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630 IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 9, SEPTEMBER 2008 Lognormal Sum Approximation with a Variant of Type IV Pearson Distribution Shaohua Chen, Hong Nie, Member, IEEE, and Benjamin Ayers-Glassey Abstract—In this paper, a variant of the Type IV Pearson distribution is proposed to approximate the distribution of the sum of lognormal random variables. Numerical and computer simulations show that independent of the statistical characteristics of the lognormal sum distribution, the Type IV Pearson variant outperforms the standard Type IV Pearson distribution and the normal variant distribution in accurately approximating the lognormal sum distribution for a whole probability range. Index Terms—Lognormal distribution, sum of lognormal random variables, Type IV Pearson variant, curve fitting. I. I NTRODUCTION T HE lognormal distribution has been widely used in wireless communication areas to model and analyze the characteristics of attenuations caused by shadowing fading in wireless channels and power fluctuations due to power control errors [1]. Therefore, frequently the knowledge about the probability density function (PDF) and cumulative distribution function (CDF) of the sum of lognormal random variables (RVs) is required so as to analyze the performance of wireless systems. During the past decades, many research efforts have been devoted to investigate the statistical characteristics of the lognormal sum RVs, and although the closed-form expressions of the PDF and CDF of the lognormal sum RVs are still unknown, various approximation methods have been developed [2-7]. The most widely used approximation method is to represent the lognormal sum RVs still with a lognormal distribution [26]. However, the numerical PDF and CDF of the lognormal sum RVs obtained by [4] have clearly shown that the lognormal sum distribution has considerable discrepancy from a lognormal distribution. In order to eliminate this discrepancy, [7] has proposed to approximate the lognormal sum RVs with the Type IV Pearson distribution, which can accurately approximate the lognormal sum RVs in a wide probability range (from 0.01 to 0.99999). However, the Type IV Pearson approximation proposed by [7] has two limitations. First, the variable domain of the Type IV Pearson distribution is (−∞, +∞), but that of the lognormal sum RVs is (0, +∞). Therefore, the Type IV Pearson approximation leads to a large discrepancy in the head portion (small values) of the lognormal sum distribution. Second, in order to use the Type IV Pearson Manuscript received April 8, 2008. The associate editor coordinating the review of this letter and approving it for publication was P. Cotae. S. Chen and B. Ayers-Glassey are with the Department of Math, Physics and Geology, Cape Breton University, Sydney, NS, B1P6L2, Canada (e-mail: george [email protected], [email protected]). H. Nie is with the Department of Industrial Technology, University of Northern Iowa, Cedar Falls, IA 50614-0178, USA (e-mail: [email protected]). Digital Object Identifier 10.1109/LCOMM.2008.080553. approximation, the selection parameter of the lognormal sum RVs must be less than one [8]. Consequently, quite a few number of lognormal sum RVs cannot be approximated with the Type IV Pearson distribution since their selection parameters are larger than one. In order to relieve the above two limitations but still maintain the advantages of the Type IV Pearson approximation, in this paper, we have developed a variant of the Type IV Pearson distribution to approximate the lognormal sum RVs. This newly developed Type IV Pearson variant has a variable domain of (0, +∞), and does not have any preclusive condition on the selection parameter of the lognormal sum RVs to be approximated. Therefore, it can accurately approximate almost all lognormal sum RVs for a whole probability range, from the head portion to the body portion (intermediate values) and the tail portion (large values). It is worth to mention that before the Type IV Pearson variant is proposed, we had investigated other types of Pearson distributions and found none of them can accurately approximate lognormal sum RVs. The paper is organized as follows: a variant of the Type IV Pearson distribution is developed in Section II. Then the parameter evaluations for the newly proposed Type IV Pearson variant is discussed in Section III. Through Numerical and computer simulations, the performance improvement of the Type IV Pearson variant approximation is also described in Section III. Finally, conclusions of the paper are given in Section IV. II. A VARIANT OF T YPE IV P EARSON D ISTRIBUTION As it is shown in [8], the PDF of the Type IV Pearson distribution is given by: −µ1 fP (z) = ν 1 + ((z − μ4 )/μ3 )2 × exp [−μ2 arctan((z − μ4 )/μ3 )] , −∞ < z < +∞, (1) where μ1 to μ4 are four independent +∞ parameters, and the function of ν is to ensure that −∞ fP (z)dz = 1. Clearly, since the