Continuous-time Process Research Articles

Thinking beyond the closure assumption: Designing surveys for estimating biological truth with occupancy models

Abstract Occupancy models estimate distributions of imperfectly detected species, but violations of the closure assumption can bias results. However, researchers working with mobile animals may find it impossible to eliminate such violations. Here, we tested the hypothesis that occupancy models fit to realistic sampling data can generate unbiased occupancy estimates for an itinerant Wood Thrush (Hylocichla mustelina) population. In 2013 and 2014, we tracked movements of 41 breeding Wood Thrush males. We modelled territory shift probabilities using logistic exposure models and within‐territory movements using continuous‐time stochastic process models. We then constructed an individual‐based model, simulated (1000 iterations) spatiotemporal locations for individuals and simulated sampling these populations using 162 different point count protocols with variable spatial (sampling radius and point placement method), and temporal (survey length, between‐survey intervals and number of surveys) characteristics. We compared occupancy estimates with true values of instantaneous, daily and seasonal occupancy from the simulations. We parameterized continuous time stochastic process models based on movements within 34 unique territories and estimated a daily territory shift probability of 0.0099 (95% CI: 0.0060, 0.0152). Simulated data indicated that estimates of occupancy ranged from 0.18 (0.06, 1.00) to 0.80 (0.71, 0.89) depending on protocol characteristics. Occupancy estimates increased with increasing survey radius, survey length and between‐survey interval. Protocols using shorter surveys and between‐survey intervals were good estimators for instantaneous occupancy (low bias and mean‐squared error) but poor estimators for daily and seasonal occupancy; longer surveys and intervals generated unbiased estimators of daily occupancy but underestimated seasonal occupancy. Logistic regression models that ignored imperfect detection outperformed occupancy models for estimating instantaneous occupancy but not daily or seasonal occupancy. For mobile animals, occupancy of sampling sites changes in space and time. Consequently, the spatial and temporal aspects of a sampling protocol have strong, but predictable, effects on occupancy model parameter estimates. Our results demonstrate that how these factors interact is critical for designing surveys that produce occupancy estimates representative of the biological process of interest to a researcher.

Defense of abortion: A response to the tacit consent objection

As abortion has been a debated moral issue, advocators and objectors held different opinions and stands. One of the most prominent defense of abortion is Judith Jarvis Thomson’s violinist analogy, which has been challenged by the tacit consent objection. The latter states that a woman, by voluntarily engaging in sexual intercourse, tacitly gives the fetus the right to use her body. In responding to this objection, this paper highlights the ambiguous nature of tacit consent and underlines that such a consent should not be seen as unconditional or irrevocable. That is, as a process in continuous time, a consent given earlier may be revoked later. Overall, this paper defends and emphasizes a woman’s right to bodily autonomy and self-determination by allowing her to withdraw her consent even in the case of a consensual pregnancy.

Methods for estimating integrated variance: Jump robustness issues in high frequency time series

Integrated variance is a volatility measure of a process in continuous time, which is applicable in financial mathematics as a means to optimize a portfolio, project dynamics of a price of a financial asset. The consistency of an estimator for integrated variance of a random process is at the core of this article. A fundamental diffusion process is extended by adding a jump component as a means of improving the descriptive function of the process. It is activity of jumps that is a factor subject to which is the consistency of the estimator for integrated variance. Due to this fact, consistency is defined as an extent to which the estimator at hand is jump robust. Two main methods for estimating integrated variance are considered and the capacity of corresponding estimators to withstand the effect of jumps while converging is briefly analyzed. The arguments indicating a necessity for further research of the effect of jumps with reference to works of the authors who have established a ground for analysis of integrated variance and those works containing main asymptotic results for consistency of integrated variance estimators are elaborated. Based on this, avenue for further research and development of asymptotic theory for consistency of an estimator for integrated variance is identified.

Mixed orthogonality graphs for continuous-time stationary processes

In this paper, we introduce different concepts of Granger causality and contemporaneous correlation for multivariate stationary continuous-time processes to model different dependencies between the component processes. Several equivalent characterisations are given for the different definitions, in particular by orthogonal projections. We then define two mixed graphs based on different definitions of Granger causality and contemporaneous correlation, the (mixed) orthogonality graph and the local (mixed) orthogonality graph. In these graphs, the components of the process are represented by vertices, directed edges between the vertices visualise Granger causal influences and undirected edges visualise contemporaneous correlation between the component processes. Further, we introduce various notions of Markov properties in analogy to Eichler (2012), which relate paths in the graphs to different dependence structures of subprocesses, and we derive sufficient criteria for the (local) orthogonality graph to satisfy them. Finally, as an example, for the popular multivariate continuous-time AR (MCAR) processes, we explicitly characterise the edges in the (local) orthogonality graph by the model parameters.

