating function value by using the Φ x) = Φx) equation for negative values of x. 3. Dombi Operators in Continuous-value Logic Here, we will introuce the Dombi operator class that can be utilize for implementing the conjunction an isjunction operations in continuous-value logic [8], [0]. Definition. The Dombi conjunction an isjunction operator in continuousvalue logic is given by o α x) = n + i= ) ) α /α an o αx) = x i x i + n n x i ) ), α /α 4) x i where x = x, x,..., x n ), an x, x,..., x n are continuous-value logic variables. If α > 0, then the Dombi operator is a conjunction operator; if α < 0, then it is a isjunction operator. Here, we will use the Dombi conjunction operators with two operans an α = : 0, if x = 0 or x = 0 cx, x ) = ox, x ) α= =, + x x + x x otherwise, 0, if x = 0 or x = 0 cx, x ) = ox, x ) α= = ), otherwise, 6) + x x + x x where x an x are two continuous-value logic variables. We call c the averaging Dombi conjunction operator. Note that operation c is not iempotent, while c may be viewe as an iempotent variant of c. i= 5) Remark. Base on the general representation theorem [9], n ) ) ox) = f fx i ) an ox) = f n fx i ) n i= i= 7)

8 836 József Dombi an Tamás Jónás are strict operators, if fx) is a strictly monotone function, where x = x, x,..., x n ), an x, x,..., x n are continuous-value logic variables. If we apply the function ) α x fx) = f α x) = 8) x to ox) an ox), then we get the operators o α x) an o α x), respectively. That is, f α x) is the generator function of Dombi conjunction an isjunction operators. In fuzzy logic, the linguistic moifiers like very, more or less, somewhat, rather an quite over fuzzy sets that have strictly monotonously increasing or ecreasing membership functions can be moele by the following unary operator calle the Kappa function []. Definition. The Kappa moifier operator Kappa function) is given by κ λ) ν,ν 0 x) = + ν0 ν ν 0 ν ) λ, 9) x x where ν, ν 0 0, ), λ R, an x is a continuous-value logic variable. In Section 3.4, we will use a special form of the unary moifier operator in 9) to construct a probability istribution function. 3.3 The Sigmoi Function an Some of Its Basic Properties Since we will use the Sigmoi function to construct probability ensity an probability istribution functions, here we will introuce it an some of its main properties. Definition 3. The Sigmoi function σ λσ) x) with the parameter λ σ is given by where λ σ R, λ σ 0, x R. σ λσ) x) =, 0) + e λσx Note that the Sigmoi function is also known as the Logistic function. The main properties, such as the range, continuity, monotonicity, limits, role of the parameter an convexity of the Sigmoi function σ λσ) x) are as follows. Range. The range of σ λσ) x) is the interval 0, ). Continuity. σ λσ) x) is a continuous function in R. Monotonicity. If λ σ > 0, then σ λσ) x) is strictly monotonously increasing If λ σ < 0, then σ λσ) x) is strictly monotonously ecreasing

9 Continuous-value Logic an Approximations to the Normal Distribution 837 Limits. Function σ λσ) x) takes neither the value zero, nor the value, as these are its limits: {, if λ σ > 0 lim x + σλσ) x) = ) 0, if λ σ < 0, lim x σλσ) x) = {, if λ σ < 0 0, if λ σ > 0. ) Role of the parameter. The parameter λ σ of σ λσ) x) has a semantic meaning relate to the shape of the function curve. The first erivative of σ λσ) x) at x = 0 is σ λσ) x) x = λ σ σ λσ) 0) x=0 ) σ λσ) 0) = λ σ 4. 3) That is, the λ σ parameter etermines the slope of σ λσ) x) at x = 0. Convexity. σ λσ) x) has a single inflection point that is at x = 0 If λ σ > 0, then σ λσ) x) changes from concave to convex at x = 0 If λ σ < 0, then σ λσ) x) changes from convex to concave at x = 0 Figure shows some examples of Sigmoi function plots λ σ = λ σ = λ σ =4 λ σ = λ σ = λ σ = x Figure : Examples of Sigmoi function plots.

