BOUNDED GENERALIZED GAMMA DISTRIBUTION

Transcription

1 BOUNDED GENERALIZED GAMMA DISTRIBUTION Osama Abdo Mohamed 1,2,, Saif Elislam Adam Abdellah 2,4 - and omran Malik Awad 2 Department of Computer Science and information technology, khulais Faculty of Computer Science and Information Technology, King Abdul Aziz University, Saudi Arabia. 2. Department of Computer Science and information technology, khulais Faculty of Computer Science and Information Technology, Jeddah University, Saudi Arabia. 3. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt. Department of Information Technology, Faculty of Computer Science and information technology, Alneelain University Abstract Generalized Gamma distribution is useful in analyzing several data sets, which arising into places at image processing, speech recognition, signal processing, statistical quality control, agricultural experimentation, light wave modeling, industrial experimentation and biological experiments. This paper present a new bounded generalized gamma model, which includes bounded Gaussian model, bounded Laplace model, and bounded Gamma model as special cases. This new distribution has a flexibility to fit different shapes of observed data such as non- Gaussian and bounded support data. The various distribution properties such as the distribution function, moments, skewness, kurtosis, hazard function, survival function, the r th order statistic and the median distributions are derived. In order to estimate the model parameters, the moment and maximum likelihood estimation are studied. We quantify the performance of bounded generalized gamma model with simulations and real data. It is shown via Kolmogorov-Smirnov test that the probability mass function based on the Bounded Generalized Gamma Model are the most suitable distributions in our simulation. Keywords :bounded generalized gamma model, BGM: bounded Gaussian model, BLM :bounded Laplace model, M:bounded Gamma model, KST:.Kolmogorov-Smirnov Test 7

2 1. Introduction Truncated distributions arise in many practical situations, particularly in numerous industrial settings [3, 4, 6,7, 8, 10, 11, 12]. Truncated distributions can also be used to model intensity statistics in the study of atomic heterogeneity [1].Field, Harder, and Harrison [9] showed that measured traffic from three locations on a state-of-the-art switched Ethernet fit closely various truncated distributions. K. Anithakumari, K. Srinivas Rao and PRS Reddy [18 ] introduced and derived a right truncated generalized Gaussian distribution and its various properties. K. Anithakumari1, K. Srinivas Rao, and P. R. S. Reddy [19 ] studied a left truncated generalized Gaussian distribution and its various properties. L. Zaninetti[16] introduced an upper and a lower boundary to truncated gamma distribution. The parameters which characterize the truncated gamma distribution are evaluated [16]. In [ 2,5,13,17 ] the truncated (below zero) normal distribution is considered, Some existing results are surveyed, and a recursive moment formula is used to derive the first four central moments in terms of the mean and variance of the underlying normal and in terms of lower moments of the truncated distribution. Bounding and monotonicity of the moments of the truncated distribution are considered and some previously unknown features of the distribution are presented[ 16 ]. This paper present a new bounded generalized gamma model ( ), which includes the previously studied distributions as special cases. The estimation of parameters of the presented distribution are derived. Some other related distributions and properties are also studied. This paper is organized as follows. Section 2 describes the two sided generalized gamma distribution. In Section 3 the various distribution properties such as the distribution function, moments, skewness, kurtosis, hazard function, survival function, the r th order statistic, the median distributions, the distribution of the r th order statistics and the median distribution are also 8

