Continuous-Time Hidden Markov Factor Model for Mobile Health Data: Application to Adverse Posttraumatic Neuropsychiatric Sequelae

Abstract

Each year, a significant portion of the 40 million individuals in the United States who seek care in emergency departments (EDs) following traumatic experiences develop adverse posttraumatic neuropsychiatric sequelae (APNS). This highlights the widespread impact of trauma and the critical need for effective interventions to address the health outcomes of these events. Despite significant research efforts, advancements in understanding the neurobiological mechanisms of APNS have been hindered, primarily due to reliance on subjective self-reports, which are susceptible to recall biases and careless responses. To overcome this limitation, we investigate the use of objective, longitudinal mobile device data to identify consistent APNS states and examine their dynamic transitions over time. To address the complexity of longitudinal mobile data, we developed a continuous-time hidden Markov factor model and applied it to mobile device data from the Advancing Understanding of Recovery after Trauma (AURORA) study. Findings from this study highlight the great potential of leveraging mobile device data for continuous health monitoring and for guiding personalized treatment approaches through mobile health initiatives. A Python implementation of the proposed method is available at https://anonymous.4open.science/r/CTHMFM.

I. Introduction

Adverse posttraumatic neuropsychiatric sequelae (APNS) (e.g., pain, depression, and PTSD) are frequently observed in civilians and military veterans who have experienced traumatic events. These APNS increase the risk of chronic illnesses, including cancer and heart disease, and substantially contribute to drug abuse, suicide, and disability. Moreover, APNS impose enduring psychosocial and financial burdens not only on individuals with the disorder but also on their families, communities, and society as a whole. However, little progress has been made in advancing APNS research over the past few decades due to unique challenges. First, APNS have been evaluated through subjective self-reported measures, which lack objective reliability. Second, the heterogeneity among patients, as recognized in traditional classification and diagnoses, complicates the study of APNS. Lastly, these APNS disorders are often studied and treated independently, despite their frequent co-occurrence [1]. These obstacles hinder the identification of objective markers, the advancement in understanding the neurobiological mechanisms of APNS, and the development of effective preventative/treatment strategies.

Identifying homogeneous states and exploring the dynamic prognosis of APNS in the immediate aftermath of trauma exposure holds promise for enhancing our understanding of APNS and identifying effective intervention options and appropriate timing at the individual level. Regrettably, due to the lack of appropriate data and effective statistical method, no large-scale studies have been conducted to investigate the onset, dynamic transitions (such as recovery and relapse), and associated risk factors of APNS. To help address the challenges, the National Institutes of Mental Health, joined by the US Army Medical Research and Material Command, developed the Advancing Understanding of RecOvery afteR traumA (AURORA) study [1]. This study gathered biobehavioral data from a large cohort of trauma survivors (n = 2,997) across the United States over a year, including self-reported surveys, web-based neurocognitive tests, digital phenotyping data (i.e., from wrist wearables and smartphones), psychophysical tests, neuroimaging assessments, and genomics data. In contrast to previous studies relying on self-report surveys or neuroimages [2], our work aims to address the challenges of APNS by utilizing the objective digital phenotyping data that tracks individuals’ behaviors, moods, and health statuses in real-time, real-life environments. Specifically, we developed a Hidden Markov Factor Model (HMFM) to analyze the mobile device data, allowing us to simultaneously identify homogeneous subtypes, investigate subtype-specific structures, and model individual progression with associated risk factors.

Hidden Markov Models (HMMs) have been widely used in various fields [3]. However, mobile device data presents two unique challenges that standard HMMs cannot handle, including the interdependent variables with unknown interrelationship structures and unevenly spaced measurements.

Mobile device sensor data, such as accelerometer and photoplethysmography (PPG) from smartwatches, involves highly intensive time series data. Instead of being directly used, these data are typically pre-processed to generate technical summaries that represent various characteristics of each time series variable, which are often highly correlated. As the number of features increases, the parameters in the covariance matrix expand exponentially, making a fully free covariance matrix impractical. Consequently, when implementing HMMs, features are usually assumed to be independent given the latent state membership, despite their correlation, potentially introducing estimation bias.

To appropriately model the association between features, factor analysis models (FMs) [4] provide an efficient and parsimonious approach and have been incorporated into HMMs in various ways. For example, the factor analyzed hidden Markov model [5] combines an FM with a discrete-time HMM (DTHMM), which assumes evenly spaced measurements. It has been extensively used in a variety of real-world applications, including speech recognition [6], environmental protection [7], and seizure detection [8]. Similarly, [9] introduced the regime-switching factor model to handle high-dimensional financial market data. However, they all assume homogeneous transition probability matrices, limiting their ability to account for the heterogeneity of transition probabilities over time and among different subjects and explore risk factors of state transition. To simultaneously capture the interrelationships among observed features and account for the variability of transition probabilities, a joint framework incorporating HMM, FM, and a feature-based transition model was recently proposed [10], [11]. However, it is not directly applicable to mobile device data. First, it employs a confirmatory factor model (CFM) with pre-specified structures for the factor loading matrices, which are often unknown a priori. Therefore, an exploratory factor model (EFM) is needed to explore the interrelationships among all observed features. Second, their framework assumes ordered states, which is inappropriate for our use case.

Another challenge posed by mobile device data is the irregular spacing of measurements. For example, activity and heart rate variability (HRV) data were collected only when the participants wore the watches, resulting in non-uniformly spaced observations and significant variation in sampling schedules between individuals. While the aforementioned methods are all based on DTHMM, neglecting the impact of time gaps between consecutive observations on transition rates, continuous-time discrete-state HMM (CTHMM) was developed to handle irregularly spaced measurements [12]. CTHMM and its extensions that incorporate covariates to characterize transition rates are widely used in medical research [13], [14]. However, none of them address the interrelationships among features.

In this paper, to simultaneously address the two challenges and examine heterogeneous transition patterns, we propose to use Continuous-Time HMFM (CT-HMFM), integrating CTHMM, EFM, and a feature-based transition model. Our contributions are as follows: First, we examine the utility of data collected in an open environment from consumer-grade mobile devices for mental health research. This contrasts with most existing studies on data collected in controlled lab environments. Second, we propose CT-HMFM to address the unique challenges introduced by mobile device data and depict the non-homogeneous state transition processes of multiple individuals. Simulation studies using synthetic data demonstrate exceptional parameter estimation and model selection performance. Finally, we analyze HRV and activity data from the AURORA study, followed by interpretations and discussions of biological findings that highlight the immense potential of mobile health data and our proposed method for mental health research.

II. AURORA Dataset

In the AURORA study [1], data were collected from multiple sources, with our analysis focusing on the accelerometer and PPG data collected by the Verily smartwatches. Given the known associations with APNS [15], [16], the accelerometer data were used to quantify physical activity features, while the PPG data were pre-processed to extract heart rate variability (HRV) metrics (for details on the raw data and pre-processing steps, refer to [17]).