domain of z in Eq. (1) is (−∞, +∞) and that of the lognormal sum RVs is (0, +∞), approximating the lognormal sum RVs with the Type IV Pearson distribution leads to a large discrepancy in the head portion of the lognormal sum distribution, i.e. when z << E[z]. In order to mitigate this discrepancy, we have made the following mapping for z: z(ξ) = ξ − (μ5 /ξ)µ6 , (2) where μ5 and μ6 are two new parameters. In this mapping, the domain of ξ is (0, +∞). Furthermore, by properly choosing the values of μ5 and μ6 so that (μ5 /ξ)µ6 becomes numerically significant only when ξ << E[ξ], we can ensure that z(ξ) ≈ ξ for all the other scenarios, and hence the advantages c 2008 IEEE 1089-7798/08$25.00 CHEN et al.: LOGNORMAL SUM APPROXIMATION WITH A VARIANT OF TYPE IV PEARSON DISTRIBUTION 0 −∞ 1−(1e−8) 1−(1e−6) N=40, σ ∈ [4, 5] i 0.9999 0.999 0.99 CDF GPV(ξ < A) of the Type IV Pearson approximation described in [7] are maintained. Based upon the above mapping, a variant of the Type IV Pearson distribution has been developed, the PDF of which can be mathematically expressed as follows: µ μ6 μ5 6 fP [z(ξ)], gP V (ξ) = z (ξ)fP [z(ξ)] = 1 + ξ ξ (3) and the CDF of which can be derived as follows: A z(A) gP V (ξ)dξ = fP (z)dz, (4) GP V (ξ < A) = 631 0.9 N=100, σ ∈ [5.5, 6.5] 0.5 N=70, σ ∈ [5, 6] 0.1 N=50, σi ∈ [4.5, 5.5] i i 0.01 Empirical PDF, mi=[−.5 .5] 0.001 Type IV Pearson distribution Type IV Pearson variant MGF method − head fitting MGF method − tail fitting Schwartz−Yeh method 1e−4 µ6 where z(A) = A − (μ5 /A) . 1e−6 1e−8 III. PARAMETER E VALUATIONS FOR THE T YPE IV P EARSON VARIANT 15 where mi and σi are the parameters of lognormal distributions in dB, and β = (ln 10)/10. In order to approximate the sum of these N lognormal RVs with the Type IV Pearson variant, the mean (MT ), variance (VT ), skewness (SkT ), and kurtosis (KuT ) of the sum of the lognormal RVs must be derived first with the method given by [7]. Then the selection parameter described in [8], denoted as KT here, can be obtained as follows: SkT2 (KuT + 3)2 . 4(4KuT − 3SkT2 )(2KuT − 3SkT2 − 6) (6) Depending on whether the value of KT is smaller than one or not, two different methods must be employed respectively to evaluate the values of μ1 to μ6 in the Type IV Pearson variant. 3.1. KT < 1 When KT < 1, as it is shown in [7], the lognormal sum distribution can be approximated with a standard Type IV Pearson distribution. Correspondingly, the values of μ1 , μ2 , μ3 , and μ4 can be derived from the following equations [8]: μ1 = (r + 2)/2, −r(r − 2)SkT , μ2 = 16(r − 1) − SkT2 (r − 2)2 μ3 = VT [r − 1 − SkT2 (r − 2)2 /16], μ4 = MT − (r − 2)SkT VT /4, 25 30 35 A(dB) Let Z1 , · · · , ZN be N independent but not necessarily identical lognormal RVs with PDFs as follows: (10 log10 z − mi )2 1 exp − fZi (z) = √ , 2σi2 2πβσi z i =1, · · · , N, (5) KT = 20 (7) (8) (9) (10) where r = 6(KuT − SkT2 − 1)/(2KuT − 3SkT2 − 6). Unfortunately, there is no close-form equation to derive the values of μ5 and μ6 , so in this paper, we have evaluated the values of μ5 and μ6 as follows: first, according to the values of mi , σi , and N , obtain the empirical CDF of the lognormal sum distribution through Monte Carlo computer simulations, and draw the empirical CDF on the so-called lognormal paper [4]; second, use the built-in least square curve fitting function in MATLAB software, lsqcurvefit, to estimate the optimal Fig. 1. CDF comparisons of the lognormal sum distributions with KT < 1 and their Type IV Pearson variant approximations. values of μ5 and μ6 , which achieve the best match between the empirical CDF of the lognormal sum distribution and the numerical CDF of the Type IV Pearson variant calculated from Eq. (4). Our further investigations on the values of μ5 and μ6 show that when the values of mi , σi , and N vary within a notvery-wide range, the values of μ5 and μ6 linearly depend on those of μ1 , μ3 , and μ4 . For example, when mi ∈ [−0.5, 0.5], N ∈ [40, 100], and σi ∈ [σ − 0.5, σ + 0.5] with σ ∈ [4.5, 6], the values of μ5 and μ6 can be calculated from the following empirical equations: μ5 = −11.8 − 0.07μ1 + 0.3μ3 + 0.95μ4, μ6 = 3.8 + 0.92μ1 − 0.04μ3 + 0.032μ4. (11) (12) Therefore, when KT < 1, for most cases the values of μ1 to μ6 can be easily evaluated from the close-form analytical equations and the empirical equations. In order to demonstrate the approximation improvement of the newly developed Type IV Pearson variant, in Fig. 1 we have plotted the CDFs of i) the lognormal sum distribution with 4 × 108 random samples, ii) the standard Type IV Pearson approximation [7], iii) the Type IV Pearson variant approximation, iv) the MGF lognormal approximation [5], and v) the S-Y lognormal approximation [3] for four different sets of N and σi . Since on the lognormal paper the CDF of the lognormal sum distribution is a convex curve and that of a lognormal distribution is a straight line, as shown in Fig. 1, the MGF approximation is good at either the head portion or the tail portion but not the both, and the S-Y approximation is good only at the body portion. Furthermore, from Fig. 1 it is clear that the standard Type IV Pearson approximation leads to a large discrepancy in the head portions of the lognormal sum distribution. Only the Type IV Pearson variant shows excellent approximation for a whole probability range, from the head portion, the body portion, to the tail portion. 