Fitting thunderstorms to the the power-law covariate self-exciting point process

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Jumps and interevents

To specify a temporal point process we can use many different approaches. In this text, we will consider two of those approaches. One rely on the distribution of the occurrence times of jumps or the time lengths between subsequent events. The second one rely on the number of events occurring in a given time-interval. The lengths of the time intervals between subsequent events are called the interevents. We can define a temporal point process by specifying the distribution of these interevents. In the case of i.i.d. exponential interevents with parameter ‚, the process is the well-known homogeneous Poisson process (HPP) with intensity ‚. The process is also obtained assuming that the number of events in a given inter-val of length t is a Poisson distribution with parameter ‚t. Another classical hypothesis for the interevents is a Weibull distribution. In this case, the in-ference for the process boils down to be the classical inference for i.i.d. Weibul r.v. Remark that in this case, the alternative definition through the numbers of events is not possible. Now, suppose that given ti, the date of the ith jump , the time to the next jump is a Weibull distribution with parameters (fi,fl), the interevents are no longer independent and has distribution: n o f (t j ti, . . . , t1) ˘ (fl/fi)(t/fi)fl¡1 exp ¡(t/fi)fl ¯(ti/fi)fl , which is a left-truncated Weibull distribution with support [ti,¯1[. This is an alternative definition of the power-law process (PLP) which is defined as we will see in the next chapter, through the distribution of the number of events in a given interval [s, t] as a Poisson distribution with pa-rameter (t/fi)fl ¡(s/fi)fl.

Conditional intensity function

The conditional intensity function is an intuitive and convenient way of spec-ifying how the present depends on the past in an evolutionary point process. The behavior of a simple temporal point process is typically modeled by speci-fying its conditional intensity ‚⁄(t), which represents the infinitesimal rate at which events are expected to occur around a particular time t, conditionally on the history of the point process prior to time t. Definition 2.2.1. – The conditional intensity associated with a temporal point process N(t) may be defined via the limiting conditional probability ‚⁄(t) ˘ lim P[N(t, t ¯¢t) ¨ 0jHt] , ¢t!0 ¢t.

Poisson processes

One of the most popular family of stochastic point processes is Poisson pro-cess, which is a simple point process {N(t), t ‚ 0} such that the number of points in any set follows a Poisson distribution and the numbers of points in disjoint sets are independent. That is, {N(t), t ‚ 0} is a Poisson process if N(A1), . . . , N(A k) are independent Poisson random variables, for any disjoint and measurable subsets A1, . . . , Ak of measure space S.
The Poisson process is characterized by the number of points in disjoint sets being independent. The conditional intensity function inherits this inde-pendence. A Poisson process is a process satisfying the following properties:
1. The numbers of changes in non-overlapping intervals are independent for all intervals.
2. The probability of exactly one change in a sufficiently small interval h ˘ 1/n is P ˘ „h ˘ „/n, where „ is the probability of one change and n is the number of trials.
3. The probability of two or more changes in a sufficiently small interval h is essentially 0.
In the limit of the number of trials becoming large, the resulting distribu-tion is called a Poisson distribution. In all settings, the Poisson point process has the property that each point is stochastically independent to all the other  points in the process, which is why it is sometimes called a purely or com-pletely random process. Despite its wide use as a stochastic model of phe-nomena representable as points, the inherent nature of the process implies that it does not adequately describe phenomena in which there is sufficiently strong interaction between the points. This has sometimes led to the overuse of the point process in mathematical models and has inspired other point processes, some of which are constructed via the Poisson process, that seek to capture this interaction.

Model Checking

In practice, after fitting a model to a data set we normally would like to as-sess whether the model provides an adequate fit. Checking the adequacy of a model is the next step of data analysis. We can only use the data in the first half of the observation interval to fit a model, and then simulate predictions of the second half to see if this corresponds to the second half of the observed data. Or we can use all of the data, and compare with simulations of the whole dataset. In addition to the model checking approaches based on sim-ulation, there are some particular kinds of model checking associated with the conditional intensity function. For some models we can build a graphical test, as for power-law process. If a graphical test is not available, residual test is a more general tool to assess the model.

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Graphical test

Graphical test is a visual tool for model checking. It gives us an immedi-ate view whether the model fit the data or not. For example, plotting time ( ti, i) on the plane can give us the first idea of the process. If all the point are nearby a straight line we can guess that the point pattern follows the homogeneous Poisson process. Duane plot is a graphical test for the power-law process. It is said that if the PLP fits the point pattern (t1, . . . , tn) then plotting (log(i), log(ti) on the plane gives an image of a straight line. Such a simple graphical test is not available for the exponential-law process and the self-exciting point process. Therefore we need residual test for those models.

