Non-stationary Poisson Process Research Articles

Temporal Characteristics of the February 6, 2023 MW7.8, Turkey Earthquake Aftershock Sequence

The devastating earthquake with moment magnitude MW7.8, (local magnitude ML=7.5) which occurred in the East Anatolian Fault Zone, westwards of Gaziantep, Turkey on February 6, 2023 was followed by an intensive aftershock activity. Studying time distribution of aftershocks is important for understanding physics of the earthquake generation process.
 In the present study temporal pattern of aftershock sequence of the 2023 MW7.8 Turkey earthquake is analyzed. Because of the factors such as location and radiation pattern and the cumulative nature of building damage, aftershocks can cause more damage than the main shock. The temporal clustering of aftershocks is a dominant non-random element of the seismicity, so when the clusters are removed, the remaining activity can be modelled (as first approximation) as a Poisson process.
 On the assumption that aftershocks are distributed in time as a non-stationary Poisson process, we use the maximum likelihood method for estimating the parameters (K, c and p) of the modified Omori formula. For testing the goodness of fit between the aftershock occurrence and different statistical models, a transformation from the time scale t to a frequency-linearized time scale τ is applied. Akaike's information criteria is used to select the best model for the temporal distribution of aftershocks.
 It has been found that temporal distribution of the 2023 Turkey earthquake aftershocks is dominated by the classic power law decay in time and suggests the existence of secondary aftershock activity.

Probabilistic modelling of substorm occurrences with an echo state network

Abstract. The relationship between solar-wind conditions and substorm activity is modelled with an approach based on an echo state network. Substorms are a fundamental physical phenomenon in the magnetosphere–ionosphere system, but the deterministic prediction of substorm onset is very difficult because the physical processes that underlie substorm occurrences are complex. To model the relationship between substorm activity and solar-wind conditions, we treat substorm onset as a stochastic phenomenon and represent the stochastic occurrences of substorms with a non-stationary Poisson process. The occurrence rate of substorms is then described with an echo state network model. We apply this approach to two kinds of substorm onset proxies. One is a sequence of substorm onsets identified from auroral electrojet intensity, and the other is onset events identified from activity of Pi2 pulsations, which are irregular geomagnetic oscillations often associated with substorm onsets. We then analyse the response of substorm activity to solar-wind conditions by feeding synthetic solar-wind data into the echo state network. The results indicate that the effect of the solar-wind speed is important, especially for Pi2 substorms. A Pi2 pulsation can often occur even if the interplanetary magnetic field (IMF) is northward, while the activity of auroral electrojets is depressed during northward IMF conditions. We also observe spiky enhancements in the occurrence rate of substorms when the solar-wind density abruptly increases, which might suggest an external triggering due to a sudden impulse of solar-wind dynamic pressure. It seems that northward turning of the IMF also contributes to substorm occurrences, though the effect is likely to be minor.

Probabilistic model of traffic scenarios for extreme load effects in long-span bridges

The traffic scenarios that may cause extreme load effects are of great importance to the safety assessment of bridge structures. The traditional simulation method of traffic flow cannot depict the distribution pattern of vehicles on the bridge deck when the maximum effect is induced. In this paper, a probabilistic Gaussian mixture model (GMM) for heavy vehicle scenarios on the bridge deck under free-flow condition is proposed for long-span bridges based on collected Weigh in Motion (WIM) data. The scenarios of extreme response under free-flow occur more frequently than congestion scenarios and are of similar value and relevance in the daily management and safety assessment of long-span bridges.A non-stationary Poisson process is utilized to simulate the uneven occurrence of heavy vehicles in different lanes, and it is assumed that they are located within the artificially defined cells on the bridge deck. Then, Nataf transformation is employed to consider the correlation of gross vehicle weights (GVWs) within close range in the same lane. The numerical study is carried out on a long-span cable-stayed bridge to investigate the effects of correlation in GVWs and stationarity of vehicle distribution location on the structural responses. The load responses calculated by the proposed model and Monte Carlo method for different effects are compared with the values derived from code model. The results show that with the increase of the correlation level of the neighboring GVWs, the simulated responses are more prone to get extreme values, which means an increasing probability of the most unfavorable spatial distribution of on-bridge vehicles. The same results are also found under the non-stationary simulation state for vehicle location. The non-stationary Poisson process provides an efficient, highly feasible method, which is also in the safe side, for simulating the vehicle spatial distribution for specific effects.

