Functional data analysis for dynamical system identification of behavioral processes

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.

Companion: A Pilot Randomized Clinical Trial to Test an Integrated Two-Way Communication and Near-Real-Time Sensing System for Detecting and Modifying Daily Inactivity among Adults >60 Years-Design and Protocol

Supervised personal training is most effective in improving the health effects of exercise in older adults. Yet, low frequency (60 min, 1-3 sessions/week) of trainer contact limits influence on behavior change outside sessions. Strategies to extend the effect of trainer contact outside of supervision and that integrate meaningful and intelligent two-way communication to provide complex and interactive problem solving may motivate older adults to "move more and sit less" and sustain positive behaviors to further improve health. This paper describes the experimental protocol of a 16-week pilot RCT (N = 46) that tests the impact of supplementing supervised exercise (i.e., control) with a technology-based behavior-aware text-based virtual "Companion" that integrates a human-in-the-loop approach with wirelessly transmitted sensor-based activity measurement to deliver behavior change strategies using socially engaging, contextually salient, and tailored text message conversations in near-real-time. Primary outcomes are total-daily and patterns of habitual physical behaviors after 16 and 24 weeks. Exploratory analyses aim to understand Companion's longitudinal behavior effects, its user engagement and relationship to behavior, and changes in cardiometabolic and cognitive outcomes. Our findings may allow the development of a more scalable hybrid AI Companion to impact the ever-growing public health epidemic of sedentariness contributing to poor health outcomes, reduced quality of life, and early death.

Streamlined variational inference for higher level group-specific curve models

A two-level group-specific curve model is such that the mean response of each member of a group is a separate smooth function of a predictor of interest. The three-level extension is such that one grouping variable is nested within another one, and higher level extensions are analogous. Streamlined variational inference for higher level group-specific curve models is a challenging problem. We confront it by systematically working through two-level and then three-level cases and making use of the higher level sparse matrix infrastructure laid down in Nolan and Wand (2019). A motivation is analysis of data from ultrasound technology for which three-level group-specific curve models are appropriate. Whilst extension to the number of levels exceeding three is not covered explicitly, the pattern established by our systematic approach sheds light on what is required for even higher level group-specific curve models.

Dynamical System Modeling of Self-Regulated Systems Undergoing Multiple Excitations: First Order Differential Equation Approach

This article proposes a dynamical system modeling approach for the analysis of longitudinal data of self-regulated homeostatic systems experiencing multiple excitations. It focuses on the evolution of a signal (e.g., heart rate) before, during, and after excitations taking the system out of its equilibrium (e.g., physical effort during cardiac stress testing). Such approach can be applied to a broad range of outcomes such as physiological processes in medicine and psychosocial processes in social sciences, and it allows to extract simple characteristics of the signal studied. The model is based on a first order linear differential equation with constant coefficients defined by three main parameters corresponding to the initial equilibrium value, the dynamic characteristic time, and the reaction to the excitation. Assuming the presence of interindividual variability (random effects) on these three parameters, we propose a two-step procedure to estimate them. We then compare the results of this analysis to several other estimation procedures in a simulation study that clarifies under which conditions parameters are accurately estimated. Finally, applications of this model are illustrated using cardiology data recorded during effort tests.

Validity of dynamical analysis to characterize heart rate and oxygen consumption during effort tests

Performance is usually assessed by simple indices stemming from cardiac and respiratory data measured during graded exercise test. The goal of this study is to characterize the indices produced by a dynamical analysis of HR and VO₂ for different effort test protocols, and to estimate the construct validity of these new dynamical indices by testing their links with their standard counterparts. Therefore, two groups of 32 and 14 athletes from two different cohorts performed two different graded exercise testing before and after a period of training or deconditioning. Heart rate (HR) and oxygen consumption (VO₂) were measured. The new dynamical indices were the value without effort, the characteristic time and the amplitude (gain) of the HR and VO₂ response to the effort. The gain of HR was moderately to strongly associated with other performance indices, while the gain for VO₂ increased with training and decreased with deconditioning with an effect size slightly higher than VO₂ max. Dynamical analysis performed on the first 2/3 of the effort tests showed similar patterns than the analysis of the entire effort tests, which could be useful to assess individuals who cannot perform full effort tests. In conclusion, the dynamical analysis of HR and VO₂ obtained during effort test, especially through the estimation of the gain, provides a good characterization of physical performance, robust to less stringent effort test conditions.