10 838 József Dombi an Tamás Jónás 3.4 Tocher s Approximation an the Averaging Dombi Conjunction Operator Applying the averaging Dombi conjunction in 6) to σ λσ) x) an σ λσ) x) yiels the following λσ x) function: ) λσ x) = c σ λσ) x), σ λσ) x) = = + σ λσ ) x) σ λσ ) x) + e λσx + e λσx ) = e λσx + e λσx ). ) = + σ λσ ) x) σ λσ ) x) Figure shows the averaging Dombi conjunction of two Sigmoi fuzzy membership functions; that is, the intersection of two fuzzy sets that are given by Sigmoi functions: by a ecreasing an an increasing Sigmoi function with the same absolute λ σ parameter values ) σ λσ) x) σ λσ) x) c σ λσ) x), σ λσ) x) ) x Figure : The averaging Dombi conjunction of two Sigmoi fuzzy membership functions. Function λσ x), like the ensity function φx), has a bell-shape curve, but since + + e λσx [ ] + λσ x)x = + e λσx ) = λ σ + e λσx =, 5) ) λ σ λσ x) is not a probability ensity function. Hence, + λ σ λ σ x)x = + λ σ e λσx + e λσx =, 6) )

11 Continuous-value Logic an Approximations to the Normal Distribution 839 an so we efine the probability ensity function φ σ x) as follows. Definition 4. The probability ensity function φ σ x) is given by where λ σ = /π. Note that setting λ σ to /π ensures that φ σ x) = λ σe λσx + e λσx ), 7) φ σ x) x=0 = φx) x=0. 8) The corresponing probability istribution function Φ σ x) is Φ σ x) = x [ φ σ t)t = + e /πt ] x = + e /πx. 9) This means that the probability istribution function Φ σ x) is a Sigmoi function that has the parameter λ σ = /π. It is worth aing here that Φ σ x) is ientical to Tocher s approximation result in 3) from 963 [9]. However, we erive the function Φ σ x) by generating the ensity function φ σ x) from Sigmoi functions by utilizing the averaging Dombi conjunction operator, an this approach is ifferent from Tocher s. Approximation accuracy. It can be shown numerically that max Φx) Φ σx) ) x R Figure 3 shows the curve of absolute error function Φx) Φ σ x). Properties of the approximation. Here, we summarize the properties of this approximation in the light of expectations that were prescribe in Section 3.. Simplicity an accuracy. Φ σ x) has a simple formula, but its maximum absolute approximation error has an orer of magnitue of -. Ientity to first orer at zero. Since Φ σ 0) = Φ0) an the parameter λ σ of φ σ x) was set such that φ σ 0) = φ0), Φ σ x) an Φx) are ientical to first orer at x = 0. Symmetry. The probability ensity function φ σ x) is an even function an so Φ σ x) = Φ σ x) hols for any x R. Direct connection between the ensity an istribution functions. There is an interesting relation between the probability ensity function φ σ x)

12 840 József Dombi an Tamás Jónás 0.05 Φx) Φ σx) x Figure 3: Absolute errors of approximation by Φ σ x). an the probability istribution function Φ σ x) that is worth mentioning here. Namely, utilizing 7) an λ σ = /π gives the following equation: φ σ x) = π Φ σx) Φ σ x)). ) That is, φ σ x) can be expresse in terms of Φ σ x) in a close form. Accoring to Hillier an Liberman [7], the Sigmoi function that matches Φx) the best, has only one parameter an the form of 0) is Φ HL x) =. ) + e.70x This approximation has a maximum absolute error of Note that although this approximation yiels a higher accuracy than the approximation by the Φ σ x) function, the first erivative of Φ HL x) at x = 0 is not / π; that is, Φ HL x) is not ientical with Φx) to first orer. Note that if we use the Dombi conjunction operator in 5) to create a probability ensity function from two Sigmoi functions, then we woul get the following probability istribution function: Φ σx) = 3 π arctan 3 3 e 6π/3x + ) ). 3) The maximum absolute error of this approximation is 0.03 an calculation of the approximation formula requires the computation of an exponential function an an arcus tangent function. We can fin simpler formulas with better precision values among the known approximations enumerate in Section.