Briefly, activity features are extracted during a 24-hour window to evaluate daily activity patterns. After converting accelerometer data to activity counts [18], meanAcc is the average activity counts. Amplitude, a cosinor rhythmometry feature, is computed to capture circadian rhythms [19]. Using the Cole-Kripke algorithm [20], accelerometry epochs are classified into wake or sleep states, and the SWCK quantifies transition rates between wake and sleep. Additionally, the average activity during the five least active hours (L5) is calculated from raw accelerometer data [21], representing nighttime activity.

HRV features were derived from PPG data by first calculating and denoising beat-to-beat (BB) interval [22] time series to obtain normal-to-normal (NN) intervals, which were then analyzed using a 5-minute sliding window. Selected HRV features for this study include average heart rate (NNmean), skewness (NNskew), and standard deviation (SDNN) of NN intervals, indicating heart rate variability and rapid changes. Lfhf is the low-frequency to high-frequency power ratio, with extremes suggesting parasympathetic or sympathetic dominance [22]. DC serves as a mortality risk indicator in cardiac conditions [23]. The lower the DC index, the higher the mortality risk. Additionally, SD1SD2 and ApEn are used to quantify the unpredictability and regularity of successive heartbeats (R-R interval). To match the activity data, daily statistical summaries of each HRV feature are used, including mean, minimum, maximum, interquartile range, and variance.

In this study, we focused on survivors of motor vehicle collision trauma. Since the data collection depends on the participants’ wearing of the devices, missing data is a common issue. To assure data quality, we include only records that have complete activity data and a positive wake percentage. For HRV data, ideally, an individual can have 2,880 records per day. We retain only those days where at least 30% of these records are available to ensure that our daily summary statistics are representative. The final dataset consists of daily summaries of activity and HRV features from 258 patients, with each providing at least 50 days of records.

III. CT-HMFM

Motivated by the structures of the processed AURORA datasets, we focus on data consisting of repeated measurements of $p$ features collected over $T_i$ time points for each individual $i$ among $N$ subjects. Let ${\mathit{y}}_{\mathit{\text{it}}}$ denote the observed values of the $p$ features for subject $i$ at time $t$, and let ${\mathit{Y}}_i=\left({\mathit{y}}_{i_1},\cdots ,{\mathit{y}}_{i{T}_i}\right)$ denote the $p\times T_i$ matrix containing all measurements for individual $i$. To analyze these longitudinal data, we adopt a HMM framework, which facilitates the inference of latent states and their dynamic transitions over time. Specifically, we assume that the latent state of individual $i$ at time $t$, denoted by $w_{\mathit{\text{it}}}$, takes values from a finite discrete set $\{1,\cdots ,J\}$. At $t=1$, the initial state is assumed to follow a multinomial distribution with probabilities $\mathit{\pi }={\left({\pi }_1,\cdots ,{\pi }_J\right)}^{\prime }$, such that ${\sum }_{i=1}^J{\pi }_i=1$. Transition probabilities between theses states at time point $t\ge 2$ are captured by a $J\times J$ transition probability matrix ${\mathit{P}}_{\mathit{\text{it}}}$, of which the $(k,j)$-th entry is ${\mathit{P}}_{\mathit{\text{it}},\mathit{\text{kj}}}=P\left(w_{\mathit{\text{it}}}=j\mid w_{i,t-1}=k\right)$, and ${\mathit{P}}_{\mathit{\text{it}},\mathit{\text{kk}}}=1-{\sum }_{j:j\ne k}{\mathit{P}}_{\mathit{\text{it}},\mathit{\text{kj}}}$. The primary goal is to estimate the $\mathit{\pi }$ and ${\mathit{P}}_{\mathit{\text{it}}}$, thereby delineating latent Markov processes underlying the observed data. To meet our specific objectives of analyzing the mobile health data, we extend the traditional HMM framework with two additional components. The first, detailed in Section III-A, is a state-specific measurement model using EFM to uncover interrelationships among variables. The second, outlined in Section III-B, is a transition model (TM) designed to capture heterogeneous transition patterns.

A. State-Specific Measurement Model

The first component is an FM, aimed at identifying the interrelationship structures between observed response variables and the underlying latent variables. For individual $i$ at time $t$, given that $w_{\mathit{\text{it}}}=j$, the FM assumes that:

$$\begin{gathered} \left[{\mathit{y}}_{\mathit{\text{it}}}\mid w_{\mathit{\text{it}}}=j\right]={\mathit{\mu }}_j+{\mathit{\Lambda }}_j{\mathit{z}}_{\mathit{\text{it}}}+{\mathit{e}}_{\mathit{\text{it}}}, \\ {\mathit{z}}_{\mathit{\text{it}}}\stackrel{\mathit{\text{i.i.d.}}}{~}𝒩\left(\mathbf{0},{\mathit{I}}_K\right),{\mathit{e}}_{\mathit{\text{it}}}\stackrel{\mathit{\text{i.i.d.}}}{~}𝒩(\mathbf{0},\mathit{\Psi }),{\mathit{z}}_{\mathit{\text{it}}}⫫{\mathit{e}}_{\mathit{\text{it}}}, \end{gathered}$$ (1)

where ${\mathit{\mu }}_j$ is a $p\times 1$ vector of state-specific expected mean response, ${\mathit{\Lambda }}_j$ is a $p\times K$ state-specific factor loading matrix, ${\mathit{z}}_{\mathit{\text{it}}}$ is a $K$-dimensional vector of latent scores assumed to be independent of $w_{\mathit{\text{it}}}$ and following a standard multivariate normal distribution, and $\mathit{\Psi }$ is a $p\times p$ diagonal covariance matrix for the error term $e_{\mathit{\text{it}}}$ with positive nonconstant diagonal entries. Alternatively, (1) can also be expressed as

$$\left[{\mathit{y}}_{\mathit{\text{it}}}\mid w_{\mathit{\text{it}}}=j\right]\stackrel{\mathit{\text{i.i.d.}}}{~}𝒩\left({\mathit{\mu }}_j,{\mathit{\Lambda }}_j{\mathit{\Lambda }}_j^{\prime }+\mathit{\Psi }\right).$$ (2)

It is crucial to emphasize that, unlike CFMs that require pre-specified structures of the factor loading matrices, our approach imposes no assumptions on ${\mathit{\Lambda }}_j$. Consequently, the structure of ${\mathit{\Lambda }}_j$ is entirely data-driven, making the first component (1) an EFM.