3.2. KT ≥ 1 When KT ≥ 1, the standard Type IV Pearson distribution can no longer be used to approximate the lognormal sum distribution. Correspondingly, there is no close-form analytical 632 IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 9, SEPTEMBER 2008 TABLE I T HE PARAMETERS OF THE T YPE IV P EARSON VARIANT FOR σ = 12 μ2 -6.47 -4.88 -2.17 μ3 62.2 647 3393 μ4 -38 403 4836 μ5 641 1287 7985 0.9999 μ6 0.95 4.57 8.43 equation to derive the values of μ1 to μ4 either. In order to approximate the lognormal sum distribution with the Type IV Pearson variant, we have evaluated all the six parameters, μ1 to μ6 , with the curve fitting method listed in the previous subsection. In previous publications, [9] also used the curve fitting method to approximate the lognormal sum distribution. However, a variant of normal distribution, instead of the Type IV Pearson variant, is employed by [9] to approximate the lognormal sum distribution. In order to compare the performance of the Type IV Pearson variant approximation with that of the normal variant approximation proposed in [9], we have evaluated the values of μ1 to μ6 with the same parameters as those in [9], i.e., mi = m = 0 and σi = σ = 12. The obtained optimal values for μ1 to μ6 when N = 20, 100, 300 are listed in Table 1. According to the values of μ1 to μ6 listed in Table 1, in Fig. 2 the numerical CDFs of the Type IV Pearson variant approximation and the normal variant approximation are plotted on the lognormal paper together with the empirical CDFs of the lognormal sum distribution obtained from computer simulations. As shown in Fig. 2, when the value of N is small, e.g. N = 20, the Type IV Pearson variant approximation has almost the same performance as the normal variant approximation. However, when the value of N is large, e.g. N = 100 and 300, the Type IV Pearson variant approximation outperforms the normal variant approximation in almost all probability ranges, especially in the tail portion of the lognormal sum distribution. Furthermore, fitting curves in the lognormal paper puts much more emphasis on minimizing the approximation discrepancy in the head and tail portions rather than in the body portion. In order to accurately illustrate the approximation discrepancy in the body portion, in Fig. 3 for N = 100 we have plotted the PDFs of the Type IV Pearson variant approximation, the normal variant approximation, and the lognormal sum distribution in a linear-scale paper. Clearly, when the value of N is large, the normal variant approximation has to sacrifice its accuracy in the body portion of the lognormal sum distribution so as to approximate the head and tail portions of the lognormal sum distribution. However, the Type IV Pearson variant can still accurately approximate all portions of the lognormal sum distribution. IV. C ONCLUSIONS In this paper, we have developed a variant of the Type IV Pearson distribution with a variable domain of (0, +∞) to approximate the lognormal sum RVs. This newly developed Type IV Pearson variant approximation maintains all the advantages of the standard Type IV Pearson approximation as well as relieves its two limitations. Therefore, independent of the statistical characteristics of the lognormal sum distribution, the Type IV Pearson variant can accurately approximate the lognormal sum distributions for a whole probability range. 0.999 0.99 0.9 CDF GPV(ξ < A) μ1 1.23 1.37 1.41 N=100 N=20 0.5 N=300 0.1 0.01 0.001 Empirical CDF Type IV Pearson variant Normal variant distribution 1e−4 1e−6 10 20 30 40 50 60 70 A(dB) CDF comparisons for σ = 12 in the lognormal paper. Fig. 2. −4 x 10 Empirical PDF Type IV Pearson variant Normal variant distribution 3 Probability Density Function N 20 100 300 1−(1e−6) 2 1 0 24 26 28 30 32 34 36 38 40 42 44 A(dB) Fig. 3. paper. PDF comparisons for σ = 12 and N = 100 in the linear-scale R EFERENCES [1] T. S. Rappaport, Wireless Communications: Principles and Practice. Prentice Hall, Inc., 1996. [2] L. F. Fenton, “The sum of lognormal probability distributions in scatter transmission systems,” IRE Trans. Commun. Syst., vol. CS-8, pp. 57–67, 1960. [3] S. Schwartz and Y. Yeh, “On the distribution function and moments of power sums with lognormal components,” Bell Syst. Tech. J., vol. 61, pp. 1441–1462, 1982. [4] N. C. Beaulieu and Q. Xie, “An optimal lognormal approximation to lognormal sum distributions,” IEEE Trans. Veh. Technol., vol. 53, pp. 479–489, Mar. 2004. [5] N. B. Mehta, J. Wu, A. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. 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