Discrete Stochastic Processes Research Articles

Moment inequalities for mixing long-span high-frequency data and strongly consistent estimation of OU integrated diffusion process

Mixing is not much used in the high-frequency literature so far. However, mixing is a common weakly dependent property of continuous and discrete stochastic processes, such as Gaussian, Ornstein–Uhlenberck (OU), Vasicek, CIR, CKLS, logistic diffusion, generalized logistic diffusion, and double-well diffusion processes. So, long-span high-frequency data typically have weak dependence, and using mixing to study them is also an alternative approach. In this paper, we give some moment inequalities for long-span high-frequency data with ϕ-mixing, ρ-mixing, and α-mixing. These inequalities are effective tools for studying asymptotic properties. Applying these inequalities, we investigate the strong consistency of parameter estimation for the OU-integrated diffusion process. We also derive the mean square error of the estimation of the OU process and the optimal interval for the drift parameter estimator.

On asymptotic convergence rate of random search

This paper presents general theoretical studies on asymptotic convergence rate (ACR) for finite dimensional optimization. Given the continuous problem function and discrete time stochastic optimization process, the ACR is the optimal constant for control of the asymptotic behaviour of the expected approximation errors. Under general assumptions, condition ACR<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$<1$$\\end{document} implies the linear behaviour of the expected time of hitting the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}- optimal sublevel set with ε→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0^+ $$\\end{document} and determines the upper bound for the convergence rate of the trajectories of the process. This paper provides general characterization of ACR and, in particular, shows that some algorithms cannot converge linearly fast for any nontrivial continuous optimization problem. The relation between asymptotic convergence rate in the objective space and asymptotic convergence rate in the search space is provided. Examples and numerical simulations with use of a (1+1) self-adaptive evolution strategy and other algorithms are presented.

A Novel Continuous-time Hidden Markov Model Based on a Bag of Features Extracted from MR Brain Images for Alzheimer's Stage Progression and Detection.

Mounting novel solutions for conspicuous neurodegenerative disorders that grow consistently, such as Alzheimer's disease, rely on tracking and identifying disease development, improvement, and progression. Compared to many clinical or survey-based detection methods, early Alzheimer's stage detection can be possible through computer-based MR brain images and discrete stochastic processes. In the case of Alzheimer's stage progression, the existing models illustrate that the learning problem comprises two issues: estimating posterior probabilities of the Alzheimer's stage and computing conditioned statistics of the Alzheimer's end-stage. The proposed model overcomes these issues by restructuring the estimation problem as EM-centered CT- HMM. This paper proposes a novel framework model with two phases; the first phase covers the feature extraction of magnetic resonance imaging based on many computer vision methods known as a collection of bag-of-features (BoF). In the second phase, the EM-centered learning method is used for the continuous-time hidden Markov model (CT-HMM), an efficient approach to modeling Alzheimer's disease progression with time and stages. The proposed CT-HMM is implemented with eight Alzheimer's stages (source: ADNI) to visualize and predict the stage progression of the ADNI MRI dataset. The proposed model reported the transition posterior probability as 0.765 (high to low stage progression) and 0.234 (low to high stage progression). The model's accuracy and F1 score are estimated as 97.13 and 96.51, respectively. The proposed model's accuracy and evaluation metrics reported higher results in the work on Alzheimer's stage progression and prediction.

Topology and Local Geometry of the Eden Model

The Eden cell growth model is a simple discrete stochastic process which produces a “blob” (aggregation of cells) in mathbb {R}^d: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers at time t grow at a rate between t^{(d-1)/d} and P_d(t), where P_d(t) is the size of the site perimeter. Assuming a widely believed conjecture, this establishes the rate of growth of the Betti numbers in every dimension. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes.

