Multiple Stochastic Processes Research Articles

Interpreting the power spectral density of a fluctuating colloidal current.

The transport of molecules through biological and synthetic nanopores is governed by multiple stochastic processes that lead to noisy, fluctuating currents. Disentangling the characteristics of different noise-generating mechanisms is central to better understanding molecular transport at a fundamental level but is extremely challenging in molecular systems due to their complexity and relative experimental inaccessibility. Here, we construct a colloidal model microfluidic system for the experimental measurement of particle currents, where the governing physical properties are directly controllable and particle dynamics directly observable, unlike in the molecular case. Currents of hard spheres fluctuate due to the random arrival times of particles into the channel and the distribution of particle speeds within the channel, which results in characteristic scalings in the power spectral density. We rationalize these scalings by quantitatively comparing to a model for shot noise with a finite transit time, extended to include the distribution of particle speeds. Particle velocity distributions sensitively reflect the confining geometry, and we interpret and model these in terms of the underlying fluid flow profiles. Finally, we explore the extent to which details of these distributions govern the form of the resulting power spectral density, thereby establishing concrete links between the power spectral density and underlying mechanisms for this experimental system. This paves the way for establishing a more systematic understanding of the links between characteristics of transport fluctuations and underlying molecular mechanisms in driven systems such as nanopores.

Abstract PR016: Topology of the tumor microenvironment: Integrating imaging, modeling, and topological data analysis

Abstract Cancer evolution involves multiple stochastic processes across many scales. In addition to genomic and transcriptomic changes, hallmarks of cancer within solid tumors include the malignant co-evolution of the tumor microenvironment. Although the evaluation of tissue architecture plays a key role in pathologist staging, the automated extraction of additional structural features remains far from translation to the clinic. However, one particularly effective novel methodology for extracting patterns from these clinical images is topological data analysis(TDA). TDA is a field that studies the shape of data and its connectedness using concepts from geometric and algebraic topology. Recent work using topological data analysis has allowed the extraction of image features. Such features have been associated with clinical outcomes in both the analysis of H&E images in prostate cancer and CT images in lung cancer. We set out to question how the topological features of tissues, including tumor and stromal composition, evolve during treatment and between original sites and metastasis. To analyze this we use a combination of tissue sections, imaging, and cell-automata models and persistent homology and cubical persistence techniques from TDA. We apply certain topological methods for the first time to tissue images (H&E) and to the output of spatial simulations. We focus specifically on analyzing the evolution of topological features in ovarian and lung cancer and compare our features to 2D and 3D simulations (HAL) with varying game theoretical interactions as well as comparing features across tissues from matched primary and metastatic sites. We find that several topological features are conserved between primary and metastatic sites and some feature evolution is replicated in simulation. Citation Format: Rowan Barker-Clarke, Jason M. Gray, Jacob G. Scott. Topology of the tumor microenvironment: Integrating imaging, modeling, and topological data analysis [abstract]. In: Proceedings of the AACR Special Conference in Cancer Research: Translating Cancer Evolution and Data Science: The Next Frontier; 2023 Dec 3-6; Boston, Massachusetts. Philadelphia (PA): AACR; Cancer Res 2024;84(3 Suppl_2):Abstract nr PR016.

A loading contribution degree analysis-based strategy for time-variant reliability analysis of structures under multiple loading stochastic processes

This paper proposes a new single-loop method based on the loading contribution degree analysis (LCDA) and decoupling strategy for analyzing the time-variant reliability of structures considering multiple loading stochastic processes and degradation processes. In this method, the loading stochastic processes are first discretized using the expansion optimal linear estimation method. Then, the LCDA is carried out to obtain a total weighted loading stochastic process used to accurately reflects the structural response, thus constructing a single loading stochastic process analysis problem. Subsequently, a decoupling strategy is applied to decouple the degradation processes and the total weighted loading stochastic process to obtain the equivalent extreme-value limit state function (EEV-LSF) of the structure. Finally, an efficient probability distribution reconstruction tool is applied to derive the probability density function of EEV-LSF. Notably, the proposed LCDA can avoid the calculation of the response time series of a structure under the input loading stochastic processes, providing a novel perspective to simplify the calculation of structural extreme-value response. Additionally, the proposed method requires only one round of deterministic analysis to obtain the structural time-variant failure probability at the specified time, constructing an efficient single-loop analysis method. Three numerical examples are used to validate the proposed method.

