# The Importance of Including Spatial Autocorrelation When Modelling Species Richness in Archipelagos: A Bayesian Approach

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{z},

^{z},

## 2. Materials and Methods

#### 2.1. Statistical Model

_{i}, is a discrete number. Two distributions are obvious candidates: the Poisson distribution, which has only one parameter, μ, the mean (which is equal to the variance),

_{i}, as a function of predictor variables, x

_{ij}. Here we assume power–law relationships,

_{j}are the parameters to be estimated. We chose power laws because this is the form often assumed to be the relationship between the area of an island and the number of species, Equation (1) (e.g., [30]). Nevertheless, as we will see later, we also assume an exponential decay for the distance to the mainland.

_{[i]}, to Equation (5) (the square brackets emphasize the hierarchical nature of this parameter), and then assuming that the terms γ

_{[i]}come from a common distribution (e.g., normal), whose parameters would also need to be estimated. To preclude μ

_{i}to become negative, γ

_{[i]}enters Equation (5) as a multiplicative term (e.g., [21]),

_{[i]}, to “absorb” some of the variation associated with each island. However, it assumes that γ

_{[i]}are discrete categories; therefore, it ignores that spatial autocorrelation is a function of distance and, thus, should be a continuous variable. One way to handle cases where the varying intercepts, γ

_{[i]}, exhibit a continuous dependence is to use a Gaussian Process (e.g., [22]). Specifically, we assume that the varying intercepts γ

_{[i]}come from a multivariate normal distribution (MVNormal)

**0**stands for the vector of the means, all equal to zero, and

**K**is a covariance matrix. The dependence of γ

_{[i]}on distance is established through the functional form of the covariance matrix

**K**. Often, γ

_{[i]}is assumed to decay exponentially with the square of the distance (e.g., [21]), with the elements of

**K**, k

_{ij}, given by

_{ij}stands for the distance between islands i and j, and η, ρ, and σ are parameters to be estimated. δ

_{ij}is the Kronecker delta, equal to 1 when i = j and zero otherwise, meaning that the term ${\delta}_{ij}{\sigma}^{2}$ corresponds to the variance within an island. However, because there is only one observation per island (its number of species), the term ${\delta}_{ij}{\sigma}^{2}$ is irrelevant (there is no variance among the values for an island); hence, following [21], we set this term equal to a constant (0.01). The above choice of the covariance matrix, Equation (7), is by no means the only possible one; see, for instance, [33]. We tested alternative formulations, but the results were similar.

**K**, plays a major role in the formulation of the model. It is through this matrix that the distances between the islands, D

_{ij}, are explicitly included in the model and its elements reflect the importance (or not) of the spatial autocorrelation among the islands. Although this is the matrix whose parameters are being estimated, Equation (7), it is easier to interpret, instead, the correlation matrix (e.g., see [21]). Given two islands, I and J, if k

_{i,j}is the corresponding element in the covariance and k

_{i,i}and k

_{j,j}are their variances, then the correlation is calculated using ${k}_{i,j}/\sqrt{{k}_{i,i}{k}_{j,j}}$; naturally, the diagonal elements of the correlation matrix are equal to 1.

_{0}, j parameters β

_{j}(to be identified), the parameter ϕ if we used a negative binomial distribution, and ρ

^{2}and η when we include the Gaussian process. To illustrate the above model, assume a Poisson distribution with the area of the island, A

_{i}, as the only explanatory variable but including a Gaussian process. From Equations (3) and (6), we obtain

#### 2.2. Case Study—The Azores and Canary Archipelagos

#### 2.2.1. Study Area

#### 2.2.2. Arthropod Data

#### 2.3. Selection of Explanatory Variables

_{i}exhibits an exponential relationship with the distance to the mainland, expression (2). Therefore, in addition to expressions (5) and (6), we also considered:

_{i}is the distance to the mainland of the island i and d is a parameter to be estimated.

#### 2.4. Choice of Priors

#### 2.5. Software Packages

## 3. Results

#### 3.1. The Traditional ISAR with and without the GAUSSIAN Process

^{z}, has played an important role in ecological theory (e.g., [6]), our analyses start with this version with and without the Gaussian process. The main purpose of this analysis is to illustrate the advantage of including a Gaussian process. For ease of comparison, we use in both cases the negative binomial likelihood. Table 1 and Table 2 show the values of the parameters and Figure 2a,b show the fitting.