Locally Typical Sampling

AbstractToday’s probabilistic language generators fall short when it comes to producing coherent and fluent text despite the fact that the underlying models perform well under standard metrics (e.g., perplexity). This discrepancy has puzzled the language generation community for the last few years. In this work, we posit that the abstraction of natural language generation as a discrete stochastic process—which allows for an information-theoretic analysis—can provide new insights into the behavior of probabilistic language generators, for example, why high-probability texts can be dull or repetitive. Humans use language as a means of communicating information, aiming to do so in a simultaneously efficient and error-minimizing manner; in fact, psycholinguistics research suggests humans choose each word in a string with this subconscious goal in mind. We formally define the set of strings that meet this criterion: Those for which each word has an information content close to the expected information content, namely, the conditional entropy of our model. We then propose a simple and efficient procedure for enforcing this criterion when generating from probabilistic models, which we call locally typical sampling. Automatic and human evaluations show that, in comparison to nucleus and top-k sampling, locally typical sampling offers competitive performance (in both abstractive summarization and story generation) in terms of quality while consistently reducing degenerate repetitions.

Topology, convergence, and reconstruction of predictive states

Predictive equivalence in discrete stochastic processes has been applied with great success to identify randomness and structure in statistical physics and chaotic dynamical systems and to inferring hidden Markov models. We examine the conditions under which predictive states can be reliably reconstructed from time-series data, showing that convergence of predictive states can be achieved from empirical samples in the weak topology of measures. Moreover, predictive states may be represented in Hilbert spaces that replicate the weak topology. We mathematically explain how these representations are particularly beneficial when reconstructing high-memory processes and connect them to reproducing kernel Hilbert spaces.

Analysis of extreme sea states under the impact of typhoon in different periods: A nested stochastic compound distribution applied in the South China Sea

• A nested stochastic compound distribution (NSCD) is introduced to analyze the extreme sea states in typhoon-prone areas. • The applicability of the NSCD model in the South China Sea is discussed. • The NSCD model is found to be a better model to consider the impact of typhoon in different time periods. • It is shown that the NSCD model has a broader scope of application than existing models. Extreme sea conditions are often caused by typhoons that occur obviously in season. How to calculate the design values of marine environmental elements based on statistical characteristic of typhoon in different periods is important. A nested stochastic compound distribution (NSCD) model is proposed based on the stochastic process theory. During a typhoon occurrence, the distributions of the marine environmental elements such as wind velocity (WV) and storm surge (SS) at any moment are described in continuous stochastic processes, and the distribution of typhoon occurrence frequency (TOF) during any period is described by a discrete stochastic process. Based on the measured data in the South China Sea (SCS), the applicability of different marginal distribution functions and Copula functions in the SCS are further discussed, and the specific form of the NSCD model suitable for the SCS is obtained. The Design values of WV and SS for different joint return periods and coincidence return periods are calculated according to the NSCD and the compound extreme value distribution (CEVD). The effects of treating TOF as a stochastic process and a random variable on the design values of the two marine environmental elements are discussed. Results show that the NSCD model contains the traditional CEVD model, which is a form of the NSCD model in a specific time situation. The design values of WV and SS obtained by the NSCD model can reflect the impact of TOF on marine environmental elements. Moreover, the probability of simultaneous occurrence of extreme WV and extreme SS is higher over longer return periods under the influence of typhoons. The NSCD model can therefore be used to make joint probability predictions for a wide range of extreme sea conditions in different time periods according to engineering needs.

FBSDE based neural network algorithms for high-dimensional quasilinear parabolic PDEs

In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial differential equations (PDEs), which is related to the FBSDEs from the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the path-wise difference between two discrete stochastic processes, which are defined by the time discretization of the FBSDEs and the DNN representation of the PDE solution, respectively. The proposed algorithms are shown to generate DNN solution for a 100-dimensional Black–Scholes–Barenblatt equation, which is accurate in a finite region in the solution space, and has a convergence rate close to that of the Euler–Maruyama scheme used for discretizing the FBSDEs.

The estimation of the mean value of a matrix-valued discrete stationary stochastic process

The estimation of the mean value of a matrix-valued discrete stationary stochastic process

Hölder regularity for stochastic processes with bounded and measurable increments

We obtain an asymptotic Holder estimate for expectations of a quite general class of discrete stochastic processes. Such expectations can also be described as solutions to a dynamic programming principle or as solutions to discretized PDEs. The result, which is also generalized to functions satisfying Pucci-type inequalities for discrete extremal operators, is a counterpart to the Krylov-Safonov regularity result in PDEs. However, the discrete step size $\varepsilon$ has some crucial effects compared to the PDE setting. The proof combines analytic and probabilistic arguments.

