JOSA LETTERS

Higher-order skewness and excess coefficients of some probability distributions applicable to optical propagatior

phenomena Arun K. Majumdar

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91103 (Received 3 August 1978)

Expressions are derived for higher-order skewness and excess coefficients using central moments and cumulants up to 8th order. These coefficients are then calculated for three probability distribu­tions: (i) Log-normal, (ii) Rice-Nakagami, and (iii) Gamma distributions. Curves are given to show the variation of skewness with excess coefficients for these distributions. These curves are independent of the particular distribution parameters. This method is useful for studying fluctuating phenomena, which obey non- Gaussian statistics.

I. INTRODUCTION To study phenomena like laser propagation through tur­

bulent atmosphere, speckle phenomena, interplanetary scintillations, etc., it is necessary to know the probability distribution and the higher-order moments of the random variables which are used to model the fluctuating phenomena. Since the characteristic function of the random variable (e.g. intensity fluctuations) contains a series of moments (or cu­mulants) terms of higher orders, the information about the probabilty distribution can be obtained from these moments (or cumultants). The higher-order moments give more in­formation concerning the contributions from the tails of the probability density function. Even- and odd-order moments give information generally about the width and the lack of symmetry of the distribution respectively. The skewness represents the measure and nature of asymmetry and the excess or flatness measures the weight in the tail. These statistical parameters are therefore convenient measurements of the departures of a particular probability distribution from a Gaussian distribution. In many practical cases one deals with random processes where the probability distribution is in general non-Gaussian and the events are irreversible. These non-Gaussian processes are the results of nonlinear transformations of Gaussian processes. Statistical parame­ters like skewness and excess of some of the fluctuating phe­nomena mentioned earlier have been described in the litera­ture1 - 4 using the moments of the intensity fluctuations up to 4th order. To understand sufficiently well the physical nature and the origin of some of these random processes, it is neces­sary to know the moments beyond the 4th order. For exam­ple, the analysis of a non-Gaussian statistics of the received intensity fluctuations containing sharp or occasional jumps or spikes may require higher-order correlation study. Any departure from the Gaussian statistics can be determined from the higher-order skewness and excess coefficients.

Higher-order correlations in a turbulent field have been studied and analyzed.5-7 They pointed out that it was nec­essary to analyze fluctuations up to 8th-order correlation to explain some experimental results. The purpose of the present paper is to define and obtain expressions for higher-order skewness and excess coefficients using central moments and cumulants up to 8th order in support of their works. It is not necessary for practical purposes to go beyond the 8th order, because moments up to the 8th order are usually suf­

ficient for the statistical characterization of the random pro­cess. These higher-order skewness and excess coefficients are computed for some recently considered probability dis­tributions of intensity fluctuations of optical wave propagating through turbulence or speckle pattern. These particular probability distributions are the following: (i) Log-normal, (ii) Rice-Nakagami and (iii) Gamma. Curves showing the variations of the higher-order skewness with the excess coef­ficients have been drawn for each of these distributions. This gives a very convenient and effective method of studying the statistical characteristics of the fluctuating phenomena and to determine which of these probability distributions is the closest for describing the random process under consider­ation.

II. DEFINITIONS AND NOTATIONS Consider the central moments μn of order n = 2, 3 , . . . 8 of

the received intensity I, the fluctuations of which are to be characterized statistically. We can write

where + , denotes the average.

In an analogy to the problem of analyzing the higher-order correlations of the velocity fluctuations in a turbulent field5-7

(see also Lumley8), we can define in a slightly different form the following nondimensional coefficients:

Note that for a Gaussian probability distribution, the odd-order central moments vanish and therefore the coeffi­cients Γ0 and Γ7 does not exist and become indeterminate. However, the even-order central moments are finite and ac­cording to the above definitions Γ4 = Γ6 = Γ8 = 0 for a Gaussian distribution. Thus any nonzero value of the even-order Γ parameters will indicate that the random process is not independent. Since the central moments and cumulants

199 J. Opt. Soc. Am., Vol. 69, No. 1, January 1979 0030-3941/79/010199-04$00.50 © 1979 Optical Society of America 199

are related,9 the above coefficients can also be expressed in terms of the cumulants, which are usually defined10 as fol­lows:

where χn is the nth cumulant and φ(t) is the characteristic function given by

where f(x) is the probability distribution of the random variable x. In many practical situations, it might be conve­nient to calculate the cumulants directly from the charac­teristic function. Therefore writing the above Eqs. (2)-(7), in terms of the cumulants χn (n = 2, 3 , . . . 8) we obtain:

Note also that the skewness and excess coefficients defined above are the same as the usual notations γ1 and γ2, i.e., Γ3 = γ1 and Γ4 = γ2.

III. COMPUTATION OF HIGHER-ORDER SKEWNESS AND EXCESS COEFFICIENTS FOR SOME DISTRIBUTIONS

The above coefficients have been calculated for three probability distributions, which are of interest to the problems of optical propagation. These distributions are: (i) Log-normal, (ii) Rice-Nakagami, and (iii) Gamma.

A. Log-normal distribution For this distribution, it is convenient to calculate the

higher-order skewness and excess coefficients in terms of the central moments. The central moments are given by11,12

where x is the random variable and w = exp (σ2), σ2 = var (log x).