13 Continuous-value Logic an Approximations to the Normal Distribution An Approximation Connecte with the Unary Moifier Operator 3.5. The Epsilon Function Here, we introuce the Epsilon function that we will utilize for constructing approximations to the stanar normal probability istribution function. Definition 5. The Epsilon function ε λ) x) is given by where λ R, λ 0, R, > 0, x, +). ) λ ε λ) x + x) =, 4) x The following theorem introuces an important asymptotic property of the Epsilon function. Theorem. For any x, +), if, Proof. Let x have a fixe value, x, +). ε λ) x) eλx. 5) lim ελ) x) = lim ) λ x + = lim x = lim + x ) ) λ. x x + x x ) ) λ = 6) Since x is fixe, if, then = x an so the previous equation can be continue as follows: = lim lim + x ) ) λ = lim x + x ) ) λ +x = + x ) lim + x ) ) λ x = e x) λ λ = e λx. 7) Base on Theorem, we can state that the asymptotic Epsilon function is just the exponential function. It is worth mentioning here that the Epsilon function is the basis of the so-calle Epsilon probability istribution, which can be utilize to approximate the exponential probability istribution [].

14 84 József Dombi an Tamás Jónás 3.5. The Kappa Function an Some of its Basic Properties Here, we efine the Kappa function that we will use to approximate the stanar normal probability istribution function. Definition 6. The Kappa function κ λκ) x) is given by κ λκ) x) = + x +x where λ κ R, λ κ > 0, R, > 0, x, +). ) λκ, 8) Note that we utilize the Kappa function κ λκ) x) solely with positive λ κ parameter values. Here, we state the most important properties of the Kappa function κ λκ) x); namely, range, continuity, monotonicity, limits, role of the parameters an convexity. Range. The range of κ λκ) x) is the interval 0, ]. Continuity. κ λκ) x) is a continuous function in, +). Monotonicity. interval, +). As λ κ > 0, κ λκ) x) is strictly monotonously increasing in the Limits. lim x + κλκ) x) = 0 9) Note that as λ κ > 0, κ λκ) x) takes the value of at. lim x + κλκ) x) = 30) Role of the parameters. Both parameters λ κ an of κ λκ) x) have a semantic meaning relate to the shape of the function curve. Parameter specifies the, +) omain of κ λκ) x). The first erivative of κ λκ) x) at x = 0 is x) x = λ κ x=0 κ λκ) κ λκ) x) ) κ λκ) x) x) + x) = λ κ x=0. 3) That is, parameter λ κ etermines the graient of function κ λκ) x) at x = 0.

15 Continuous-value Logic an Approximations to the Normal Distribution 843 Convexity. It can be shown that the Kappa function κ λκ) x) has a single inflection point at x = 0, where it changes its shape from convex to concave Connection with the Sigmoi function The Kappa function has the following asymptotic property that allows us to use it for approximating the Sigmoi function an through that the stanar normal probability istribution function. Lemma. If σ λσ) x) is a Sigmoi function with the parameter λ σ > 0, κ λκ) x) is a Kappa function with parameters λ κ, > 0 an then for any x, +), if, then λ κ = λ σ, 3) κ λκ) x) σ λσ) x). 33) Proof. Let x have a fixe value. If the conitions of the lemma are satisfie, then the Kappa function κ λκ) x) may be written as κ λκ) x) = + x +x ) λκ = + +x x ) λσ an base on Theorem, ε λσ) x) e λσx, if ; that is, κ λκ) x) = + ε λσ) x) = + ε λσ) x), 34) + e λσx = σλσ) x). 35) Corollary. In the interval, +), the probability istribution function Φ σ x) = can be approximate by the Kappa function κ λκ) x) = + e λσx 36) + x +x where R, > 0, λ σ = /π, λ κ = /π. Proof. The corollary follows from Lemma. ) λκ, 37)

16 844 József Dombi an Tamás Jónás The Quasi Logistic Probability Distribution Function Now, we will we efine the Quasi Logistic probability istribution function by utilizing the Kappa function given in 8). Definition 7. The Quasi Logistic probability istribution function is given by 0, if x Φ κ, x) = κ λκ) x), if x, +) 38), if x +, where R, > 0, λ κ = /π. It is worth mentioning here that there is an interesting relation between the Quasi Logistic probability ensity function φ κ, x) an the probability istribution function Φ κ, x). Lemma. If x, +), then where R, > 0. φ κ, x) = π Φ κ, x) Φ κ, x)), 39) x) + x) Proof. Base on the efinition of Φ κ, x) in 38), if x, +), then Utilizing this equation, 3) an λ κ = /π, we get Φ κ, x) = κ λκ) x). 40) φ κ, x) = Φ κ,x) x ) κ λκ) x) κ λκ) x) = λ κ = x) + x) π = κλκ) x) = x Φ κ, x) Φ κ, x)). x) + x) 4) Utilizing Lemma, the Quasi Logistic probability ensity function φ κ, x) for x R is ) φ κ, x) = κ λκ) x) κ λκ) x), if x, +) π x) + x) 4) 0, otherwise, where R, > 0. Note that base on the properties of κ λκ) x), it can be shown that Φ κ, x) is in fact a probability istribution function an φ κ, x) is its probability ensity function. Therefore, the following criteria are met:

