# Spatial Statistical Models: An Overview under the Bayesian Approach

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## Abstract

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## 1. Introduction

#### 1.1. Aim

#### 1.2. Outline

#### 1.3. Spatial Modeling Overview

**x**is the fixed-effect design vector. $\mathit{\beta}$ is the unknown fixed-effect vector. $\mathit{\gamma}$ is a random field, whose uncertainty is to be quantified. The choice of the probability distribution assumed for $f\left(y|\mathit{\theta}\right)$ guides the researcher on the choice of $g(.)$. Examples of commonly used canonical link functions include the identity ($g\left(\theta \right)=\theta $), logit ($g\left(\theta \right)=log(\theta /(1-\theta \left)\right)$), and log ($g\left(\theta \right)=log\left(\theta \right)$).

## 2. Research Methodology

- Search results that were written in English and articles published in peer-reviewed journals available online. We excluded books, dissertations/theses, conference proceedings, and reviews (or any other form that was not an article);
- Articles that specifically implemented Bayesian spatial models, excluding the ones that only mentioned Bayesian spatial models.

- Names of all authors;
- Publication year;
- Journal title;
- Response to the ten items of the conceptual classification scheme on Bayesian spatial models.

## 3. Conceptual Scheme for Spatial Models

#### 3.1. Spatial Statistics Fields of Application

#### 3.2. Spatial Domains

- Area or lattice data: This is a simple way to represent spatial data in the domain $\mathcal{D}$. In this type of spatial domain, $z\left(s\right)$ is a random aggregated realization across an area s of distinct boundaries. For area data, the boundaries are irregular, such as administrative divisions, whereas for the lattice, the boundaries are a regular division of $\mathcal{D}$. For simplicity, it may be necessary to aggregate other types of spatial domain realizations to form area or lattice data. This process may sometimes be referred to as a discretization of $\mathcal{D}$;
- Geostatistical or point-reference data: $z\left(s\right)$ is a realization at a specific location s in a continuous spatial domain $\mathcal{D}$. Location s is considered to be a coordinate made up of longitudes and latitudes and sometimes includes altitudes. Location s could also be represented in Cartesian coordinates;
- Spatial point pattern: Realization $z\left(s\right)$ represents the occurrence or nonoccurrence of an event at location s. In this case, the location itself is considered to be random. The random realization is a location indicator of the presence or absence of a phenomenon of interest in the domain $\mathcal{D}$. In agriculture, for example, the interest may be the distribution of a specific tree species, in which each realization is the presence or absence of the tree species in domain $\mathcal{D}$. In epidemiology, the realization may be the house address of a patient that has a particular disease [70,71].