Similar properties between gamma-ray emission of 3C 454.3 and solar GeV flares

ABSTRACTBased on the survey data observed by Fermi-Large Area Telescope (LAT), we elaborate the statistical characteristics of gamma-ray flares from 3C 454.3 and solar GeV flares among flare parameters, such as isotropic energy (Eγ), peak luminosity (LP), and duration time (TDuration). We find two significant correlations as: $T_{\rm Duration} \propto E_{\gamma }^{0.31\pm 0.03}$ and $L_{\rm P} \propto E_{\gamma }^{0.61\pm 0.03}$ for 3C 454.3. The exponents are in a better agreement those of the Sun from the first Fermi-LAT solar flare catalogue, namely $T_{\rm Duration,\odot } \propto E_{\gamma ,\odot }^{0.38\pm 0.08}$ and $L_{\rm P,\odot } \propto E_{\gamma ,\odot }^{0.81\pm 0.08}$. The relationship of TDuration–Eγ and LP–Eγ could be interpreted naturally as due to magnetic dissipation through reconnection. On top of that the frequency distributions of isotropic energy, peak luminosity, and duration time of gamma-ray emission of 3C 454.3 show power-law forms, and the waiting time distribution can be described by a non-stationary Poisson process. These distribution behaviours are comparable to those of the Sun, Sagittarius A*, and M87, and follow the prediction of a self-organized criticality model. All statistical properties suggest that similar to the physical process accounting for solar GeV flares and X-ray flares in supermassive black hole systems, magnetic reconnection cloud govern the energy-release, and particle acceleration process for gamma-ray flares of 3C 454.3.

Clustering Behavior in Solar Flare Dynamics

The solar magnetic activity cycle provides energy input that is released in intense bursts of radiation known as solar flares. As such, the dynamics of the activity cycle is embedded in the sequence of times between the flare events. Recent analysis shows that solar flares exhibit memory on different timescales. These previous studies showed that the time ordering of flare events is not random, but rather there is dependence between successive flares. In the present work, the clustering of flares is demonstrated through a straightforward nonparametric method where the cumulative distribution function of successive flares is compared with the cumulative distribution function of surrogate sequences of flares obtained by random permutation of flares. The random permutation is performed within rate-variable Bayesian blocks during which the flare rate is assumed to be constant. Differences between the cumulative distribution functions are substantial on a timescale around 3 hr, suggesting that flare recurrence on that timescale is more likely than would be expected if the waiting time were drawn from a nonstationary Poisson process.

Decomposable Product Systems Associated to Non-stationary Poisson Processes

Abstract Let $P$ be a closed convex cone in $\mathbb {R}^{d}$, which we assume is pointed and spanning, i.e, $P \cap -P=\{0\}$ and $P-P=\mathbb {R}^{d}$. We demonstrate that, when $d \geq 2$, in contrast to the one-parameter situation, Poisson processes on $\mathbb {R}^{d}$, with intensity measure absolutely continuous with respect to the Lebesgue measure, restricted to $P$-invariant closed subsets, provide us with a source of examples of decomposable $E_0$-semigroups that are not always CCR flows.

An explicit compound Poisson process-based shock deterioration model for reliability assessment of aging structures

Existing structures may suffer from resistance deterioration due to repeated attacks. The modeling of resistance deterioration is a critical ingredient in the reliability assessment and service life prediction of these degraded structures. In this paper, an explicit compound Poisson process-based model is developed to describe the shock deterioration of structural resistance, where the magnitude of each shock deterioration increment is modeled by a Gamma-distributed random variable. The moments (mean value and variance) and the distribution function of the cumulative shock deterioration are derived in a closed form, based on a proposed W-function. A method for the efficient calculation of the W-function is presented, which reduces to the Bessel type I function if the shock deterioration increment is exponentially distributed (a special case of Gamma distribution). The proposed shock deterioration model is applicable to either a stationary or a nonstationary Poisson process of random jumps. Subsequently, the overall resistance deterioration is modeled as the linear combination of gradual and shock deteriorations, based on which the proposed model can be used in the time-dependent reliability assessment of aging structures efficiently. A numerical example is presented to demonstrate the applicability of the proposed deterioration model by estimating the time-dependent reliability of an aging bridge. It is found that a smaller threshold for the degraded resistance leads to greater mean value and standard deviation of the time to failure, and this effect is enhanced by a smaller occurrence rate of the shock deterioration.