Scalar-on-function regression for predicting distal outcomes from intensively gathered longitudinal data: Interpretability for applied scientists

Researchers are sometimes interested in predicting a distal or external outcome (such as smoking cessation at follow-up) from the trajectory of an intensively recorded longitudinal variable (such as urge to smoke). This can be done in a semiparametric way via scalar-on-function regression. However, the resulting fitted coefficient regression function requires special care for correct interpretation, as it represents the joint relationship of time points to the outcome, rather than a marginal or cross-sectional relationship. We provide practical guidelines, based on experience with scientific applications, for helping practitioners interpret their results and illustrate these ideas using data from a smoking cessation study.

Just-in-Time Adaptive Interventions (JITAIs) in Mobile Health: Key Components and Design Principles for Ongoing Health Behavior Support

Background

The just-in-time adaptive intervention (JITAI) is an intervention design aiming to provide the right type/amount of support, at the right time, by adapting to an individual's changing internal and contextual state. The availability of increasingly powerful mobile and sensing technologies underpins the use of JITAIs to support health behavior, as in such a setting an individual's state can change rapidly, unexpectedly, and in his/her natural environment.

Purpose

Despite the increasing use and appeal of JITAIs, a major gap exists between the growing technological capabilities for delivering JITAIs and research on the development and evaluation of these interventions. Many JITAIs have been developed with minimal use of empirical evidence, theory, or accepted treatment guidelines. Here, we take an essential first step towards bridging this gap.

Methods

Building on health behavior theories and the extant literature on JITAIs, we clarify the scientific motivation for JITAIs, define their fundamental components, and highlight design principles related to these components. Examples of JITAIs from various domains of health behavior research are used for illustration.

Conclusions

As we enter a new era of technological capacity for delivering JITAIs, it is critical that researchers develop sophisticated and nuanced health behavior theories capable of guiding the construction of such interventions. Particular attention has to be given to better understanding the implications of providing timely and ecologically sound support for intervention adherence and retention.

Practical Tools and Guidelines for Exploring and Fitting Linear and Nonlinear Dynamical Systems Models

A dynamical system is a system of variables that show some regularity in how they evolve over time. Change concepts described in most dynamical systems models are by no means novel to social and behavioral scientists, but most applications of dynamic modeling techniques in these disciplines are grounded on a narrow subset of-typically linear-theories of change. I provide practical guidelines, recommendations, and software code for exploring and fitting dynamical systems models with linear and nonlinear change functions in the context of four illustrative examples. Cautionary notes, challenges, and unresolved issues in utilizing these techniques are discussed.

A one-step method for modelling longitudinal data with differential equations

Differential equation models are frequently used to describe non-linear trajectories of longitudinal data. This study proposes a new approach to estimate the parameters in differential equation models. Instead of estimating derivatives from the observed data first and then fitting a differential equation to the derivatives, our new approach directly fits the analytic solution of a differential equation to the observed data, and therefore simplifies the procedure and avoids bias from derivative estimations. A simulation study indicates that the analytic solutions of differential equations (ASDE) approach obtains unbiased estimates of parameters and their standard errors. Compared with other approaches that estimate derivatives first, ASDE has smaller standard error, larger statistical power and accurate Type I error. Although ASDE obtains biased estimation when the system has sudden phase change, the bias is not serious and a solution is also provided to solve the phase problem. The ASDE method is illustrated and applied to a two-week study on consumers' shopping behaviour after a sale promotion, and to a set of public data tracking participants' grammatical facial expression in sign language. R codes for ASDE, recommendations for sample size and starting values are provided. Limitations and several possible expansions of ASDE are also discussed.

