Zero-Inflated Regime-Switching Stochastic Differential Equation Models for Highly Unbalanced Multivariate, Multi-Subject Time-Series Data

Abstract

In the study of human dynamics, the behavior under study is often operationalized by tallying the frequencies and intensities of a collection of lower-order processes. For instance, the higher-order construct of negative affect may be indicated by the occurrence of crying, frowning, and other verbal and nonverbal expressions of distress, fear, anger, and other negative feelings. However, because of idiosyncratic differences in how negative affect is expressed, some of the lower-order processes may be characterized by sparse occurrences in some individuals. To aid the recovery of the true dynamics of a system in cases where there may be an inflation of such “zero responses,” we propose adding a regime (unobserved phase) of “non-occurrence” to a bivariate Ornstein–Uhlenbeck (OU) model to account for the high instances of non-occurrence in some individuals while simultaneously allowing for multivariate dynamic representation of the processes of interest under nonzero responses. The transition between the occurrence (i.e., active) and non-occurrence (i.e., inactive) regimes is represented using a novel latent Markovian transition model with dependencies on latent variables and person-specific covariates to account for inter-individual heterogeneity of the processes. Bayesian estimation and inference are based on Markov chain Monte Carlo algorithms implemented using the JAGS software. We demonstrate the utility of the proposed zero-inflated regime-switching OU model to a study of young children’s self-regulation at 36 and 48 months.

Appendices

Appendix A: JAGS Code

JAGS selects an appropriate MCMC sampler for the parameters automatically from a list of possible samplers, including Gibbs sampler (Geman & Geman, 1984), slice sampler (Neal, 2003), Metropolis–Hastings (Hastings, 1970), and several other possibilities.

Appendix B: Full Conditional Distributions of Parameters

Let $\mathbf{x}_i=(\mathbf{x}_{i,t_{i,1}},\ldots ,\mathbf{x}_{i,t_{i,T_i}})$, $\mathbf{x}=(\mathbf{x}_{1},\ldots ,\mathbf{x}_{n})$, $\mathbf{l}_i=(\mathbf{l}_{t_{i,1}},\ldots ,\mathbf{l}_{t_{i,T_i}})$, $\mathbf{l}=(\mathbf{l}_{1},\ldots ,\mathbf{l}_{n})$, ${\varvec{\theta }}=\{{\varvec{\theta }}_1,\ldots ,{\varvec{\theta }}_R\}$. Let ${\varvec{\alpha }}$ consist of the free parameters in all ${\varvec{\alpha }}^{\mathrm{EP}}_{r_1s_1s_2}$ and ${\varvec{\alpha }}^{\mathrm{PR}}_{r_2s_1s_2}$ in Eq. (8). The posterior distribution of ${\varvec{\theta }}$, ${\varvec{\alpha }}$, and $\mathbf{l}$ is

$$\begin{aligned} P(\mathbf{l},{\varvec{\theta }},{\varvec{\alpha }}|\mathbf{x})\propto & {} P(\mathbf{x}|\mathbf{l},{\varvec{\theta }})P(\mathbf{l}|{\varvec{\alpha }})p({\varvec{\theta }})p({\varvec{\alpha }}). \end{aligned}$$

The distribution of the process under the Euler approximation and given the latent regime indicator and the parameters, $P(\mathbf{x}|\mathbf{l},{\varvec{\theta }})$, is

$$\begin{aligned} P(\mathbf{x}|\mathbf{l},{\varvec{\theta }})= & {} \prod _{i=1}^{n}\prod _{k=0}^{T_n-1}P(\mathbf{x}_{i,{t_{i,k+1}}}|\mathbf{x}_{i,{t_{i,k}}},\mathbf{l},{\varvec{\theta }})\nonumber \\&=\prod _{i=1}^{n}\prod _{k=0}^{T_n-1} (2\pi )^{-q/2}|{\Delta {t_{i,k}}}\mathbf{S}\left( \mathbf{x}_{i,{t_{i,k}}},{\varvec{\theta }}_{\mathbf{l}_{{t_{i,k}}}}\right) \mathbf{S}\left( \mathbf{x}_{i,{t_{i,k}}},{\varvec{\theta }}_{\mathbf{l}_{{t_{i,k}}}}\right) ^T|^{-1/2} \nonumber \\&\quad \exp \Bigg \{-\frac{1}{2}\bigg (\Delta \mathbf{x}_{i,{t_{i,k}}} - \mathbf{f}(\mathbf{x}_{i,{t_{i,k}}},{\varvec{\theta }}_{\mathbf{l}_{{t_{i,k}}}})\Delta {t_{i,k}}\bigg )^T\big ({\Delta {t_{i,k}}}\mathbf{S}(\mathbf{x}_{i,{t_{i,k}}},{\varvec{\theta }}_{\mathbf{l}_{{t_{i,k}}}})\mathbf{S}(\mathbf{x}_{i,{t_{i,k}}},{\varvec{\theta }}_{\mathbf{l}_{{t_{i,k}}}})^T\big )^{-1}\nonumber \\&\qquad \left( \Delta \mathbf{x}_{i,{t_{i,k}}} - \mathbf{f}\left( \mathbf{x}_{i,{t_{i,k}}},{\varvec{\theta }}_{\mathbf{l}_{{t_{i,k}}}}\right) \Delta {t_{i,k}}\right) \Bigg \}. \end{aligned}$$

