| Term | Mean | 2.5% | 97.5% |
|---|---|---|---|
| (Intercept) | -1.45e+01 | -1.5e+01 | -1.4e+01 |
| dist_mrt | -1.50e-03 | -2.0e-03 | -1.1e-03 |
| pop_den | 4.50e-05 | -1.3e-05 | 7.8e-05 |
Spatial Point Patterns of Hawker Centres in Singapore
Alvin Bong Jia Lok
VSRP, BAYESCOMP @ CEMSE-KAUST
https://alvinbjl.github.io/hawker-spatial-point/presentation/
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Introduction
Hawker centres are a key feature of Singapore’s food landscape, offering affordable, diverse, and convenient meals.
They cater especially to working adults and reflect Singapore’s cultural diversity.
Project objective
Examine hawker centre accessibility via proximity to MRT stations.
Test spatial randomness vs clustering using quadrat counts and Ripley’s K-function.
Identify hotspots with KDE and model-based LGCP incorporating population density and MRT proximity.
Data & Covariates
- Nearest-MRT distances: 59–2418 meters
- Median & mean distance: 656 & 721 meters (walkable)
- 129 hawker centres across mainland Singapore
- Population density by planning area
- MRT stations for accessibility analysis
Methods
Complete Spatial Randomness: Quadrat Method
- Study region divided into 24 sub-regions
- Compare observed counts \(O_i\) to expected counts \(E_i\) under CSR
- Chi-square and Monte Carlo approaches applied (
quadrad.test)
\[ E_i = \frac{\text{total count of hawker centres}}{\text{number of sub-regions}} = \frac{129}{24} \approx 5.4 \]
\[ X^2 = \sum_{i=1}^{24} \frac{(O_i - E_i)^2}{E_i} \sim \chi^2_{23} \]
- \(H_0: \text{Complete Spatial Randomness (CSR)}, \quad H_1: \text{Clustered pattern}\)
Complete Spatial Randomness: Quadrat Method Ripley’s K-function
- Measures point interactions within distance \(s\)
- Compare empirical \(K(s)\) with CSR expectation \(K_{\text{CSR}}(s) = \pi s^2\) \[ K(s) = \lambda^{-1} \, \mathbb{E}\left[ \text{number of further events within distance } s \text{ of an arbitrary event} \right], \]
- Monte Carlo simulations (99) generate envelopes to account for stochastic variability
Intensity: Kernel Density Estimate (KDE)
- Smoothed surface highlighting dense hawker regions
- Masked airports, water catchments, and mountainous areas
- Gaussian kernel, bandwidth 1200 meters
- Provides baseline for comparison with LGCP model \[ \hat{\lambda}(x) = \sum_{i=1}^{n} \frac{1}{h^2} K\Big(\frac{x - x_i}{h}\Big) \]
Intensity: Log-Gaussian Cox Process (LGCP)
Let \(y_{ij}\) be the count of hawker centers in cell \(ij\) with area of cell denoted as \(|s_{ij}|\). The model is as follow:
Model
\[ y_{ij} | \eta_{ij} \sim \text{Poisson}(|s_{ij}| \cdot \exp(\eta_{ij})) \]
\[ \eta_{ij} = \mathbf{x}_{ij}^\top \boldsymbol{\beta} + f_s(s_{ij}) + f_u(s_{ij}) \]
Intensity: Log-Gaussian Cox Process (LGCP)
Model
\[ y_{ij} | \eta_{ij} \sim \text{Poisson}(|s_{ij}| \cdot \exp(\eta_{ij})) \]
\[ \eta_{ij} = \mathbf{x}_{ij}^\top \boldsymbol{\beta} + f_s(s_{ij}) + f_u(s_{ij}) \]
\(\beta_0\) is the intercept
\(\beta_1\), \(\beta_2\) are regression coefficients for the covariates
\(f_s(s_{ij})\) is a second-order two-dimensional CAR-model on a regular lattice
\(f_u(s_{ij})\) is a unstructured random effect
Model fitting was performed in a Bayesian framework using the Integrated Nested Laplace Approximation (INLA) (Rue, Martino, and Chopin 2009).
Results
Spatial Randomness
Quadrat test: \(X^2 = 113.4\), p < 0.05 → clustered
K-function: above CSR envelope → significant clustering
Locations of clusters
KDE hotspots concentrated in southern (near downtown Singapore)
Model
- Distance to MRT:: ➖ fewer hawker centers further away
- Population density: ➕ nore hawkers in dense areas
Prediction
- Clustering near downtown reflects central commercial hubs
- Hotspots identified: Chinatown, Bukit Merah, Braddell, Joo Chiat, Rochor
- Posterior unstructured effect shows higher values in south → unobserved factors present
Conclusion
- MRT stations are located near to hawker centers
- Hawker centres are clustered, not random
- Accessibility and population density drive spatial distribution
Limitations
- Edge effects in K-function and KDE
- Limited covariates; missing socio-economic and land-use factors
- Static population data; does not capture daily mobility patterns
- Future work can explore SPDE spatial models
References
Rue, Håvard, Sara Martino, and Nicolas Chopin. 2009. “Approximate Bayesian Inference for Latent Gaussian Models by Using Integrated Nested Laplace Approximations.” Journal of the Royal Statistical Society Series B: Statistical Methodology 71 (2): 319–92.
