European International Journal of Science and Technology Vol. 4 No. 9 December, 2015
On the Approximation of the Step function by Raised-Cosine and Laplace Cumulative Distribution Functions
Vesselin Kyurkchiev 1 Faculty of Mathematics and Informatics Institute of Mathematics and Informatics, Paisii Hilendarski University of Plovdiv, Bulgarian Academy of Sciences, 24 Tsar Assen Str., 4000 Plovdiv, Bulgaria Email: [email protected]
Nikolay Kyurkchiev Faculty of Mathematics and Informatics Institute of Mathematics and Informatics, Paisii Hilendarski University of Plovdiv, Bulgarian Academy of Sciences, Acad.G.Bonchev Str.,Bl.8 1113Sofia, Bulgaria Email: [email protected]
1 Corresponding Author
Abstract In this note the Hausdorff approximation of the Heaviside step function by Raised-Cosine and Laplace Cumulative Distribution Functions arising from lifetime analysis, financial mathematics, signal theory and communications systems is considered and precise upper and lower bounds for the Hausdorff distance are obtained. Numerical examples, that illustrate our results are given, too.
Key words: Heaviside step function, Raised-Cosine Cumulative Distribution Function, Laplace Cumulative Distribution Function, Alpha–Skew–Laplace cumulative distribution function, Hausdorff distance, upper and lower bounds
2010 Mathematics Subject Classification: 41A46
75 European International Journal of Science and Technology ISSN: 2304-9693 www.eijst.org.uk
1. Introduction The Cosine Distribution is sometimes used as a simple, and more computationally tractable, approximation to the Normal Distribution. The Raised-Cosine Distribution Function (RC.pdf) and Raised-Cosine Cumulative Distribution Function (RC.cdf) are functions commonly used to avoid inter symbol interference in communications systems [1], [2], [3]. The Laplace distribution function (L.pdf) and Laplace Cumulative Distribution function (L.cdf) is used for modeling in signal processing, various biological processes, finance, and economics. Examples of events that may be modeled by (L.pdf) include: credit risk and exotic options in financial engineering, uncurance claims and structural changes in switching–regime model and Kalman filter [4].
Definition 1. The RC.cdf is defined for by: ( ; , ) − < < + , ∈ , >0 = . ( ) (1) ( ; , ) 1 + + sin Special case. For a=0 we obtain the special RC.cdf:
= (2) for ( ;which 0, ) 1 + + sin , (0; 0, ) = .
Definition2.The L.cdf f1(t;a,b) is defined for b>0 by:
ƒ = , < , (3) ( ; , ) where is1 a − location , parameter. ≥ , > 0 Definition3. The Heaviside step function h0(t) is defined by:
= 0, < 0, (4) ℎ ( ) 0, 1 , = 0 1, > 0 2. Approximation of the Heaviside function by RC.cdf.
We study the Hausdorff approximation [5] of the step function h0(t) by RC.cdf of the form (2) and find an expression for the error of the best approximation.
The Hausdorff distance d=d (b) between the function h0(t) and the function f(t;0,b) satisfies the relation
ƒ = (5) or (d; 0, b) 1 + + sin = 1 − , . (6) sin + − + = 0
76 European International Journal of Science and Technology Vol. 4 No. 9 December, 2015
The following proposition give supper and lower bounds for d(b).
Proposition1. The Hausdorff distance d=d(b) between the step function h0 and the RC.cdf(2) can be expressed interms of the parameter for any real 0 (7) 1 ln 2.5 1 + < < Proof. Let us examine the function 2.5(see,(6) 1 + 2.5 1 + 1 1 From F’ ‘(d)>0 we conclude that the function( ) = F sinis strictly+ monotonically− + . increasing. Consider function 2 2 2 1 1 ( ) = − + 1 + . 2 2 Figure1: The cumulative function (2) with b =0.5; Hausdorff distance d =0.178351. Figure 2: The cumulative function (2) with b =0.1; Hausdorff distance d=0.0575502. European International Journal of Science and Technology ISSN: 2304-9693 www.eijst.org.uk Figure 3: The functions F(d) and G(d) with b=0.5. From Taylor expansion 1 1 1 1 − + 3 + sin = − + 1 + + ( ) We obtain G(d)−F(d)=2O(d2 ). 2 2 Hence G(d) approximates F(d) with d→0asO(d3)(see,Fig.3). In addition G’ (d)>0. Further, for 0 1 1 1 = − + < 0, 2.5 1 + 2 2.5 ln 2.5 1 + 1 1 1 = − + ln 2.5 1 + > 0. 2 2.5 2.5 1 + This completes the proof of the proposition. Some computational examples using relation (6) are presented in Table1 for various b. 3. Approximation of the schifted Heaviside function by L.cdf. We study the Hausdorff approximation [5] of the schifted step function by L.cdf of the form (3) and find an expression for the error of the best approximation. 78 European International Journal of Science and Technology Vol. 4 No. 9 December, 2015 Figure 4: Simple module implemented in programming environment Mathematica for calculation of the value of the Hausdorff distanced between the Heaviside step function and the sigmoidal RC.cdf function. Figure 5: An example of the usage of dynamical and graphical representation. The plots are prepared using CAS Mathematica. 