3 derived, estimate of the model parameters using the moment and maximum likelihood estimation are studied. Simulation illustrating the reliable performance of the method are given in section 4. Finally, section 5 concludes the paper. 2. Two sided Generalized Gamma Distribution The probability density function of the two sided generalized gamma is[14] (1) Where denotes the gamma function and are positive real valued parameters. When or the generalized gamma reduces to the standard gamma distribution. Furthermore, if, it becomes the Gaussian pdf, and if it representsthe Laplacian pdf. When one obtains the weibull distribution, and as the distribution approaches the lognormal. In fact, cannot be specified when the argument equals zero exactly. For that reason, zero inputs are ignored during the parameter estimation procedure and also in the calculation of measures of fit. 3. Bounded two sided Generalized Gamma Model Suppose we have a continuous distribution with probability density function (pdf) and cumulative distribution function (cdf) specified by g( ) and G( ), respectively. Let X be a random variable representing the truncated version of this distribution over the interval [a, b]. The pdf of X are given by (2) Where g(x) is the two sided generalized gamma defined in equation (1) and G(z) is the cumulative distribution function of two sided generalized gamma distribution defined as follows: ) (3) Where is incomplete gamma function defined by 9

4 f(x) International Journal of Computer Application ( ) (4) Substituting g(x), G(b) and G(a) from equations (1) and (3) in equation (2) we get (5) Hence the commutative distribution of X is given by (6) 3.1Distributional Properties The various distributional properties of the right truncated generalized Gaussian distribution are discussed in this section Graph of bounded generalized gamma f(x,eta, beta, gam) with gamm*eta=1, beta=1 0.7 f(x,-5,5,2,1,0.5) f(x,-5,5,1,1,1) 0.6 f(x,-5,5,2/3,1,1.5) f(x,-5,5,0.5,1,2) 0.5 f(x,-5,5,1/3,1,3) f(x,-5,5,0.2,1,5) =< x <= 5 Figure 1:Bounded generalized gamma with 10

5 f(x) f(x) International Journal of Computer Application ( ) Graph of bounded generalized gamma with gamma=2 beta=0.5,1,2, eta=2 f(x,-5,5,2,0.5,2) f(x,-5,5,2,1,2) f(x,-5,5,2,2, =< x <= 5 Figure2:Bounded generalized gamma with Graph of bounded generalized gamma with gamma=0.7,1,2 beta=2, eta=2 f(x,-5,5,0.7,2,2) f(x,-5,5,1,2,2) f(x,-5,5,2,2, =< x <= 5 Figure3: bounded generalized gamma with 11

6 f(x) International Journal of Computer Application ( ) Graph of bounded generalized gamma with gamma=2 beta=2 and eta= 0.7,1, f(x,-5,5,2,2,0.7) f(x,-5,5,2,2,1) f(x,-5,5,2,2, =< x <= 5 Figure4: Bounded generalized gamma with From figure1 it is observed that this distribution is unimodal distribution at x=0 when. From figures 2, 3, 4 it is observed that this distribution is two modal distribution hen. The r th moment of the distribution is given by The mean is given by: (7) (8) The second moment is given by 12

7 (9) In addition, the third moment is given by (10) Moreover, the fourth moment is given by (11) The variance is given by Substituting of Skewness and the Kurtosis are given by [ 13] from equations (8), (9), (10) and (11) The (13) (14) The median M of the distribution can be obtained by solving the equation 13

8 (15) The mode of the distribution can be obtained by solving the following equation (10) for x The hazard rate function of the distribution is (16) The survival rate function (17) (18) 3.2 Order Statistics of two sided Bounded Generalized Gamma distribution Let denote the order statistics obtained from a random sample of size n from two sided bounded generalized gamma distribution having the probability density function of the form given in (5). The probability density function of s th order statistics is given by [15], (19) The probability density function of the first order statistics is obtained by substituting s =1 in the equation (19) (20) If n is odd. The distribution of the median is obtained by substituting in equation (19 ) 14

9 (21) 3.3 Distribution parameter estimation Method of Moments: In this method, the theoretical moments of the population and the sample moments are equated which leads to the following equations. (22) (23) (24) By solving the equations (22), (23) and (24) using fsolve in matlab one can obtain the estimated parameters Maximum Likelihood Method of Estimation The parameters should be estimated to test the goodness of fit or to take advantage of the assumed pdf for various applications. Here, we apply the ML criterion to estimate the parameters of Given n data, with the assumption that the data are mutually independent, the log-likelihood function is given as follows: (25) 15