B. Transition Model

To account for the effects of varying time intervals between observations, a continuous-time Markov process is used. Instead of directly relying on the transition probability matrix $\mathit{P}$, the continuous-time Markov process is defined by a transition intensity matrix $\mathit{Q}$ [24], which is the limit of $\mathit{P}$ as the time interval approaches zero. Specifically, let ${\delta }_{\mathit{\text{it}}}$ denote the number of pre-specified time units between $t^{\mathit{\text{th}}}$ and $(t-1{)}^{\mathit{\text{th}}}$ observation, the transition intensity from state $j$ to state $k$ for subject $i$ at time $t$ is given by:

$$q_{\mathit{\text{jk}}}=\underset{{\delta }_{\mathit{\text{it}}}\to 0}{\text{lim}}\frac{P\left(w_{\mathit{\text{it}}}=k\mid w_{i,t-1}=j\right)}{{\delta }_{\mathit{\text{it}}}}>0,j\ne k,$$ (3)

and $q_{\mathit{\text{jj}}}=-\sum _{k\ne j}{q}_{\mathit{\text{jk}}}$. The corresponding transition probability $\mathit{P}\left({\delta }_{\mathit{\text{it}}}\right)$ can then be calculated as the matrix exponential of ${\delta }_{\mathit{\text{it}}}*\mathit{Q}$. Since computing the matrix exponential directly can be computationally intensive, in practice, we approximate the $\text{exp}\left(\mathit{Q}\right)$ using $(\mathit{I}+\mathit{Q}/a{)}^a$ for a sufficiently large $a$ [25].

To investigate the impact of covariates on transition rates, the $\mathit{Q}$ is modeled with a log-linear model [26], such that

$$\mathit{\text{log}}\left(q_{\mathit{\text{jk}}}\mid {\mathit{x}}_{\mathit{\text{it}}}\right)={\mathit{x}}_{\mathit{\text{it}}}^{\prime }{\mathit{B}}_{\mathit{\text{jk}}},$$ (4)

where ${\mathit{x}}_{\mathit{\text{it}}}$ is a $d\times 1$ vector of covariates, and ${\mathit{B}}_{\mathit{\text{jk}}}$ is a $d\times 1$ vector of state-specific fixed effects coefficients, quantifying the effect of covariates on the transition intensity from state $j$ to state $k$.

IV. Expectation-Maximization Algorithm

Let $\mathit{\lambda }=\left({\left\{{\mathit{\mu }}_j\right\}}_{j=1}^J,{\left\{{\mathit{\Lambda }}_j\right\}}_{j=1}^J,\mathit{\Psi },{\left\{{\mathit{B}}_{\mathit{\text{kj}}}\right\}}_{k,j=1}^J,\mathit{\pi }\right),{\mathit{W}}_i=\left(w_{i1},\cdots ,w_{i{T}_i}\right)$, and ${\mathit{Z}}_i=\left({\mathit{z}}_{i1},\cdots ,{\mathit{z}}_{\mathit{\text{it}}}\right)$. Under the standard assumptions of HMMs–(1) given the current state $w_{\mathit{\text{it}}}$, the observations ${\mathit{y}}_{\mathit{\text{it}}}$ are conditionally independent, and (2) given $w_{\mathit{\text{it}}}$ and the subject’s contextual features, the subsequent state $w_{i,t+1}$ is independent of any historical information–the joint probability distribution of the observations and all latent variables, $L_{\mathit{\text{ci}}}\left(\mathit{\lambda }\right)$, is constructed as follows:

$$P\left(w_{i1}\right)\cdot \prod _{t=2}^{T_i}P\left(w_{\mathit{\text{it}}}\mid w_{i,t-1},{\mathit{x}}_{\mathit{\text{it}}}\right)\cdot \prod _{t=1}^{T_i}P\left({\mathit{y}}_{\mathit{\text{it}}}\mid w_{\mathit{\text{it}}},{\mathit{z}}_{\mathit{\text{it}}}\right)P\left({\mathit{z}}_{\mathit{\text{it}}}\right).$$ (5)

Assuming the independence of ${\mathit{Y}}_i,{\mathit{W}}_i$, and ${\mathit{Z}}_i$ across individuals, the complete likelihood function $\left(L_c\right)$ for the entire sample is obtained as the product of (5) over $i$. To estimate $\mathit{\lambda }$, the expectation-maximization (EM) algorithm is used, as both ${\mathit{W}}_i$ and ${\mathit{Z}}_i$ are unobserved. Specifically, the EM algorithm iteratively applies the expectation and maximization steps to find a local maximum of the marginal likelihood, as detailed below.

A. Expectation Step (E-step)

The E-step computes the expected value of the logarithm of $L_c$, given the observations and the current parameter estimates ${\mathit{\lambda }}^v$. Directly computing this expectation, $\left.E_{{\mathit{\lambda }}^v}\left[\mathit{\text{log}}{L}_c\left(\mathit{\lambda }\right)\mid \mathit{Y},\mathit{X}\right]\right)$, is challenging due to the computational intensity in calculating the involved conditional state probabilities, including

$${ϵ}_{i\mathit{\text{kj}}}^v(t)≔P\left(w_{i,t}=j,w_{i,t-1}=k\mid {\mathit{Y}}_i,{\mathit{\lambda }}^v\right)$$ (6)

and

$${\gamma }_{\mathit{\text{ij}}}^v\left(t\right)≔P\left(w_{\mathit{\text{it}}}=j\mid {\mathit{Y}}_i,{\mathit{\lambda }}^v\right).$$ (7)

To efficiently compute these probabilities, we utilize a scaled version of the forward-backward algorithm (FBA) [27].

Specifically, we first calculate the forward probability ${\alpha }_{\mathit{\text{ij}}}\left(t\right)=P\left(w_{\mathit{\text{it}}}=j\mid {\mathit{y}}_{i1},\cdots ,{\mathit{y}}_{\mathit{\text{it}}}\right)$. Let $P_j\left({\mathit{y}}_{\mathit{\text{it}}}\right)$ be the probability density of ${\mathit{y}}_{\mathit{\text{it}}}$ given $w_{\mathit{\text{it}}}=j$, and $c_i\left(t\right)$ the conditional probability of observation ${\mathit{y}}_{\mathit{\text{it}}}$ given all past observations. Omitting the dependence on ${\delta }_{\mathit{\text{it}}}$ for brevity, the forward probabilities are computed recursively as:

$${\alpha }_{\mathit{\text{ij}}}\left(1\right)=\frac{{\pi }_j{P}_j\left({\mathit{y}}_{i1}\right)}{\sum _{j=1}^J{\pi }_j{P}_j\left({\mathit{y}}_{i1}\right)}=\frac{{\pi }_j{P}_j\left({\mathit{y}}_{i1}\right)}{c_i\left(1\right)};$$ (8)

$${\alpha }_{\mathit{\text{ij}}}(t)=\frac{P_j\left({\mathit{y}}_{\mathit{\text{it}}}\right)\left[\sum _{k=1}^J{\alpha }_{\mathit{\text{ik}}}(t-1){\mathit{P}}_{\mathit{\text{itkj}}}\right]}{c_i\left(t\right)},$$ (9)

where $c_i(t)=\sum _{j=1}^J{P}_j\left({\mathit{y}}_{\mathit{\text{it}}}\right)\left[\sum _{k=1}^J{\alpha }_{\mathit{\text{ik}}}(t-1){\mathit{P}}_{\mathit{\text{itkj}}}\right]$. Similarly, the backward probability ${\beta }_{\mathit{\text{ij}}}\left(t\right)$, defined as $\frac{P\left({\mathit{y}}_{i,t+1},\cdots ,{\mathit{y}}_{i,T_i}\mid w_{\mathit{\text{it}}}=j,\mathit{\lambda }\right)}{c_i(t+1)}$, is also obtained recursively as:

$${\beta }_{\mathit{\text{ij}}}\left(T_i\right)=1,$$ (10)

$${\beta }_{\mathit{\text{ij}}}(t)=\frac{\sum _{k=1}^J{\mathit{P}}_{i,t+1,\mathit{\text{jk}}}{P}_k\left({\mathit{y}}_{i,t+1}\right){\beta }_{\mathit{\text{ik}}}(t+1)}{c_i(t+1)}.$$ (11)