79 European International Journal of Science and Technology ISSN: 2304-9693 www.eijst.org.uk b d(b) 0.5 0.178351 0.4 0.155721 0.3 0.129224 0.2 0.0974216 0.1 0.0575502 0.05 0.0326076 0.01 0.00786171 0.001 0.00089689 Table1: Bounds for d computed by (6) for various b. Figure 6: The cumulative function (3) with a=1, b=0.05; Hausdorff distance d=0.0872764. The Hausdorff distance d=d(b) between the schifted step function and the function f1(t;a,b) satisfies the relation (8) 1 ( + ; 0, ) =or 1 − = 1 − 2 (9) 1 The following proposition give supper − and =lower 0. bounds for d(b). 2 Proposition 2. The Hausdorff distance d=d(b) between the step function h0 and the L.cdf(3)can be expressed interms of the parameter b for any real 0 European International Journal of Science and Technology Vol. 4 No. 9 December, 2015 1 ln 3 1 + < < 3 1 + 3 1 + Proof. Let us examine the function (see,(9) 1 ’ ( ) = − . From L (d)>0 we conclude that the function L is strictly2 monotonically increasing. Consider function 1 1 ( ) = − + 1 +3 . From Taylor expansion we obtain M(d)–L2(d)=O(d ).2 Hence M(d) approximates L(d) with d→0 as O(d3). Inaddition M’(d)>0. Further, for 0 1 1 1 = − + < 0, 3 1 + 2 3 ln 3 1 + 1 1 1 = − + ln 3 1 + > 0. 2 3 2 3 1 + This completes the proof of the proposition. Some computational examples using relation (9) are presented in Table 2 for various b. b d(b) 0.1 0.1326722 0.05 0.0872764 0.01 0.0286089 0.001 0.00467284 Table 2: Bounds for d computed by (9) for various b and a=1. 4. Remarks. 1. The Semi–elliptical distribution [14] is known as Wigner’s semicircle distribution. The Semi–elliptical cumulative distribution function is defined for −b 1 1 (11) ( ; ) = + 1 − + arcsin . 2 81 European International Journal of Science and Technology ISSN: 2304-9693 www.eijst.org.uk Figure 7: The cumulative function (11) with b=1; Hausdorff distanced=0.307419. Figure 8: The Alpha–Skew–Laplace cumulative distribution function with α=0.1 in [−100,100]. Proposition 3. The Hausdorff distance d=d(b) between the step function h0 and the Semi–elliptical cumulative distribution function (11) can be expressed interms of the parameter b for any real 0 (12) 1 ln 3 1 + < < . 3 1 + 3 1 + The proof of the Proposition 3 follows the ideas given in this note and will be omitted. European International Journal of Science and Technology Vol. 4 No. 9 December, 2015 2. The Alpha–Skew–Laplace cumulative distribution function (t;α) is defined by [12](see,Fig.7): 1 + (1 − ) 1 + (1 − ) (13) + , < 0, 4 (1 + ) 2 (1 + ) ( ; ) = 1 + (1 − ) 1 − (1 + ) 1 − + , ≥ 0. 4 (1 + ) 2 (1 + ) ≡ We note that f1(t;0,1) f3(t;0). In the present paper we do not consider sigmoid functions generated as cumulative functions of other probabilistic distributions, such as the Skew–Laplace, α–Skew Laplace, and multimodal–Skew Laplace distribution [11]–[13]. Based on the methodology proposed in the present note, the reader may formulate the corresponding approximation problems on his/her own. For other results, see [6]–[13]. References [1] Turner C.Raised Cosine and Root Raized Cosine Formulae, Wiereless Systems Engineering, Inc., May 29, 2007, V1.2; www.claysturner.com [2] Surhone L.,M.Tennoe, S.Henssonow. Raized Cosine Distribution, VDM Publishing, 2010, ISBN 978-6-1313-6436-5 [3] SunE., R.Yang, Y.Zhang. Raized Cosine–Like Companding Scheme for Peak–to–Averadge Power Ratio Reduction of SCFDMA Signals, Wireless Pers Commun (2012) 67,913–291; DOI:10.1007/S11277-011-0418-0 [4] Kotz,S.,T.Kozubowski, K.Podgorski. The Laplace Distribution and Generalizations: A Revis it with Applications to Communications, Economics, Engineering, and Finance, 2001, Birkha¨ user, Basel. [5] Sendov Bl.(1990) Hausdorff Approximations, Kluwer, Boston, ISBN: 978–94–010–6787–4. [6] Kyurkchiev N.,S.Markov. On the Hausdorff distance between the Heaviside step function and Verhulst logistic function, J.Math.Chem., 2015; DOI:10.1007/S10910-015-0552-0 [7] A.Iliev, N.Kyurkchiev, S.Markov, On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions, BIOMATH 4(2)(2015) (http://dx.doi.org/10.11145/j.biomath.2015.10.101). [8] N.Kyurkchiev, S.Markov, Sigmoidal functions: some computational and modelling aspects, Biomath Communications 1(2)(2014)30–48;doi:10.11145/j.bmc.2015.03.081. 83 European International Journal of Science and Technology ISSN: 2304-9693 www.eijst.org.uk [9] N.Kyurkchiev, S.Markov, On the approximation of the generalized cut function of degree p+1by smooth sigmoid functions, Serdica J. Computing, (2015) (submitted). [10] N.Kyurkchiev, S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects, LAPLAMBERT Academic Publishing, S aarbrucken, 2015; ISBN 978-3-659-76045-7. [11] Chakraborty, P. Hazarika, M. Ali. A multi modal skew Laplace distribution, Pak.J. Statist. (2014)30(2),253–264 [12] Harandi, S.,M.Alamatsaz. Alpha–Skew–Laplace distribution, Statistics and Probability Letters (2013)83,774–782 [13] Aryal, G.,C.T sokos. Application of the Truncated Skew Laplace Probability Distribution in Maintenance System, J. of Modern Applied Statistical Methods (2009)8(2),409–422 [14] Rinne,H. Location–Scale Distribution: Linear Estimation and Probability Plotting Using MATLAB, Copyright: Prof.em. Dr Hors Rinne, Department of Economics and Management Science, Justus–Lieblig– University, Giessen, Germany, 2010 84