Stochastic Point Research Articles

A doubly stochastic point process approach for spatio-temporal dynamics of crime data

Crime, in general, is at the base of crucial issues for many societies living in large cities worldwide. Indeed, crime and neighbourhood disorder may negatively impact the health of urban residents. Thus, the crime rate reduction is at the core of many local policies driven by active plans supported by police action and local authorities. Considering crime reports as a spatio-temporal point pattern, we propose spatio-temporal log-Gaussian Cox processes as a modelling framework for crimes in space and time. We model the spatial and temporal variation through generalized parametric additive and linear models, and a Gaussian space-time process approximates the residual variation. The inference is performed via Markov chain Monte Carlo through MALA algorithms. We provide short-term forecasts of future crimes and suggest a surveillance system that operates by reporting predictive probabilities. Our data come from the reported crimes in the locality of Kennedy (Bogota) over several years and several types of crimes. The police department may use our method to help allocate police resources and design crime prevention strategies and policies, such as surveillance, in specific zonal planning units.

Predicting the Probability of Abrupt Changes to Wave‐Generated Seafloor Sand Ripples

AbstractA new, non‐dimensional ripple reset parameter and a stochastic point process model is used to estimate the likelihood of propagating ocean waves to form ripples on sandy seabeds. The ripple reset parameter is a function only of water depth, significant wave height, and mean grain size. Ripple formation is estimated by the magnitude of an intensity function based on a time series of the ripple reset parameter. The point process model is trained with a time series of observed waves and ripple change, and is then applied to predict the probability that a ripple field with a different geometry will form within a given time interval from another time series of wave data. The model is trained and tested with four field deployments at three field sites to determine its skill in predicting the ripple formation (a) at one field site over one time period after being trained with observations from the same site over a different time period, and (b) at one field site after being trained with observations from another field site. Results show that while the model is sufficient at predicting ripple formation in both scenarios, it is sensitive to the quality and quantity of the training data. Increasing the amount of training data greatly improves model performance. Employing a stochastic model based on a simple ripple reset parameter reduces tunable model parameters and provides a prediction of the probability for ripple formation given only a water depth, grain size, and time series of wave heights.

Spatio-Temporal Hawkes Point Processes: A Review

AbstractHawkes processes are a particularly interesting class of stochastic point processes that were introduced in the early seventies by Alan Hawkes, notably to model the occurrence of seismic events. They are also called self-exciting point processes, in which the occurrence of an event increases the probability of occurrence of another event. The Hawkes process is characterized by a stochastic intensity, which represents the conditional probability density of the occurrence of an event in the immediate future, given the observations in the past. In this paper, we present some background and all major aspects of Hawkes processes, with a particular focus on simulation methods, and estimation techniques widely used in practical modeling aspects. We aim to provide a rich and self-contained overview of these stochastic processes as a way to have an overall vision of Hawkes processes in only one piece of paper. We also discuss possibilities for future research in the area of self-exciting processes.

Stochastic differential equations for non-analog Monte Carlo model of point kinetics equations

The stochastic form of one-point and two-point reactor kinetics models have been extensively investigated in the literature. We have noticed that, in some studies, the probabilistic nature of neutron interaction mechanisms was not incorporated truly. For instance, in the case of negative reactivity insertions, one of the presented probability functions becomes negative, making the Monte Carlo (MC) simulation impossible. In this manuscript, the non-analog stochastic point reactor kinetics models are developed to simulate the stochastic behavior of nuclear systems. To reduce the imposed variance from the leakage and capture of neurons, both capture and leakage events are sampled implicitly. Precursors are also forced to decay in each simulation step. The true variances of the neutron’s and delayed neutron precursor’s populations are determined from the solution of the developed non-analog stochastic models. Also, they are compared with those of both analog and non-analog Monte Carlo methods. The results show that the analog Monte Carlo (MC) has a larger variance than the non-analog MC.

