Chapter 34 - Hierarchical continuous time modeling

In the theoretical part of this article, we provide a brief introduction to differenttypes of repeated measure designs and methods to analyze repeatedly measured data, with aparticular focus on continuous time modelling of intensive longitudinal data (ILD) withN≥1 analysis. We built on the distinction betweenwithin-person versus between-person effects, and how this is addressed in static versusdynamic models. Further, we elaborate on the distinction between discrete time dynamicmodels versus continuous time dynamic models. In particular, we deal with continuous timestructural equation models (CTSEM), and we provide a brief introduction into theunderlying math. Since smart devices have become useful tools in monitoring health, we usethe applied part of this article for explaining how to retrieveN=1 bivariate ILD from popular smart watches and how toprepare them for CTSEM (including N>1 multivariateextensions). We show how to specify a cross-lagged panel CTSEM using the R package ctsem,how to fit the specified model to the retrieved data, and how to interpret the results.Limitations of CTSEM are discussed, too. Monitoring and forecasting industrial healthrepresent important issues for organizations.

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

A sequential refinement approach for parameter optimization in continuous dynamic models

In this essay we review some of the emerging trends in modeling repeated measures data. Three general forms of longitudinal models are discussed: panel model designs, growth curve models, and intensive within-person assessments. Each section discusses design elements that should be considered when using each of these types of longitudinal models, and introduces some emerging trends. In the section on panel designs, continuous time models and planned missing data models are introduced; these ideas will revolutionize the modeling and collection of panel data. In the section on growth curve models, the necessity of separately evaluating mean and covariance model fit is discussed. This section also introduces methods being used to carefully consider the time of measurements in temporal designs. Finally, the budding analysis of intensive within individual observations is considered, including recent work from mathematics that limits the generalizability of interindividual studies to individual outcomes. Keywords: longitudinal; structural equation modeling; latent variable modeling; cross-lagged and auto-regressive panel modeling; growth curve modeling; continuous time modeling; dynamical systems modeling

System identification of the human thermoregulatory system using continuous-time block-oriented predictive modeling

Well-being and smartphone use are thought to influence each other. However, previous studies mainly focused on one direction (looking at the effects of smartphone use on well-being) and considered between-person effects, with self-reported measures of smartphone use. By using 2548 assessments of well-being and trace data of smartphone use collected for 45 consecutive days in 82 adolescent participants (Mage = 13.47, SDage = 1.62, 54% females), the present study disentangled the reciprocal and individual dynamics of well-being and smartphone use. Hierarchical Bayesian Continuous Time Dynamic Models were used to estimate how a change in frequency and duration of smartphone use predicted a later change in well-being, and vice versa. Results revealed that (i) when participants used the smartphone frequently and for a longer period, they also reported higher levels of well-being; (ii) well-being positively predicted subsequent duration of smartphone use; (iii) usage patterns and system dynamics showed heterogeneity, with many subjects showing reciprocal effects close to zero; finally, (iv) changes in well-being tend to persist longer than changes in the frequency and duration of smartphone use.

Limitations record on a variable which have continuous variable changes make the 'time' attached to the data is discrete. That underlies development of continuous time model. Compared to discrete time model, the continuous time model has advantages that can make prediction at different lag. In addition, it still can perform continuous time model parameter estimation, although the objects observations do not always have the same time interval (irregular time). Thus, the purpose of this study is to examine the consistency of continuous time model coefficient parameter with structural equation model (SEM) approach in irregular time data. The method used is the Exact Discrete Model (EDM) with SEM approach. EDM is a model that connects the parameter to value which is underlying parameter of the continuous time model with non-linear relationships. The result analysis of the research indicated that the coefficients of continuous time model parameter are still consistent until 20% of irregular time with 5% consistency limits.