The Solar Memory from Hours to Decades

Abstract Waiting-time distributions allow us to distinguish at least three different types of dynamical systems, including (i) linear random processes (with no memory); (ii) nonlinear, avalanche-type, nonstationary Poisson processes (with memory during the exponential growth of the avalanche rise time); and (iii) chaotic systems in the state of a nonlinear limit cycle (with memory during the oscillatory phase). We describe the temporal evolution of the flare rate λ(t) ∝ t p with a polynomial function, which allows us to distinguish linear (p ≈ 1) from nonlinear (p ≳ 2) events. The power-law slopes α of the observed waiting times (with full solar cycle coverage) cover a range of α = 2.1–2.4, which agrees well with our prediction of α = 2.0 + 1/p = 2.3–2.6. The memory time can also be defined with the time evolution of the logistic equation, for which we find a relationship between the nonlinear growth time τ G = τ rise/(4p) and the nonlinearity index p. We find a nonlinear evolution for most events, in particular for the clustering of solar flares (p = 2.2 ± 0.1), partially occulted flare events (p = 1.8 ± 0.2), and the solar dynamo (p = 2.8 ± 0.5). The Sun exhibits memory on timescales of ≲2 hr to 3 days (for solar flare clustering), 6–23 days (for partially occulted flare events), and 1.5 month to 1 yr (for the rise time of the solar dynamo).

Role of the Solar Minimum in the Waiting Time Distribution Throughout the Heliosphere

AbstractMany processes throughout the heliosphere such as flares, coronal mass ejections (CMEs), storms and substorms have abrupt onsets. The waiting time between these onsets provides key insights as to the underlying dynamical processes. We explore the tail of these waiting time distributions (WTDs) in the context of random processes driven by the solar magnetic activity cycle, which we approximate by a sinusoidal driver. Analytically, we find that the distribution of large waiting times of such a process approaches a power law slope of −2.5, which is primarily controlled by the conditions when the driving is minimum. We find that the asymptotic behavior of WTDs of solar flares, CMEs, geomagnetic storms, and substorms exhibit power laws that are in reasonable agreement with a sinusoidally driven nonstationary Poisson process. However, the WTD of substorms during solar minimum may be more consistent with prolonged periods of weak driving followed by abrupt increase in the rate.

Simulation of non-stationary event flow with a nested stationary component

A method for constructing an ensemble of time series trajectories with a nonstationary flow of events and a non-stationary empirical distribution of the values of the observed random variable is described. We consider a special model that is similar in properties to some real processes, such as changes in the price of a financial instrument on the exchange. It is assumed that a random process is represented as an attachment of two processes - stationary and non-stationary. That is, the length of a series of elements in the sequence of the most likely event (the most likely price change in the sequence of transactions) forms a non-stationary time series, and the length of a series of other events is a stationary random process. It is considered that the flow of events is non-stationary Poisson process. A software package that solves the problem of modeling an ensemble of trajectories of an observed random variable is described. Both the values of a random variable and the time of occurrence of the event are modeled. An example of practical application of the model is given.

Simulation of non-stationary event flow with a nested stationary component

A method for constructing an ensemble of time series trajectories with a nonstationary flow of events and a non-stationary empirical distribution of the values of the observed random variable is described. We consider a special model that is similar in properties to some real processes, such as changes in the price of a financial instrument on the exchange. It is assumed that a random process is represented as an attachment of two processes - stationary and non-stationary. That is, the length of a series of elements in the sequence of the most likely event (the most likely price change in the sequence of transactions) forms a non-stationary time series, and the length of a series of other events is a stationary random process. It is considered that the flow of events is non-stationary Poisson process. A software package that solves the problem of modeling an ensemble of trajectories of an observed random variable is described. Both the values of a random variable and the time of occurrence of the event are modeled. An example of practical application of the model is given.