A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Nonlinear independent component analysis for discrete-time and continuous-time signals

We study the classical problem of recovering a multidimensional source signal from observations of nonlinear mixtures of this signal. We show that this recovery is possible (up to a permutation and monotone scaling of the source’s original component signals) if the mixture is due to a sufficiently differentiable and invertible but otherwise arbitrarily nonlinear function and the component signals of the source are statistically independent with ‘nondegenerate’ second-order statistics. The latter assumption requires the source signal to meet one of three regularity conditions which essentially ensure that the source is sufficiently far away from the nonrecoverable extremes of being deterministic or constant in time. These assumptions, which cover many popular time series models and stochastic processes, allow us to reformulate the initial problem of nonlinear blind source separation as a simple-to-state problem of optimisation-based function approximation. We propose to solve this approximation problem by minimizing a novel type of objective function that efficiently quantifies the mutual statistical dependence between multiple stochastic processes via cumulant-like statistics. This yields a scalable and direct new method for nonlinear Independent Component Analysis with widely applicable theoretical guarantees and for which our experiments indicate good performance.

MGDGAN: Multiple Generator and Discriminator Generative Adversarial Networks for Solving Stochastic Partial Differential Equations

We propose novel structures of generator and discriminator in physics-informed generative adversarial networks called multiple-generator-and-discriminator generative adversarial networks (MGDGANs), that are designed to solve stochastic partial differential equations (SPDEs). MGDGANs for SPDEs consist of three steps: a generator that samples a solution to the SPDEs, a physics-informed operator that enforces the governing equation, and a discriminator that distinguishes between samples from the generator and training samples. Inspired by the polynomial chaos, we represent the solution by the inner product of functions in spatial and random variables, and model each function by a separate generator. We show that the proposed multiple generator structure offers huge computational savings in training and prediction. If multiple stochastic processes exist in the system, then a distinct discriminator is used for each of them. We show that the loss function obtained by these distinct discriminators provides an equivalent metric to the Wasserstein distance loss by a single discriminator, and provide numerical examples to demonstrate that these multiple discriminators enhance the training accuracy. Numerical examples are demonstrated to verify that the proposed model is efficient in computation and memory; the model reduces computing time by more than a factor of 10 and relative l2 error by about one-third in the SPDE example.

Low-Cost Active Anomaly Detection with Switching Latency

Consider the problem of detecting anomalies among multiple stochastic processes. Each anomaly incurs a cost per unit time until it is identified. Due to the resource constraints, the decision-maker can select one process to probe and obtain a noisy observation. Each observation and switching across processes accompany a certain time delay. Our objective is to find a sequential inference strategy that minimizes the expected cumulative cost incurred by all the anomalies during the entire detection procedure under the error constraints. We develop a deterministic policy to solve the problem within the framework of the active hypothesis testing model. We prove that the proposed algorithm is asymptotic optimal in terms of minimizing the expected cumulative costs when the ratio of the single-switching delay to the single-observation delay is much smaller than the declaration threshold and is order-optimal when the ratio is comparable to the threshold. Not only is the proposed policy optimal in the asymptotic regime, but numerical simulations also demonstrate its excellent performance in the finite regime.

Time Variant Interval Linear Programming for Environmental Management Systems

Optimization technology is widely applied to maximize economic profit under ecology constrains in environmental management systems. To tackle the inherent uncertainties, inexact optimization methods have been proposed. Interval linear program- ming (ILP) model has drawn increasing scholarly attention. ILP model describe uncertainty by one coarse scaled stochastic process. However, uncertainty often involves multiple stochastic processes when zooming into high resolution. ILP model may not satisfied fine scale constraints. A time variant interval linear programming (TVILP) model is developed to implement temporal downscaling, and likewise, a heuristic algorithm integrating dynamic programming is proposed for Markov chained TVILP. Dynamic programming can converts time complexity exponential to polynomial. In the current paper, the performance of TVILP model is analyzed based on the following three metrics: maximal profit (M_profit), constraint violation risk (CVR), and maximal profit path risk (MPR). The perfor- mance of TVILP is further compared with the performance of Best and Worst method, the classic ILP model, Interval linear programming contractor, and Interval-parameter multi-stage stochastic linear programming. Experimental results reveal that TVILP provides refined solutions on a smaller granularity whose decision space contracts based on the most possible transition paths. Unable to obtain the maxi- mum profit, though, TVILP does pose decreased constraint violation risk and maximal profit path risk, facilitating more feasible and reliable decision-making on environmental management.