**K**; however, as we previously mentioned, the correlation matrix is easier to interpret [21]. To obtain the correlation matrix, because both parameters have skewed distributions, we use the median of the η

^{2}and ρ

^{2}posteriors. The correlation matrices are shown in Table 4 and Table 5 for the Azores and the Canary Islands, respectively. Figure 2c,d shows the archipelagos and the intensity of the color of the line between the islands reflects the value of the correlation; for some islands, the correlation is so weak that the lines are not easily discerned. Although the model does not consider the species’ identity, only species richness, visual inspection of Figure 2c,d reveals clusters of islands in the Azores and in the Canary Islands. These clusters are clearly based on proximity among the islands; thus, independently of species identity, species richness alone is highly influenced by proximity.

#### 3.2. Considering All Explanatory Variables

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The archipelagos of the Azores (

**a**) and of the Canary Islands (

**b**); the total number of species in each island is shown in brackets.

**Figure 2.**Plots (

**a**,

**b**) show the fitting with the Gaussian process, GP, (blue line) and without (red line) for the island species–area relationship (ISAR) model. The shadow areas correspond to 95% credible intervals. The fitting lines were obtained with the mean values of the posterior of the parameters c and z of S = cA

^{z}. For the model with the Gaussian process, the fitting line corresponds only to S = cA

^{z}, i.e., without the Gaussian process term. Plots (

**c**,

**d**) are geographical locations of the archipelagos and the darkness of the lines among the islands represents the strength of the correlations (for some pairs of islands, the correlation value is so small that the line is not visible). The size of the dots is proportional to the area of the islands.

**Table 1.**Stan output for the ISAR model with and without the Gaussian process (GP) for the Azores. “SD” stands for standard deviation, “CV” for coefficient of variation, “2.50%” and “97.50%” correspond to the width of the credible interval, “n_eff” is the effective sample size, and “Rhat” stands for $\widehat{R}$ (see main text).

Mean | SD | CV | 2.50% | 97.50% | n_eff | Rhat | ||
---|---|---|---|---|---|---|---|---|

Without GP | c | 100.64 | 20.23 | 0.20 | 63.33 | 139.77 | 223 | 1.00 |

z | 0.41 | 0.05 | 0.12 | 0.32 | 0.52 | 172 | 1.00 | |

ϕ | 4.53 | 1.99 | 0.44 | 1.66 | 9.36 | 195 | 1.00 | |

With GP | c | 99.53 | 19.16 | 0.19 | 64.45 | 136.28 | 219 | 1.01 |

z | 0.42 | 0.06 | 0.14 | 0.31 | 0.56 | 123 | 1.00 | |

ϕ | 6.47 | 3.17 | 0.49 | 2.05 | 14.54 | 232 | 1.02 | |

η^{2} | 0.23 | 0.48 | 2.09 | 0.00 | 1.39 | 363 | 1.01 | |

ρ^{2} | 1.56 | 1.83 | 1.17 | 0.02 | 6.22 | 406 | 1.00 |

**Table 2.**Stan output for the ISAR model with and without the Gaussian process (GP) for the Canary Islands. “SD” stands for standard deviation, “CV” for coefficient of variation, “2.50%” and “97.50%” correspond to the width of the credible interval, “n_eff” is the effective sample size and “Rhat” stands for $\widehat{R}$ (see main text).

Mean | SD | CV | 2.50% | 97.50% | n_eff | Rhat | ||
---|---|---|---|---|---|---|---|---|

Without GP | c | 101.91 | 20.31 | 0.20 | 61.69 | 139.74 | 230 | 1.00 |

z | 0.47 | 0.05 | 0.11 | 0.38 | 0.59 | 172 | 1.01 | |

ϕ | 2.93 | 1.47 | 0.50 | 0.89 | 6.00 | 224 | 1.00 | |

With GP | c | 99.31 | 20.4 | 0.20 | 57.9 | 136.31 | 248 | 1.00 |

z | 0.46 | 0.10 | 0.22 | 0.18 | 0.63 | 54 | 1.00 | |

ϕ | 5.03 | 2.64 | 0.52 | 1.30 | 11.49 | 292 | 1.00 | |

η^{2} | 0.58 | 0.80 | 1.34 | 0.02 | 2.69 | 122 | 1.00 | |

ρ^{2} | 1.37 | 1.96 | 1.43 | 0.02 | 7.79 | 204 | 1.00 |

**Table 3.**Comparison of ISAR models using the “Widely Applicable Information Criterion” (WAIC). ΔWAIC for model i is calculated as WAIC

_{i}—WAIC

_{min}, and “weight” is calculated as $\mathrm{exp}\left(-0.5{\Delta \mathrm{WAIC}}_{i}\right)/{\displaystyle \sum}_{Allmodels}\mathrm{exp}\left(-0.5{\Delta \mathrm{WAIC}}_{i}\right)$. “GP” stands for Gaussian Process.