10 By differentiating the log-likelihood function with respect to we obtain the following three equations and setting them to zero, (26) (27) (28) Where (29) Where is the digamma function, which denotes the first order derivative of. Where the function T is defined by the meiger-g function which is defined by the contour integral [5] (30) (31) By solving the equations (26), (27) and (28) Using fsolve in matlab one can obtain the estimated parameters 4. Experiment In this experiment, two artificial random samples and one real life data are studied. The artificial random samples are generated from the truncated generalized gamma distribution with parameters (1,2,1 ), (1,2, 2). The real life data represents an image. Which shown in figure 8. The distribution of both the artificial 16

11 Cumulative probability Cumulative probability International Journal of Computer Application ( ) and the real life data are estimated. Figures 5,6 and 7 show the hypothesis and empirical cumulative distributions for the examined data. Table (1) shows the goodness of fit statistics. From table (1) one can observe that h=1 and p-value=0 that means that the bounded generalized gamma is a good distribution for the studied artificial and real life data R data bounded generalized gamma fit Data Figure5. The empirical and hypothesis distribution for random sample 1 generated from the bounded generalized gamma R data bounded generalized gamma fit Data Figure 6. The empirical and hypothesis distribution for random sample 2, 17

12 Cumulative probability International Journal of Computer Application ( ) which generated from the bounded generalized gamma R data bounded generalized gamma fit Data Figure 7. The empirical and hypothesis distribution for real data,whichrepresent an image. Figure 8: the real life data H P KS statistics Cv random samples random samples Image e Table 1: represent the goodness of fit statistics for the artificial and real life data 5. Conclusion This paper present a new bounded generalized gamma model ( ),which includes bounded Gaussian model (BGM), bounded Laplace model(blm), and bounded Gamma model ( M) as special cases. This new distribution has a flexibility to fit different shapes of observed data such as non-gaussianand bounded support data. The variousdistribution properties such as the 18

13 distribution function, moments, skewness, kurtosis, hazard function, survival function, the r th order statistic and the median distributions are derived. In order to estimate the model parameters,the moment and maximum likelihood estimation are studied. We quantify the performance of the BGGM with simulations and real data. It is shown via Kolmogorov-Smirnov (KST) test that the PMFs based on the Bounded Generalized Gamma Model ( the most suitable distributions in our simulation. ) PDF are References 1. Bhowmick, K.,Mukhopadhyay, A.and Mitra, G.B. Edgeworth Series Expansion of thetruncated Cauchy Function and its Effectiveness in the Study of Atomic Heterogeneity. Zeitschriftf urkristallographie, 2000, 215, pp C edric F, Denis Al and Philippe Na. Truncated skew-normal distributions: estimation by weighted moments and application to climaticdata, metron- International journal of statistics, 2010,,vol LXVIII, n.3, pp Cho BR, Govindaluri MS. Optimal Screening Limits in Multi-Stage Assemblies. International Journal Production Research, 2002, 40, pp Field, T., Harder, U., Harrison, P.,. Network Traffic Behaviour in Switched EthernetSystems. Performance Evaluation, 2004, 58, pp Geddes, K.O.,Glasser, M.L., Moore, R.A. and Scott, T.C., Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions., Applicable Algebra in Engineering, Communication and Computing, Nov 1990, 1 (2),, pp Jeang A. An Approach of Tolerance Design for Quality Improvement and Cost Reduction. International Journal Production Research, 1997, 35, pp Kapur KC, Cho BR Economic Design and Development of Specification. QualityEngineering, 1994, 6, pp Kapur KC, Cho BR. Economic Design of the Specification Region for Multiple QualityCharacteristics. IIE Transactions, 1996, 28, pp Khasawneh MT, Bowling SR, Kaewkuekool S, Cho BR. Tables of a Truncated StandardNormal Distribution: A Singly Truncated Case., Quality Engineering, 2004, 17, pp

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