Online reinforcement learning for adaptive interference coordination

AbstractHeterogeneous networks (HetNets), in which small cells overlay macro cells, are a cost‐effective approach to increase the capacity of cellular networks. However, HetNets have raised new issues related to cell association and interference management. In particular, the optimal configuration of interference coordination (IC) parameters is a challenging task because it depends on multiple stochastic processes such as the locations of the users, the traffic demands, or the strength of the received signals. This work proposes a self‐optimization algorithm capable of finding the optimal configuration in an operating network. We address the problem using a reinforcement learning (RL) approach, in which the actions are the IC configurations, whose performances are initially unknown. The main difficulty is that, due to the variable network conditions, the performance of each action may change over time. Our proposal is based on two main elements: the sequential exploration of subsets of actions (exploration regions), and an optimal stopping (OS) strategy for deciding when to end current exploration and start a new one. For our algorithm, referred to as local exploration with optimal stopping (LEOS), we provide theoretical bounds on its long‐term regret per sample and its convergence time. We compare LEOS to state‐of‐the‐art learning algorithms based on multiarmed bandits and policy gradient RL. Considering different changing rates in the network conditions, our numerical results show that LEOS outperforms the first alternative by 22%, and the second one by 48% in terms of average regret per sample.

Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations

We developed a new class of physics-informed generative adversarial networks (PI-GANs) to solve forward, inverse, and mixed stochastic problems in a unified manner based on a limited number of scattered measurements. Unlike standard GANs relying solely on data for training, here we encoded into the architecture of GANs the governing physical laws in the form of stochastic differential equations (SDEs) using automatic differentiation. In particular, we applied Wasserstein GANs with gradient penalty (WGAN-GP) for its enhanced stability compared to vanilla GANs. We first tested WGAN-GP in approximating Gaussian processes of different correlation lengths based on data realizations collected from simultaneous reads at sparsely placed sensors. We obtained good approximation of the generated stochastic processes to the target ones even if there is a mismatch between the input noise dimensionality and the effective dimensionality of the target stochastic processes. We also studied the overfitting issue for both the discriminator and the generator, and we found that overfitting occurs also in the generator in addition to the discriminator as previously reported. Subsequently, we considered the solution of elliptic SDEs requiring approximations of three stochastic processes, namely the solution, the forcing, and the diffusion coefficient. Here again, we assumed data collected from simultaneous reads at a limited number of sensors for the multiple stochastic processes. Three generators were used for the PI-GANs: two of them were feed forward deep neural networks (DNNs), while the other one was the neural network induced by the SDE. For the case where we have one group of data, we employed one feed forward DNN as the discriminator, while for the case of multiple groups of data we employed multiple discriminators in PI-GANs. We solved forward, inverse, and mixed problems without changing the framework of PI-GANs, obtaining both the means and the standard deviations of the stochastic solution and the diffusion coefficient in good agreement with benchmarks. In this work, we have demonstrated the effectiveness of PI-GANs in solving SDEs for about 120 dimensions. In principle, PI-GANs could tackle very high dimensional problems given more sensor data with low-polynomial growth in computational cost.

Switching Stochastic Approximation and Applications to Networked Systems

This paper investigates the interaction between control and communications in networked systems by studying a class of stochastic approximation algorithms that accommodate random network topology switching processes, time-varying functions, nonlinear dynamics, additive and nonadditive noises, and other uncertainties. Interaction among control strategy and the multiple stochastic processes introduces critical challenges in such problems. By modeling the random switching as a discrete-time Markov chain and studying multiple stochastic uncertainties in a unified framework, it is shown that under broad conditions, the algorithms are convergent. The performance of the algorithms is further analyzed by establishing their rate of convergence and asymptotic characterizations. Simulation case studies are conducted to evaluate the performance of the procedures in various aspects.

LearnPopGen: An R package for population genetic simulation and numerical analysis.