(17)

The distribution of the latent regime indicator given the parameters, $P(\mathbf{l}|{\varvec{\alpha }})$, is

$$\begin{aligned} P(\mathbf{l}|{\varvec{\alpha }})= & {} \prod _{i=1}^{n}\prod _{k=0}^{T_n-1}\prod _{j=1}^{q}p\left( l_{j{t_{i,k+1}}}|\mathbf{l}_{{t_{i,k}}}\right) \nonumber \\= & {} \prod _{i=1}^{n}\prod _{k=0}^{T_n-1}\prod _{j=1}^{q}\frac{\exp \big \{{\varvec{\alpha }}^{(j)T}_{l_{j{t_{i,k+1}}}\mathbf{l}_{{t_{i,k}}}}\mathbf{u}_i\big \}}{\sum _{r=1}^R\exp \big \{{\varvec{\alpha }}^{(j)T}_{r,\mathbf{l}_{{t_{i,k}}}}\mathbf{u}_i\big \}}. \end{aligned}$$

(18)

The full conditional distribution of ${l^{\mathrm{EP}}_{{t_{i,k}}}}$ is

$$\begin{aligned} P\left( {l^{\mathrm{EP}}_{{t_{i,k}}}}|\cdot \right)\propto & {} P\left( \mathrm{EP}_{i,{t_{i,k+1}}}|\mathrm{EP}_{i,{t_{i,k}}},{l^{\mathrm{EP}}_{{t_{i,k}}}},{\varvec{\theta }}\right) P\left( {l^{\mathrm{EP}}_{{t_{i,k+1}}}}|\mathbf{l}_{{t_{i,k}}},{\varvec{\alpha }}\right) P\left( {l^{\mathrm{EP}}_{{t_{i,k}}}}|\mathbf{l}_{{t_{i,k-1}}},{\varvec{\alpha }}\right) , \end{aligned}$$

(19)

which is a categorical distribution. The probabilities of all categories can be calculated by standardizing the right-hand side of Eq. (19).

The full conditional distribution of ${\varvec{\alpha }}$ is proportional to $P(\mathbf{l}|{\varvec{\alpha }})P({\varvec{\alpha }})$, where $P({\varvec{\alpha }})$ is the prior distribution of ${\varvec{\alpha }}$ defined in Eq. (8). The full conditional distribution of ${\varvec{\alpha }}$ is not standard. MCMC algorithms may be employed to generate samples from the distribution, for instance, via Metropolis–Hastings algorithm (Hastings, 1970) or the slice sampler (Neal, 2003).

The full conditional distribution of ${\varvec{\theta }}$ is model-dependent. In the ZI-OU model, given the prior distributions in Eq. (7), the full conditional distributions of the parameters in the OU model are as follows.

$$\begin{aligned}&\beta _{\mathrm{EP},l}|\cdot \sim N\left( {\tilde{\beta }}_{\mathrm{EP},0l},{\tilde{\sigma }}^2_{\beta _{\mathrm{EP},l}}\right) I(0,\infty ), ~~~~ \beta _{\mathrm{PR},l}|\cdot \sim N\left( {\tilde{\beta }}_{\mathrm{PR},0l},{\tilde{\sigma }}^2_{\beta _{\mathrm{PR},l}}\right) I(0,\infty ),\\&\mu _{\mathrm{EP},l}|\cdot \sim N\left( {\tilde{\mu }}_{\mathrm{EP},0l},{\tilde{\sigma }}^2_{\mu _{\mathrm{EP},l}}\right) , ~~~~ \mu _{\mathrm{PR},l}|\cdot \sim N\left( {\tilde{\mu }}_{\mathrm{PR},0l},{\tilde{\sigma }}^2_{\mu _{\mathrm{PR},l}}\right) ,\\&\sigma _{\mathrm{EP},l}|\cdot \sim IG({\tilde{a}}_{\mathrm{EP},1l},{\tilde{a}}_{\mathrm{EP},2l}), ~~~~ \sigma _{\mathrm{PR},l}|\cdot \sim IG({\tilde{a}}_{\mathrm{PR},1l},\tilde{a}_{\mathrm{PR},2l}), \end{aligned}$$

