Dynamic model-based clustering for spatio-temporal data

Abstract

In many research fields, scientific questions are investigated by analyzing data collected over space and time, usually at fixed spatial locations and time steps and resulting in geo-referenced time series. In this context, it is of interest to identify potential partitions of the space and study their evolution over time. A finite space-time mixture model is proposed to identify level-based clusters in spatio-temporal data and study their temporal evolution along the time frame. We anticipate space-time dependence by introducing spatio-temporally varying mixing weights to allocate observations at nearby locations and consecutive time points with similar cluster’s membership probabilities. As a result, a clustering varying over time and space is accomplished. Conditionally on the cluster’s membership, a state-space model is deployed to describe the temporal evolution of the sites belonging to each group. Fully posterior inference is provided under a Bayesian framework through Monte Carlo Markov chain algorithms. Also, a strategy to select the suitable number of clusters based upon the posterior temporal patterns of the clusters is offered. We evaluate our approach through simulation experiments, and we illustrate using air quality data collected across Europe from 2001 to 2012, showing the benefit of borrowing strength of information across space and time.

Appendix

The full conditional distribution of the variances $\lambda ^2_k$, for $k=2,\dots ,K$, is

$$\begin{aligned} \lambda ^2_k&\mid \text {rest} \sim \mathcal {IG} \Biggl (a+\frac{Tn}{2}, b+\frac{1}{2}\sum _{t=1}^T\left( \varvec{\phi }_{t,k}-\rho _k \varvec{\phi }_{t-1,k}\right) ^\prime \\&\times C(\theta )^{-1} \left( \varvec{\phi }_{t,k}-\rho _k \varvec{\phi }_{t-1,k}\right) \Biggr ). \end{aligned}$$

The full conditional distribution of the variances $\tau ^2_k$, for $k=1,\dots ,K$, is

$$\begin{aligned} \tau ^2_k \mid \text {rest} \sim \mathcal {IG} \left( a+\frac{T}{2}, b+ \frac{1}{2} \sum _{t=1}^T{\left( z_{t,k}-g_k z_{t-1,k}\right) ^2} \right) . \end{aligned}$$

The full conditional distribution of the error variance $\sigma ^2$ is given by

$$\begin{aligned} \sigma ^2 \mid \text {rest}\! \sim \! \mathcal {IG}\left( a\!+\!\frac{Tn}{2}, b\!+\!\frac{1}{2} \sum _{t=1}^T{\left( \mathbf {y}_t\!-\!\mathbf {H}_t\mathbf {z}_t\right) ^\prime \left( \mathbf {y}_t\!-\!\mathbf {H}_t\mathbf {z}_t\right) } \right) . \end{aligned}$$

The full conditional distribution of $\rho _k$, $k=2,\dots ,K$, is a univariate normal distribution $\mathcal {N}(vd,v)$ restricted in the interval $I(-1<\rho _k<1)$, where

$$\begin{aligned}&v^{-1} = \frac{1}{\lambda _k^2}\varvec{\phi }_{t-1,k}^\prime C(\theta )^{-1} \varvec{\phi }_{t-1,k} + 10^{-4}\\&d = \frac{1}{\lambda _k^2}\varvec{\phi }_{t-1,k}^\prime C(\theta )^{-1} \varvec{\phi }_{t,k} . \end{aligned}$$

The full conditional distribution of $g_k$, $k=1,\dots , K$, is a univariate normal distribution¹ $\mathcal {N}(vd,v)$ truncated in the interval $I(-1<g_k<1)$, where

$$\begin{aligned}&v^{-1} = \frac{1}{\tau ^2_k}\sum _{t=1}^T{z_{t-1,k}^2} + 10^{-4}\\&d = \frac{1}{\tau ^2_k} \sum _{t=1}^T{z_{t-1,k}z_{t,k}} . \end{aligned}$$

The full conditional distribution of the allocation variables $w_t(\mathbf {s})$ is given by

$$\begin{aligned} w_t(\mathbf {s}) \mid \text {rest} \sim \text {Multinomial}\left( \pi _{t,1}(\mathbf {s})^\star , \dots , \pi _{t,K}(\mathbf {s})^\star \right) \end{aligned}$$

where the posterior probabilities are

$$\begin{aligned} \pi ^\star _{t,k}(\mathbf {s})=\frac{\pi _{t,k}(\mathbf {s}) N(z_{t,k}, \sigma ^2)}{\sum _{l=1}^K{\pi _{t,l}(\mathbf {s}) N(z_{t,l}, \sigma ^2)}} . \end{aligned}$$

The full conditional distribution of the latent states $\mathbf {z}_{t}$ is a K-dimensional multivariate normal distribution $\mathcal {N}_K(VD,V)$, where

• $t=1$

  $$\begin{aligned}&V^{-1} = \frac{1}{\sigma ^2}\mathbf {H}_t^\prime \mathbf {H}_t + \mathbf {G}^\prime \varSigma _\eta ^{-1} \mathbf {G} + 10^{-4}I_K \\&D = \frac{1}{\sigma ^2}\mathbf {H}_t^\prime \mathbf {y}_t + \mathbf {G}^\prime \varSigma _\eta ^{-1} \mathbf {z}_{t+1} \end{aligned}$$

• $t=2,\dots ,T-1$

  $$\begin{aligned}&V^{-1} = \frac{1}{\sigma ^2}\mathbf {H}_t^\prime \mathbf {H}_t + \mathbf {G}^\prime \varSigma _\eta ^{-1} \mathbf {G} + \varSigma _\eta ^{-1} \\&D = \frac{1}{\sigma ^2}\mathbf {H}_t^\prime \mathbf {y}_t + \mathbf {G}^\prime \varSigma _\eta ^{-1} \mathbf {z}_{t+1} + \varSigma _\eta ^{-1}\mathbf {G} \mathbf {z}_{t-1} \end{aligned}$$

• $t=T$

  $$\begin{aligned}&V^{-1} = \frac{1}{\sigma ^2}\mathbf {H}_t^\prime \mathbf {H}_t + \varSigma _\eta ^{-1} \\&D = \frac{1}{\sigma ^2}\mathbf {H}_t^\prime \mathbf {y}_t + \varSigma _\eta ^{-1}\mathbf {G} \mathbf {z}_{t-1}. \end{aligned}$$