Here, I briefly present a new R package called learnPopGen that has been designed primarily for the purposes of teaching evolutionary biology, population genetics, and evolutionary theory. Functions of the package can be used to conduct simulations and numerical analyses of a wide range of evolutionary phenomena that would typically be covered in advanced undergraduate through graduate‐level curricula in population genetics or evolution. For instance, learnPopGen functions can be used to visualize gene frequency changes through time under multiple deterministic and stochastic processes, to compute and animate the changes in phenotypic trait values or distributions under natural selection, to numerically analyze and graph the outcome of simple game theory models, and to plot coalescence within a population experiencing genetic drift, along with a number of other things. Functions have been designed to be maximally didactic and frequently employ compelling animated visualizations. Furthermore, it is straightforward to export plots and animations from R in the form of flat or animated graphics, or as videos. For maximum flexibility, students working with the package can run functions directly in R; however, instructors may choose to guide students less adept in the R environment to one of various web interfaces that I have built for a number of the functions of the package and that are already available online.

A squeeze in the suitable fire interval: Simulating the persistence of fire-killed plants in a Mediterranean-type ecosystem under drier conditions

Mediterranean-type ecosystems (MTEs) harbor an exceptionally high biodiversity of vascular plants. At the same time, climatic conditions in many MTE regions are projected to become both drier and hotter, and fire intervals shorter. The Interval Squeeze conceptual model integrates the potential effects of a changing climate and fire regimes on perennial plant population persistence and postulates that warmer, drier conditions will negatively affect multiple plant demographic processes. Dependent on species-specific traits, the required fire intervals that allow for population persistence might become longer, while projected future fire intervals are shorter, leading to a potential mismatch. However, conceptual models are per se not able to quantify outcomes of multiple stochastic processes or to simulate temporal dynamics. Here, we develop a simple, process-based model for a fire-sensitive woody plant species to evaluate the response of demographic processes to future climatic conditions and to quantify the potential impact also of future changes in fire interval. This allowed us to assess key assumptions of the interval squeeze model, particularly in relation to demographic drivers.We simulated populations of Banksia hookeriana, a typical fire-killed shrub found in MTEs of South-West Australia which stores its seeds in a canopy (serotinous) seedbank and shows strong cohort recruitment in the first year after fire. We estimated suitable fixed fire intervals for population persistence under historic climatic conditions and evaluated impacts of a higher dry year frequency (as projected under future climate by two representative concentration pathways, RCP4.5 and RCP8.5). Our findings support the Interval Squeeze Model: the fire interval allowing plant population persistence is squeezed from currently 10–28 years to 13–28 years for RCP4.5. For RCP8.5 population persistence is not possible under any of the tested fire intervals because of low seed production and low survival probability of both seedlings and adult plants. The results show that projected drier conditions alone will cause a higher extinction risk for fire-sensitive perennial plant populations in MTEs, which is further pronounced in combination with shorter fire intervals. This will likely lead to a strong shift in community composition and a loss of biodiversity. Fire management practices may need to be modified to attempt to counteract prospective biodiversity loss and ecosystem structure change.

Patterns for the waiting time in the context of discrete-time stochastic processes.

The aim of this study is to extend the scope and applicability of the level-crossing method to discrete-time stochastic processes and generalize it to enable us to study multiple discrete-time stochastic processes. In previous versions of the level-crossing method, problems with it correspond to the fact that this method had been developed for analyzing a continuous-time process or at most a multiple continuous-time process in an individual manner. However, since all empirical processes are discrete in time, the already-established level-crossing method may not prove adequate for studying empirical processes. Beyond this, due to the fact that most empirical processes are coupled; their individual study could lead to vague results. To achieve the objectives of this study, we first find an analytical expression for the average frequency of crossing a level in a discrete-time process, giving the measure of the time experienced for two consecutive crossings named as the "waiting time." We then introduce the generalized level-crossing method by which the consideration of coupling between the components of a multiple process becomes possible. Finally, we provide an analytic solution when the components of a multiple stochastic process are independent Gaussian white noises. The comparison of the results obtained for coupled and uncoupled processes measures the strength and efficiency of the coupling, justifying our model and analysis. The advantage of the proposed method is its sensitivity to the slightest coupling and shortest correlation length.

Remarks on the Optimal Portfolio Problem in Discrete Variables with Multiple Stochastic Processes

Remarks on the Optimal Portfolio Problem in Discrete Variables with Multiple Stochastic Processes

Gene regulation and noise reduction by coupling of stochastic processes.