#### 3.3. Spatial Priors

#### 3.4. Computational Techniques

#### 3.5. Simulation Study and Validation

## 4. Analyses

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1

QUESTION 1 | Is it only an application? |

1.1 | Yes |

1.2 | Both |

1.3 | No (only method) |

QUESTION 2 | What is the field of application? |

2.1 | Medical science |

2.2 | Economics and humanities |

2.3 | Physical science and engineering |

2.4 | Agricultural and environmental science |

2.5 | Sports |

QUESTION 3 | What spatial domain was employed? |

3.1 | Area or lattice |

3.2 | Geostatistical data |

3.3 | Spatial point patterns |

3.4 | Area and geostatistical data |

QUESTION 4 | What type of spatial priors are used? |

4.1 | Conditional Autoregressive (CAR) |

4.2 | Besag–York–Mollié (BYM) |

4.3 | Leroux CAR |

4.4 | Gaussian Markov random field (other specifications) |

4.5 | Covariance function (Not GMRF) |

4.6 | Other (new methodology/proposed) |

QUESTION 5 | What type of response variable is used? |

5.1 | Discrete (countable) |

5.2 | Continuous |

5.3 | Combined (mixed) |

5.4 | Ordinal |

QUESTION 6 | What is the statistical model used? |

6.1 | Generalize linear (mixed) model (or hierarchical models) |

6.2 | Survival and longitudinal models |

6.3 | Nonparametric models (machine-learning models) |

6.4 | Spatial econometrics |

6.5 | Proposed |

6.6 | Not stated |

6.7 | Other |

QUESTION 7 | How are model prior specified? |

7.1 | Vague prior (noninformative) |

7.2 | Used verbatim from the literature |

7.3 | Elicited from experts or from the problem |

7.4 | No explicit use or reference/not applicable |

QUESTION 8 | What is the estimation method applied? |

8.1 | Markov Chain Monte Carlo (MCMC) |

8.2 | Integrated Nested Laplace Approximation (INLA) |

8.3 | Expectation–Maximization (EM) |

8.4 | Maximum (penalized quasi-) likelihood method |

8.5 | Not stated |

8.6 | Other |

QUESTION 9 | Is the model validated through simulation? |

9.1 | Yes |

9.2 | No |

QUESTION 10 | Is the application validated through data-driven procedures? |

10.1 | Cross-validation and data splitting (K-fold/holdout) |

10.2 | Leave-One-Out Cross-Validation (LOOCV) |

10.3 | Posterior predictive check |

10.4 | Other |

10.5 | None or not applicable |

#### Appendix A.2. Class of the Gaussian Markov Random Field

**–Conditional Autoregressive (CAR)**

**–Besag–York–Mollie (BYM)**

**–Dean’s Conditional Autoregressive**

**–Simpson Conditional Autoregressive**

**–Leroux Conditional Autoregressive (Leroux CAR)**

**–Conditional Autoregressive dissimilarity**

#### Appendix A.3. Class of Gaussian Non-Markov Random Field Models

**–Spatial Weight Matrix**

**–Spatial Covariance Function**

**–Skewed Gaussian random field**

**–Geostatistical model**

**–Stochastic Partial differential Equation (SPDE)**

#### Appendix A.4. Class of Non-Gaussian Random Fields Models

**–Asymmetric Laplace Process**

**–Multivariate Log-Gamma process**

**–Student-t Process**

**–Spatial Mixture model**

**–Spatial Partition Model**

**–Global Spline Mode**

**–Dirichlet Process (DP)**

#### Appendix A.5. Statistical Models

**–Generalized Linear Mixed Model**

**–Survival Model**

**–Bayesian Spatial Econometrics**

#### Appendix A.6. Computation Technique

**–Maximum Likelihood Estimation**

**–Expectation–Maximization (EM Algorithm)**

**–Markov Chain Monte Carlo (MCMC)**

**–Integrated Nested Laplace Approximation (INLA)**

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**Figure 1.**(

**a**) A 2-dimensional topology divided into an $8\times 16$ regular rectangular spatial domain $\mathcal{D}$. Each square box is represented by a latent variable z. At these locations, we intend to determine ${z}_{i},i=1,2,\cdots ,(8\times 16)$, which explains the overall pattern in the spatial domain. To do so, we observe noisy samples ${y}_{i}$ at each spatial site. The observed samples are denoised and used to predict z. Rectangular lattice data of this nature are common in agriculture science, environmental science, neuroimaging, etc. (

**b**) A 1st-order rook type of spatial dependence for the field ${z}_{1}$. The illustrative diagram shows the relationship between ${z}_{1}$ and its four neighbors (${z}_{2},{z}_{3},{z}_{4},{z}_{5}$). In the neighborhood structure, the $j$th entry of the binary neighborhood matrix is set to 1, $j\in \{2,3,4,5\}$. The total number of neighbors is four $\forall i$, except those at the edges, which have either two or three neighbors.

**Figure 2.**Flowchart of the systematic review search procedure in the Scopus, Science Direct, Web of Science, and MathSciNet databases. From 1280 articles, based on the queries words, 728 articles were removed (duplicate papers, non-English written, not peer-reviewed, nor Bayesian spatial modeling), and 552 remained to be analyzed. Then, information such as the authors’ names, journal titles, publication year, and the conceptual classification scheme were explored.

**Figure 3.**Distribution of the response variable type. Most published studies presented discrete (countable) response variables. The discrete response variable is frequently used in disease prevalence, wildlife population studies, accident analysis, crime analysis, etc.

**Figure 4.**Class of models often using spatial modeling and its numerical estimation method distribution. Given the class of GLMM/hierarchical models, Markov Chain Monte Carlo (MCMC) is the most used intensive computation technique.

**Figure 5.**Most frequent authors on the Bayesian spatial models. The top graph is a tag cloud for the 50 most frequent authors over the past 20 years. The bottom graph is a bar plot displaying the Top 10 authors and their relative frequencies. The most frequently appearing authors are Jane Law and Archie C. A. Clements. These authors are followed, in order, by Wenbiao Hu, M. Grazia Pennino, Brian J. Reich, Andrew B. Lawson, Antonio López-Quílez, Montserrat Fuentes, and Kerrie Mengersen.

