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2012 International Conference on Lightning Protection (ICLP), Vienna, Austria Stochastic Modeling of Lightning Occurrence by Nonhomogeneous Poisson Process Soiram Ernesto Silva Artigas Doctoral Program in Statistics and Actuarial Science-Universidad Central de Venezuela (UCV) General Academic Department - Universidad Bolivariana de Venezuela (UBV) Caracas, Venezuela soiram@cantv.net silva.soiram@gmail.com Abstract—The characterization of lightning flash occurrence processes is key to risk analysis. Based on it, the degree of exposure of structures to direct lightning strikes can be determined, and thus, the level of protection required by the protection system can be properly set. Several studies attest that lightning flash occurrence is a natural phenomenon whose parameters vary in time and space. Looking in detail the process of occurrence leads to identification of features that allow classifying it as a member of the family of stochastic processes called counting processes. This paper offers a proposal to conceptualize the lightning flash occurrence as a Non Homogeneous Poisson type counting process, in contrast to other existing approaches which are based on the statistical concept of independent and identically distributed samples (iid) that is used in statistical inference of random variables. To justify the approach, a number of considerations is set out, including formal theoretical aspects in the field of probability and statistics, along with the analysis of the variability measure in the approach involving iid. Additionally, the paper also considers the question of statistical estimation and inference process of occurrence of lightning from the perspective of stochastic processes, offering methodological approaches that are considered useful in modeling as-Nonhomogeneous Poisson Process. Keywords- Lightning Ocurrence; Stochastic Modeling of Lightning Occurrence; Poisson Process and Lightnings. I. INTRODUCTION An exploratory analysis of data sets generated by Lightning Detection and Location Networks, allows to establish that the process of occurrence of Cloud to Ground (CG) lightning and its parameters vary temporally and spatially. The temporal variation of the process of occurrence is expressed in terms of the existence of periods and times when in a specific region an increase or a decrease in the frequency of occurrence of CG lightning flashes may take place. This is determined by the presence or absence of storm clouds which, in turn, is associated with cyclical fluctuations in annual weather conditions. Lightning itself is a random phenomenon whose occurrence in time and space cannot be accurately predicted, however with the statistical analysis of available information it is possible to measure and quantify the uncertainty associated with the occurrence of the phenomenon. Based on this, and a final approximation model as the Electrogeometric Model (EGM), is that may be performed risk assessments and exposure degrees of structures against lightning strikes. The approach used to assess the uncertainty has a direct impact on risk estimates. Its assessment is based on the analysis of the variability present in the available information on the occurrence of the phenomenon. This paper proposes the adoption of Non Homogeneous Poisson Stochastic Process, as a suitable alternative model for the analysis and characterization of lightning occurrence, which reduce the uncertainty in the occurrence estimate in comparison with others models that are based on statistical inference of iid samples. II. LIGHTNING FLASH AND COUNTING PROCESS A. Counting Variables The occurrence of a CG lightning flash is a discrete event that can be counted. So, given a finite time interval I=(0,t], and by observing the process, it is possible to count the number N(I) of events which occur in it. This feature allows us to classify the occurrence of CG lightning flashes process like a counting process, and it can be defined by a counting variable such as [1]. Sn=T1+T2 . . .+ Tn. (1) Where Sn represents the total time elapsed until the occurrence of the “n” event, and T1, T2,.., Tn is a sequence of times. Ti is the time from the event (i–1) to event “i” Ti=Si–S(i–1) for all i=1,2,…,n. (2) Based on Sn is defined the counting function of the process in the time interval (0,t] as: Nt=N((0,t])=max {n>0 : Sn ≤ t}. (3) n Where Nt stands for the number of events that occur up to the instant of time "t" and is called the counting process. 