Total-current population-dependent branching processes: analysis via stochastic approximation

We consider continuous-time two-type population size-dependent Markov Branching Processes. The offspring distribution can depend on the current (alive) and total (dead and alive) populations. We assume finite second-moment conditions and use the stochastic approximation technique for the analysis. In particular, we identify an appropriate ordinary differential equation (ODE) and show that either of the two events occurs with a certain probability: (a) the time-asymptotic proportion of the populations converges to the attractors or saddle points of the ODE, (b) it enters every neighbourhood and exits some neighbourhood of a saddle point infinitely often. We also prove a finite time approximation result for the stochastic trajectory. Further, we analyse a branching process with attack and acquisition, which captures the competition in online viral markets; for this case, the probability of approximation equals one.

Entanglement buffering with two quantum memories

Quantum networks crucially rely on the availability of high-quality entangled pairs of qubits, known as entangled links, distributed across distant nodes. Maintaining the quality of these links is a challenging task due to the presence of time-dependent noise, also known as decoherence. Entanglement purification protocols offer a solution by converting multiple low-quality entangled states into a smaller number of higher-quality ones. In this work, we introduce a framework to analyse the performance of entanglement buffering setups that combine entanglement consumption, decoherence, and entanglement purification. We propose two key metrics: the availability, which is the steady-state probability that an entangled link is present, and the average consumed fidelity, which quantifies the steady-state quality of consumed links. We then investigate a two-node system, where each node possesses two quantum memories: one for long-term entanglement storage, and another for entanglement generation. We model this setup as a continuous-time stochastic process and derive analytical expressions for the performance metrics. Our findings unveil a trade-off between the availability and the average consumed fidelity. We also bound these performance metrics for a buffering system that employs the well-known bilocal Clifford purification protocols. Importantly, our analysis demonstrates that, in the presence of noise, consistently purifying the buffered entanglement increases the average consumed fidelity, even when some buffered entanglement is discarded due to purification failures.

Competitive networked bi-virus spread: Existence of coexistence equilibria

The paper studies multi-competitive continuous-time epidemic processes. We consider the setting where two viruses are simultaneously prevalent, and the spread occurs due to individual-to-individual interaction. In such a setting, an individual is either not affected by any of the viruses, or infected by one and exactly one of the two viruses. One of the equilibrium points is the coexistence equilibrium, i.e., multiple viruses simultaneously infect separate fractions of the population. We provide a sufficient condition for the existence of a coexistence equilibrium. We identify a condition such that for certain pairs of spread matrices either every coexistence equilibrium lies on a line that is locally exponentially attractive, or there is no coexistence equilibrium. We then provide a condition that, for certain pairs of spread matrices, rules out the possibility of the existence of a coexistence equilibrium, and, as a consequence, establishes global asymptotic convergence to the endemic equilibrium of the dominant virus. Finally, we provide a mitigation strategy that employs one virus to ensure that the other virus is eradicated. The theoretical results are illustrated using simulations.

Analytic Formula of Schreiber’s Transfer Entropy Rate of Continuous-time Processes

Analytic Formula of Schreiber’s Transfer Entropy Rate of Continuous-time Processes

Random telegraph processes with nonlocal memory.

We study two-state (dichotomous, telegraph) random ergodic continuous-time processes with dynamics depending on their past. We take into account the history of the process in an explicit form by introducing integral nonlocal memory term into conditional probability function. We start from an expression for the conditional transition probability function describing additive multistep binary random chain and show that the telegraph processes can be considered as continuous-time interpolations of discrete-time dichotomous random sequences. An equationinvolving the memory function and the two-point correlation function of the telegraph process is analytically obtained. This integral equationdefines the correlation properties of the processes with given memory functions. It also serves as a tool for solving the inverse problem, namely for generation of a telegraph process with a prescribed pair correlation function. We obtain analytically the correlation functions of the telegraph processes with two exactly solvable examples of memory functions and support these results by numerical simulations of the corresponding telegraph processes.

Stochastic modeling of stationary scalar Gaussian processes in continuous time from autocorrelation data

We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the AAA algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function’s real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data.

