Compound multivariate Hawkes processes: Large deviations and rare event simulation

When simulating small probabilities, say of order 10 -6 or less, by importance sampling, an established principle is to choose the importance sampling distribution as close to the conditional distribution given the rare event as possible. Implementing this often leads into large deviations calculations and exponential change of measure. We survey some of the standard examples where this approach works and supplement existing counterexamples with new ones. Difficulties often arise as consequence of reflecting barriers and we present an algorithm which at least in simple cases is able to deal with this problem. Also the case of heavy-tailed distributions is considered

The Flexible Rare Event Sampling Harness System (FRESHS)

We consider stable open Jackson networks and study the rare events of excessive backlogs. Although, these events occur rarely they can be critical, since they can impair the functioning of the network. We attempt to estimate the probability of these events by simulations. Since, the direct simulation of rare events takes a very long time, this procedure is very costly. Instead, we devise a method for changing the network to speed up the simulation of rare events. We try to pursue this idea with the help of Large Deviation theory. This approach, under certain assumptions, results in a system of differential equations which may be difficult to solve. To circumvent this, we develop a heuristic method which gives the rule for changing the network for the purpose of simulations. We illustrate, by examples, that our method of simulations can be several orders of magnitude faster than direct simulations.

Extreme climate events have major impacts on human societies and ecosystems. The most detrimental events are often extremely rare, with return time of centuries or even millennia. Studying these events in the context of climate change is crucial to help adaptation efforts globally. Yet the study of these extremely rare events is extremely challenging due to the lack of data. Indeed, such events have likely not been observed in the instrumental period. An alternative is to use a global climate model to simulate these extremely rare events. However, this comes at a huge computational cost : gathering good statistics on centennial events would require to run a few thousand years of simulation. Rare event algorithms have recently been introduced in the field of climate science to tackle this difficulty [1]. By concentrating the computational effort on the trajectories most susceptible to lead to the extreme event of interest, they allow for the sampling of extremely rare events at a much lower computational cost than standard simulations. In this study, we run a rare event algorithm to sample extreme heatwave seasons in a heatwave hotspot of South Asia, using the intermediate complexity model Plasim. We compare the outcome of the algorithm against an extremely long – 8000 years – control run. This comparison allows us to demonstrate that the algorithm not only estimates return times with high precision (as shown in previous work), but also exhibits high precision in the estimation of composite statistics: composite maps conditioned on centennial heatwave seasons estimated from the algorithm, are in very good agreement with the ones from the 8000-year long control simulation. Our results suggest that extreme heatwave seasons in the studied region are associated with a quasi-stationary atmospheric wave-pattern stretching from the North Atlantic towards South Asia. We also show that the algorithm correctly estimates the intensity-duration-frequency statistics of subseasonal heatwaves occuring within centennial heatwave seasons. Thus rare event algorithms could, for instance, be combined with seasonal forecasts to provide information regarding expected number of heatwave days and the distribution of the duration and intensity in an extreme heatwave season, which could be useful for adaptation planning.  References[1] F. Ragone, J. Wouters, and F. Bouchet, “Computation of extreme heat waves in climate models using a large deviation algorithm,” Proc Natl Acad Sci USA, vol. 115, pp. 24–29, Jan. 2018. 

The occurrence patterns of important words found in six texts (one historical pamphlet and five renowned academic books) are analyzed using both univariate and multivariate Hawkes processes. By treating the occurrence patterns as binary time-series data along the texts, we investigate how effectively univariate and multivariate Hawkes processes capture the characteristics of these word occurrence signals. Through maximum likelihood estimation and subsequent simulations, we found that the multivariate Hawkes process clearly outperforms the univariate Hawkes process in modeling word occurrence signals. Moreover, we found that the multivariate Hawkes process can provide a Hawkes graph, which serves as an intuitive representation of the relationships between concepts appearing in the analyzed text. Furthermore, our study demonstrates that the importance of concepts within a given text can be quantitatively estimated based on the optimized parameter values of the multivariate Hawkes process.