Simulation of doubly stochastic Poisson point processes and application to nucleation of nanocrystals and evaluation of exciton fluxes

Abstract In this study we solve the following problem: Simulate random 2D Poisson point processes with a desired correlation function. To solve this problem we suggest the following algorithm: (1) simulate a positive valued random process with the desired correlation function, (2) use this process as an intensity of the doubly stochastic Poisson random point process. We apply this algorithm to simulate random distribution of nanocrystals on a plane. Then we apply the developed methods to calculate excitonic fluxes to the family of generated nanocrystals.

Variance Reduction Techniques for Stochastic Proximal Point Algorithms

Variance Reduction Techniques for Stochastic Proximal Point Algorithms

Vapor bubble nucleation in flowing liquids

In this work, a stochastic diffuse interface model coupled with the Navier–Stokes equations has been exploited to numerically investigate vapor nucleation in a non-equilibrium system (a flowing liquid). Both homogeneous and heterogeneous nucleation is addressed and the influence of macroscopic flows on nucleation observables is discussed. The extended mesoscale simulations allow us to infer the spatial distributions of the nucleated bubbles via Voronoi tessellation analysis and to represent the nucleation phenomenon as a stochastic Random Poisson Point process. The findings open the way to design multiscale fluid simulations experiencing phase change.

General Procedure to Provide High-Probability Guarantees for Stochastic Saddle Point Problems

General Procedure to Provide High-Probability Guarantees for Stochastic Saddle Point Problems

A novel and efficient stochastic collocation method for estimating failure probability function in one-dimensional reduced space

Failure probability function (FPF) is an important index when measuring the effect of random input distribution parameters on the safety of structures. It can decouple reliability-based design optimization. However, efficient estimation of the FPF is challenging. This study proposes a novel stochastic collocation method for estimating the FPF. The novelty of the proposed method lies in three aspects: First, an equivalent integral expression is proposed for the FPF in one-dimensional reduced space, enabling FPF estimation using an efficient stochastic collocation method. Secondly, a unified sampling density function is constructed to remove the coupling between the computational cost of FPF estimation and the number of distribution parameter realizations. Thirdly, a stochastic collocation point set is derived and shared to estimate the whole FPF using arbitrary distribution parameter realization. The proposed method is particularly suitable for the strength-stress and resistance-load models used in the field of engineering. The efficiency and accuracy of the proposed method are verified by several examples.

Using Stochastic Point Pattern Analysis to Track Regional Orientations of Magmatism During the Transition to Cenozoic Extension and Rio Grande Rifting, Southern Rocky Mountains

AbstractThe southern Rocky Mountains in Colorado and northern New Mexico hosted intracontinental magmatism that developed during a tectonic transition from shortening (Laramide orogeny, ca. 75 to 40 Ma) through extension and rifting. We present a novel approach that uses stochastic weighted bootstrap simulations of a large set of new and historical geochronology data to better understand how regional anisotropies responsible for focusing magma emplacement evolved through time. This technique can detect subtle trends in directional distributions, including multi‐modal orientations, and can be filtered from regional to local scales. Our results indicate that magmatism followed first the northeast trend of the Colorado mineral belt between 75 and 40 Ma and deviated afterward. These deviations vary depending on the scale of the analysis. At the smallest scale we evaluated (<75 km), the orientation of magmatism from 45 to 30 Ma rotated counter‐clockwise before aligning with the north‐south trend of the modern Rio Grande rift. Larger, regional‐scale analyses indicate magma centers between 40 to 35 Ma and 25 to 20 Ma were dominantly oriented southwest‐northeast, whereas magmatism between 35 and 25 Ma had north‐south orientation. The large areal footprint of magmatism and shifting regional patterns suggest that ancient zones of weakness in the North American lithosphere accommodated magma flow at different moments in time, rather than controlled by a retreating interface of the Farallon and North American plates.