Getting the most out of intensive longitudinal data: a methodological review of workload-injury studies

OBJECTIVES

To systematically identify and qualitatively review the statistical approaches used in prospective cohort studies of team sports that reported intensive longitudinal data (ILD) (>20 observations per athlete) and examined the relationship between athletic workloads and injuries. Since longitudinal research can be improved by aligning the (1) theoretical model, (2) temporal design and (3) statistical approach, we reviewed the statistical approaches used in these studies to evaluate how closely they aligned these three components.

DESIGN

Methodological review.

METHODS

After finding 6 systematic reviews and 1 consensus statement in our systematic search, we extracted 34 original prospective cohort studies of team sports that reported ILD (>20 observations per athlete) and examined the relationship between athletic workloads and injuries. Using Professor Linda Collins' three-part framework of aligning the theoretical model, temporal design and statistical approach, we qualitatively assessed how well the statistical approaches aligned with the intensive longitudinal nature of the data, and with the underlying theoretical model. Finally, we discussed the implications of each statistical approach and provide recommendations for future research.

RESULTS

Statistical methods such as correlations, t-tests and simple linear/logistic regression were commonly used. However, these methods did not adequately address the (1) themes of theoretical models underlying workloads and injury, nor the (2) temporal design challenges (ILD). Although time-to-event analyses (eg, Cox proportional hazards and frailty models) and multilevel modelling are better-suited for ILD, these were used in fewer than a 10% of the studies (n=3).

CONCLUSIONS

Rapidly accelerating availability of ILD is the norm in many fields of healthcare delivery and thus health research. These data present an opportunity to better address research questions, especially when appropriate statistical analyses are chosen.

Representing Sudden Shifts in Intensive Dyadic Interaction Data Using Differential Equation Models with Regime Switching

A growing number of social scientists have turned to differential equations as a tool for capturing the dynamic interdependence among a system of variables. Current tools for fitting differential equation models do not provide a straightforward mechanism for diagnosing evidence for qualitative shifts in dynamics, nor do they provide ways of identifying the timing and possible determinants of such shifts. In this paper, we discuss regime-switching differential equation models, a novel modeling framework for representing abrupt changes in a system of differential equation models. Estimation was performed by combining the Kim filter (Kim and Nelson State-space models with regime switching: classical and Gibbs-sampling approaches with applications, MIT Press, Cambridge, 1999) and a numerical differential equation solver that can handle both ordinary and stochastic differential equations. The proposed approach was motivated by the need to represent discrete shifts in the movement dynamics of [Formula: see text] mother-infant dyads during the Strange Situation Procedure (SSP), a behavioral assessment where the infant is separated from and reunited with the mother twice. We illustrate the utility of a novel regime-switching differential equation model in representing children's tendency to exhibit shifts between the goal of staying close to their mothers and intermittent interest in moving away from their mothers to explore the room during the SSP. Results from empirical model fitting were supplemented with a Monte Carlo simulation study to evaluate the use of information criterion measures to diagnose sudden shifts in dynamics.

Dynamical Systems Modeling of Couple Interaction: a New Method for Assessing Intervention Impact Across the Transition to Parenthood

This study explored the use of dynamical systems modeling techniques to evaluate self- and co-regulation of affect in couples' interactions before and after the transition to parenthood, and the impact of the Family Foundations program on these processes. Thirty-four heterosexual couples, randomized to intervention and control conditions, participated in videotaped dyadic interaction tasks at pretest (during pregnancy) and posttest (1 year after birth). Husbands' and wives' positivity and negativity were micro-coded throughout interactions. Individual negativity set-points, self-regulation, and partner co-regulatory processes during interactions were examined using a coupled oscillators model. Regarding self-regulatory processes, men exhibited amplification of negativity at the prenatal assessment that did not change at the postnatal assessment; women demonstrated no significant damping or amplification at pretest and a marginally significant change towards greater amplification at the postnatal assessment. In terms of partner-influenced regulatory dynamics, men's positive behaviors changed from damping to amplifying women's negative behaviors in the control group following the transition to parenthood, but exerted an even stronger damping effect on women's negative behaviors in the intervention group. The study highlights the advantages of dynamic modeling approaches in testing specific hypotheses in the study of self- and co-regulatory couple dynamics and demonstrates the potential of studying dynamic processes to further understanding of developmental and intervention-related change mechanisms.