Rare events are an important limitation of discrete-event simulation in some areas of applications. The existing rare-event simulation (RES) techniques, such as importance sampling (IS), splitting, and RESTART, have been used for RES so far. However, these techniques not only have some limitations, but also are not general or easy to use. In this paper, the partition of the region (PoR) method is used as a variance reduction technique in a new RES technique for usage with a wide range of modeling and simulation languages. Three variants of the proposed technique will be introduced: simple PoR, recursive PoR, and advanced PoR. Simple and recursive PoR methods are powerful techniques that have some important features such as bounded relative error; however, they are restricted to some specific models. Advanced PoR is a more general technique that integrates the PoR method and the IS technique. To evaluate the efficiency of the proposed technique and its variants, the comparative simulation results of some illustrative examples using stochastic activity networks will be presented.

<p>Rare events, such as heat waves, floods, or hurricanes, play a crucial role in climate dynamics mainly due to the large impact they have. Predicting the occurrence of such events is thus a major challenge.<span> </span></p><p>In this talk, we introduce the relevant mathematical object for predicting a future event: the committor function is the probability that an event will occur, conditioned on the current state of the system. Computing this quantity from observations is an extremely difficult task since rare events have a very low probability of occurring and may not even have been observed in measurements made to date. Similarly, direct simulation of such events with comprehensive climate models comes at a prohibitive computational cost. Hence, rare event algorithms have been devised to simulate rare events efficiently, avoiding the computation of long periods of typical fluctuations.</p><p>The effectiveness of these algorithms strongly relies on the knowledge of a measure of how close the event of interest is to occur, called the “score function”. The main difficulty is that the optimal score function is the committor function which is exactly the quantity to be computed. Therefore, it is very natural to consider an iterative procedure where the data produced by the algorithm is used to improve the score function, which in turn improves the algorithm, and so on.</p><p>In this presentation, we propose a data-driven approach for computing the committor function, based on a Markov chain approximation of the dynamics of the system (the analogue method). We first illustrate this approach for a paradigmatic toy model of multistability for atmospheric dynamics with six variables (the Charney-Devore model). Secondly, we apply this methodology to data generated from a climate model, in order to study and predict the occurrence of extreme heat waves. In both cases, we show that it is possible to obtain fairly precise estimates of the committor function, even when few observations are available.</p><p>In the second part of the talk, we show the advantage of coupling the analogue Markov chain with a rare event algorithm. Indeed, the committor learned with the analogue Markov chain can be used as a score function performing better than user-defined score functions, as we show for the Charney-Devore model.<span> </span></p><p>This new approach is promising for studying rare events in complex dynamics: the rare events can be simulated with a minimal prior knowledge and the results are much more precise than those obtained with a user-designed score function.</p>

Birth of a new distinct phase is a phenomenon encountered in a myriad of processes, and has wide ranging consequences in material processing, biological self-assembly, separations and several other processes. Several phase transitions are nucleation driven. The nucleation events occur over nanosecond timescales and involve hundreds to thousands of molecules. These length and timescales are difficult to access in experiments, thereby making experimental studies of nucleation challenging. On the other hand, molecular simulations sample the nanosecond and nanometer scales making them ideal to study nucleation. However, nucleation is a rare event, meaning that the waiting time to observe one nucleation event is significant. This makes simulation studies of rare events challenging. The project focused on a multi-pronged approach to address such challenges to develop the next generation rare event sampling methods for molecular simulations. The key outcomes of our work include developing more effective methods for sampling rare events, utilizing machine learning to better elucidate nucleation mechanisms, development of software for easy implementation of the methodologies, and applications of the methods to realistic systems to push the method applicability beyond model systems. Overall, this work has enabled pushing the frontiers of molecular simulations to study rare events with a focus on nucleation in aqueous solutions.

In this dissertation we present several papers about two optimization methods that are derived from rare-event simulation techniques, i.e., the splitting method and the cross-entropy (CE) method. The splitting method is a well-known method for rare-event simulation, where sample paths of a Markov process are split into multiple copies during the simulation, so as to make the occurrence of a rare event more frequent. Motivated by the splitting algorithm we introduce a novel global optimization method for continuous optimization (SCO) that is both very fast and accurate. We also introduce a new multi-objective optimization (MOO) methodology based the SCO algorithm. The method, called MOS, generalizes the elite set selection of the traditional splitting framework, and uses both local and global sampling to sample in the decision space. Both SCO and MOS were compared with state-of-the art algorithms using a prevailing set of benchmark problems. Numerical experiments demonstrate that the new splitting methods are competitive with famous methods in the fields. The thesis also contains two new applications of the CE method on Laguerre fitting problems. The Laguerre fitting problems are about recovering the weighted generator points from a given Laguerre tessellation. When the description of the Laguerre tessellation is exact, the solutions are not unique and different weighted generator points can create the same tessellation. To recover pertinent generator points we formulate the problem as an optimization problem and apply the modified CE method to solve it. However, the representation of a Laguerre tessellation is typically not exact when working with real data, such as tomographic image data. In this case, we formulate an optimization problem where we minimize the discrepancy between the image data and our approximated tessellations. We then solve the optimization problem using the CE method. Last, we introduce a new R package that implements the general CE methodology for optimization and well as some useful modifications. The usage and efficacy of CEoptim is demonstrated through a variety of optimization examples, including model fitting, combinatorial optimization, and maximum likelihood estimation.

