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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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