The coefficients of skewness γ1 and excess γ2 (which are Γ3 and Γ4 respectively according to our present notations) are given in the literature.2,11,12 Using the above relation (16) and substituting in Eqs. (4)-(7) we obtain the following expres­sions for the different coefficients:

200 J. Opt. Soc. Am., Vol. 69, No. 1, January 1979

B. Rice-Nakagami distribution The Rice-Nakagami distribution of the random variable

I is given by13

In terms of mean +I, and β ≜ Ic/σ2,

where I0 is the modified Bessel function of order zero. The expressions for γ1 and γ2 (corresponding to our Γ3 and

Γ4 respectively) are given in the literature.2 The expressions for higher-order coefficients are calculated as follows:

From Eq. (8) we get

From the expression of the characteristic function φ(t) of the Rice-Nakagami distribution and equating the corre­sponding coefficients of (it)n from Eq. (23), we get

Substituting Eq. (24) into Eqs. (12)-(15) we get the fol­lowing expressions:

C. Gamma distribution The probability density function of the Gamma distribution

can be written as

JOSA Letters 200

FIG. 1. Variation of the coefficient of superexcess with superskewness for three probability distribution functions.

where m equals the parameter of the Gamma distribution and r(m) is the Gamma function with parameter m.

Following the same procedure of equating the coefficients of (it)n from the Eq. (23), the cumulants of the Gamma dis­tribution are given by

where α = 1/β. Using this expression we obtain the higher-order coeffi­

cients of this distribution. These are given by

As a special case for m = 1, the Gamma distribution be­comes an "Exponential" distribution for which Γ5 = 12, Γ6 = 250, Γ7 = 722, and 8 = 14 728.

IV. RESULTS AND DISCUSSIONS Using the equations derived in the Sec. III, the higher-order

skewness and excess coefficients were calculated. In order

FIG. 2. Variation of the coefficient of hyperexcess with hyperskewness for three probability distribution functions.

201 J. Opt. Soc. Am., Vol. 69, No. 1, January 1979

to show the variation of the skewness with the excess factors, curves were drawn for the three probability distributions and are shown in Figs. 1 and 2 in logarithmic scale. Figure 1 shows the variation of superexcess Γ6 with superskewness Γ5, and in Fig. 2 hyperexcess Γ8 versus hyperskewness Γ7 is plotted for these distributions. These curves are independent of the particular parameters of the distribution. Note that in both cases, excess factors increase with the skewness and these higher-order coefficients increase with the order of correla­tions. Most of these skewness and excess usually come from a small part of the signal representing the random process and describe the most asymmetric part. In an experiment, these coefficients, Γ5-Γ8, indicate the occurrence of sharp jumps or occasional spikes in a random signal.

V. CONCLUSIONS In this paper, the higher-order skewness and excess coef­

ficients of non-Gaussian fluctuating phenomena are defined and the corresponding expressions are derived for the three probability distributions of interest to optical propagation. To describe the physical nature of the process, it may be necessary and significant to measure the correlations of order five and higher. When measured values of the skewness and excess coefficients are plotted in Figs. 1 and 2, it gives a con­venient and efficient tool for determining which of the prob­ability distributions is the closest to describe the random process under consideration. Furthermore it is also possible to establish a realistic model of the random process taking into account of these higher-order non-dimensional coefficients. The method of calculating the coefficients can be applied to study other non-Gaussian distributions.

ACKNOWLEDGMENTS The author wishes to thank Professor H. Gamo of the

University of California, Irvine for his encouragement and interest. This paper presents the results of one phase of re­search carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract NAS 7-100, sponsored by the National Aeronautics and Space Administration. 1P. Ciolli, A. Consortini, F. Pasqualetti, L. Ronchi, and R. Vanni,

"Intensity fluctuations of an atmospherically degraded beam at the focus of a large collecting lens," Appl. Opt. 16, 1128-1130 (1977).

2G. Bourgois and C. Cheynet, "Observations of the interplanetary medium and of the structure of radio sources using higher moments of interplanetary scintillations," Astron. & Astrophys. 21, 25-31 (1972).

3G. C. Valley and D. L. Knepp, "Application of joint gaussian statistics to interplanetary scintillation," J. Geophys. Res. 81, 4723-4730 (1976).

4A. Consortini and L. Ronchi, "Probability distribution of the sum of N complex random variables," J. Opt. Soc. Am. 67, 181-185 (1977).

5F. N. Frenkiel and P. S. Klebanoff, "Higher-order correlations in a turbulent field," Phys. Fluids. 10, 507-520 (1967).

6C. W. Van Atta and W. Y. Chen, "Correlation measurements in grid turbulence using digital harmonic analysis," J. Fluid Mech. 34, 497-515 (1968).

7R. Betchov and C. Lorenzen, "Phase relations in isotropic turbu­lence," Phys. Fluids. 17, 1503-1508 (1974).

8J. L. Lumley, Stochastic Tools in Turbulence (Academic, New York, 1970), p. 20.

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9M. G. Kendall and A. Stuart, The Advanced Theory of Statistics (Hafner, New York, 1969), Vol. 1.

10H.C ramér, Mathematical Methods of Statistics (Princeton Uni­versity, Princeton, 1963).

πR. Barákat, "Sums of independent log-normally distributed random

variables," J. Opt. Soc. Am. 66, 211-216 (1976). 12R. L. Mitchell, "Permanence of the log-normal distribution," J. Opt.

Soc. Am. 58, 1267-1272 (1968). 13P. Beckman, Probability in communication engineering (Harcourt,

Brace and World, New York, 1967).

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