In this chapter we review continuous time series modeling and estimation by extended structural equation models (SEM) for single subjects and N > 1. First-order as well as higher-order models will be dealt with. Both will be handled by the general state space approach which reformulates higher-order models as first-order models. In addition to the basic model, the extensions of exogenous variables and traits (random intercepts) will be introduced. The connection between continuous time and discrete time for estimating the model by SEM will be made by the exact discrete model (EDM). It is by the EDM that the exact estimation procedure in this chapter differentiates from many approximate procedures found in the literature. The proposed analysis procedure will be applied to the well-known Wolfer sunspot data, an N = 1 time series that has been analyzed by several continuous time analysts in the past. The analysis will be carried out by ctsem, an R-package for continuous time modeling that interfaces to OpenMx, and the results will be compared to those reported in the previous studies.

This paper describes a case study of event and continuous hydrologic modeling in the Kelani River basin in Sri Lanka using the Hydrologic Engineering Center—Hydrologic Modeling System (HEC–HMS). An extremely high rainfall event in November 2005 was used to calibrate model parameters, and extremely high rainfall events in April–May 2008, May–June 2008, and May 2010 were used to validate the event model. The calibrated, direct runoff and base flow parameters were then used in the continuous hydrologic model. The Green and Ampt infiltration loss method was used to account for infiltration loss in event-based modeling and a five-layer soil moisture accounting loss method was employed in continuous modeling. The Clark unit hydrograph method and the recession base flow method were used to simulate direct runoff and base flow, respectively. The results depict the capability of HEC–HMS to reproduce streamflows in the basin to a high accuracy with averaged computed Nash–Sutcliffe efficiencies of 0.91 for e...

Hierarchical model updating strategy of complex assembled structures with uncorrelated dynamic modes

In this article, a mathematical model of controlling the system in conditions of fuzzy processes is presented. Such a system can also be a power supply system. Analytical approaches that can be used to describe the mutual impact of output and stocks (additional capacities) on hierarchically distributed occurrence/usage/variation or demand already exist. We add dynamics to the system with the use of continuous and discrete dynamic processes, which are of a random (stochastic) form. The dynamic discrete model of control for this system is built with a system of difference equations, and the dynamic continuous model is built with a system of differential equations. These systems of equations can be solved with a one-part z-transform in discrete situations in with Laplace transform in the continuous systems. Fuzzy system control follows a continuous and discrete stochastic mathematical closed-loop model of control of stocks (additional capacities) in production systems. The fuzzy model is demonstrated with a numerical example.

BackgroundGiven the heterogeneity of stroke, it is important to determine the best course of motor therapy for each patient, i.e., to personalize rehabilitation based on predictions of long-term outcomes. Here, we propose a hierarchical Bayesian dynamic (i.e., state-space) model (HBDM) to forecast long-term changes in a motor outcome due to rehabilitation in the chronic phase post-stroke.MethodsThe model incorporates the effects of clinician-supervised training, self-training, and forgetting. In addition, to improve forecasting early in rehabilitation, when data are sparse or unavailable, we use the Bayesian hierarchical modeling technique to incorporate prior information from similar patients. We use HBDM to re-analyze the Motor Activity Log (MAL) data of participants with chronic stroke included in two clinical trials: (1) the DOSE trial, in which participants were assigned to a 0, 15, 30, or 60-h dose condition (data of 40 participants analyzed), and (2) the EXCITE trial, in which participants were assigned a 60-h dose, in either an immediate or a delayed condition (95 participants analyzed).ResultsFor both datasets, HBDM accounts well for individual dynamics in the MAL during and outside of training: mean RMSE = 0.28 for all 40 DOSE participants (participant-level RMSE 0.26 ± 0.19—95% CI) and mean RMSE = 0.325 for all 95 EXCITE participants (participant-level RMSE 0.32 ± 0.31), which are small compared to the 0-5 range of the MAL. Bayesian leave-one-out cross-validation shows that the model has better predictive accuracy than static regression models and simpler dynamic models that do not account for the effect of supervised training, self-training, or forgetting. We then showcase model’s ability to forecast the MAL of “new” participants up to 8 months ahead. The mean RMSE at 6 months post-training was 1.36 using only the baseline MAL and then decreased to 0.91, 0.79, and 0.69 (respectively) with the MAL following the 1st, 2nd, and 3rd bouts of training. In addition, hierarchical modeling improves prediction for a patient early in training. Finally, we verify that this model, despite its simplicity, can reproduce previous findings of the DOSE trial on the efficiency, efficacy, and retention of motor therapy.ConclusionsIn future work, such forecasting models can be used to simulate different stages of recovery, dosages, and training schedules to optimize rehabilitation for each person.Trial registration This study contains a re-analysis of data from the DOSE clinical trial ID NCT01749358 and the EXCITE clinical trial ID NCT00057018