Stochastic Modeling of Wind Derivatives with Application to the Alberta Energy Market

Wind-power generators around the world face two risks, one due to changes in wind intensity impacting energy production, and the second due to changes in electricity retail prices. To hedge these risks simultaneously, the quanto option is an ideal financial tool. The natural logarithm of electricity prices of the study will be modeled with a variance gamma (VG) and normal inverse Gaussian (NIG) processes, while wind speed and power series will be modeled with an Ornstein–Uhlenbeck (OU) process. Since the risk from changing wind-power production and spot prices is highly correlated, we must model this correlation as well. This is reproduced by replacing the small jumps of the Lévy process with a Brownian component and correlating it with wind power and speed OU processes. Then, we will study the income of the wind-energy company from a stochastic point of view, and finally, we will price the quanto option of European put style for the wind-energy producer. We will compare quanto option prices obtained from the VG process and NIG process. The novelty brought into this study is the use of a new dataset in a new geographic location and a new Lévy process, VG, apart from NIG.

On Gaussian Markov processes and Polya processes

In previous work we characterized Gaussian Markov processes with stationary increments and showed that they arise as asymptotic approximations for stochastic point processes with a random rate such as Polya processes, which can be useful to model over-dispersion and path-dependent behavior in service system arrival processes. Here we provide additional insight into these stochastic processes.

CanStoc: A Hybrid Stochastic–GCM System for Monthly, Seasonal and Interannual Predictions

Beyond their deterministic predictability limits of ≈10 days and 6 months, the atmosphere and ocean become effectively stochastic. This has led to the development of stochastic models specifically for this macroweather regime. A particularly promising approach is based on the Fractional Energy Balance Equation (FEBE), an update of the classical Budyko–Sellers energy balance approach. The FEBE has scaling symmetries that imply long memories, and these are exploited in the Stochastic Seasonal and Interannual Prediction System (StocSIPS). Whereas classical long-range forecast systems are initial value problems based on spatial information, StocSIPS is a past value problem based on (long) series at each pixel. We show how to combine StocSIPS with a classical coupled GCM system (CanSIPS) into a hybrid system (CanStoc), the skill of which is better than either. We show that for one-month lead times, CanStoc’s skill is particularly enhanced over either CanSIPS or StocSIPS, whereas for 2–3-month lead times, CanSIPS provides little extra skill. As expected, the CanStoc skill is higher over ocean than over land with some seasonal dependence. From the classical point of view, CanStoc could be regarded as a post-processing technique. From the stochastic point of view, CanStoc could be regarded as a way of harnessing extra skill at the submonthly scales in which StocSIPS is not expected to apply.

Continuous‐time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

AbstractWe develop a new continuous‐time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events.

On spare parts demand and the installed base concept: A theoretical approach

Original Equipment Manufacturers (OEMs) aim to design their service supply chain before the introduction of their products to maximize their aftersales business revenues, reduce waste and achieve sustainability. In this study, we develop a stochastic model that unifies the installed base, i.e., the number of products in use, spare parts demand, and the number of discarded products within a single modeling framework based on three product characteristics: sales rate, usage time, and failure rate. Our model describes the installed base and spare part demand evolution over the entire life cycle of a parent product using stochastic point processes. At the same time we propose under very general assumptions on the cdf of the usage time and the mean arrival functions of the sales and failure processes an easy bisection procedure to compute the time at which the expected installed base and rate of the expected demand for spare parts is maximal. Our numerical experiments show that the volume of aftersales services increases in the expected usage time if the products face an increasing failure rate. The same experiments also reveal a 20 percent shift of the time at which the expected installed base is maximal in case the expected usage time is increased threefold. At the same time, we observe a boosting effect of the intensity of the sales process on this point in time.

Extension of Stochastic Point Location for Multimodal Problems.

Stochastic point location (SPL) involves a learning mechanism (LM) determining an optimal point on the line when the only inputs LM receives are stochastic information about the direction in which LM should move. The complexity of SPL comes from the stochastic responses of the environment, which may lead LM completely astray. SPL is a fundamental problem in optimization and was studied by many researchers during the last two decades, including improvement of its solution and all-pervasive applications. However, all existing SPL studies assume that the whole search space contains only one optimal point. Since a multimodal optimization problem (MMOP) contains multiple optimal solutions, it is significant to develop SPL's multimodal version. This article extends it from a unimodal problem to a multimodal one and proposes a parallel partition search (PPS) solution to address this issue. The heart of the proposed solution involves extracting the feature of the historical sampling information to distinguish the subintervals that contain the optimal points or not. Specifically, it divides the whole search space into multiple subintervals and samples them parallelly, then utilizes the feature of the historical sampling information to adjust the subintervals adaptively and to find the subintervals containing the optimal points. Finally, the optimal points are located within these subintervals according to their respective sampling statistics. The proof of the ε-optimal property for the proposed solution is presented. The numerical testing results demonstrate the power of the scheme.