The Brownian motion, Wiener process, and Pu migration as examples of random walk

The theory of random walks holds significance across various domains. To this end, this paper discusses the application of random walk theory in the modeling and analysis of complex systems, including the Brownian motion, Wiener process, and Pu migration. The first part mainly explains the concepts of Brownian motion and random walks, as well as their relationship. The core characteristic is the independence and normal distribution of increments, which makes it a special model of random walks. The next part is the Wiener process, which is a continuous time stochastic process. It can describe random phenomena and solve complex problems in finance, encompassing areas such as stock price prediction, option pricing, and risk management. The last part is about Pu migration, with the clear introduction of Pu migration in biology. The authors discuss why the random walk theory can be regarded as the better approach to analysis this problem. Thus, this paper highlights the importance of the random walk in solving many scientific issues.

Stable predictive control of continuous stirred-tank reactors using deep learning

In this work, a Lyapunov-based predictive control method utilizing deep learning techniques is proposed for driving continuous-time nonlinear processes towards the desired equilibrium point. Initially, a neural ordinary differential equation model is developed to learn the continuous-time dynamics from discrete-time sampling data. Subsequently, the theoretical conditions that the control Lyapunov function (CLF) developed on the learned continuous-time system is feasible to act on the actual system through a zero-order-holder are analyzed. Building upon the theoretical analysis and the constructed continuous-time dynamics, a deep neural network control Lyapunov function (DNN-CLF) controller is developed. Importantly, the DNN-CLF is independent of the affine system formulation, effectively addressing the challenge of designing the CLF. Benefiting from the stability guarantee provided by the established DNN-CLF, it is integrated into the model predictive control (MPC) framework to develop Lyapunov-based MPC to further improve the control performance. To ensure safety, considering error propagation and associated risks with enlarging the feasible region of the CLF, tubes are constructed and integrated into the MPC scheme. Finally, a numerical simulation of a chemical process is conducted to validate the efficiency of the proposed method.

Prediction problem for continuous time stochastic processes with periodically correlated increments observed with noise

We propose solution of the problem of the mean square optimal estimation of linear functionals which depend on the unobserved values of a continuous time stochastic process with periodically correlated increments based on observations of this process with periodically stationary noise. To solve the problem, we transform the processes to the sequences of stochastic functions which form an infinite dimensional vector stationary sequences. In the case of known spectral densities of these sequences, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

P-adic Brownian motion is a scaling limit

A p-adic Brownian motion is a continuous time stochastic process in a p-adic state space that has a Vladimirov operator as its infinitesimal generator. The current work shows that any such process is the scaling limit of a discrete time random walk on a discrete group. Earlier work required the exponent of the Vladimirov operator to be in (1,∞) , and the convergence was the weak convergence of probability measures on the Skorohod space of paths on a compact time interval. The current approach simplifies the earlier approach, allows for any positive exponent, eliminates the restriction to compact time intervals, and establishes some moment estimates for the discrete time processes that are of independent interest.

LFT: Neural Ordinary Differential Equations With Learnable Final-Time.

Since the last decade, deep neural networks have shown remarkable capability in learning representations. The recently proposed neural ordinary differential equations (NODEs) can be viewed as the continuous-time equivalence of residual neural networks. It has been shown that NODEs have a tremendous advantage over the conventional counterparts in terms of spatial complexity for modeling continuous-time processes. However, existing NODEs methods entail their final time to be specified in advance, precluding the models from choosing a desirable final time and limiting their expressive capabilities. In this article, we propose learnable final-time (LFT) NODEs to overcome this limitation. LFT rebuilds the NODEs learning process as a final-time-free optimal control problem and employs the calculus of variations to derive the learning algorithm of NODEs. In contrast to existing NODEs methods, the new approach empowers the NODEs models to choose their suitable final time, thus being more flexible in adjusting the model depth for given tasks. Additionally, we analyze the gradient estimation errors caused by numerical ordinary differential equations (ODEs) solvers and employ checkpoint-based methods to obtain accurate gradients. We demonstrate the effectiveness of the proposed method with experimental results on continuous normalizing flows (CNFs) and feedforward models.

Viscosity solutions approach to finite-horizon continuous-time Markov decision process

This paper investigates the optimal control problems for the finite-horizon continuous-time Markov decision processes with delay-dependent control policies. We develop compactification methods in decision processes and show that the existence of optimal policies. Subsequently, through the dynamic programming principle of the delay-dependent control policies, the differential-difference Hamilton-Jacobi-Bellman (HJB) equation in the setting of discrete space is established. Under certain conditions, we give the comparison principle and further prove that the value function is the unique viscosity solution to this HJB equation. Based on this, we show that among the class of delay-dependent control policies, there is an optimal one which is Markovian.

