A hidden Markov approach to the analysis of space–time environmental data with linear and circular components

Abstract

The analysis of bivariate space–time series with linear and circular components is complicated by (1) multiple correlations, across time, space and between variables, (2) different supports on which the variables are observed, the real line and the circle, and (3) the periodic nature of circular data. We describe a multivariate hidden Markov model that includes these features of the data within a single framework. The model integrates a circular von Mises Markov field and a Gaussian Markov field, with parameters that evolve in time according to a latent (hidden) Markov chain. It allows to describe the data by means of a finite number of time-varying latent regimes, associated with easily interpretable components of large-scale and small-scale spatial variation. It can be estimated by a computationally feasible expectation–maximization algorithm. In a case study of sea currents in the Northern Adriatic Sea, it provides a parsimonious representation of the sea surface in terms of alternating environmental states.

Appendix

For $n>2$, the normalizing constant of the multivariate von Mises density $f(\varvec{x}_t; \varvec{\theta }^{\mathrm{circ}}_k)$ is unknown. However, the conditional log-likelihood $l_k(\varvec{\theta }^{\mathrm{circ}}\mid \varvec{x}_t)=\log f(\varvec{x}_t; \varvec{\theta }^{\mathrm{circ}}_k)$ is well approximated about it maximum point by the pseudo-loglikelihood

$$\begin{aligned} {\mathrm pl} _{k}(\varvec{\theta }^{\mathrm{circ} }\mid \varvec{x}_t)=\sum _{i=1}^{n}\log f_{\mathrm{vm} }(x_{it};\nu _{ik},\kappa _{ik}), \end{aligned}$$

obtained by taking the logarithm of the product of the conditional von Mises distributions (Mardia et al. 2008). We take advantage of this result to compute the posterior probabilities of the Markov chain

$$\begin{aligned} \hat{p}_{tk}(\hat{\varvec{\theta }}_s)={\mathbb {E}}(\xi _{tk}|\varvec{z},\hat{\varvec{\theta }}_s) \hat{p}_{t-1,t,hk}(\hat{\varvec{\theta }}_s)={\mathbb {E}}(\xi _{t-1,h}\xi _{tk}|\varvec{z},\hat{\varvec{\theta }}_s) \end{aligned}$$

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from the estimate $\hat{\varvec{\theta }}_s$ provided by the $s$th step of the EM algorithm. This task is generally referred to as the HMM-smoothing numerical issue and it is typically solved by specifying the posterior probabilities in terms of suitably normalized functions, which can be computed recursively, avoiding unpractical summations over the state space of latent Markov chain and numerical under- and over-flows. In the literature, this approach is known as the Forward-Backward (FB) recursion and it can be implemented in a number of different ways (Cappé et al. 2005; ch. 3). We describe below the FB recursion that we have exploited in this paper.

Let

$$\begin{aligned} L_{tk}(\hat{\varvec{\theta }}^{\mathrm{circ}}_s,\hat{\varvec{\theta }}^{\mathrm{lin}}_s)= N(\varvec{y}_t;\varvec{\mu }_k,\sigma ^2_k(\varvec{I}-\rho _k\varvec{I})^{-1})\prod _{i=1}^{n} f_{\mathrm{vm}}(x_{it};\nu _{iks},\kappa _{iks}), \end{aligned}$$

be the conditional contribution of the bivariate spatial series $\varvec{z}_{t}$ to the likelihood function, under state $k$. In addition, let

$$\begin{aligned} L_{0:t}(\hat{\varvec{\theta }}_s)=\sum _{\varvec{\xi }_0}\ldots \sum _{\varvec{\xi }_t} p(\varvec{\xi }_{0:t};\hat{\varvec{\pi }}_s) \prod _{\tau =0}^{t}\prod _{k=1}^{K}\left( L_{\tau k}(\hat{\varvec{\theta }}^{\mathrm{circ}}_s,\hat{\varvec{\theta }}^{\mathrm{lin}}_s)\right) ^{\xi _{\tau k}} \end{aligned}$$

be the contribution of the first $t$ profiles to the likelihood and let $\varvec{z}_{0:t}$ be the space–time series, observed up to time $t$. We run a forward and a backward iteration.