Simulation of non-stationary event flow with a nested stationary component

A method for constructing an ensemble of time series trajectories with a nonstationary flow of events and a non-stationary empirical distribution of the values of the observed random variable is described. We consider a special model that is similar in properties to some real processes, such as changes in the price of a financial instrument on the exchange. It is assumed that a random process is represented as an attachment of two processes - stationary and non-stationary. That is, the length of a series of elements in the sequence of the most likely event (the most likely price change in the sequence of transactions) forms a non-stationary time series, and the length of a series of other events is a stationary random process. It is considered that the flow of events is non-stationary Poisson process. A software package that solves the problem of modeling an ensemble of trajectories of an observed random variable is described. Both the values of a random variable and the time of occurrence of the event are modeled. An example of practical application of the model is given.

Information Theoretical Approach to Understanding Flare Waiting Times

Abstract There is not currently a consensus on the process responsible for producing the waiting time distribution of solar flares. This study presents an information theoretical approach to determining whether solar flare data are significantly distinguishable from a nonstationary Poisson process. A study of solar flares stronger than C1 class detected by the Geostationary Operational Environmental Satellite from 1975 to 2017 was performed. A sequence of waiting times (time elapsed between adjacent X-ray flare peaks) was constructed from the data. Surrogate waiting time sequences were produced using a time-varying Poisson firing rate from the Bayesian block procedure. Utilizing Shannon entropy, the mutual information of time-lagged waiting time distributions was computed for both the original data and the surrogates using a method of discretization by binning. When the entire period is considered, we see that when compared to carefully constructed surrogates, there is a significant elevation of mutual information on a timescale of approximately 30 hr, demonstrating that flares are confidently related to subsequent flares, contradicting the null hypothesis that flares are produced by a nonstationary Poisson process. When only 4 yr subsets of the data are considered, we see that at relatively small timescales (on the order of 10–30 hr), solar flare waiting times have a significant impact on subsequent flares. When corrected for the number of points in each considered time window, there is no correlation between the magnitude of significance and position in the solar cycle.

Simulating Nonstationary Spatio-Temporal Poisson Processes using the Inversion Method

We study the problem of simulating a class of nonstationary spatio-temporal Poisson processes. The Poisson intensity function is non-stationary and piecewise linear in both the time dimension and the spatial location dimensions. We propose an exact simulation algorithm based on the inversion method. This simulation algorithm adopts three advantages. First, the entire procedure involves only closed-form computation with no need for numerical integration or numerical inversion of any function. Each step in the algorithm only requires exact arithmetic operations. Second, the proposed algorithm is sample efficient, especially compared to the thinning method when the maximum intensity value is much larger than the minimum intensity value. Third, the algorithm generates arrivals sequentially, one at a time in ascending order, so that they can be conveniently fed into real-time or online decision-making tools.

Self-excitation in the solar flare waiting time distribution

Solar flares release high amounts of energy into the solar system and can negatively impact earth based systems through their effects on satellites and power systems. It is hence important to understand and forecast their occurrence. The solar flare waiting time distribution (WTD) defines the amount of time which elapses between the occurrence of successive flares and hence provides a starting point for forecasts and risk assessment. Previous research has hypothesized that the observed WTD can be derived from a simple model which posits that flares follow a nonstationary Poisson process. This Poissonian assumption has implications for fundamental physical theories about the origin of flares, since it is a direct consequence of the widely studied avalanche model. However in this paper we call the Poissonian assumption into question, by showing that the occurrence of solar flares seems to have a substantial amount of burstiness and self-excitation that continues to exist even when controlling for the solar cycle. This leads to a strong non-Poissonian dependence between the occurrence time of successive flares.

Online Learning and Matching for Resource Allocation Problems

In order for an e-commerce platform to maximize its revenue, it must recommend customers items they are most likely to purchase. However, the company often has business constraints on these items, such as the number of each item in stock. In this work, our goal is to recommend items to users as they arrive on a webpage sequentially, in an online manner, in order to maximize reward for a company, but also satisfy budget constraints. We first approach the simpler online problem in which the customers arrive as a stationary Poisson process, and present an integrated algorithm that performs online optimization and online learning together. We then make the model more complicated but more realistic, treating the arrival processes as non-stationary Poisson processes. To deal with heterogeneous customer arrivals, we propose a time segmentation algorithm that converts a non-stationary problem into a series of stationary problems. Experiments conducted on large-scale synthetic data demonstrate the effectiveness and efficiency of our proposed approaches on solving constrained resource allocation problems.