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

NIRK-based mixed-type accurate continuous–discrete Gaussian filters with deterministically sampled expectation and covariance for state estimation in continuous-time stochastic process models with discrete measurements

This paper advances further the idea of overall continuous–discrete Gaussian filtering with deterministically sampled expectation and covariance towards efficient mixed-type accurate Kalman-like schemes based on self-regulated Ordinary Differential Equation (ODE) solvers with automatic local and global error controls intended for treating continuous-time nonlinear stochastic process models with discrete measurements. The universal state estimation algorithms of such sort designed recently by Kulikov and Kulikova (2022) possess three potential limitations, which might affect their successful applications in practice. First, these deal with complicated Moment Differential Equations (MDEs) and, hence, can be rather time-consuming in estimating stochastic process models of large size, that compromises their efficiency for online computations. Second, the aforementioned filters do not employ any kind of global error regulation and, hence, can be insufficiently accurate in solving difficult MDEs arisen. Third, these are hardly effective in treating stiff continuous–discrete stochastic systems. Below, all the listed issues are addressed and solved via utilization of the much simpler but more robust MDEs arisen in the usual continuous–discrete Extended Kalman Filter (EKF) within the mixed-type accurate filtering approach initiated by Kulikova and Kulikov (2015), which is rooted in the variable-stepsize Nested Implicit Runge–Kutta (NIRK) methods. For the sake of stability of our novel methods to the numerical integration and round-off errors, we devise square-root versions of the mixed-type Gaussian filters, which are based either on hyperbolic QR factorizations or on hyperbolic Singular Value Decompositions (SVDs). Performances of these new mixed-type square-root Gaussian filters are assessed and compared to their original (non-mixed-type) counterparts as well as to their non-square-root versions in a simulated non-stiff ill-conditioned radar tracking scenario and on a stiff stochastic Oregonator reaction.

Integration of rapid bioburden testing into production quality management systems and process control.

The move to integrated continuous bioprocessing (ICB), while providing a means for process intensification, can put added strain on process analytics when conventional methods are used. For instance, traditional microbial methods provide minimal value to ICB processes given that the time required for data to become available is much longer than a typical full cycle of the manufacturing process. Although rapid microbial detection has been in discussion for over 30 years, it is still not routinely deployed in commercial biopharmaceutical manufacturing. One contributing factor is the ability to integrate this technology into a process control strategy and existing quality systems. An understanding of the capability of microbial detection technology available today can be leveraged to implement a control strategy for bioburden monitoring in real time for process intermediates. One key tenet of this proposed control strategy is the use of a "two-tiered approach" wherein a fast (but possibly less sensitive) test is used to monitor the process and trigger further action for a second, longer duration test which is used to confirm and quantify the presence of bioburden and identify the organism. This approach, presented here alongside several case studies for microbial monitoring, can have broader application for other process analytical technologies where fit for purpose methods could be employed to establish process control alongside real time continuous processes.

White-tailed deer detection rates increase when coyotes are present.

Predator species can indirectly affect prey species through the cost of anti-predator behavior responses, which may involve shifts in occupancy, space use, or movement. Quantifying the various strategies implemented by prey species to avoid adverse interactions with predators can lead to a better understanding of potential population-level repercussions. Therefore, the purpose of this study was to examine predator-prey interactions by quantifying the effect of predator species presence on detection rates of prey species, using coyotes (Canis latrans) and white-tailed deer (Odocoileus virginianus) in Central Appalachian forests of the eastern United States as a model predator-prey system. To test two competing hypotheses related to interspecific interactions, we modeled species detections from 319 camera traps with a two-species occupancy model that incorporated a continuous-time detection process. We found that white-tailed deer occupancy was independent of coyote occupancy, but white-tailed deer were more frequently detectable and had greater detection intensity at sites where coyotes were present, regardless of vegetation-related covariates. In addition, white-tailed deer detection rates at sites with coyotes were highest when presumed forage availability was relatively low. These findings suggest that white-tailed deer may be exhibiting an active avoidance behavioral response to predators by increasing movement rates when coyotes are present in an area, perhaps due to reactive evasive maneuvers and/or proactive attempts to reduce adverse encounters with them. Concurrently, coyotes could be occupying sites with higher white-tailed deer densities. Because white-tailed deer did not exhibit significant shifts in daily activity patterns based on coyote occupancy, we further suggest that white-tailed deer in our study system generally do not use temporal partitioning as their primary strategy for avoiding encounters with coyotes. Overall, our study implements a recently developed analytical approach for modeling multi-species occupancy from camera traps and provides novel ecological insight into the complex relationships between predator and prey species.