Stochastic Saddle Point Problems with Decision-Dependent Distributions

Stochastic Saddle Point Problems with Decision-Dependent Distributions

Spatial metapopulation dynamics with local and global colonization

We revisit a spatial metapopulation model on continuous space as a stochastic point pattern dynamics. In the model, local patches as points are distributed with a certain spatial configuration and status of each patch changes stochastically between occupied and empty: an occupied patch becomes empty by local extinction and an empty patch becomes occupied both by local and global colonization. We carry out simulation analysis and derive an analytical model in terms of singlet, pair and triplet probabilities that describe the stochastic dynamics. Using a simple closure that approximates triplet probabilities by singlet and pair probabilities, we show that equilibrium singlet and pair probabilities can be analytically derived. The derived equilibrium properties successfully describe simulation results under a certain condition where the range of local colonization and the proportion of global colonization play key roles. Our model is an extension of the classical non-spatial Levins model to a spatially explicit metapopulation model. We appeal the advantage of point pattern approach to study spatial dynamics in general ecology and call for the need to deepen our understanding of mathematical tools to explore point pattern dynamics.

Stochastic Exceptional Points for Noise-Assisted Sensing.

Noise is a fundamental challenge for sensors deployed in daily environments for ambient sensing, health monitoring, and wireless networking. Current strategies for noise mitigation rely primarily on reducing or removing noise. Here, we introduce stochastic exceptional points and show the utility to reverse the detrimental effect of noise. The stochastic process theory illustrates that the stochastic exceptional points manifest as fluctuating sensory thresholds that give rise to stochastic resonance, a counterintuitive phenomenon in which the added noise increases the system's ability to detect weak signals. Demonstrations using a wearable wireless sensor show that the stochastic exceptional points lead to more accurate tracking of a person's vital signs during exercise. Our results may lead to a distinct class of sensors that overcome and are enhanced by ambient noise for applications ranging from healthcare to the internet of things.

Observer-Based Control for a New Stochastic Maximum Power Point Tracking for Photovoltaic Systems With Networked Control System

This study discusses the new stochastic maximum power point tracking (MPPT) control approach towards the photovoltaic cells (PCs). PC generator is isolated from the grid, resulting in a direct current (DC) microgrid that can provide changing loads. In the course of the nonlinear systems through the time-varying delays, we proposed a Networked Control Systems (NCSs) beneath an event-triggered approach basically in the fuzzy system. In this scenario, we look at how random, variable loads impact the PC generator's stability and efficiency. The basic premise of this article is to load changes and the value matching to a Markov chain. PC generators are complicated nonlinear systems that pose a modeling problem. Transforming this nonlinear PC generator model into the Takagi–Sugeno (T– S) fuzzy model is another option. Takagi–Sugeno (T–S) fuzzy model is presented in a unified framework, for which 1) the fuzzy observer–based on this premise variables can be used for approximately in the infinite states to the present system, 2) the fuzzy observer–based controller can be created using this same premises be the observer, and 3) to reduce the impact of transmission burden, an event-triggered method can be investigated. Simulating in the PC generator model for the realtime climate data obtained in China demonstrates the importance of our method. In addition, by using a new Lyapunov–Krasovskii functional (LKF) for combining to the allowed weighting matrices incorporating mode-dependent integral terms, the developed model can be stochastically stable and achieves the required performances. Based on the T-P transformation, a new depiction of the nonlinear system is derived in two separate steps in which an adequate controller input is guaranteed in the first step and an adequate vertex polytope is ensured in the second step. To present the potential of our proposed method, we simulate it for PC generators.