Dynamical systems modeling of early childhood self-regulation

Self-regulation can be conceptualized in terms of dynamic tension between highly probable reactions (prepotent responses) and use of strategies that can modulate those reactions (executive processes). This study investigated the value of a dynamical systems approach to the study of early childhood self-regulation. Specifically, ordinary differential equations (ODEs) were used to model the interactive influences of 115 36-month-olds' executive processes (strategy use) and prepotent responses to waiting to open a gift (desire for the gift and frustration about waiting to open it). Using a pair of coupled second-order ODEs in a nonlinear mixed effects framework, the study tested predictions for specific within- and between-child patterns of prepotent response-executive process coupling. Dynamic modeling results articulated the limits of 36-month olds' strategic efforts. They engaged executive processes when their prepotent responding levels were high, which delayed the resurgence of prepotent responses, but ultimately did not damp prepotent responding over the course of the wait. There was, however, preliminary evidence that the effectiveness of 36-month-olds' self-regulation depended upon child characteristics. Externalizing behavior problems were associated with more regulatory interference. Temperamental negative affectivity was marginally associated with more regulatory inefficiency. Compared with conventional methods of studying self-regulation, dynamic modeling yielded complementary and unique findings, suggesting its potential. (PsycINFO Database Record

As Good as GOLD: Gram-Schmidt Orthogonalization by Another Name

Generalized orthogonal linear derivative (GOLD) estimates were proposed to correct a problem of correlated estimation errors in generalized local linear approximation (GLLA). This paper shows that GOLD estimates are related to GLLA estimates by the Gram-Schmidt orthogonalization process. Analytical work suggests that GLLA estimates are derivatives of an approximating polynomial and GOLD estimates are linear combinations of these derivatives. A series of simulation studies then further investigates and tests the analytical properties derived. The first study shows that when approximating or smoothing noisy data, GLLA outperforms GOLD, but when interpolating noisy data GOLD outperforms GLLA. The second study shows that when data are not noisy, GLLA always outperforms GOLD in terms of derivative estimation. Thus, when data can be smoothed or are not noisy, GLLA is preferred whereas when they cannot then GOLD is preferred. The last studies show situations where GOLD can produce biased estimates. In spite of these possible shortcomings of GOLD to produce accurate and unbiased estimates, GOLD may still provide adequate or improved model estimation because of its orthogonal error structure. However, GOLD should not be used purely for derivative estimation because the error covariance structure is irrelevant in this case. Future research should attempt to find orthogonal polynomial derivative estimators that produce accurate and unbiased derivatives with an orthogonal error structure.

A Comparison of Two-Stage Approaches for Fitting Nonlinear Ordinary Differential Equation Models with Mixed Effects

Several approaches exist for estimating the derivatives of observed data for model exploration purposes, including functional data analysis (FDA; Ramsay & Silverman, 2005 ), generalized local linear approximation (GLLA; Boker, Deboeck, Edler, & Peel, 2010 ), and generalized orthogonal local derivative approximation (GOLD; Deboeck, 2010 ). These derivative estimation procedures can be used in a two-stage process to fit mixed effects ordinary differential equation (ODE) models. While the performance and utility of these routines for estimating linear ODEs have been established, they have not yet been evaluated in the context of nonlinear ODEs with mixed effects. We compared properties of the GLLA and GOLD to an FDA-based two-stage approach denoted herein as functional ordinary differential equation with mixed effects (FODEmixed) in a Monte Carlo (MC) study using a nonlinear coupled oscillators model with mixed effects. Simulation results showed that overall, the FODEmixed outperformed both the GLLA and GOLD across all the embedding dimensions considered, but a novel use of a fourth-order GLLA approach combined with very high embedding dimensions yielded estimation results that almost paralleled those from the FODEmixed. We discuss the strengths and limitations of each approach and demonstrate how output from each stage of FODEmixed may be used to inform empirical modeling of young children's self-regulation.

