Continuous-time Dynamic Model Research Articles

The dynamics of self-esteem and depressive symptoms across days, months, and years.

Self-esteem and depressive symptoms are important predictors of a range of societally relevant outcomes and are theorized to influence each other reciprocally over time. However, existing research offers only a limited understanding of how their dynamics unfold across different timescales. Using three data sets with different temporal resolutions, we aimed to advance our understanding of the temporal unfolding of the reciprocal dynamics between self-esteem and depressive symptoms. Across these data sets, participants (Ntotal = 6,210) rated their self-esteem and depressive symptoms between 6 and 14 times across days, months, and years, respectively. Using continuous time dynamic models, we found limited evidence for significant within-person cross-lagged effects between self-esteem and depressive symptoms. Only in the yearly data set, a cross-lagged effect from depressive symptoms to self-esteem emerged quite consistently. However, in all data sets, cross-lagged effects were small in size (-0.04 ≤ β ≤ -0.01). These findings suggest that the reciprocal dynamics between self-esteem and depressive symptoms may be less robust than commonly thought. Furthermore, exploratory analyses indicated that these effects depended on people's overall levels of depressive symptoms, suggesting that theoretical frameworks that highlight transactions between self-esteem and depression may not generalize across all levels of depressive symptoms. Finally, self-esteem and depressive symptoms were strongly correlated within measurements, similarly stable over time, and changed similarly in response to negative life events, provoking questions as to their conceptual distinctiveness and measurement approaches. (PsycInfo Database Record (c) 2025 APA, all rights reserved).

SHORT-TERM ASSOCIATIONS OF ACTIVITY ENGAGEMENT WITH COGNITIVE PERFORMANCE

Abstract Prior research showed that activity engagement had short-term effects on numerical working memory over hours in German-speaking older adults. However, it is unclear whether these effects could be generalized to other cognitive outcomes and populations. This study examined short-term associations of activity engagement with cognitive performance using data from two studies that included adults of different demographic backgrounds in the US. That is, 16 days’ data from 282 older adults aged 70 to 90 years (M = 76.62, SD = 4.68; 66% women; 44% White, 40% Black, 11% Hispanic White, 3% Hispanic Black) from the Einstein Aging Study (EAS) and 14 days’ data from 260 participants aged 25 to 65 (M = 46.49, SD =11.09; 65% women; 8% White, 63% Black, 18% Hispanic White, 6% Hispanic Black) from the Effects of Stress on Cognitive Aging, Physiology, and Emotion Project (ESCAPE). For 4 or 5 times per day, participants reported whether they engaged in different social and cognitive activities (e.g., social interactions, TV watching) and completed smartphone cognitive assessments (i.e., processing speed, spatial working memory). Using hierarchical continuous-time dynamic modeling, we did not find any time-lagged associations between activity engagement and cognitive performance. Additionally, age, sex, and race-ethnicity did not moderate the time-lagged associations. Our results suggest that activity engagement did not have short-term effects on processing speed and spatial working memory in young and older populations with ethnic and racial diversity.

Within-person, longitudinal associations between neighborhood cohesion and adult mental health: A test of bidirectional relations.

There is theory and evidence supporting a relationship between neighborhood cohesion and mental health among adult people. However, most studies have used a cross-sectional design, and longitudinal studies have provided mixed support for this hypothesis. Moreover, while neighborhood cohesion is assumed to be a consistent predictor of mental health, the possibility of a reciprocal relation has been overlooked. The aim of the current study was to investigate the within-person reciprocal associations between neighborhood cohesion and adult mental health. This study used data from Understanding Society: the UK Household Longitudinal Study including waves 1, 3, 6, 9, and 12 (n = 81,895). A short version of Buckner's Neighborhood Cohesion Instrument was used, along with two well-established measures of mental health: the 12-item General Health Questionnaire and the mental component summary score of the 12-item Short-Form Health Survey. Hierarchical Bayesian continuous time dynamic modeling revealed significant reciprocal within-person cross-effects (i.e. deterministic effects) between neighborhood cohesion and adult mental health, a larger standardized effect of mental health on neighborhood cohesion than than vice versa. Moreover, peak standardized cross-lagged effects were found for a time interval of approximately 1 year. Finally, the combined stochastic and deterministic interpretation revealed effects of neighborhood cohesion on mental health that were opposite to what was expected, suggesting a faster dissipation of some initially correlated change. Although neighborhood cohesion has traditionally been conceptualized in terms of its contribution to mental health, there is greater support for the view that mental health precedes neighborhood cohesion rather than the reverse.

The dynamic relationship between state extraversion and the subjectively perceived situational sociality in continuous time

This study examined the dynamic interplay between extraversion and subjectively perceived sociality of situations using continuous-time dynamic modeling. In a week-long experience sampling study, we collected 4,694 reports of state extraversion and perceived sociality from a sample of university students. Results showed that changes in state extraversion lasted longer and took longer to return to baseline than changes in perceived sociality. Increases in state extraversion predicted subsequent increases in perceived sociality, whereas increases in perceived sociality predicted subsequent decreases in state extraversion. Cross-lagged effects reached their maximum at 1.5 h lag. The negative cross-lagged effect of sociality on state extraversion suggests a compensatory mechanism of temporal reduction of extraverted behavior in an effort to balance the level of social engagement.

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.

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.

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.