17 Continuous-value Logic an Approximations to the Normal Distribution 845. φ κ, x) 0 for any x R x φ κ, x)x = φ κ, t)t = Φ κ, x). Corollary. The stanar normal probability istribution function Φx) can be approximate by the Quasi Logistic probability istribution function Φ κ, x). Proof. The corollary follows from the fact that Φx) can be approximate by the Sigmoi function σ λσ) x) that has the parameter λ σ = /π an from Corollary an from the efinition of the Quasi Logistic probability istribution function. It is worth mentioning that φ κ, x) can be erive from the Kappa function also in the following way. Utilizing the fact that is a probability ensity function, κ λκ) x) f λκ,x) =, if x, +) x 0, otherwise κ λκ) x) = λ κ x κ λκ) x) κ λκ) x) + x) ) x) 43), 44) an setting the requirement f λκ,0) = φ0) results in the following equation: ) κ λκ) x) κ λκ) x) λ κ x) + x) = e x x=0 π. 45) x=0 Using 3), this equation leas to λ κ λ κ = /π. = /π; that is, f λκ,x) = φ κ, x), if Approximation Accuracy It can be shown numerically that Φx) Φ κ, x) is approximately minimal, if = 3.5. In this case, the maximum absolute approximation error is Consiering the fact that 3.5 is close to π, using = π instea of = 3.5 oes not worsen significantly the approximation accuracy. If = π, then the maximum absolute approximation error is Although the parameter with value of = 3.5 yiels the least maximum absolute approximation error among the Quasi Logistic probability istribution functions, we propose the use of function

18 846 József Dombi an Tamás Jónás Φ κ,π x) as it has a very simple form an its maximum absolute approximation error is just slightly greater than that of function Φ κ, x) with = , if x π Φ κ,π x) = ), if x π, +π) π 46) π x + π+x, if x +π We call the Quasi Logistic probability istribution function with = π; that is, the function Φ κ,π x), the Dombi-Jónás probability istribution function. The absolute errors Φx) Φ κ, x) for = 3.5 an = π are shown in Figure Φx) Φ κ,x),= π.5 Φx) Φ κ, x),= x Figure 4: Absolute errors of approximations by using Quasi Logistic probability istribution functions Properties of the Approximation Here, we summarize the properties of the Φ κ,π x) approximation. Simplicity an accuracy. The maximum absolute error of approximation Φ κ,π x) is an at the same time function Φ κ,π x) has a very simple form. In this accuracy range, there is no other known approximation that has such a simple form. The known approximations that yiel higher accuracy have more complex forms, while the ones with similarly complex formulas o not give greater accuracy. Ientity to first orer at zero. Since Φ κ,π 0) = Φ0) an the probability ensity function φ κ,π x) was constructe such that φ κ,π 0) = φ0), Φ κ,π x) an Φx) are ientical to first orer at x = 0.

19 Continuous-value Logic an Approximations to the Normal Distribution 847 Symmetric absolute error function. It can be shown that the probability ensity function φ κ,π x) is an even function an so Φ κ,π x) = Φ κ,π x) hols for any x R. Direct connection between the ensity an istribution functions. Base on Lemma, the ensity function φ κ,π x) can be irectly expresse in terms of the istribution function Φ κ,π x) in a close form Connections with Dombi Operators Next, we will show how the Epsilon function ε λ) x) an the Kappa function x) are connecte with the Dombi operators. κ λκ) Lemma 3. The generator function f α x) of Dombi conjunction an isjunction operators can be erive from the Epsilon function ε λ) x) by a linear function transformation. Proof. Let us apply the x = x + )/) linear transformation to the variable x, where x, ), > 0. After this transformation, the omain of x is the interval 0, ), x = x, an ε λ) x) = ) λ x + x x + = x + x = x ) λ = fα x ), ) λ = x x ) λ = 47) where α = λ/. Base on this result, the generator function of the Dombi operators may be viewe as a special case of the Epsilon function. Lemma 4. If ν = ν 0 = /, then the Kappa function κ λκ) x) can be erive from the Kappa function κ λ) ν,ν 0 x) in 9) by applying a linear function transformation. Proof. The lemma can be proven by setting λ = λ κ an applying the x = x linear transformation > 0). Base on this lemma, we can state that the Kappa function κ λκ) x), which we utilize to construct the Quasi Logistic probability istribution function, is a special case of the general fuzzy moifier operators. 4 Conclusions an Future Work Table summarizes the maximum absolute errors of the approximations presente earlier. From this table, we can see that the approximation by function Φ κ,π x) has a one orer of magnitue less maximum absolute error than the approximation by