Modeling intensive longitudinal data with mixtures of nonparametric trajectories and time-varying effects

Behavioral scientists increasingly collect intensive longitudinal data (ILD), in which phenomena are measured at high frequency and in real time. In many such studies, it is of interest to describe the pattern of change over time in important variables as well as the changing nature of the relationship between variables. Individuals' trajectories on variables of interest may be far from linear, and the predictive relationship between variables of interest and related covariates may also change over time in a nonlinear way. Time-varying effect models (TVEMs; see Tan, Shiyko, Li, Li, & Dierker, 2012) address these needs by allowing regression coefficients to be smooth, nonlinear functions of time rather than constants. However, it is possible that not only observed covariates but also unknown, latent variables may be related to the outcome. That is, regression coefficients may change over time and also vary for different kinds of individuals. Therefore, we describe a finite mixture version of TVEM for situations in which the population is heterogeneous and in which a single trajectory would conceal important, interindividual differences. This extended approach, MixTVEM, combines finite mixture modeling with non- or semiparametric regression modeling, to describe a complex pattern of change over time for distinct latent classes of individuals. The usefulness of the method is demonstrated in an empirical example from a smoking cessation study. We provide a versatile SAS macro and R function for fitting MixTVEMs.

A control systems engineering approach for adaptive behavioral interventions: illustration with a fibromyalgia intervention

The term adaptive intervention has been used in behavioral medicine to describe operationalized and individually tailored strategies for prevention and treatment of chronic, relapsing disorders. Control systems engineering offers an attractive means for designing and implementing adaptive behavioral interventions that feature intensive measurement and frequent decision-making over time. This is illustrated in this paper for the case of a low-dose naltrexone treatment intervention for fibromyalgia. System identification methods from engineering are used to estimate dynamical models from daily diary reports completed by participants. These dynamical models then form part of a model predictive control algorithm which systematically decides on treatment dosages based on measurements obtained under real-life conditions involving noise, disturbances, and uncertainty. The effectiveness and implications of this approach for behavioral interventions (in general) and pain treatment (in particular) are demonstrated using informative simulations.

A dynamical systems approach to understanding self-regulation in smoking cessation behavior change

INTRODUCTION

Self-regulation, a key component of the addiction process, has been challenging to model precisely in smoking cessation settings, largely due to the limitations of traditional methodological approaches in measuring behavior over time. However, increased availability of intensive longitudinal data (ILD) measured through ecological momentary assessment facilitates the novel use of an engineering modeling approach to better understand self-regulation.

METHODS

Dynamical systems modeling is a mature engineering methodology that can represent smoking cessation as a self-regulation process. This article shows how a dynamical systems approach effectively captures the reciprocal relationship between day-to-day changes in craving and smoking. Models are estimated using ILD from a smoking cessation randomized clinical trial.

RESULTS

A system of low-order differential equations is presented that models cessation as a self-regulatory process. It explains 87.32% and 89.16% of the variance observed in craving and smoking levels, respectively, for an active treatment group and 62.25% and 84.12% of the variance in a control group. The models quantify the initial increase and subsequent gradual decrease in craving occurring postquit as well as the dramatic quit-induced smoking reduction and postquit smoking resumption observed in both groups. Comparing the estimated parameters for the group models suggests that active treatment facilitates craving reduction and slows postquit smoking resumption.

CONCLUSIONS

This article illustrates that dynamical systems modeling can effectively leverage ILD in order to understand self-regulation within smoking cessation. Such models quantify group-level dynamic responses in smoking cessation and can inform the development of more effective interventions in the future.