Using the forward and backward probabilities, the conditional state probabilities are derived as:

$${\gamma }_{\mathit{\text{ij}}}^v\left(t\right)={\alpha }_{\mathit{\text{ij}}}\left(t\right){\beta }_{\mathit{\text{ij}}}\left(t\right),$$ (12)

$${ϵ}_{i\mathit{\text{kj}}}^v(t)=\frac{{\alpha }_{\mathit{\text{ik}}}^v(t-1){\mathit{P}}_{\mathit{\text{itkj}}}{P}_j\left({\mathit{y}}_{\mathit{\text{it}}}\right){\beta }_{\mathit{\text{ij}}}^v\left(t\right)}{c_i\left(t\right)}.$$ (13)

With these, $\left.E_{{\mathit{\lambda }}^v}\left[\mathit{\text{log}}{L}_c\left(\mathit{\lambda }\right)\mid \mathit{Y},\mathit{X}\right]\right)$ can be expressed explicitly. While detailed expressions are provided in Appendix I-A, the resulted expression can be summarized as:

$$\mathit{\text{constant}}+h\left(\mathit{\pi }\right)+h\left({\left\{{\mathit{B}}_{\mathit{\text{kj}}}\right\}}_{k,j=1}^J\right)-\frac{1}{2}h\left(\mathit{\Psi },{\left\{{\mathit{\Lambda }}_j\right\}}_{j=1}^J,{\left\{{\mathit{\mu }}_j\right\}}_{j=1}^J\right),$$ (14)

where $h\left(\mathit{\pi }\right)$ depends on the initial state distribution, $h\left({\left\{{\mathit{B}}_{\mathit{\text{kj}}}\right\}}_{k,j=1}^J\right)$ depends on the probability transition matrix, and $h\left(\mathit{\Psi },{\left\{{\mathit{\Lambda }}_j\right\}}_{j=1}^J,{\left\{{\mathit{\mu }}_j\right\}}_{j=1}^J\right)$ is a function of parameters $\mathit{\Psi },{\mathit{\Lambda }}_j$, and ${\mathit{\mu }}_j$.

B. Maximization (M-step)

Within each M-step, since $h\left(\mathit{\Psi },{\left\{{\mathit{\Lambda }}_j\right\}}_{j=1}^J,{\left\{{\mathit{\mu }}_j\right\}}_{j=1}^J\right),h\left(\mathit{\pi }\right)$, and $h\left({\left\{{\mathit{B}}_{\mathit{\text{kj}}}\right\}}_{k,j=1}^J\right)$ do not share parameters, we maximize each of them separately. The estimator of $\mathit{\pi },{\mathit{\Lambda }}_j,{\mathit{\mu }}_j$, and $\mathit{\Psi }$ are directly derived by setting $h\left(\mathit{\pi }\right)=0$ and $h\left(\mathit{\Psi },{\left\{{\mathit{\Lambda }}_j\right\}}_{j=1}^J,{\left\{{\mathit{\mu }}_j\right\}}_{j=1}^J\right)=0$ (see Appendix I-B for details). For ${\left\{{\mathit{B}}_{\mathit{\text{kj}}}\right\}}_{k,j=1}^J$, a one-step Fisher scoring (FS) [28] algorithm is implemented to optimize the parameters.

Let $\mathit{\theta }$ be the ordered vector of all transition model parameters, such that $\mathit{\theta }=\mathit{\text{vec}}\left(\left\{{\mathit{B}}_{\mathit{\text{kj}}}^{\prime }\right\},k\ne j\right)$, with ${\theta }_u$ representing the $u^{\mathit{\text{th}}}$ entry of $\mathit{\theta }$. Recalling the first derivative of the matrix exponential [14] and using Theorem 1 in [29], the derivation of $\mathit{\text{exp}}\left(\mathit{A}\left({\theta }_u\right)\right)$ with respect to ${\theta }_u$ is given by:

$$\frac{\partial }{\partial {\theta }_u}\mathit{\text{ex}}p\left(\mathit{A}\left({\theta }_u\right)\right)=\mathit{\text{ex}}p{\left(\left[\begin{array}{cc}\mathit{A}\left({\theta }_u\right)& \tilde{\mathit{A}}\left({\theta }_u\right)\\ 0& \mathit{A}\left({\theta }_u\right)\end{array}\right]\right)}_{0:J,J:2J},$$ (15)

where $\tilde{\mathit{A}}\left({\theta }_u\right)=\left({\tilde{A}}_{\mathit{\text{ij}}}\left({\theta }_u\right)\right)=\left(\frac{\partial A_{\mathit{\text{ij}}}\left({\theta }_u\right)}{\partial {\theta }_u}\right)$. Let $\frac{\partial {\mathit{P}}_{\mathit{\text{kj}}}\left({\delta }_{\mathit{\text{it}}}\right)}{\partial {\theta }_u}$ denote the $(k,j)$ entry of the first derivative of $\mathit{P}\left({\delta }_{\mathit{\text{it}}}\right)$ with respect to ${\theta }_u$. Having the first derivative of $\mathit{P}\left({\delta }_{\mathit{\text{it}}}\right)=\mathit{\text{exp}}\left({\delta }_{\mathit{\text{it}}}*\mathit{Q}\right)$ with respect to each component of $\mathit{\theta }$ calculated accordingly, the FS can then be directly implemented to update $\mathit{\theta }$ without requiring the second derivative of the matrix exponential. Specifically, let ${\mathit{S}}^{*}$ be the score function, defined as

$${\mathit{S}}_u^{*}\left(\mathit{\theta }\right)=\sum _{i=1}^N\sum _{t=2}^{T_i}\sum _{j=1}^J\sum _{k=1}^J\frac{{ϵ}_{i\mathit{\text{kj}}}^v\left(t\right)}{{\mathit{P}}_{\mathit{\text{kj}}}\left({\delta }_{\mathit{\text{it}}}\right)}\frac{\partial {\mathit{P}}_{\mathit{\text{kj}}}\left({\delta }_{\mathit{\text{it}}}\right)}{\partial {\theta }_u}.$$ (16)