Extreme rainless periods in Pannonian Basin

AbstractThis paper analyses extreme meteorological droughts in the territory of the Pannonian Basin in central and southeastern Europe. Droughts are treated from an agricultural perspective as periods of consecutive days with very little or no rainfall at all. By using data from the historical period 1981–2010, time intervals longer than 20 days with less than 3 mm per day during the growing season April–September are examined to determine theoretical and empirical distributions of their number and durations. Related discrete stochastic process of extremes is analysed by the Zelenhasic–Todorovic (ZT) method, successfully applied in hydrology and meteorology (e.g., peak flows, low flows, meteorological droughts, etc.). The method is applied to 52 locations across Pannonia Basin. At each location, the Poisson distribution is determined for the number of droughts, exponential distribution for the duration of droughts, and double exponential distribution for the duration of the (annually) longest droughts. To validate the quality of modelling and the results obtained, standard statistical goodness of fit tests are used. Historical data from the period 1981–2010 indicate that the central and northern part of Hungary and the entire territory of Vojvodina Province in the north of Serbia experienced the longest droughts, on average 55–60 days. In statistical terms, across Pannonia Basin once in 100 years drought may occur with an average duration of 56 days. Theoretically longest 100‐year droughts could be expected in the northern part of Hungary (up to 70 days), the eastern part of Romania (up to 68 days), and Serbia (up to 69 days). The 2‐year droughts are on the average duration of 3 weeks. Generated maps of the longest droughts in the basin may help to reduce agricultural risks due to global warming and other negative impacts of climate change.

Age Analysis of Status Updating System with Probabilistic Packet Preemption.

The age of information (AoI) metric was proposed to measure the freshness of messages obtained at the terminal node of a status updating system. In this paper, the AoI of a discrete time status updating system with probabilistic packet preemption is investigated by analyzing the steady state of a three-dimensional discrete stochastic process. We assume that the queue used in the system is , which represents that the system size is 2 and the packet in the buffer can be preempted by a fresher packet with probability . Instead of considering the system’s AoI separately, we use a three-dimensional state vector to simultaneously track the real-time changes of the AoI, the age of a packet in the server, and the age of a packet waiting in the buffer. We give the explicit expression of the system’s average AoI and show that the average AoI of the system without packet preemption is obtained by letting . When is set to 1, the mean of the AoI of the system with a queue is obtained as well. Combining the results we have obtained and comparing them with corresponding average continuous AoIs, we propose a possible relationship between the average discrete AoI with the queue and the average continuous AoI with the queue. For each of two extreme cases where and , we also determine the stationary distribution of AoI using the probability generation function (PGF) method. The relations between the average AoI and the packet preemption probability , as well as the AoI’s distribution curves in two extreme cases, are illustrated by numerical simulations. Notice that the probabilistic packet preemption may occur, for example, in an energy harvest (EH) node of a wireless sensor network, where the packet in the buffer can be replaced only when the node collects enough energy. In particular, to exhibit the usefulness of our idea and methods and highlight the merits of considering discrete time systems, in this paper, we provide detailed discussions showing how the results about continuous AoI are derived by analyzing the corresponding discrete time system and how the discrete age analysis is generalized to the system with multiple sources. In terms of packet service process, we also propose an idea to analyze the AoI of a system when the service time distribution is arbitrary.

An Optimal Mass Transport Method for Random Genetic Drift

We propose and analyze an optimal mass transport method for a random genetic drift problem driven by a Moran process under weak-selection. The continuum limit, formulated as a reaction-advection-diffusion equation known as the Kimura equation, inherits degenerate diffusion from the discrete stochastic process that conveys to the blow-up into Dirac-delta singularities hence brings great challenges to both the analytical and numerical studies. The proposed numerical method can quantitatively capture to the fullest possible extent the development of Dirac-delta singularities for genetic segregation on one hand, and preserves several sets of biologically relevant and computationally favored properties of the random genetic drift on the other. Moreover, the numerical scheme exponentially converges to the unique numerical stationary state in time at a rate independent of the mesh size up to a mesh error. Numerical evidence is given to illustrate and support these properties, and to demonstrate the spatio-temporal dynamics of random generic drift.

