Continuous-time Dynamic Model Research Articles

Robust dynamic contracts with multiple agents

We develop a continuous-time dynamic multi-agent contracting model in which the principal is unsure about the distributions of the project's terminal payoffs and worries about model misspecification. With model uncertainty, workers' wages depend on the outputs of other unrelated projects and the optimal contracts exhibit overdetermination. We demonstrate an inverse U-shaped relationship between the extent of overdetermination and group size. Moreover, model uncertainty induces wage compression, especially in small firms as the empirical evidence demonstrates. Finally, expanding the group size increases the average project value by mitigating the negative impacts of ambiguity.

Impact of government’s support policy on decision-making of platform participants under ESG

Blockchain-based token platform economy is a new branch of digital platform economics. Constructing a continuous time dynamic model of token platform economy, this paper analyzes what kind of ESG policy is appropriate for the government, meanwhile the token platform participants (developers, users and speculators) make optimal investments and decisions under ESG policy. Simulation result shows neutral ESG policy is optimal. Based on the given neutral ESG policy, we have done the research on ESG investment and decision strategies for platform participants. Our research shows that the tokens selling rate and efforts of green platform (ESG score greater than 0) developers are lower than the ones of brown platform (ESG score less than 0). Consequently, when developers’ token retention is about half of the initial amount, users should invest more brown tokens. Speculators should invest brown tokens for developers’ high token retention. Green token investments of speculators and users are needed in other cases. Next, the impact of the government’s three ESG policies on the maturity or termination of the platform also been analyzed. An important conclusion occurred: the government’s aggressive or conservative ESG policy cannot make the development of the green platform better; Therefore, we suggest a neutral ESG policy which means that the government could adopt high tax incentive and high tax burden on the green and brown platform while it is not necessary to implement the extra subsidy and punishment policy on the green and brown platform

Outcome prediction in psychological therapy with continuous time dynamic modeling of affective states and emotion regulation.

To date, many prediction studies in psychotherapy research have used cross-sectional data to predict treatment outcome. The present study used intensive longitudinal assessments and continuous time dynamic modeling (CTDM) to investigate the temporal dynamics of affective states and emotion regulation in the early phase of therapy and their ability to predict treatment outcome. Ninety-one patients undergoing psychological treatment at a university outpatient clinic took part in a 2-week ecological momentary assessment (EMA) period. Participants answered self-report measures on positive affect (PA), negative affect, and emotion regulation (ER) four times a day. Hierarchical Bayesian CTDM was conducted to identify temporal effects within (autoregressive) and between (cross-regressive) PA, negative affect, and ER. The resulting CTDM parameters, simple EMA parameters (e.g., mean), and cross-sectional predictors were entered into a LASSO model to be examined as predictors of treatment outcome at Session 15. Two significant predictors were identified: initial impairment and the continuous time cross-effect of PA on ER. The final model explained 40% of variance in treatment outcome, with the cross-effect (PA-ER) accounting for 4% of variance beyond initial impairment. The results demonstrate that temporal patterns of affective EMA data are valuable for the mapping of individual differences and the prediction of treatment outcome. This information can be used to provide therapists with feedback to personalize treatments. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

MTdyg: Multi-scale transformers with continuous time dynamic graph model for link prediction

Link prediction through continuous dynamic graph neural networks is a challenging endeavour. Previous studies have considered historic interaction sequences among pairs of nodes. However, this approach does not sufficiently model the links between them. Thus, the frequency of historical common neighbours was proposed to obtain more information about node pairs. Nonetheless, this measure fails to consider the timing of interactions because the relative importance of different interactions shifts over time.Multi-scale transformers with a continuous time dynamic graph model (MTdyg) are our response to this challenge to enhance link prediction. The novel MTdyg model employs a multi-scale patching strategy that dynamically adjusts based on the interaction frequency, feeding segmented interaction sequences into the transformer. This methodology significantly improves the model’s ability to assimilate historical data. Furthermore, we developed a temporal attention-based historical common neighbour encoding technique to effectively identify linkages between source and target nodes in interaction sequences. Extensive testing on eight public datasets confirmed that MTdyg delivers superior performance on most datasets, demonstrating its effectiveness in capturing the nuances of node interactions over time.