Let ${\mathit{M}}^{*}$ denote the negative Fisher information matrix, whose $(u,v)$ entry is given by:

$${\mathit{M}}_{\mathit{\text{uv}}}^{*}\left(\mathit{\theta }\right)=\sum _{i=1}^N\sum _{t=2}^{T_i}\sum _{j=1}^J\sum _{k=1}^J\frac{{\gamma }_{\mathit{\text{ik}}}^v\left(t-1\right)}{{\mathit{P}}_{\mathit{\text{kj}}}\left({\delta }_{\mathit{\text{it}}}\right)}\frac{\partial {\mathit{P}}_{\mathit{\text{kj}}}\left({\delta }_{\mathit{\text{it}}}\right)}{\partial {\theta }_u}\frac{\partial {\mathit{P}}_{\mathit{\text{kj}}}\left({\delta }_{\mathit{\text{it}}}\right)}{\partial {\theta }_v}.$$ (17)

With the score function and the Fisher information matrix, the parameters $\mathit{\theta }$ can be updated as

$${\mathit{\theta }}^{v+1}={\mathit{\theta }}^v+{\mathit{M}}^{*}{\left({\mathit{\theta }}^v\right)}^{-1}{\mathit{S}}^{*}\left({\mathit{\theta }}^v\right).$$ (18)

To ensure numerical stability, we control the learning rate in practice by updating

$${\mathit{\theta }}^{v+1}={\mathit{\theta }}^v+{\left\{{\mathit{M}}^{*}\left({\mathit{\theta }}^v\right)+{\mathit{S}}^{*}{\left({\mathit{\theta }}^v\right)}^T{\mathit{S}}^{*}\left({\mathit{\theta }}^v\right)\right\}}^{-1}{\mathit{S}}^{*}\left({\mathit{\theta }}^v\right).$$ (19)

Note that the algorithm requires specification of $(K,J)$, which are typically unknown in practice. In this study, we use information criteria to determine $(K,J)$, the efficacy of which is evaluated in Section V-C.

V. Simulation Study

This section assesses the proposed methods through simulation studies using synthetic data resembling the AURORA dataset. We generate data with $N=200,p=23,d=3,J=3$, and $K=3$. Each individual’s number of observations, $T_i$, is uniformly sampled from [50, 100]. From this, we randomly select $T_i$ time points from $\{1,\cdots ,100\}$ to create sequences of ${\delta }_{\mathit{\text{it}}}$. Each individual’s initial state is drawn from a multinomial distribution with probabilities $\mathit{\pi }=\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$. Latent state trajectories are then generated based on transition probabilities ${\mathit{P}}_{\mathit{\text{it}}}\left({\delta }_{\mathit{\text{it}}}\right)$, given individuals’ features. Observation vectors ${\mathit{y}}_{\mathit{\text{it}}}$ for an individual $i$ in state $j$ at time $t$ are then sampled from a normal distribution with mean ${\mathit{\mu }}_j$ and covariance ${\mathit{\Lambda }}_j{\mathit{\Lambda }}_j^{\prime }+\mathit{\Psi }$, where $\mathit{\Psi }=\mathit{I}$.

In the following, subsection V-A assesses model reliability by comparing empirical parameter estimates against true values; subsection V-B compares the effectiveness of our method against baseline methods; and subsection V-C explores the performance of information criteria in model selection.

A. Simulation 1

To validate the estimation procedure, we implement the EM algorithm with true $J$ and $K$. Parameter initialization involves first fitting Gaussian Mixture Models to estimate groups, followed by EFM for each group. Guided by the insights from a pilot study, we set the maximum number of iterations for each replication at 100. The reliability and precision of the proposed methods are then evaluated from two perspectives: i) the accuracy of each individual parameter estimate and ii) the misclassification rate $\left(C_{\mathit{\text{mis}}}\right)$, which quantifies the proportion of estimated states that diverge from the actual states.

The accuracy of parameters $\mathit{\pi },\mathit{\mu },\mathit{\Lambda }$, and $\mathit{\Psi }$ is assessed by calculating the average absolute difference (AAD) between the estimates and their true values, defined as $\mathit{\text{AAD}}(\mathit{o})=\frac{{\sum }_{i=1}^r\left|\widehat{o_i}-o_i\right|}{r},o_i$ is an individual entry in matrix $\mathit{o}$ and $r$ is the total number of free parameters. The mean of AADs (standard errors in the parentheses) aggregated over 100 random seeds are presented in Table I. These mean AADs for all parameter matrices are sufficiently close to zero with small standard errors, indicating effective parameter recovery. In Table II, we report the mean bias (standard error) for each parameter in the transition model, where the biases are all close to zero. Moreover, we present the mean (standard error) of $C_{\mathit{\text{mis}}}$ in Table I. On average, misclassification rates are only .24% (0.0005), highlighting the exceptional accuracy of the proposed EM algorithm in estimating latent states.

TABLE I.

The Mean (standard error) AADs of $\mathit{\pi },\mathit{\mu },\mathit{\Lambda },\mathit{\Psi }$, and ${\mathit{C}}_{\mathit{\text{mis}}}$.

$\mathit{\pi }$	$\mathit{\mu }$	$\mathit{\Lambda }$	$\mathit{\Psi }$	${\mathit{C}}_{\mathit{\text{mis}}}$
.026 (.013)	.015 (.002)	.014 (.001)	.011 (.002)	.0024 (.0005)

TABLE II.

Bias (standard error) of the parameter estimates for Each transition model parameter ${\mathit{B}}_{\mathit{j}\mathit{k}\mathit{l}}$.

${\mathit{B}}_{\mathit{j}\mathit{k}}$	${\mathit{B}}_{\mathit{j}\mathit{k}0}$	${\mathit{B}}_{\mathit{j}\mathit{k}1}$	${\mathit{B}}_{\mathit{j}\mathit{k}2}$
${\mathit{B}}_{12}$	.014(.142)	.017(.150)	−.080(.219)
${\mathit{B}}_{13}$	−.021(.126)	.010(.125)	.016(.182)
${\mathit{B}}_{21}$	−.018(.147)	−.004(.139)	.014(.214)
${\mathit{B}}_{23}$	−.003(.112)	.004(.097)	.005(.200)
${\mathit{B}}_{31}$	.003(.112)	.002(.107)	−.006(.192)
${\mathit{B}}_{32}$	.006(.131)	−.032(.091)	.028(.232)

Intuitively, factors such as N, $T_i$, sizes of $J$ and $K$, variance $\mathit{\Psi }$, differences in ${\mathit{\mu }}_j$ and ${\mathit{\Lambda }}_j$ between states, and frequency of state transitions, can affect the performance of parameter estimation. Additional simulations in Appendix II reveals that estimation accuracy for $\mathit{\mu },\mathit{\Lambda },\mathit{\Psi }$, and $\mathit{B}$, as well as classification accuracy, improve when (i) common variances decrease, (ii) differences in ${\mathit{\mu }}_j$ and ${\mathit{\Lambda }}_j$ between states increase, (iii) $J$ decreases, or (iv) sample size $\left(N\right)$ or panel length $\left(T_i\right)$ increases. Increasing $K$ or using a $\mathit{B}$ that induces infrequent transitions slightly affects most parameters but enhances the precision of transition probability estimates, thereby reducing misclassification rates. Estimation of $\mathit{\pi }$ improves solely with increases in N or state-to-state differences in ${\mathit{\mu }}_j$.