Drift criteria for persistence of discrete stochastic processes on the line

We provide sufficient conditions for the persistence or transience of stochastic processes on the real line based on the behavior of the first and second moment of their conditional increments at the boundaries. Our findings extend previous results in the literature (Lamperti, 1960) to the large class of discrete-time processes with bounded increments. We present some examples of application by studying survival and dominance of agents trading in complete financial markets.

Bayesian nonparametric dynamic hazard rates in evolutionary life tables.

In the study of life tables the random variable of interest is usually assumed discrete since mortality rates are studied for integer ages. In dynamic life tables a time domain is included to account for the evolution effect of the hazard rates in time. In this article we follow a survival analysis approach and use a nonparametric description of the hazard rates. We construct a discrete time stochastic processes that reflects dependence across age as well as in time. This process is used as a bayesian nonparametric prior distribution for the hazard rates for the study of evolutionary life tables. Prior properties of the process are studied and posterior distributions are derived. We present a simulation study, with the inclusion of right censored observations, as well as a real data analysis to show the performance of our model.

Stochastic Theory of Discrete Binary Fragmentation-Kinetics and Thermodynamics.

We formulate binary fragmentation as a discrete stochastic process in which an integer mass k splits into two integer fragments j, , with rate proportional to the fragmentation kernel . We construct the ensemble of all distributions that can form in fixed number of steps from initial mass M and obtain their probabilities in terms of the fragmentation kernel. We obtain its partition function, the mean distribution and its evolution in time, and determine its stability using standard thermodynamic tools. We show that shattering is a phase transition that takes place when the stability conditions of the partition function are violated. We further discuss the close analogy between shattering and gelation, and between fragmentation and aggregation in general.

Modeling Stock Price of Transnational companies using Brownian motion

This article is concerned with creating a model for stock price using the Brownian motion. At first, we consider the notion of a discrete time stochastic process, simple random walk, and then move on to its continuous analogue, the Brownian motion. Next we identify the problem with using regular Calculus for stochastic differential equations and derive Ito’s Lemma. After that we derive a model for stock prices and use lognormal distribution to determine its expected value and variance. Finally, we use sample volatility to make predictions for Apple and Gazprom stock prices.

Quantum-enhanced analysis of discrete stochastic processes

Discrete stochastic processes (DSP) are instrumental for modeling the dynamics of probabilistic systems and have a wide spectrum of applications in science and engineering. DSPs are usually analyzed via Monte-Carlo methods since the number of realizations increases exponentially with the number of time steps, and importance sampling is often required to reduce the variance. We propose a quantum algorithm for calculating the characteristic function of a DSP, which completely defines its probability distribution, using the number of quantum circuit elements that grows only linearly with the number of time steps. The quantum algorithm reduces the Monte-Carlo sampling to a Bernoulli trial while taking all stochastic trajectories into account. This approach guarantees the optimal variance without the need for importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. The Fourier approximation can be used to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented. Proof-of-principle experiments are performed using the IBM quantum cloud platform.

The tail process and tail measure of continuous time regularly varying stochastic processes

The goal of this paper is to investigate the tools of extreme value theory originally introduced for discrete time stationary stochastic processes (time series), namely the tail process and the tail measure, in the framework of continuous time stochastic processes with paths in the space \(\mathcal {D}\) of càdlàg functions indexed by \(\mathbb {R}\), endowed with Skorohod’s J1 topology. We prove that the essential properties of these objects are preserved, with some minor (though interesting) differences arising. We first obtain structural results which provide representation for homogeneous shift-invariant measures on \(\mathcal {D}\) and then study regular variation of random elements in \(\mathcal {D}\). We give practical conditions and study several examples, recovering and extending known results.

A Sequential Response Model for Analyzing Process Data on Technology-Based Problem-Solving Tasks

Students’ response sequences to a technology-based problem-solving task can be treated as a discrete time stochastic process with a conditional Markov property—after conditioning on the students’ abilities of problem solving, the next state only depends on the current state. This article proposes a sequential response model (SRM) with a Bayesian approach for parameter estimation that incorporates comprehensive information from the response process to infer problem-solving ability more effectively. A Monte Carlo simulation study showed that parameters were well-recovered. An illustrated example is provided to showcase additional gains using our model for understanding the response process with a real-world interactive assessment item “Tickets” in the programme for international student assessment (PISA) 2012.