A Day-to-Day Dynamical Approach to the Most Likely User Equilibrium Problem

The lack of a unique user equilibrium (UE) route flow in traffic assignment has posed a significant challenge to many transportation applications. The maximum-entropy principle, which advocates for the consistent selection of the most likely solution, is often used to address the challenge. Built on a recently proposed day-to-day discrete-time dynamical model called cumulative logit (CumLog), this study provides a new behavioral underpinning for the maximum-entropy user equilibrium (MEUE) route flow. It has been proven that CumLog can reach a UE state without presuming that travelers are perfectly rational. Here, we further establish that CumLog always converges to the MEUE route flow if (i) travelers have no prior information about routes and thus, are forced to give all routes an equal initial choice probability or if (ii) all travelers gather information from the same source such that the general proportionality condition is satisfied. Thus, CumLog may be used as a practical solution algorithm for the MEUE problem. To put this idea into practice, we propose to eliminate the route enumeration requirement of the original CumLog model through an iterative route discovery scheme. We also examine the discrete-time versions of four popular continuous-time dynamical models and compare them with CumLog. The analysis shows that the replicator dynamic is the only one that has the potential to reach the MEUE solution with some regularity. The analytical results are confirmed through numerical experiments. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25 Conference. Funding: This research was funded by the United States National Science Foundation’s Division of Civil, Mechanical and Manufacturing Innovation [Grant 2225087]. The work of J. Xie was funded by the National Natural Science Foundation of China [Grant 72371205]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0525 .

Interplay of physical and recognition performance using hierarchical continuous-time dynamic modeling and a dual-task training regime in Alzheimer's patients.

Hierarchical Bayesian continuous-time dynamic modeling approachA total of 72 repeated physical exercise (PP) and integrated recognition speed-accuracy (IRSA) measurementsPP is dynamically linked to session-to-session variability of IRSAHigher PP improved IRSA in subsequent sessions in subjects with Alzheimer's dementiaShort-term effect: lasting up to 4 days after training session.

Coordinated discrete-time super-twisting sliding mode controller coupled with time-delay estimator for PWR-based nuclear steam supply system

The coordination control problem for the core thermal power of the nuclear reactor (NR) and the water level of the nuclear steam generator (SG) is a crucial challenge for real-time model-based control of a pressurized-water reactor (PWR)-based nuclear steam supply system (NSSS). In this paper, a new coordinated discrete-time super-twisting sliding mode controller (CDTSTSMC) coupled with time-delay estimators (TDEs) is proposed, tackling the coordination control problem for the core thermal power of the NR and the water level of the nuclear SG in the presence of unknown dynamics, such as unmeasured system states, model uncertainties, parameter perturbations, and disturbances. First of all, the continuous-time dynamical models of both the NR and the SG in the PWR-based NSSS are described with consideration of unknown dynamics and then are transformed into discrete-time forms using the Euler approximation method. Two TDEs capable of estimating the unknown dynamics existing in the NR and the SG, respectively, are developed, which rely merely on the measurements of the PWR-based NSSS without the upper bounds of the uncertain dynamics being known. After that, a CDTSTSMC strategy, based on the discrete-time dynamical models describing both the NR and the SG, is proposed, allowing the core thermal power to be driven to a desired value while at the same time maintaining the desired water level in the SG with the help of TDEs. It is proved from stability analysis using a Lyapunov approach that under the proposed CDTSTSMC strategy coupled with the TDEs, the stability of the controlled PWR-based NSSS can be ensured in a wide range of load, where the control errors of both the NR and the SG are driven to a neighbor of zero. Finally, the proposed CDTSTSMC strategy coupled with the TDEs is compared with the previous discrete-time optimized proportional–integral–derivative controller (DTOPIDC) and fuzzy weighting controller (FWC). The simulation results reveal that the proposed CDTSTSMC strategy coupled with the TDEs outperforms the DTOPIDC and the FWC, particularly in rejecting unknown dynamics while maintaining both the core thermal power in the NR and the water level in the SG in their desired ranges.

Continuous-Time Graph Representation with Sequential Survival Process

Over the past two decades, there has been a tremendous increase in the growth of representation learning methods for graphs, with numerous applications across various fields, including bioinformatics, chemistry, and the social sciences. However, current dynamic network approaches focus on discrete-time networks or treat links in continuous-time networks as instantaneous events. Therefore, these approaches have limitations in capturing the persistence or absence of links that continuously emerge and disappear over time for particular durations. To address this, we propose a novel stochastic process relying on survival functions to model the durations of links and their absences over time. This forms a generic new likelihood specification explicitly accounting for intermittent edge-persistent networks, namely GraSSP: Graph Representation with Sequential Survival Process. We apply the developed framework to a recent continuous time dynamic latent distance model characterizing network dynamics in terms of a sequence of piecewise linear movements of nodes in latent space. We quantitatively assess the developed framework in various downstream tasks, such as link prediction and network completion, demonstrating that the developed modeling framework accounting for link persistence and absence well tracks the intrinsic trajectories of nodes in a latent space and captures the underlying characteristics of evolving network structure.