**Figure 6.**Growth in scientific publications related to topics in Bayesian spatial models from 2001 to June 2020. There was a positive growth over the 20 years considered. The growth could be associated with the improvement in computational tools and data collection.

Spatial Smoothing | Gaussian Process | Non-Gaussian Process | ||||||
---|---|---|---|---|---|---|---|---|

Spatial Model | Article | Global | Local | GMRF | Non-GMRF | Parametric | Semiparametric | Nonparametric |

CAR dissimilarity | Lee and Mitchell, 2012 [31] | ✔ | ✔ | ✔ | ||||

Intrinsic CAR/BYM | Besag et al., 1991 [80] | ✔ | ✔ | ✔ | ||||

Proper CAR | Besag, 1974 [64] | ✔ | ✔ | ✔ | ||||

Leroux | Leroux et al., 2000 [81] | ✔ | ✔ | ✔ | ||||

Geostatistical | Clements et al., 2006 [49] | ✔ | ✔ | ✔ | ||||

Globalspline | Lee and Durbán (2009) [82] | ✔ | ✔ | ✔ | ||||

Simpson CAR | Simpson et al. [78] | ✔ | ✔ | ✔ | ||||

Dean’s CAR | Dean et al. [83] | ✔ | ✔ | ✔ | ||||

SPDE | Lindgren, Rue and Lindström, 2011 [17] | ✔ | ✔ | ✔ | ||||

Mixture Model | Green and Richardson [84] | ✔ | ✔ | ✔ | ||||

Spatial Partition Model | Leonhard and Raßer [18] | ✔ | ✔ | ✔ | ||||

Asymmetric Laplace | Kuzobowski and Pogorski [85] | ✔ | ✔ | ✔ | ||||

Student-t | Fonseca [86] | ✔ | ✔ | ✔ | ||||

Log-Gamma | Bradley et al. [87] | ✔ | ✔ | ✔ | ||||

Dirichlet | Gelfand et al., 2005 [88] | ✔ | ✔ | ✔ |

**Table 2.**Crosstab spatial priors used versus the statistical model adopted. The GLMM with a CAR spatial prior family for the spatial component is the most frequently used modeling structure in the literature, though some alternatives have been growing in the past decade such as the GLMM framework combined with non-GMRF, GLMM with SPDE, and spatial autoregressive model define dependence matrices.

GLMM | Nonparametric | Spatial Autoregressive | Proposed | Survival Models | Not Stated | Other | Total | |
---|---|---|---|---|---|---|---|---|

Conditional Autoregressive models (CARs) | 227 | 1 | 1 | 0 | 9 | 4 | 3 | 245 |

Non-Gaussian Markov Random Field (non-GMRF) | 101 | 0 | 49 | 5 | 1 | 1 | 16 | 173 |

Gaussian Markov Random Field (GMRF) | 19 | 0 | 0 | 0 | 0 | 0 | 3 | 22 |

Stochastic Partial Differential Equations (SPDE) | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 20 |

Nonparametric | 4 | 0 | 0 | 0 | 0 | 1 | 2 | 7 |

Not Stated | 30 | 0 | 2 | 0 | 0 | 9 | 4 | 45 |

New Methodology | 17 | 0 | 0 | 0 | 0 | 0 | 23 | 40 |

Total | 418 | 1 | 52 | 5 | 10 | 15 | 51 | 552 |

Prior Specified | Freq. |
---|---|

Elicited from experts or from the problem | 168 |

No explicit use or reference/not applicable | 101 |

Used verbatim from the literature | 166 |

Vague prior (noninformative) | 119 |

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**MDPI and ACS Style**

Louzada, F.; Nascimento, D.C.d.; Egbon, O.A.
Spatial Statistical Models: An Overview under the Bayesian Approach. *Axioms* **2021**, *10*, 307.
https://doi.org/10.3390/axioms10040307

**AMA Style**

Louzada F, Nascimento DCd, Egbon OA.
Spatial Statistical Models: An Overview under the Bayesian Approach. *Axioms*. 2021; 10(4):307.
https://doi.org/10.3390/axioms10040307

**Chicago/Turabian Style**

Louzada, Francisco, Diego Carvalho do Nascimento, and Osafu Augustine Egbon.
2021. "Spatial Statistical Models: An Overview under the Bayesian Approach" *Axioms* 10, no. 4: 307.
https://doi.org/10.3390/axioms10040307