We study a scenario in which n mobile robots with a limited viewing range are distributed in the Euclidean plane and have to solve a formation problem. The formation problems we consider are the G athering problem and the C hain-Formation problem. In the G athering problem, the robots have to gather in one (not predefined) point, while in the C hain -F ormation problem they have to form a connected communication chain of minimal length between two stationary base stations. Each robot may base its decisions where to move only on the current relative positions of neighboring robots (that are within its viewing range); that is, besides having a limited viewing range, the robots are oblivious (they do not use information from the past), have none or only very limited identities, and they do not have a common sense of direction. Variants of these problems (especially for the G athering problem) have been studied extensively in different discrete time models. In contrast, our work focuses on a continuous time model; that is, the robots continuously sense the positions of other robots within their viewing range and continuously adapt their speed and direction according to some simple, local rules. Hereby, we assume that the robots have a maximum movement speed of one. We show that this idealized idea of continuous sensing allows us to solve the mentioned formation problems in linear time O( n ) (which, given the maximum speed of one, immediately yields a maximum traveled distance of O( n )). Note that in the more classical discrete time models, the best known strategies need at least Θ( n 2 ) or even Θ( n 2 log n ) timesteps to solve these problems. For the G athering problem, our analysis solves a problem left open by Gordon et al. [2004], where the authors could prove that gathering in a continuous model is possible in finite time, but were not able to give runtime bounds. Apart from these linear bounds, we also provide runtime bounds for both formation problems that relate the runtime of our strategies to the runtime of an optimal, global algorithm. Specifically, we show that our strategy for the G athering problem is log OPT-competitive and the strategy for the C hain -F ormation problem is log n -competitive. Here, by c -competitive, we mean that our (local) strategy is asymptotically by at most a factor of c slower than an optimal, global strategy.

In this article, we provide empirical evidence of the recent financial crisis over 2007–2009 using discrete time multivariate GARCH (MGARCH) models and continuous time modelling approaches. Using daily data for 14 countries, we investigate the return and volatility spillovers among the US and other international markets. The MGARCH results reveal positive return spillovers from the US to a number of markets, and volatility transmission is verified. The US market is prone to return and volatility transmission from a limited number of markets. The continuous time analysis finds evidence of feedback effects in some cases. Evidence shows that spillover effects intensified during the financial crisis.

Open-pit mine vehicle scheduling problem is mainly about allocating and designing the schedule of trucks and electric forklift under the constraints, which includes the decision of transportation route, distribution of electric forklifts and trucks number at different mining area with the objective of maximizing equipment utilization rate and reducing production cost. In addition, the various real-time practical changes in mining lead open-pit mine vehicle scheduling problem a dynamic scheduling problem. In this paper, the mathematical model of open-pit mines vehicle scheduling problem using continuous time modeling is established. The transport process, characteristics and requirements of vehicle scheduling problem in open-pit mines are analyzed. For large-scale examples, an approximate dynamic programming model is established by ADP algorithm based on Q-Learning. Numerical experiments of different extraction methods of feature vector and update methods of coefficient vector are carried out, and the results are compared with the results by using solver. The experimental results present that the ADP algorithm designed in this paper can effectively solve the large scale open-pit mine vehicle scheduling problem.