Optimal Consumption and Investment with Independent Stochastic Labor Income

We develop a new dynamic continuous-time model of optimal consumption and investment to include independent stochastic labor income. We reduce the problem of solving the Bellman equation to a problem of solving an integral equation. We then explicitly characterize the optimal consumption and investment strategy as a function of income-to-wealth ratio. We provide some analytical comparative statics associated with the value function and optimal strategies. We also develop a quite general numerical algorithm for control iteration and solve the Bellman equation as a sequence of solutions to ordinary differential equations. This numerical algorithm can be readily applied to many other optimal consumption and investment problems especially with extra nondiversifiable Brownian risks, resulting in nonlinear Bellman equations. Finally, our numerical analysis illustrates how the presence of stochastic labor income affects the optimal consumption and investment strategy. Funding: A. Bensoussan was supported by the National Science Foundation under grant [DMS-2204795]. S. Park was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea, South Korea [NRF-2022S1A3A2A02089950].

A generalized model for the population dynamics of a two stage species with recruitment and capture using a nonstandard finite difference scheme

The aim of this work is to formulate and analyze a new and generalized discrete-time population dynamics model for a two-stage species with recruitment and capture factors. This model is derived from a well-known continuous-time population dynamics model of a two-stage species with recruitment and capture developed by Ladino and Valverde and the nonstandard finite difference (NSFD) methodology proposed by Mickens. We establish positivity and asymptotic stability of the proposed discrete-time population dynamics model. As an important consequence, the population dynamics of the new discrete-time model is determined fully. Also, a set of numerical examples is conducted to illustrate the theoretical results and to demonstrate advantages of the new model. The theoretical results and numerical examples show that the proposed discrete-time model not only preserves correctly the population dynamics of the continuous one but is also easy to be implemented. However, some discrete-time models based on the standard Runge–Kutta methods fail to preserve the population dynamics of the continuous-time model. As a result, they generate numerical approximations which are not only non-negative but also unstable.

Budget Deficit in a Growing Economy and Impossibility of Fiscal Collapse: A Continuous Time Analysis

Abstract Using a continuous time dynamic model of growing economy we will show the following results. 1) When people derive utility from their money holding (or government bond holding) along with their consumption, a budget deficit is essential to achieve and maintain full employment under stable prices or inflation in a growing economy. 2) If we take into account that government spending due to budget deficits increases financial assets held by the private sector, and then consumption will occur from assets in addition to consumption from income, even when the interest rate on government bonds is higher than the real economic growth rate, the ratio of government debt to GDP can not diverge and the divergence is naturally prevented by mild inflation. The required inflation rate is such that the interest rate of the government bonds is smaller than the weighted average of the rate of return on capital and the nominal growth rate. Since the interest rate of the government bonds is usually considered smaller than the rate of return on capital, this is not a very demanding requirement. Thus, we need not worry at all about the accumulation of government debt or about the divergence of the debt to GDP ratio, which is often taken as an indicator of fiscal collapse.

Beyond the average: embracing speaker individuality in the dynamic modeling of the acoustic-articulatory relationship

This paper explores the acoustic-articulatory relationship while considering individual differences in speech production. We aimed to determine whether there is a causal relationship between tongue movements and the contours of the first and second formant frequencies (F 1 and F 2) employing a hierarchical Bayesian continuous-time dynamic model, which allows for a more direct connection between the acoustic and articulatory measured variables and theories involving dynamicity. The results show predictive tendencies for both formants, where the anteroposterior and vertical tongue movements may predict changes in F 1, with rising predicting an increase and retraction a decrease; and with tongue fronting and tongue height inversely predicting F 2. Further, the modeled individual differences showed similar global tendencies, except for the rate of change of F 2. Overall, this study provides valuable insights into the relationship between tongue articulatory variables and formant contours, while accounting for between-speaker variability. The results show predictive tendencies for both formants, where the anteroposterior and vertical tongue movements may predict changes in f1, with rising predicting an increase and retraction a decrease; and with tongue fronting and tongue height inversely predicting f2. Further, the modeled individual differences showed similar global tendencies, however with larger error estimates. Overall, this study provides valuable insights into the relationship between tongue articulatory variables and formant contours, while accounting for between-speaker variability.