B. Simulation 2

This section compares the performance of the proposed methods and the baseline approaches in correctly identifying latent states. Three benchmark methods are under our consideration: i) TM+independent HMM, which assumes independence among observed features given the states; ii) CFM+TM+HMM, which addresses interrelationships but inaccurately pre-specifies the latent factor structure by setting certain loading matrix entries to zero; and iii) EFM+HMM, which assumes a homogeneous transition probabilities matrix for all subjects. We first repeat the data generation process of Simulation 1. Then, we consider three additional scenarios by adjusting the state-to-state differences in ${\mathit{\mu }}_j$ to be closer ($\mathit{\mu }$: medium diff), increasing the similarity of the ${\mathit{\Lambda }}_j$ at different states ($\mathit{\Lambda }$: medium diff), and increasing the significance of the covariance matrix $\mathit{\Psi }(\mathit{\Psi }=2\times I)$, respectively.

As depicted in Figure 1, our proposed methods (HMFM) consistently outperform the benchmark methodologies in all settings. Regardless of sample size, our methods consistently achieve the lowest misclassification rate, nearly approximating zero, thereby emphasizing the importance of each component in our proposed models. Specifically, the comparison with TM+independent HMM shows the importance of accounting for the interrelationship between observed features; the comparison with CFM+TM+HMM reveals the risk of incorrectly specifying the interrelationship structure; and the comparison with EFM+HMM demonstrates the inadequacy of assuming homogeneous transition probabilities.

Fig. 1.

${\mathit{C}}_{\mathit{m}\mathit{i}\mathit{s}}$ of various methods. The error bars represent the 95% CI. The first column shows the results under the settings we used in simulation 1. The last three columns summarize the results under different settings by varying the true value of $\mathit{\mu },\mathit{\Lambda }$, and $\mathit{\Psi }$, respectively.

C. Simulation 3

Information criteria such as the Akaike information criteria (AIC) and the Bayesian information criteria (BIC) have been widely used in model selection [10], [30]. Within this simulation study, we investigate whether the AIC or BIC is reliable for determining J and K simultaneously. We repeat the data generation process of Simulation 1, but implement the proposed methods with a different set of $(J,K)$ for each replicate when fitting the generated data. Let $J=\{\mathrm{2,3},4\}$ and $K=\{\mathrm{2,3},4\}$. We consider all possible combinations of $J$ and $K$, yielding a total of nine fitted candidate models for each replicate. For 100 replications, both AIC and BIC consistently recommend the model with accurate $J$ and $K$. Therefore, we believe that the sample size and the number of observations per individual in the processed AURORA data will yield reliable information criteria-based model selection results and, consequently, reliable parameter estimation.

VI. Analysis of the AURORA Data

Given the irregular data collection from mobile devices, we applied the CT-HMFM to smartwatch data from the AURORA study. Considering $J=\mathrm{1,2},\cdots ,6$ and $K=\mathrm{1,2},\cdots ,9$, we evaluate 54 candidate models. For each candidate model, the EM algorithm is implemented with multiple random seeds, and the seed yielding the highest estimated likelihood is selected. Finally, model comparison using AIC and BIC led to the selection of a model with $J=3$ and $K=8$. Subsequent subsections will discuss the interpretation of parameter estimates and biological insights from three perspectives: i) the interpretation of the three estimated states, ii) symptom co-occurrence patterns, and iii) the influence of demographic factors on transition probabilities.

A. Interpretation of Hidden States

To investigate the biological differences between states, we first focus on the selected features. Figure 2 depicts scaled sample means for each feature across states, with a 99% confidence interval (CI). Further pairwise Tukey tests indicate significant differences between states for nearly all features, except for amplitude, SWCK, L5, and NNskew.q3 between state 1 and state 2. Specifically, features related to average heart rate (NNmean), heart rate variability (SDNN), and heart deceleration capacity (dc) vary significantly across the three latent states, decreasing sequentially from state 1 to state 3. According to previous research, lower heart rate variability and deceleration capacity are associated with a higher mortality rate [23], [31], suggesting that states 1 through 3 represent decreasing levels of health, with state 1 being the healthiest. Additionally, activity levels are similar and higher in states 1 and 2 compared to state 3, indicating better overall health for participants in the first two states. Among features related to heart rate unpredictability (lfhf, ApEn, and SD1SD2), state 3 demonstrates significantly higher values for SD1SD2-related features but lower values for lfhf-related features compared to states 1 and 2, suggesting a different interpretation of the estimated states than our previous interpretation. However, it is important to note that the relationship between these features and the psychological or physiological state is neither straightforward nor unique [32].

Fig. 2.

Relative sample mean for features in each estimated state, with 99% CI error bars that are too small to distinguish.

To confirm the validity of the three states, we further compare their differences regarding self-reported symptoms from a flash survey. Based on the RDoC framework, ten latent constructs associated with APNS were developed using survey items selected by domain experts: Pain, Loss, Sleep Discontinuity, Nightmare, Somatic Symptoms, Mental Fatigue, Avoidance, Re-experience, and Anxious. Retaining only observations for each individual whose estimated states are known on the same day they submitted survey responses, we summarized the flash survey data with means and 95% CIs in Figure 3. Overall, state 1 exhibits the lowest severity level for all ten symptoms, while state 3 has the highest severity level. Tukey tests reveal no significant differences between states 1 and 2 in hyperarousal, re-experience, anxiety, and somatic symptoms, but both are significantly different from state 3 in these constructs. While the differences in nightmare and sleep discontinuity between states 3 and 2 are not significant, they are statistically more severe than in state 1. For mental fatigue and depression, only the difference between state 1 and state 3 is statistically significant.

Fig. 3.

Sample mean for each symptom in each estimated state. The error bars represent the 95% CI. $0$ is the least severity, $1$ is the greatest.

In summary, both the flash survey data and the AURORA data (HRV, Activity) support our interpretation of the three latent states. State 1 is the healthiest, while state 3 indicates having the most severe APNS symptoms.

B. Co-occurring Pattern of Symptoms

When studying the co-occurring of symptoms within each hidden state, we limit our attention to observations collected during the first week. For each state, the correlations between all ten symptoms are calculated. In state 1 (relative health state), there is a high degree of correlation (.782) between hyperarousal and anxiety, suggesting that patients in state 1 experiencing severe hyperarousal symptoms are also likely to suffer from severe anxiety. State 2 shows no highly correlated symptoms. In state 3 (the state with more severe disorders), symptoms such as depression, hyperarousal, anxiety, and re-experience are more likely to co-occur with pairwise correlation ranging from .716 to .91.

C. Transition Probability

This section investigates the heterogeneity of 1-day transition probabilities among subjects, focusing on transitions with a time interval ${\delta }_{\mathit{\text{it}}}=1$. We estimated the transition probabilities for males and females within the sample age range, as depicted in Figure 4. Lines with circles illustrate the probability of remaining in the same state, lines with stars indicate transitions to a more severe state, and lines with ‘x’ reflect the chance of improvement in psychological conditions.