where

$$\begin{aligned} {\tilde{\sigma }}^2_{\beta _{\mathrm{EP},l}}= & {} \Bigg (\frac{1}{\sigma ^2}\sum _{i=1}^n\sum _{{l^{\mathrm{EP}}_{{t_{i,k}}}}=l}{(\mu _{\mathrm{EP},{l^{\mathrm{EP}}_{{t_{i,k}}}}}-\mathrm{EP}_{i,{t_{i,k}}})^2\Delta {t_{i,k}}}+\frac{1}{\sigma ^2_{\beta _{\mathrm{EP},l}}}\Bigg )^{-1},\\ {\tilde{\beta }}_{\mathrm{EP},0l}= & {} {\tilde{\sigma }}^2_{\beta _{\mathrm{EP},l}}\Bigg (\frac{1}{\sigma ^2}\sum _{i=1}^n\sum _{{l^{\mathrm{EP}}_{{t_{i,k}}}}=l}{(\mu _{\mathrm{EP},{l^{\mathrm{EP}}_{{t_{i,k}}}}}-\mathrm{EP}_{i,{t_{i,k}}})(\mathrm{EP}_{i,{t_{i,k+1}}} -\mathrm{EP}_{i,{t_{i,k}}})}+\frac{\beta _{\mathrm{EP},0l}}{\sigma ^2_{\beta _{\mathrm{EP},l}}}\Bigg ),\\ {\tilde{\sigma }}^2_{\mu _{\mathrm{EP},l}}= & {} \Bigg (\frac{\beta _{\mathrm{EP},l}^2}{\sigma ^2}\sum _{i=1}^n\sum _{{l^{\mathrm{EP}}_{{t_{i,k}}}}=l}{\Delta {t_{i,k}}}+\frac{1}{\sigma ^2_{\mu _{\mathrm{EP},l}}}\Bigg )^{-1},\\ {\tilde{\mu }}_{\mathrm{EP},0l}= & {} {\tilde{\sigma }}^2_{\mu _{\mathrm{EP},l}}\Bigg (\frac{\beta _{\mathrm{EP},l}}{\sigma ^2}\sum _{i=1}^n\sum _{{l^{\mathrm{EP}}_{{t_{i,k}}}}=l}{(\mathrm{EP}_{i,{t_{i,k+1}}} -\mathrm{EP}_{i,{t_{i,k}}}+\beta _{\mathrm{EP},l}\mathrm{EP}_{i,{t_{i,k}}}\Delta {t_{i,k}})}+\frac{\mu _{\mathrm{EP},0l}}{\sigma ^2_{\mu _{\mathrm{EP},l}}}\Bigg ),\\ \tilde{a}_{\mathrm{EP},1l}= & {} a_{\mathrm{EP},1l}+\frac{1}{2}\sum _{i=1}^n\sum _{k=0}^{T_i-1}I({l^{\mathrm{EP}}_{{t_{i,k}}}}=l), \text {and}\\ \tilde{a}_{\mathrm{EP},2l}= & {} a_{\mathrm{EP},2l}+\frac{1}{2}\sum _{i=1}^n\sum _{{l^{\mathrm{EP}}_{{t_{i,k}}}}=l} \frac{\big ((\mathrm{EP}_{i,{t_{i,k+1}}} - \mathrm{EP}_{i,{t_{i,k}}}) - \beta _{\mathrm{EP},{l^{\mathrm{EP}}_{{t_{i,k}}}}}(\mu _{\mathrm{EP},{l^{\mathrm{EP}}_{{t_{i,k}}}}}-\mathrm{EP}_{i,{t_{i,k}}})\Delta {t_{i,k}}\big )^2}{\Delta {t_{i,k}}}, \end{aligned}$$

and the parameters for the PR process can be derived similarly.