Here we characterize the low-noise regime of a stochastic model for a negative self-regulating binary gene. The model has two stochastic variables, the protein number and the state of the gene. Each state of the gene behaves as a protein source governed by a Poisson process. The coupling between the two gene states depends on protein number. This fact has a very important implication: There exist protein production regimes characterized by sub-Poissonian noise because of negative covariance between the two stochastic variables of the model. Hence the protein numbers obey a probability distribution that has a peak that is sharper than those of the two coupled Poisson processes that are combined to produce it. Biochemically, the noise reduction in protein number occurs when the switching of the genetic state is more rapid than protein synthesis or degradation. We consider the chemical reaction rates necessary for Poisson and sub-Poisson processes in prokaryotes and eucaryotes. Our results suggest that the coupling of multiple stochastic processes in a negative covariance regime might be a widespread mechanism for noise reduction.

Communication: Dominance of extreme statistics in a prototype many-body Brownian ratchet.

Many forms of cell motility rely on Brownian ratchet mechanisms that involve multiple stochastic processes. We present a computational and theoretical study of the nonequilibrium statistical dynamics of such a many-body ratchet, in the specific form of a growing polymer gel that pushes a diffusing obstacle. We find that oft-neglected correlations among constituent filaments impact steady-state kinetics and significantly deplete the gel's density within molecular distances of its leading edge. These behaviors are captured quantitatively by a self-consistent theory for extreme fluctuations in filaments' spatial distribution.

Identifying Long-Run Risks: A Bayesian Mixed-Frequency Approach

We develop a nonlinear state-space model that captures the joint dynamics of consumption, dividend growth, and asset returns. Building on Bansal and Yaron (2004), our model consists of an economy containing a common predictable component for consumption and dividend growth and multiple stochastic volatility processes. The estimation is based on annual consumption data from 1929 to 1959, monthly consumption data after 1959, and monthly asset return data throughout. We maximize the span of the sample to recover the predictable component and use high-frequency data, whenever available, to efficiently identify the volatility processes. Our Bayesian estimation provides strong evidence for a small predictable component in consumption growth (even if asset return data are omitted from the estimation). Three independent volatility processes capture different frequency dynamics; our measurement error specification implies that consumption is measured much more precisely at an annual than monthly frequency; and the estimated model is able to capture key asset-pricing facts of the data.

Bridging Paradigms: Hybrid Mechanistic-Discriminative Predictive Models

Many disease processes are extremely complex and characterized by multiple stochastic processes interacting simultaneously. Current analytical approaches have included mechanistic models and machine learning (ML), which are often treated as orthogonal viewpoints. However, to facilitate truly personalized medicine, new perspectives may be required. This paper reviews the use of both mechanistic models and ML in healthcare as well as emerging hybrid methods, which are an exciting and promising approach for biologically based, yet data-driven advanced intelligent systems.

Stochastic and deterministic drivers of spatial and temporal turnover in breeding bird communities

AbstractAimA long‐standing challenge in ecology is to identify the suite of factors that lead to turnover in species composition in both space and time. These factors might be stochastic (e.g. sampling and priority effects) or deterministic (e.g. competition and environmental filtering). While numerous studies have examined the relationship between turnover and individual drivers of interest (e.g. primary productivity, habitat heterogeneity, or regional – ‘gamma’ – diversity), few studies have disentangled the simultaneous influences of multiple stochastic and deterministic processes on both temporal and spatial turnover. If turnover is governed primarily by stochastic sampling processes, removing the sampling effects of gamma diversity should result in non‐significant relationships between turnover and environmental variables. Conversely, if deterministic processes govern turnover patterns, removing sampling effects will have little influence on turnover gradients. Here, we test these predictions.LocationThe United States.MethodsContinental‐scale, multidecadal data were used to quantify spatial and temporal turnover in avian community composition within 295 survey routes. A series of regression and structural equation models were coupled with a null model to construct statistical models describing turnover patterns.ResultsExamining explanatory variables alone or in combination showed that spatial and temporal turnover increased together, decreased with primary productivity and increased with habitat heterogeneity. The relationships between turnover and all variables became weaker when sampling effects were removed, but relationships with primary productivity and habitat heterogeneity remained relatively strong. In addition, spatial turnover increased strongly with spatial gamma diversity after sampling effects were removed.Main conclusionsOur results show that spatial and temporal turnover are related to each other through a stochastic sampling process, but that each type of turnover is further influenced by deterministic processes. The relative influence of deterministic processes appears, however, to decrease with primary productivity and increase with habitat heterogeneity across an east–west longitudinal gradient in North America.