Non-parametric dynamics modeling for unmanned surface vehicle using spectral metric multi-output Gaussian processes learning

High-precision dynamic models are crucial for various marine missions and the control and navigation of unmanned surface vehicles. However, the nonlinear, strong coupling, complex structure of these vehicles poses challenges in building such models. Traditional single-output models have limitations in data utilization and high computational complexity. Therefore, this paper proposes a spectral metric multi-output Gaussian process for continuous-time dynamic modeling of unmanned surface vehicles. Specifically, a spectral covariance matrix is constructed in this scheme to establish a spectral metric multi-output Gaussian processes learning model that captures correlations under different degrees of freedom. Meanwhile, an improved particle swarm optimization algorithm is used to dynamically tuning the correlation matrix for improved model accuracy, and utilizing a subspace index to enhance computational efficiency in multi-output Gaussian processes learning. Moreover, the scheme can conduct real-time simulation under environmental disturbances. Finally, the proposed scheme is applied to an unmanned surface vehicle, and the results demonstrate that the scheme is an effective modeling tool for multi degree-of-freedom coupled dynamics.

Budget deficit and money holding when consumers live forever in an endogenous growth model

<p class="MsoNormal" style="margin-top: 12pt; text-align: justify;"><span lang="EN-US" style="font-family: arial, helvetica, sans-serif;">In this paper I will show that budget deficit (or fiscal deficit) is necessary to achieve full employment under constant prices or inflation, using a model of endogenous growth in which consumers hold money for the reason of liquidity and live forever. Budget deficit need not be offset by future budget surpluses. I consider the continuous time case by taking the limit of the discrete time case when the time interval approaches zero. A continuous time dynamic model seems to be more general than a discrete time model. When the actual budget deficit is greater (smaller) than the value which is necessary and sufficient for full employment under constant prices, an inflation (a recession) occurs. The main argument of this paper is that a growing economy requires the continuation of budget deficit, and that we should not think of paying off the resulting government debt with taxes.</span></p>

Regularized continuous time structural equation models: A network perspective.

Regularized continuous time structural equation models are proposed to address two recent challenges in longitudinal research: Unequally spaced measurement occasions and high model complexity. Unequally spaced measurement occasions are part of most longitudinal studies, sometimes intentionally (e.g., in experience sampling methods) sometimes unintentionally (e.g., due to missing data). Yet, prominent dynamic models, such as the autoregressive cross-lagged model, assume equally spaced measurement occasions. If this assumption is violated parameter estimates can be biased, potentially leading to false conclusions. Continuous time structural equation models (CTSEM) resolve this problem by taking the exact time point of a measurement into account. This allows for any arbitrary measurement scheme. We combine CTSEM with LASSO and adaptive LASSO regularization. Such regularization techniques are especially promising for the increasingly complex models in psychological research, the most prominent example being network models with often dozens or hundreds of parameters. Here, LASSO regularization can reduce the risk of overfitting and simplify the model interpretation. In this article we highlight unique challenges in regularizing continuous time dynamic models, such as standardization or the optimization of the objective function, and offer different solutions. Our approach is implemented in the R (R Core Team, 2022) package regCtsem. We demonstrate the use of regCtsem in a simulation study, showing that the proposed regularization improves the parameter estimates, especially in small samples. The approach correctly eliminates true-zero parameters while retaining true-nonzero parameters. We present two empirical examples and end with a discussion on current limitations and future research directions. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Nested Pseudo likelihood estimation of continuous-time dynamic discrete games

We introduce a sequential estimator for continuous time dynamic discrete choice models (single-agent models and games) by adapting the nested pseudo likelihood (NPL) estimator of Aguirregabiria and Mira (2002, 2007), developed for discrete time models with discrete time data, to the continuous time case with data sampled either discretely (i.e., uniformly-spaced snapshot data) or continuously. We establish conditions for consistency and asymptotic normality of the estimator, a local convergence condition, and, for single agent models, a zero Jacobian property assuring local convergence. We carry out a series of Monte Carlo experiments using an entry-exit game with five heterogeneous firms to confirm the large-sample properties and demonstrate finite-sample bias reduction via iteration. In our simulations we show that the convergence issues documented for the NPL estimator in discrete time models are less likely to affect comparable continuous-time models. We also show that there can be large bias in economically-relevant parameters, such as the competitive effect and entry cost, from estimating a misspecified discrete time model when in fact the data generating process is a continuous time model.