978-1-4673-1897-6/12/$31.00 ©2012 IEEE B. Poisson Model Counting Process One of the main models used to characterize this type of stochastic processes is the Poisson model, which is defined in general terms as follows [1]. Definition: Let N be a point process with state space €. Suppose that Є is a class of reasonable subsets of € called σ-algebra. Then N is a Poisson Process with mean measure “μ” or equivalently a Poisson Random Measure (MAP (μ)) if. For all A ⊂ Є i. e–µ(A)(µ(A))k /k! if μ(A)<∞ if μ(A)= ∞. P[N(A)=k]= (4) 0 ii. If A1,A2,…,Ak are mutually exclusive subsets (Ai∩Aj=ϕ for all i≠j) of € on Є, them N(A1), N(A2),…,N(Ak) are independent random variables. Thus N is Poisson, if the points in the set A, are distributed with Poisson distribution of parameter μ(A) and also the number of points in mutually exclusive regions are independent random variables. Property ii. is called complete randomization. When the state space is € =R, ii. is called the property of independent increments, since for any sequence of times, T1 <…< Tk its follows that {N((Ti; Ti+1]); i = 1, 2,…,k)} are independent random variables. C. Homogenious Poisson Process When the mean is a multiple of the Lebesgue measure (that is, length when € =[0,∞) or R, area when € =R2, volume when € =R3, etc.), the process is called Homogeneous Poisson Process (HPP). Thus in the homogeneous case, there is a parameter λh > 0 such that for any set A, N(A) is Poisson distributed with mean E[N(A)] = λh |A|, where |A| is the Lebesgue measure of A. When €=[0,∞), ¨λh¨ parameter is called the rate of HPP. If the set A is the time interval I=(0,t] and the process is HPP, is follows that E[N(I)]=E[N((0,t])= λh|(0,t]|= λh t = µh(t). (5) The mean value is a linear function of time ¨t¨, known as the mean value function of the HPP (µh(t)). By replacing (5) in (4) is obtained the well known form of the Poisson distribution given by [1]: –λh t P(Nt = k) = e k (λh t) /k!. (6) It can be shown that the inter-arrival times associated with the points of Homogeneous Poisson Process (HPP), are independent and identically distributed random variables with common exponential distribution of parameter ¨λh¨. Ti ~E(λh). (7) This fact gives to the HPP the property known as ¨lack of memory¨ [1]. In these cases probabilistic-statistical analysis is quite simple, being reduced to estimate the ¨λh¨ parameter, for which it is used as statistical estimator n ͞λh=1/͞T where T ͞ =(ΣTi)/n. (8) i=1 D. Lightning Occurrence and Homogeneous Poisson Processes In risk analysis, the occurrence of CG lightning flash is characterized by the Ground Flash Density (GFD), which is assumed to be constant value, which accounts for the average number of CG lightning flashes (expected value) occurring in a given region per unit time (year) and unit area (km2). This parameter is used to determine the degree of exposure to the direct lightning strikes on structures, in particular is used in estimating the expected value of lightning strikes per year on a given structure (E[Nd]). This is calculated as. E[Nd]=Ae GFD. (9) Where Ae is the equivalent attractive area of the structure which is determined according to the procedure described in [2]. Nd is a counting random variable which accounts for the number of lightning strikes per year who collects the structure. According to the accepted calculation method, to determine the expected value of lightning strikes in an arbitrary time interval (0,t], we multiply (9) by the time “t” in years E[Nt]=E[Nd]t=(Ae GFD)t. (10) Comparing (10) with (5) we can see that the calculation method implies modeling the process of occurrence of CG lightning flash as a HPP, where the mean value is proportional to the size of the considered time interval |(0,t]|=t (the Lebesgue measure) and is given by µh(t)=E[Nt]= (Ae GFD)t=λh t. (11) The proportionality factor is the rate of HPP, which is given by the constant λh =Ae GFD. (12) From (12) it can be seen that except for factor Ae, the GFD is a measure of the rate of occurrence λh of CG lightning flashes in a given region. Ae is a constant value which depends of the structure geometry where risk analysis is done. E. Implications of Lightning Occurrence Modeling, by Homogeneous Poisson Processes The homogeneity implicitly in modeling the process of occurrence of CG lightning flashes, under the calculation conditions described in the previous section (Section II-D), leads to a constant and invariant rate of occurrence, which is associated to a constant value of GFD according to (12). This analysis approach assumes that the rate of occurrence of the phenomenon λh is independent of time. In a tropical region such as the Bolivarian Republic of Venezuela where there are two seasons of similar length (summer and winter), it should be expected from (11) that a similar number of lightning flashes occurs -on mean value- both in winter time (most of the storms) and summer (low number of storms). Or equivalently, the expected value of lightning strikes on the structure is similar in both climatic periods. This obviously does not happen. What will happen is an overestimation of this value in the cycle of summer and underestimates it in the rainy season. If the analysis of the lightning occurrence process is stochastic type, is needed to look beyond the mean value characterization and analyze the behavior of the variability of the process, which is done by the study of its variance given by σ2N(t)=E[(Nt –µ(t))2]. (13) In this context, take a constant value in the GFD has important implications for