During the forward iteration, we exploit the output of the $s$th step of the EM algorithm to compute the probabilities $\psi ^{(t)}(k)=P(\xi _{tk}=1|\varvec{z}_{0:t-1})$, the likelihood ratios $c_t=\frac{L_{0:t}(\hat{\varvec{\theta }}_s)}{L_{0:t-1}(\hat{\varvec{\theta }}_s)}$ and the forward probabilities $\bar{\alpha }_t(k)=P(\xi _{tk}=1|\varvec{z}_{0:t})$, as follows.

Forward recursion:

• initialization:

  $$\begin{aligned}\psi ^{(0)}(k)=\hat{\pi }_{ks} c_0=\sum _{k=1}^{K}\psi ^{(0)}(k)L_0(\hat{\varvec{\theta }}^{\mathrm{circ}}_{s},\hat{\varvec{\theta }}^{\mathrm{lin}}_{s}) \bar{\alpha }_0(k)=\frac{\psi ^{(0)}(k)L_0(\hat{\varvec{\theta }}^{\mathrm{circ}}_{s},\hat{\varvec{\theta }}^{\mathrm{lin}}_{s})}{c_0} \end{aligned}$$

• for $t=1, \ldots T$

  $$\begin{aligned} \psi ^{(t)}(k)=\sum _{h=1}^{K}\bar{\alpha }_{t-1}(h)\hat{\pi }_{hks} c_t=\sum _{k=1}^{K}\psi ^{(t)}(k)L_t(\hat{\varvec{\theta }}^{\mathrm{circ}}_{ks},\hat{\varvec{\theta }}^{\mathrm{lin}}_{s}) \bar{\alpha }_t(k)=\frac{\psi ^{(t)}(k)L_t(\hat{\varvec{\theta }}^{\mathrm{circ}}_{ks},\hat{\varvec{\theta }}^{\mathrm{lin}}_{s})}{c_t} \end{aligned}$$

At the end the forward recursion, we store the values $c_0 \ldots c_T$ and $\bar{\alpha }_0(k) \ldots \bar{\alpha }_T(k)$. The sequence $c_0 \ldots c_T$ can be exploited to compute the value taken by the log-likelihood at the $s$th step of the EM algorithm, as follows:

$$\begin{aligned} \log L(\hat{\varvec{\theta }}_s)=\sum _{t=0}^{t}\log c_t. \end{aligned}$$

We then run a backward recursion, by computing the ratios $\bar{\varphi }_t(k)=\frac{f(\varvec{z}_{t+1:T}|\xi _{tk}=1)}{\prod _{l=t}^{T}c_l}$, as follows.

Backward recursion:

• initialization: $\bar{\varphi }_T(k)=\frac{1}{c_T}$

• for $t= T-1, T-2, \ldots 0$

  $$\begin{aligned} \bar{\varphi }_t(k)=\frac{\sum _{h=1}^{K}\hat{\pi }_{khs}L_{t+1}(\hat{\varvec{\theta }}^{\mathrm{circ}}_{ks},\hat{\varvec{\theta }}^{\mathrm{lin}}_{s})\bar{\varphi }_{t+1}(h)}{c_t}. \end{aligned}$$

At the end of the backward recursion, we store the values of $\bar{\varphi }_0(k) \ldots \bar{\varphi }_T(k)$ and compute the posterior univariate state probabilities as

$$\begin{aligned} \hat{p}_{tk}(\hat{\varvec{\theta }}_s)=\frac{\bar{\alpha }_t(k)\bar{\varphi }_{t}(k)}{\sum _{k=1}^{K}\bar{\alpha }_t(k)\bar{\varphi }_{t}(k)}. \end{aligned}$$

The bivariate posterior probabilities can be instead computed as

$$\begin{aligned} \hat{p}_{t-1,t,hk}(\hat{\varvec{\theta }}_s)=\bar{\alpha }_t(k)\hat{\pi }_{hks}L_{t+1}(\hat{\varvec{\theta }}^{\mathrm{circ}}_{ks},\hat{\varvec{\gamma }}^{\mathrm{lin}}_{ks})\bar{\varphi }_{t+1}(k)). \end{aligned}$$