Residual test

A useful technique for evaluating point process models relies on proposition 2.4.1 which can considered as a re-scaling method. The method essentially involves re-scaling the time axis of the observed point process N at time t by a factor of the intensity ‚(t). More specifically, if the point pattern (t1, . . . , tn) are observed from time 0 to time C, then each point ti is moved to the new time ⁄(ti), where ⁄ is the compensator. The resulting process M is a stationary Poisson process with unit rate, provided the original point process is simple. Similarly, one may inspect residuals obtained by randomly thinning the pro-cess: that is, keeping each point ti independently with probability inversely proportional to ‚(ti). As with rescaled residuals, the resulting thinned resid-uals will be distributed according to a stationary Poisson process ([43]). In practice, one may use the estimated intensity or compensator in place of the true intensity or compensator and inspect the rescaled or thinned process for uniformity. Several tests exist for this purpose, with different uses depend-ing on the alternative hypotheses. Some of the most useful are tests based on second and higher order properties. There are also more general second-order tests for point processes that do not rely on a stationary Poisson null hypothesis.

Table of contents :

1 General Introduction 
2 Preliminary: Temporal Point Processes 
2.1 Introduction
2.2 Temporal Point Processes
2.2.1 Jumps and interevents
2.2.2 Conditional intensity function
2.2.3 Poisson processes
2.2.4 Cox processes
2.2.5 Marked point processes
2.3 Inference
2.3.1 Likelihood
2.3.2 Maximum likelihood estimation
2.3.3 Bayesian estimation
2.4 Simulation
2.4.1 Inverse Method
2.4.2 Thinning method
2.5 Model Checking
2.5.1 Graphical test
2.5.2 Residual test
2.5.3 Model selection: AIC and BIC
3 Power-Law Process 
3.1 Introduction
3.2 Graphical Test
3.3 Maximum Likelihood Method
3.3.1 Likelihood
3.3.2 Maximum likelihood estimation
3.3.3 Properties of maximum likelihood estimators
3.4 Bayesian Approach
3.4.1 Noninformative prior
3.4.2 Independent conjugate priors (Oliveira et al. [45])
3.5 Conjugate Prior: the H-B Distribution
3.5.1 Prior information and conjugate priors
3.5.2 H-B distribution
3.5.3 The H-B distribution as a conjugate prior
3.5.4 Prior elicitation
3.5.5 Application
3.6 Concluding Remarks
4 Exponential-Law Process 
4.1 Introduction
4.2 Maximum Likelihood Method
4.2.1 Likelihood
4.2.2 Maximum likelihood estimation
4.3 Bayesian Approach
4.3.1 Fisher information matrix
4.3.2 Non-informative prior
4.3.3 Modified-Gumbel distribution
4.3.4 Independent conjugate priors
4.4 Conjugate Prior: the G-M-G Distribution
4.4.1 Prior information and conjugate priors
4.4.2 G-M-G distribution
4.4.3 Conjugate priors
4.4.4 Prior elicitation
4.4.5 Application
4.5 Concluding Remarks
5 Self-Exciting Point Processes 
5.1 Introduction
5.2 Maximum Likelihood
5.2.1 Reduced maximum likelihood procedure for the Hawkes process
5.2.2 Inference for the PLC-SEPP
5.3 Bayesian approach
6 Application: Thunderstorms 
6.1 The Dataset
6.2 Inference on Some Specific Thunderstorms
6.2.1 How to define a thunderstorm?
6.2.2 Grouping step: groups of impacts
6.2.3 Clustering step: clusters of impacts
6.2.4 Thinning step: thunderstorms
6.2.5 Inference on the selected thunderstorms
6.2.6 Fitting thunderstorms to the Hawkes process
6.2.7 Fitting thunderstorms to the the power-law covariate self-exciting point process
6.3 Concluding Remarks
7 General Conclusion 
Appendices
A Goodness-of-fit test for the power-law process
B Non-informative prior for PLP
C Distribution of the PLP shape parameter ¯
D Convergence of the integrals for the exponential-law process
E Likelikood function for a stochastic process
F Likelihood for the Hawkes process
G Likelihood for the PLP-SEPP
H A reduced Newton-Raphson algorithm for the Hawkes process
I Simulation Agorithms for Some Stochastic Processes 165
I.1 Simulation Algorithms for the Homogeneous Poisson Process
I.2 Simulation Algorithms for the Power-Law Process
I.3 Simulation Algorithms for the Exponential-Law Process
I.4 Simulation Algorithms for the Hawkes Process
I.5 Simulation Algorithms for the Power-law Covariate Self-Exciting Point Process
J Event Truncation and Time Truncation

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