estimating the variability of number of lightning strikes on the object structure of risk analysis. Analyzing the probabilistic behavior of the inter-CG lightning flash times, Ti, given by (2), follows by (6) and (7) that if the process is homogeneous, they are generated from an exponential distribution with common parameter λh. In this case there is a succession of times between CG lightning flash, which are independent and identically distributed with mean λh. The analysis of the variability of the homogeneous process involves taking as random errors, the observed deviations with respect to the mean value function µh(t) given by (11). So the variance of the process under the assumptions of homogeneity denoted as σ2Nh(t) is given by σ2Nh(t)=E[(Nt – λht)2]. (14) The calculation of process´s variance through (14), leads to an overestimation of random variability and hence attributed a greater uncertainty of which the process owns. This may be important for forecasting the CG lightning flashes occurrence. This increase in random variability is due to a long term trend and seasonally behavior that exhibit the observed rate of occurrence that are not random in nature. From a probabilistic point of view this implies that in between CG lightning flashes times, although they are statistically independent, they do not have a common distribution, are not identically distributed. The mean value of the Ti (λ) change thought the evolution of the long-term trend and seasonal cycles. This implies that the parameter λ is not a constant value and is taken as a deterministic function of time λ = λ(t). (15) Before proving these statements about the variance of the counting process under the assumptions of homogeneous behavior, the next section provides a mathematical characterization of the Poisson process when the rate of occurrence has a time-varying deterministic behavior. F. Nonhomogeneous Poisson Procees Aproach The assumption of a HPP for the occurrence of lightning, implies deny the possible influence of seasonal conditions in the rate of occurrence. This situation not corresponds to that observed when analyzing the evolution in time of occurrence of the phenomenon. In these cases there is temporal variation in the rate of occurrence of CG lightning flash given by (15) and in addition, the rate can contain one or more components of cyclic seasonal variation. Is to be hoped that significant differences exist between the rate of occurrence of lightning during the rainy season and dry season for the Venezuelan case. It is also possible that for this aspect it will overlapped with other effects which have its own cyclic behavior, for example, in the rainy season, the tendency of storms to occur during certain hours of day. Take into account this type of behavior involves abandoning the assumption of homogeneity and raise that the mean value function µ(t) of the Poisson process does not meet (5) (µ(t) ≠ λh t). In this case a rate varying in time (15) is related to µ(t) by [1] [3] t µ(t)=∫λ(t)dt. (16) 0 When the mean value function µ(t) exhibits this behavior, the resulting random process is called Nonhomogeneous Poisson Process (NHPP). The estimation of µ(t) is the main objective on the modeling of these processes. G. Estimating Lighthning Ocurrence as Nonhomogeneous Poisson Process To characterize the occurrence of lightning, through NHPP under the assumptions of existence of multiple cyclic behavior, is proposed as suitable method, the one developed by Kurhl and Wilson [3] who is known as Multiresolution Analysis. This method proposes that the estimation of the mean value function µ(t) begins with the observation of the occurrence process Nt in a time interval (0,S]. Is then established the decomposition of µ(t) in terms of an component of long-term trend and "p" components that characterize the behavior of µ(t) within each cycle or resolution. The existence of "p" cycles implies the presence of "p" distinct cycle lengths that are related in increasing order (b1>b2>…> bp). They also comply with bi is an integer multiple of bi+1 for i = 1,2,…,p–1. In addition its assumed the time horizon (0,S], in a way that S be an integer of b1, and is assigned b0 ≡ S. The number of periodicities present is determined by making a spectral analysis of the cumulative number of events through the application of Fast Fourier Transform (FFT) and identifying amplitude peaks in the resulting spectrum. The Multiresolution Analysis Method is semi-parametric; it’s not assuming any functional form to describe µ(t) in the levels of resolution. Kuhl and Wilson argue that in each resolution must be estimated a basis increasing monotone function Ri(s) that characterizes it, where s ϵ (0,bi] and Ri(0)=0 Ri(bi)=1. The estimation is performed by fitting of polynomials of degree ¨r¨ known as r-polynomials [4] that have the following structure s/bi Ri(s)= r –1 (17) r –1 Σβ (s/bi) + (1–Σβk,i)(s/bi) k,i k=1 if r=1 k k=1 where βk,i are the regression coefficients. r if r>1, The degree of the polynomial is determined