20 848 József Dombi an Tamás Jónás Table : Gooness of the approximations given previously F x) Φ σ x) Φ κ,π x) max Φx) F x) x R the function Φ σ x). Figure 5 an Figure 6 show the approximating function curves an the absolute errors of the approximations, respectively. Base on comparisons of these approximations with the ones given in the literature, the following finings shoul be emphasize Φx) Φσx) Φκ,πx) Φx) Φσx) Φx) Φκ,πx) x Figure 5: Approximations x Figure 6: Absolute errors. Simplicity an accuracy. The first of the approximations liste earlier, which is the same as Tocher s approximation [9], has a maximum absolute approximation error of Although this approximation has a simple form, its accuracy is lower than the accuracy of some known approximations that have similar complex formulas e.g. [6], [0], [], [3]). The maximum absolute error of approximation by function Φ κ,π x) is This error is one orer of magnitue less than that of the first approximation. At the same time, function Φ κ,π x) has a very simple formula with only one constant parameter which is the constant π. It shoul be ae here that there are only a few known approximations with a single constant parameter in this accuracy range e.g. [6], [0], [], [3]), an all these approximations inclue exponential terms, while Φ κ,π x) oes not contain any. That is, to the best of our knowlege, in this accuracy range, there is no other known approximation that has such a simple formula as Φ κ,π x). The known approximations that yiel a higher accuracy have more complex formulas, while the ones with similar complex formulas o not give a higher accuracy.

21 Continuous-value Logic an Approximations to the Normal Distribution 849 Ientity to first orer at zero. The presente approximations of Φx) are ientical with Φx) to first orer at x = 0. Symmetric absolute error function. It is worth noting that both of the above approximations meet the F x) = F x) criterion for any x R, an so their absolute error function curves are symmetric with respect to the vertical axis, as can be seen in Figure 6. Direct connection between the ensity an istribution functions. It is the case both for the Sigmoi approximation Φ σ x) an the Quasi Logistic approximation Φ κ, x) that the probability ensity function can be irectly expresse in terms of the probability istribution function in a close form. That is, the ensity function can be erive from the istribution function without ifferentiating it. This property of the of our approximations can be very useful in practice. Connections with the possibilistic approach. The given approximators are connecte with continuous logic. Namely, the approximation Φ σ x) is erive from Sigmoi fuzzy membership functions by applying the averaging Dombi conjunction operator, while the Quasi Logistic approximation is a linearly transforme form of the Kappa function that is a well-known moifier operator in fuzzy theory. Applicability. For any x R argument, the stanar normal probability istribution function Φx) takes a value in the interval 0, ). In other wors, it associates positive probabilities with arguments that are much less than 0, an gives probabilities less than for those arguments that are much greater than zero. In many practical applications, the probabilities for arguments that are much less or much greater than the expecte value of a normally istribute ranom variable are consiere to be zero an one, respectively, although the exact probabilities for these arguments lie in the interval 0, ). The probability istribution function Φ κ,π x) takes a value from the interval 0, ) only if its argument is greater than π an less than +π. Noting that Φ π) = , Φπ) = an max Φx) Φ κ,πx) , 48) x π,+π) the Dombi-Jónás probability istribution may be viewe as an alternative, with boune omain, to the stanar normal probability istribution. Plans for future work. The Kappa function that we use to construct the probability istribution function Φ κ,π x) is symmetric about the point 0, 0.5). In certain economic an technological applications, asymmetric probability istributions with boune omains are neee for moeling an simulation purposes. As part of our future research work, we woul like to stuy how a generalize, asymmetric version of the Kappa function, which is efine over the boune omain a, b), can be utilize for constructing asymmetric probability istribution functions.

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