Spatio-temporal modelling of property prices in Brunei Darussalam
LSE Social Statistics Seminar
Haziq Jamil 
Assistant Professor of Statistics
Universiti Brunei Darussalam
October 26, 2023
The team
Statistical modelling of spatio-temporal data in Brunei Darussalam
Types of spatial data
Location of Starbucks® around London.
Data source: Kaggle
Transport for London (TFL) Tube, Overground, Elizabeth line, DLR & Tram lines (including Crossrail alignment)
Data source: github/oobrien
London Middle Super Output Area (MSOA) population density (2011).
Data source: London Datastore
- Geospatial points vs point processes
- Point processes: Locations are random variables (e.g. earthquakes, crime, diseases, etc.)
- Geospatial points: Locations are fixed
- Raster/Grid data
- Pixelated (or gridded) data where each pixel is associated with a specific geographical location
- E.g. satellite imagery, digital elevation models, etc.
- Network data.
- E.g. road networks, social networks, etc.
Spatial data in R
- Simple features refers to a formal standard (ISO 19125-1:2004) that describes how objects in the real world can be represented in computers, with emphasis on the spatial geometry of these objects.
- The
{sf}package (Pebesma and Bivand 2023) provides support for handling spatial data in R.
Types of spatial patterns
Values are randomly distributed in space, independently of each other (and particularly its neighbours)
Lack of a spatial pattern.
Values are non-randomly distributed in space.
Large values are “attracting” other large values, and vice versa.
A clustering effect is present.
Values are non-randomly distributed in space.
Large values are “repelling” other large values, and vice versa.
A dispersion effect is present.
Neighbour definitions
Adjacency matrix
For areal data it is quite common to implement \(W=A\) where \(A = (a_{ij})\) is an adjacency matrix with the following definition: \[ a_{ij} = \begin{cases} 1 & i \sim j \text{ where } i \neq j \\ 0 & \text{otherwise.} \end{cases} \]
A B C D E
A 0 1 1 0 0
B 1 0 1 1 1
C 1 1 0 0 1
D 0 1 0 0 1
E 0 1 1 1 0Other useful definitions:
\(D = \operatorname{diag}(A1)\) is a diagonal matrix containing number of neighbours.
\(\tilde W = D^{-1}A\) is the scaled adjacency matrix (row sums are 1).
Conditionally autoregressive (CAR) priors
Let \(\psi_i=\phi_i\). Then, the CAR prior (Leroux, Lei, and Breslow 2000) is defined as \[ \phi_i \mid \phi_{-i},W,\rho,\tau^2 \sim \operatorname{N}\left(\frac{\rho\sum_{j} w_{ij} \phi_j}{\rho\sum_{j} w_{ij}+1-\rho}, \frac{\tau^2}{\rho\sum_{j} w_{ij}+1-\rho}\right), \] where \(\rho\in[0,1]\) is a spatial dependence parameter; \(\tau^2\) is a variance parameter; and \(W\) is a known weight matrix (e.g. \(W=A \in \{0,1\}^{M\times M}\) adjacency matrix).
Special cases:
- \(\rho = 0\) corresponds to independence;
- \(\rho = 1\) corresponds to the Intrinsic CAR;
- \(\rho=1\) and \(\psi_i = \phi_i + \theta_i\), where \(\theta_i\sim\operatorname{N}(0,\sigma^2)\) – this is the BYM (1991) model for binomial, Poisson and ZIP likelihoods.
Aside
Simultaneously autoregressive models
Adding the temporal element
Now suppose data at each spatial unit is recorded for \(t=1,\dots,T\) time periods. Extend the model: \[ \begin{gather} y_{it} \sim p(y_{it} \mid \mu_{it}, \nu^2) \\ g(\mu_{it}) = x_{it}^\top\beta + \psi_{it}. \end{gather} \]
We’ll discuss three ways to model the temporal dependence:
- Linear time effects
- ANOVA decomposition
- Time autoregressive process