by an algorithmic procedure based on a likelihood ratio test (LRT) developed by the authors of the method [4]. The LRT is applied to events that are observing due to the process realization in the time interval (0,S] which are superimposed by subdividing the interval (0,S] in successive subintervals of the form ((j–1)bi, jbi] for the i-th resolution, where j = 1,2, ...., S/bi and S/bi is the number of sub-intervals of length bi in (0,S]. The overlapping is achieved by shifting the times of occurrence that fall within the j-th sub interval ((j–1)bi, jbi], to the pattern interval (0,bi] via transformation s=t–(j–1)bi, where s is the time variable stated in (17). Is set as initial null hypothesis H0 that r=1 and through the implementation of the LRT with a significance level "α" is set to its acceptance or rejection. If H0 is rejected increases the degree r=2 and repeat the LRT process with the new H0 that r=2. The process is repeated until accept H0 for some "r". Finally, is constructs the mean value function µ(t) by means of a recursive procedure starting from resolution b0 corresponding to the long-term trend, and on it are added the effects of each resolution from i=1, ... , p. H. Aplication of Proposal Aproach on Rehabylity Evaluation of Electric Power Systems The original motivation of Multiresolution Analysis Method developed by Khul and Wilson, has been the analysis of data from the area of public health, particularly to observe and study the arrival process [3] of organ donor patients to hospitals emergency network in the U.S. In the proposal method by these authors, the data indicated are used as a platform to demonstrate the application of the analysis technique, but does not establish that it applies only to health data. This general feature has allowed the author of this paper extend the application of the analysis technique to the field of power systems. The method has been applied initially to characterize the behavior of permanent interruption in the transmission system CORPOELEC1 D.C. (District Capital) of the Bolivarian Republic of Venezuela [5]. In this case, has been analyzed temporal record of permanent interruptions due to events in the transmission system, between years 2001 and 2006. From Fig. 1a) it can be seen the realization of permanent interruption occurrence process NIt, due to the transmission system, which shows a fitted line which corresponds to the HPP assuming a constant rate of occurrence λhI. It can be identified the nonhomogeneous behavior as a deviation from the fitted line. Fig. 1b) shows the basis function (R(s)) of NIt in the resolution that corresponds to the annual duration, showing a tendency to increase the rate of occurrence in the second half of the year. Fig. 1c) shows the evolution of basis function for the resolution of weekly duration, where there is a tendency to a difference in behavior between working and nonworking days. 1 Corporación Eléctrica Nacional (National Electric Corporation). Is the nation-wide electricity operator in the Bolivarian Republic of Venezuela. a) b) c) d) Figure 1. Interruption Process on CORPOELEC D.C. a) Total Process years 2001-2006 b) Annual Resolution c) Weekly Resolution d) Daily Resolution. Finally Fig. 1d) provides the evolution of basis function in daily resolution, which shows a different pattern between the 23h of a day and 9 o'clock the next, where the rate of occurrence of interruptions tends to fall, and between 9 o'clock and 22h where on the contrary tends to increase. I. Inicial Aproach of Lighning Data Modeling by Non Homogeneous Poisson Processes To illustrate the method of analysis in the characterization of the occurrence of lightning, is provide a preliminary analysis of the data available on the web about the occurrence of CG lightning flash in Austria through public statistical reports presented by the Austrian Lightning Detection and Information Systems ALDIS [6]. The analysis is preliminary because it is based on monthly consolidated information of the behavior of the phenomenon and not specific temporal location of events. Fig. 2a) shows the behavior of the overall cumulative CG lightning flashes in Austria since January 1992 to December 2011 which shows a fitted line which corresponds to the HPP assuming a constant rate of occurrence λh. It can seen the existence of a long trend and seasonal behavior of approximately 12 months. The long trend is not analyzed because the unhomogenized data explained by ALDIS [6]. The following Fig. 2b), shows the basis function (R(s)) corresponding to the resolution of annual duration (12 m ...
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In his paper on “Stochastic Modeling of Lightning Occurrence by Nonhomogeneous
Poisson Process”, Artigas (2012) proposes that adopting non-homogeneous Poisson processes
(NHPP) could serve as an alternative to existing statistical models in the characterization of
lightning occurrence. The paper is built on the framework that an...

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