Although the Rare event has a very wide application range, it has limitations in that it requires mathematical knowledge and the calculation process is complicated for the Bayesian approach, which was mainly used in previous studies. Therefore, rare events need to be easily analyzed. In this study, the critical failures of the ROK Naval ship were analyzed. Bayesian inference was not applied. The long-term trend (B-spline) and short-term probability (phase-type distribution) were fitted and combined. Phase-type distribution is a useful technique that can estimate the probability distribution of a desired shape with one parameter. This study is meaningful in that only an easy methodology was used, Phase-type distribution was applied to the analysis of rare events, and only a very small number of data were analyzed.

Rare event simulation in the context of queueing networks has been an active area of research for more than two decades. A commonly used technique to increase the efficiency of Monte Carlo simulation is importance sampling. However, there are few rigorous results on the design of efficient or asymptotically optimal importance sampling schemes for queueing networks. Using a recently developed game/subsolution approach, we construct simple and efficient state-dependent importance sampling schemes for simulating buffer overflows in stable open Jackson networks. The sampling distributions do not depend on the particular event of interest, and hence overflow probabilities for different events can be estimated simultaneously. A by-product of the analysis is the identification of the minimizing trajectory for the calculus of variation problem that is associated with the sample-path large deviation rate function.

The two-node tandem Jackson network serves as a convenient reference model for the analysis and testing of different methodologies and techniques in rare event simulation. In this paper we consider a new approach to efficiently estimate the probability that the content of the second buffer exceeds some high level L before it becomes empty, starting from a given state. The approach is based on a Markov additive process representation of the buffer processes, leading to an exponential change of measure to be used in an importance sampling procedure. Unlike changes of measures proposed and studied in recent literature, the one derived here is a function of the content of the first buffer. We prove that when the first buffer is finite, this method yields asymptotically efficient simulation for any set of arrival and service rates. In fact, the relative error is bounded independent of the level L ; a new result which is not established for any other known method. When the first buffer is infinite, we propose a natural extension of the exponential change of measure for the finite buffer case. In this case, the relative error is shown to be bounded (independent of L ) only when the second server is the bottleneck; a result which is known to hold for some other methods derived through large deviations analysis. When the first server is the bottleneck, experimental results using our method seem to suggest that the relative error is bounded linearly in L .

Stochastic Petri Net Package (SPNP) is a software package whose goal is to compute performance, availability or performability measures from Stochastic Petri Nets (SPN) and Fluid Stochastic Petri nets (FSPN). This software can use either analytic numeric methods, or simulation methods. Unfortunately, the standard discrete event simulation is inefficient to estimate the probabilities of rare events. For such rare event simulations, importance splitting technique is a good method to speedup the simulation. In the literature, two diffierent importance splitting techniques are known: RESTART and splitting. In this paper, we describe the application of these methods to (both fluid and discrete) Petri nets, their implementation in SPNP and we give some illustrations of the speed-up. The RESTART technique has already been applied in another Petri net package, TimeNet, but here we implement both RESTART and splitting, and we apply them to a more general class of Petri nets including the fluid ones.

We review two types of adaptive Monte Carlo methods for rare event simulations. These methods are based on importance sampling. The first approach selects importance sampling distributions by minimizing the variance of importance sampling estimator. The second approach selects importance sampling distributions by minimizing the cross entropy to the optimal importance sampling distribution. We also review the basic concepts of importance sampling in the rare event simulation context. To make the basic concepts concrete, we introduce these ideas via the study of rare events of M/M/1 queues.

An importance sampling technique in Monte Carlo methods for SDEs with a.s. stable and mean-square unstable equilibrium