A time-dependent extension of the projected normal regression model for longitudinal circular data based on a hidden Markov heterogeneity structure

Abstract

The modelling of animal movement is an important ecological and environmental issue. It is well-known that animals change their movement patterns over time, according to observable and unobservable factors. To trace the dynamics of behaviors, to identify factors influencing these dynamics and unobserved characteristics driving intra-subjects correlations, we introduce a time-dependent mixed effects projected normal regression model. A set of animal-specific parameters following a hidden Markov chain is introduced to deal with unobserved heterogeneity. For the maximum likelihood estimation of the model parameters, we outline an expectation–maximization algorithm. A large-scale simulation study provides evidence on model behavior. The data analysis approach based on the proposed model is finally illustrated by an application to a dataset, which derives from a population of Talitrus saltator from the beach of Castiglione della Pescaia (Italy).

Appendices

Appendix 1: The mixed projected normal model

The mixed project normal model

$$\mu _{itj} = {\mathbf{x}}_{it}'{\varvec{\beta }}_j + b_{ij},\quad j=1,2,$$

relaxes the independence assumption on repeated measurements and may account for correlations between projections. Indeed, by assuming that projections are conditionally independent given the covariates and the random effects, for the i-th unit, the likelihood contribution would be

$$\begin{aligned} \int \int \prod _{t=1}^Tf({\mathbf{y}}_{it}\mid {\mathbf{x}}_{it},{\mathbf{b}}_{i})g({\mathbf{b}}_{i})d{\mathbf{b}}_{i}=\int \int \prod _{t=1}^Tf(y_{it1}\mid {\mathbf{x}}_{it}, b_{i1})f(y_{it2}\mid {\mathbf{x}}_{it}, b_{i2})g(b_{i1},b_{i2})db_{i1}db_{i2}. \end{aligned}$$

where $g(\cdot )$ is the random effects density function. Nevertheless, in practice, $g(b_{i1},b_{i2}) = g_1(b_{i1})g_2(b_{i2})$, i.e. random effects are assumed independent and, accordingly, projections can be separately modelled and are independent as well. Thus, although theoretically the mixed projected normal model could allow for conditional independence, this feature is not investigated or modelled in the literature. Formally, if $g(b_{i1},b_{i2}) = g_1(b_{i1})g_2(b_{i2})$, we have

$$\begin{aligned} \int \int \prod _{t=1}^Tf(y_{it1}\mid {\mathbf{x}}_{it}, b_{i1})f(y_{it2}\mid {\mathbf{x}}_{it},b_{i2})g(b_{i1},b_{i2})db_{i1}db_{i2}\\ = \int \prod _{t=1}^Tf(y_{it1}\mid {\mathbf{x}}_{it},b_{i1})g_1(b_{i1})db_{i1}\int \prod _{t=1}^Tf(y_{it2}\mid {\mathbf{x}}_{it},b_{i2})g_2(b_{i2})db_{i2} \end{aligned}$$

and, thus, the projections are independent.

Appendix 2: Computational details

As it stands the likelihood function of the hidden Markov projected normal model is of little or no computational use, because it involves a sum over $K^T$ terms for each unit i and cannot be directly evaluated. It quickly becomes infeasible to compute even for small values of K as T grows to moderate size. Clearly, a more efficient procedure is needed to perform the calculation of the likelihood function. This issue may be addressed via the so-called forward variables (Baum et al. 1970). Let us start defining

$$\alpha _{itk}= f({\mathbf{y}}_{i1},\ldots ,{\mathbf{y}}_{it},{\mathbf{b}}_{it}={\mathbf{b}}_k),\quad i =1,\ldots ,I, \ t = 1,\ldots ,T,$$

which represents the probability of seeing the partial sequence ending up in state k at time t for a generic unit i. We can compute $\alpha _{itk}$ recursively by

$$\alpha _{i1k} = \pi _k f({\mathbf{y}}_{i1}\mid {\mathbf{b}}_{i1}={\mathbf{b}}_k)$$

$$\alpha _{it+1k} = \sum _{h=1}^K\alpha _{ith}\pi _{k|h} f({\mathbf{y}}_{it+1}\mid {\mathbf{b}}_{it+1}={\mathbf{b}}_k).$$

As a by-product of the forward procedure we find that the likelihood can be written as

$$\ell ({\varvec{\lambda }}) = \sum _{i=1}^I\log \sum _{k=1}^K\alpha _{iTk}.$$

Let us further define

$$\tau _{itk} = f({\mathbf{y}}_{it+1},\ldots ,{\mathbf{y}}_{iT}\mid {\mathbf{b}}_{it}={\mathbf{b}}_k),$$

i.e. the probability of the partial sequence $({\mathbf{y}}_{it+1},\ldots ,{\mathbf{y}}_{iT})$ given that the i-th unit started in state k at time t. The backward recursion is given by

$$\tau _{iTk}=1$$

$$\tau _{itk} = \sum _{k=1}^K\pi _{k|h} f({\mathbf{y}}_{it+1}\mid {\mathbf{b}}_{it+1}={\mathbf{b}}_k)\tau _{it+1k}.$$

We can express $\hat{\xi }_{itk}$ and $\hat{\zeta }_{itjk}$ in terms of forward and backward variables by

$$\hat{\xi }_{itk} = \frac{\alpha _{itk}\tau _{itk}}{\sum _{k=1}^K\alpha _{itk}\tau _{itk}},$$

$$\begin{aligned} \hat{\zeta }_{ithk} = \frac{\alpha _{it-1h}\pi _{k|h} f({\mathbf{y}}_{it}\mid {\mathbf{b}}_{it}={\mathbf{b}}_k)\tau _{itk}}{\sum _{h,k=1}^K\alpha _{it-1h}\pi _{k\mid h} f({\mathbf{y}}_{it}\mid {\mathbf{b}}_{it}={\mathbf{b}}_k)\tau _{itk}}. \end{aligned}$$

At last, we have

$$\begin{aligned} {\mathtt{E}}(r_{itk}\mid \cdot )=\hat{r}_{itk} = {\mathbf{u}}_{it}{\varvec{\mu }}_{itk}+\frac{{\varvec{{{\varPhi}}}} ({\mathbf{u}}_{it}{\varvec{\mu }}_{itk})}{\phi ({\mathbf{u}}_{it}{\varvec{\mu }}_{itk})+{\mathbf{u}}_{it}{\varvec{\mu }}_{itk}{\varvec{{{\varPhi}}}} ({\mathbf{u}}_{it}{\varvec{\mu }}_{itk})}, \end{aligned}$$

where ${\varvec{\mu }}_{itk} = (\mu _{it1k},\mu _{it2k})$ is the state-specific vector of projections’ means and ${\varvec{{{\varPhi}}}} (\cdot )$ is the cdf of a bivariate standard normal distribution.