Fig. 4.

Estimated transition probability. Fix ${\mathit{\delta }}_{\mathit{i}\mathit{t}}=1$. (a, b) indicates a transition from state a to state b.

Overall, both males and females have a tendency to remain in their current state, with infrequent state transitions, aligning with most literature. For males, the probability of staying at states 3 and 2 increases with age, while it decreases for state 1. Moreover, while the likelihood of psychological deterioration increases with age, the chance of improvement decreases. Specifically, while the probability of transitioning from the most severe state (state 3) to the healthiest state (state 1) approaches zero as age increases, the likelihood of the reverse transition increases as age decreases, with direct transitions between state 1 and state 3 being particularly rare. The female group exhibits a similar trend to the male group, but with a higher likelihood of remaining in the most severe state (state 3) compared to males.

In summary, our analysis of the AURORA data suggests that older patients are more likely to transition to more severe psychological states. Moreover, achieving psychological improvement becomes increasingly challenging as one ages.

D. State Specific Associations Between Activity and HRV Features

The model provides a deeper understanding of the latent states, not only in terms of their means but also through the associations between activity and HRV features. These associations offer new insights into the underlying physiology of APNS symptoms. The state-specific associations are captured in the structure of the factor loading matrix, which reveals both shared patterns and distinct differences across states, reflecting heterogeneous relationships. For example, the analysis shows a consistent positive correlation between greater daytime activity and better sleep quality across all states. Additionally, heart rate variability and irregularity (measured by SDNN and ApEn) are correlated across all states. However, their association with heart deceleration capacity (dc-related features) is notably stronger in states 2 and 3, highlighting state-specific physiological patterns.

VII. Conclusions and Discussion

This paper explores the unique challenges and potential of using intensive mobile device data to provide objective measures and study the dynamic transitions of APNS symptoms after trauma exposure. The study addresses challenges such as interdependent variables with unknown relationships, heterogeneous transition probabilities, and irregular measurement intervals. The findings suggest that mobile device data from consumer-grade devices, combined with the proposed analytical methods, hold immense promise for assessing, monitoring, and understanding the dynamic progression of APNS symptoms post-trauma.

Traditionally, APNS assessments rely heavily on self-reported measures, which are inherently subjective and prone to recall biases or careless responses. By contrast, mobile device data collected through consumer-grade wearables like smartwatches and fitness trackers offer an objective and continuous perspective on individuals’ physical and mental health. While previous studies have largely relied on data collected in controlled environments, such as sleep labs or clinical settings, data obtained in natural, real-world environments introduces additional noise and presents unique analytical challenges. Extracting meaningful health-related insights from these data remains an open question.

This study demonstrates the immense potential of mobile device data as a secondary objective measure for a broad spectrum of APNS symptoms. These real-time data not only allow clinicians to monitor patients continuously in their natural environments but also provide more timely insights compared to periodic self-reports or clinic visits. By mapping how individuals transition between health states over time and identifying high-risk groups and recovery factors, clinicians gain a better understanding of the natural progression of trauma recovery. This knowledge enables the design of personalized, preventive interventions tailored to individual needs.

Furthermore, the proposed model is adaptable to data from other mobile devices, such as smartphones. Given the widespread ownership and daily use of mobile devices, this approach offers a scalable solution for continuous health monitoring. It also holds significant potential for personalized treatment through mobile health initiatives, particularly in under-resourced areas where access to traditional clinic-based care is limited.

Appendix I

Technical Details

A. Supplement for E-step

Denote $\widetilde{{\mathit{\Lambda }}_j}=\left({\mathit{\Lambda }}_j,{\mathit{\mu }}_j\right)\in {ℛ}^{p\times (K+1)}$ and $\stackrel{~}{{\mathit{z}}_{\mathit{\text{it}}}}={\left({\mathit{z}}_{\mathit{\text{it}}}^T,1\right)}^T\in {ℛ}^{(K+1)}$. Each of the three parts has an explicit form:

$$h\left(\mathit{\pi }\right)=\sum _{i=1}^N\sum _{j=1}^J{\gamma }_{\mathit{\text{ij}}}^v\left(1\right)\mathit{\text{log}}\left({\pi }_j\right),$$ (20)

$$h\left({\left\{{\mathit{B}}_{\mathit{\text{kj}}}\right\}}_{k,j=1}^J\right)=\sum _{i=1}^N\sum _{t=2}^{T_i}\sum _{j,k=1}^J{ϵ}_{i\mathit{\text{kj}}}^v\left(t\right)\mathit{\text{log}}\left({\mathit{P}}_{\mathit{\text{itkj}}}\right),$$ (21)

$$h\left(\mathit{\Psi },{\left\{{\mathit{\Lambda }}_j\right\}}_{j=1}^J,{\left\{{\mathit{\mu }}_j\right\}}_{j=1}^J\right)=\sum _{i=1}^N\sum _{t=1}^{T_i}\sum _{j=1}^J{\gamma }_{\mathit{\text{ij}}}^v\left(t\right)\mathit{\text{log}}\left|\mathit{\Psi }\right|+{\gamma }_{\mathit{\text{ij}}}^v\left(t\right){\mathit{\text{y}}}_{\mathit{\text{it}}}^{\prime }{\mathit{\Psi }}^{-1}{\mathit{\text{y}}}_{\mathit{\text{it}}}-2{\gamma }_{\mathit{\text{ij}}}^v\left(t\right){\mathit{\text{y}}}_{\mathit{\text{it}}}^{\prime }{\mathit{\Psi }}^{-1}\widetilde{{\mathit{\Lambda }}_j}{E}_{{\mathit{\lambda }}^v}\left(\tilde{{\mathit{z}}_{\mathit{\text{it}}}}\mid {\mathit{\text{y}}}_{\mathit{\text{it}}},w_{\mathit{\text{it}}}\right)+{\gamma }_{\mathit{\text{ij}}}^v\left(t\right)\mathit{\text{tr}}\left(\tilde{{{\mathit{\Lambda }}_j}^{\prime }}{\mathit{\Psi }}^{-1}\widetilde{{\mathit{\Lambda }}_j}{E}_{{\mathit{\lambda }}^v}(\tilde{{\mathit{z}}_{\mathit{\text{it}}}}\tilde{{{\mathit{z}}_{\mathit{\text{it}}}}^{\prime }}\mid {\mathit{\text{y}}}_{\mathit{\text{it}}},w_{\mathit{\text{it}}}\right)).$$ (22)

B. Supplement for M-step

By setting the first derivative of $h\left(\mathit{\pi }\right)$ to zero, the parameters related to the initial state distribution are estimated as:

$${\pi }_j^{\mathit{\text{new}}}=\frac{\sum _{i=1}^N{\gamma }_{\mathit{\text{ij}}}^v\left(1\right)}{\sum _{i=1}^N\sum _{k=1}^J{\gamma }_{\mathit{\text{ik}}}^v\left(1\right)}.$$ (23)