A new time-adjustable model-based method for fast open-circuit voltage estimation of Lithium-ion cells

Any system ensuring the safety and the intelligent management of Li-ion batteries requires a thorough knowledge of the curve relating their open circuit voltage to their state-of-charge. However, classical industrial methods used to characterize this relationship necessitate a complete charge/discharge cycle at low current, or with very long relaxation times, to limit the influence of the lithium diffusion. Indeed, the latter induces voltage drops of the same order of magnitude as those induced by the conduction and charge transfer phenomena, but with much higher time constants, generally of several hours. These methods are therefore time-consuming and costly. In this paper, we propose to take advantage of both a continuous-time dynamic model of the Li-ion cell and a dedicated moving horizon estimation algorithm to significantly reduce the characterization time, while simultaneously providing the incremental capacity analysis health indicator. The proposed model structure offers the best trade-off between accuracy of the system simulation and number of parameters that can be sufficiently sensitized regarding the limited duration of the characterization protocol. The originality of the moving horizon estimation algorithm consists in centering the moving observation windows on the current time, improving de facto the estimation accuracy. When using real experimental signals of LFP and NCA cells, we obtain an estimation error lower than 5 mV, close to the best-in-class relaxation method classically used by industrial actors, while dividing the whole experiment time by more than 30.

Digital deployment control of a tethered satellite system with constrained tension

This paper investigates the dynamics and digital control issues of deploying a tethered satellite using constrained tether tension. To cater to the requirements of the digital techniques applied in engineering applications, unlike existing results, this study develops a digital feedback control strategy to achieve the rapid and stable tether deployment. The proposed digital control law consists of discretized and quantized tether states and is derived based on continuous-time dynamical model instead of the discrete-time one. To keep tether tension at a positive value when implementing control action, the constraint on tether tension is explicitly accounted for during controller design. Furthermore, the discretization and quantization errors are treated as the uncertain terms in the closed-loop system which can be handled by robust analysis and synthesis techniques. A Lyapunov functional, together with an integral performance index, is made to prove the stability of the system, and the desired controller is obtained by solving an optimization problem. Finally, the effectiveness of the proposed controller is demonstrated by numerical simulation.

Leveraging Observation Timing Variability to Understand Intervention Effects in Panel Studies: An Empirical Illustration and Simulation Study

To examine developmental processes, intervention effects, or both, longitudinal studies often aim to include measurement intervals that are equally spaced for all participants. In reality, however, this goal is hardly ever met. Although different approaches have been proposed to deal with this issue, few studies have investigated the potential benefits of individual variation in time intervals. In the present paper, we examine how continuous time dynamic models can be used to study nonexperimental intervention effects in longitudinal studies where measurement intervals vary between and within participants. We empirically illustrate this method by using panel data (N = 2,877) to study the effect of the transition from primary to secondary school on students’ motivation. Results of a simulation study also show that the precision and recovery of the estimate of the effect improves with individual variation in time intervals.

A Hybrid Continuous-Time Dynamic Graph Representation Learning Model by Exploring Both Temporal and Repetitive Information

Recently, dynamic graph representation learning has attracted more and more attention from both academic and industrial communities due to its capabilities of capturing different real-world phenomena. For a dynamic graph represented as a sequence of timestamped events, there are two kinds of evolutionary essences: temporal and repetitive information. At present, the temporal information of interactions (e.g., timestamps) have been deeply explored. However, as another vital nature of dynamic graphs, the repetitive information of interactions between two nodes is neglected, which may lead to inaccurate node representation. To address this issue, we propose a novel continuous-time dynamic graph representation learning model, which consists of a node-level-memory based module, a historical high-order neighborhood based vertical aggregation module and a repetitive-topological information based horizontal aggregation module. In particular, to characterize the evolving pattern of the repetitive information of interactions between a pair of nodes, we put forward a repetitive-interaction based attention mechanism to integrate the two key attributes (i.e., the content and the number of interactions) of repetitive interactions at different moments, based on the insight that the repetitive behaviors of nodes are widespread and essential. We conduct extensive experiments including future link prediction tasks (for transductive and inductive learning) and dynamic node classification task, and results on three real-life dynamic graph datasets demonstrate that the proposed method significantly outperforms state-of-the-art baselines, for both observed nodes and new ones.