Statistical properties of magnetar bursts and FRB 121102

ABSTRACT In this paper, we present statistics of soft gamma repeater (SGR) bursts from SGR J1550−5418, SGR 1806−20, and SGR 1900+14 by adding new bursts from Kırmızıbayrak et al. detected with the Rossi X-ray Timing Explorer. We find that the fluence distributions of magnetar bursts are well described by power-law functions with indices 1.84, 1.68, and 1.65 for SGR J1550−5418, SGR 1806−20, and SGR 1900+14, respectively. The duration distributions of magnetar bursts also show power-law forms. Meanwhile, the waiting time distribution can be described by a non-stationary Poisson process with an exponentially growing occurrence rate. These distributive features indicate that magnetar bursts can be regarded as a self-organizing critical process. We also compare these distributions with the repeating fast radio burst (FRB) 121102. The statistical properties of repeating FRB 121102 are similar with magnetar bursts, combining with the large required magnetic field (B ≥ 1014 G) of neutron star for FRB 121102, which indicates that the central engine of FRB 121102 may be a magnetar.

SGR-like behaviour of the repeating FRB 121102

Fast radio bursts (FRBs) are millisecond-duration radio signals occurring at cosmological distances. However the physical model of FRBs is mystery, many models have been proposed. Here we study the frequency distributions of peak flux, fluence, duration and waiting time for the repeating FRB 121102. The cumulative distributions of peak flux, fluence and duration show power-law forms. The waiting time distribution also shows power-law distribution, and is consistent with a non-stationary Poisson process. These distributions are similar as those of soft gamma repeaters (SGRs). We also use the statistical results to test the proposed models for FRBs. These distributions are consistent with the predictions from avalanche models of slowly driven nonlinear dissipative systems.

OUP accepted manuscript

Type I X-ray bursts in a low-mass X-ray binary (LMXB) are caused by unstable nuclear burning of accreted materials. Semi-analytical and numerical studies of unstable nuclear burning have successfully reproduced partial properties of this kind of burst. However, some other properties (e.g. the waiting time) are not well explained. In this paper, we find that the probability distributions of fluence, peak count, rise time, duration and waiting time can be described as power-law-like distributions. This indicates that type I X-ray bursts may be governed by a self-organized criticality (SOC) process. The power-law index of waiting time distribution (WTD) is around $-1$, which is not predicted by any current waiting time model. We propose a physical burst rate model, in which the mean occurrence rate is inversely proportional to time $\lambda\propto t^{-1}$. In this case, the WTD is well explained by a non-stationary Poisson process within the SOC theory. In this theory, the burst size is also predicted to follow a power-law distribution, which requires that the emission area possesses only part of the neutron star surface. Furthermore, we find that the WTDs of some astrophysical phenomena can also be described by similar occurrence rate models.

Assessing hurricane damage costs in the presence of vulnerability model uncertainty

Probability-based assessment of hurricane damage costs for coastal communities is vital for policy-makers and insurers. The uncertainties associated with hurricane damage costs include both the inherent uncertainty due to the random nature of hurricane process and the model uncertainty of the mathematical representation of hurricane damage (vulnerability model). The hurricane vulnerability model has traditionally been modeled as a deterministic function of hurricane wind speed in the literature, without considering the effect of vulnerability model uncertainty on hurricane damage assessment. This paper develops two methods to assess the hurricane damage costs in the presence of vulnerability model uncertainty. To account for the non-stationarity in hurricane actions due to the potential impact of climate change, the hurricane occurrence process is modeled as a non-stationary Poisson process and the hurricane intensity is assumed to vary in time with time-variant statistical parameters of hurricane wind speed. A case study of Miami-Dade County, Florida, is conducted to illustrate the proposed methods and to investigate the impact of vulnerability model uncertainty on hurricane damage costs.