Similarly, the parameters used to characterize the conditional distribution of ${\mathit{y}}_{\mathit{\text{it}}}$ given $w_{\mathit{\text{it}}}$ are estimated by setting the first derivative of $h\left(\mathit{\Psi },{\left\{{\mathit{\Lambda }}_j\right\}}_{j=1}^J,{\left\{{\mathit{\mu }}_j\right\}}_{j=1}^J\right)$ equal to 0, with

$${\tilde{\Lambda }}_j^{\mathit{\text{new}}}=\frac{{\sum }_{i=1}^N{\sum }_{t=1}^{T_i}{\gamma }_{\mathit{\text{ij}}}^v\left(t\right)y_{\mathit{\text{it}}}{E}_{{\lambda }^v}{\left(\tilde{z_{\mathit{\text{it}}}}\mid y_{\mathit{\text{it}}},w_{\mathit{\text{it}}}\right)}^{\prime }}{{\sum }_{i=1}^N{\sum }_{t=1}^{T_i}{\gamma }_{\mathit{\text{ij}}}^v\left(t\right)E_{{\lambda }^v}\left(\tilde{z_{\mathit{\text{it}}}}\tilde{{z_{\mathit{\text{it}}}}^{\prime }}\mid y_{\mathit{\text{it}}},w_{\mathit{\text{it}}}\right)}.$$ (24)

Meanwhile, we got the updated estimation of $\mathit{\Psi },{\mathit{\Psi }}^{\mathit{\text{new}}}$, equals

$$\mathit{\text{diag}}\left\{\sum _{i=1}^N\sum _{t=1}^{T_i}\sum _{j=1}^J{\gamma }_{\mathit{\text{ij}}}^v(t)\left\{{\mathit{y}}_{\mathit{\text{it}}}-{\tilde{\mathit{\Lambda }}}_j^{\mathit{\text{new}}}{E}_{{\mathit{\lambda }}^v}\left(\stackrel{~}{{\mathit{z}}_{\mathit{\text{it}}}}\right)\right\}{\mathit{y}}_{\mathit{\text{it}}}^{\prime }\right\}/\sum _{i=1}^N{T}_i.$$ (25)

Appendix II

Additional Simulation Results

This section presents additional simulation results that investigate the impact of various factors on estimation performance. Using the settings for simulation 1 in Section V-A as the baseline, we conducted eight additional sets of simulations, each varying one component while maintaining rest components as the baseline setup. These components include: i) sample size (N), ii) number of measurements for each individual $\left(T_i\right)$, iii) J, iv) K, v) size of common variance $\mathit{\Psi }$, vi) state-to-state difference in ${\mathit{\mu }}_j$, vii) state-to-state difference in ${\mathit{\Lambda }}_j$, and viii) transition frequency. For common variance $\mathit{\Psi }$, we evaluated scenarios with $.1\mathit{I}$, $.5\mathit{I}$, and $1\mathit{I}$ (baseline). For ${\mathit{\mu }}_j$ and ${\mathit{\Lambda }}_j$, we adjusted state-to-state differences in two additional settings. For the test evaluating the effect of transition frequency, a frequent transition is defined as the probability of remaining in the same state being less than 0.70. The $\mathit{B}$ in the baseline setting corresponds to infrequent transition. For tests evaluating the effects of J, K, N, and $T_i$, the baseline setups are modified as indicated in Table III.

TABLE III.

The Mean (standard error) AADs of $\mathit{\pi },\mathit{\mu },\mathit{\Lambda },\mathit{\Psi }$, and $\mathit{B}$, and ${\mathit{C}}_{\mathit{\text{mis}}}$ of estimations under different CT settings.

ADD	$\mathit{\pi }$	$\mathit{\mu }$	$\mathit{\Lambda }$	$\mathit{\Psi }$	$\mathit{B}$	${\mathit{C}}_{\mathit{\text{mis}}}$
$\mathit{\Psi }=1*\mathit{I}$	.026(.013)	.015(.002)	.014(.001)	.011(.002)	.120(.027)	.0024(.0005)
$\mathit{\Psi }=.5*\mathit{I}$	.026(.013)	.012(.002)	.011(.001)	.005(.001)	.119(.027)	.0003(.0001)
$\mathit{\Psi }=.1*\mathit{I}$	.026(.013)	.010(.003)	.007(.001)	.001(.000)	.119(.027)	.0000(.0000)
$\mathit{\mu }$: large diff	.026(.013)	.015(.002)	.014(.001)	.011(.002)	.120(.027)	.002(.0005)
$\mathit{\mu }$: medium diff	.027(.013)	.015(.001)	.015(.001)	.011(.002)	.124(.028)	.007(.001)
$\mathit{\mu }$: minor diff	.034(.017)	.017(.003)	.478(.057)	.011(.002)	.161(.040)	.085(.004)
$\mathit{\Lambda }$: large diff	.026(.013)	.015(.002)	.014(.001)	.011(.002)	.120(.027)	.002(.0005)
$\mathit{\Lambda }$: medium diff	.026(.013)	.014(.002)	.014(.001)	.011(.002)	.122(.029)	.006(.001)
$\mathit{\Lambda }$: minor diff	.026(.013)	.014(.002)	.014(.001)	.011(.002)	.122(.028)	.004(.001)
$\mathit{B}$: infreq transit	.026(.013)	.015(.002)	.014(.001)	.011(.002)	.120(.027)	.002(.0005)
$\mathit{B}$: freq transit	.027(.013)	.015(.002)	.014(.001)	.011(.002)	.137(.043)	.007(.001)
$J=2$	.027(.021)	.012(.002)	.012(.001)	.011(.002)	.087(.026)	.0015(.0003)
$J=3$	.026(.013)	.015(.002)	.014(.001)	.011(.002)	.120(.027)	.0024(.0005)
$J=4$	.024(.011)	.017(.002)	.016(.001)	.011(.001)	.150(.028)	.0032(.0005)
$K=2$	.026(.013)	.015(.002)	.014(.001)	.010(.002)	.124(.027)	.0069(.0008)
$K=3$	.026(.013)	.015(.002)	.014(.001)	.011(.002)	.120(.027)	.0024(.0005)
$K=5$	.026(.013)	.015(.002)	.015(.001)	.012(.002)	.119(.027)	.0005(.0002)
$N=50$	.052(.029)	.030(.004)	.028(.002)	.022(.003)	.254(.058)	.0027(.0008)
$N=100$	.042(.019)	.021(.003)	.020(.001)	.015(.002)	.175(.038)	.0025(.0006)
$N=500$	.018(.010)	.009(.001)	.009(.001)	.007(.001)	.076(.017)	.0024(.0003)
$10\le T_i\le 30$	.029(.014)	.029(.004)	.027(.002)	.021(.004)	.251(.061)	.0028(.0009)
$30\le T_i\le 50$	.027(.014)	.020(.003)	.019(.001)	.015(.003)	.180(.043)	.0025(.0005)
$100\le T_i\le 150$	.028(.015)	.012(.001)	.011(.001)	.008(.001)	.098(.021)	.0023(.0003)