ISAR Model | WAIC | ΔWAIC | Weight | |
---|---|---|---|---|

Azores | With GP | 127.00 | 0.00 | 0.56 |

Without GP | 127.50 | 0.50 | 0.44 | |

Canary Islands | With GP | 116.70 | 0.00 | 0.76 |

Without GP | 119.00 | 2.30 | 0.24 |

**Table 4.**Correlation matrix of the ISAR model with a Gaussian process for the Azores (median values). The grey shading areas highlight the three major groups of islands: the western, central, and eastern groups. The abbreviations mean: C—Corvo, Fl—Flores, Fa—Faial, P—Pico, G—Graciosa, SJ—São Jorge, T—Terceira, SM—São Miguel, SMa—Santa Maria.

C | Fl | Fa | P | G | SJ | T | SM | SMa | |

C | 1.000 | 0.831 | 0.004 | 0.001 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 |

Fl | 0.831 | 1.000 | 0.005 | 0.001 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 |

Fa | 0.004 | 0.005 | 1.000 | 0.810 | 0.499 | 0.654 | 0.185 | 0.000 | 0.000 |

P | 0.001 | 0.001 | 0.810 | 1.000 | 0.567 | 0.819 | 0.346 | 0.002 | 0.000 |

G | 0.001 | 0.001 | 0.499 | 0.567 | 1.000 | 0.740 | 0.510 | 0.002 | 0.000 |

SJ | 0.000 | 0.000 | 0.654 | 0.819 | 0.740 | 1.000 | 0.560 | 0.004 | 0.000 |

T | 0.000 | 0.000 | 0.185 | 0.346 | 0.510 | 0.560 | 1.000 | 0.040 | 0.001 |

SM | 0.000 | 0.000 | 0.000 | 0.002 | 0.002 | 0.004 | 0.040 | 1.000 | 0.371 |

SMa | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.371 | 1.000 |

**Table 5.**Correlation matrix of the ISAR model with a Gaussian process for the Canary Islands (median values). The grey shading areas highlight the two major groups of islands: the western and eastern groups. The abbreviations mean: F—Fuerteventura, L—Lanzarote, GC—Gran Canaria, T—Tenerife, LG—La Gomera, EH—El Hierro, LP—La Palma.

F | L | GC | T | LG | EH | LP | |

F | 1.000 | 0.506 | 0.081 | 0.003 | 0.000 | 0.000 | 0.000 |

L | 0.506 | 1.000 | 0.008 | 0.000 | 0.000 | 0.000 | 0.000 |

GC | 0.081 | 0.008 | 1.000 | 0.345 | 0.078 | 0.004 | 0.005 |

T | 0.003 | 0.000 | 0.345 | 1.000 | 0.579 | 0.097 | 0.166 |

LG | 0.000 | 0.000 | 0.078 | 0.579 | 1.000 | 0.449 | 0.436 |

EH | 0.000 | 0.000 | 0.004 | 0.097 | 0.449 | 1.000 | 0.317 |

LP | 0.000 | 0.000 | 0.005 | 0.166 | 0.436 | 0.317 | 1.000 |

**Table 6.**Comparison of the several models using the “Widely Applicable Information Criterion” (WAIC) criterion for all species. Only models with ‘weight’ greater than 0.01 are shown. See the caption of Table 3 for the explanation of the acronyms “ΔWAIC” and “weight”. “poisson” means that the model assumes a Poisson likelihood and “gp” means it included a Gaussian process (thus necessarily the distance among islands). “area”, “dist” and “elev” stand for the models that (in addition to the Gaussian process) also consider area, distance to the mainland, or elevation, respectively, while “full” stands for a model with all these three explanatory variables included. The “e” in “diste” and “fulle” means that the distance to the mainland entered the model through an exponential relationship, and “dist” and “full” through a power–law relationship.

Archipelago | Model | WAIC | ΔWAIC | Weight |
---|---|---|---|---|

Azores | fulle.poisson.gp | 91.6 | 0.0 | 0.23 |

area.poisson.gp | 91.7 | 0.1 | 0.21 | |

dist.poisson.gp | 91.9 | 0.3 | 0.20 | |

diste.poisson.gp | 92.0 | 0.4 | 0.19 | |

elev.poisson.gp | 92.2 | 0.5 | 0.17 | |

Canary Islands | area.poisson.gp | 78.3 | 0.0 | 0.20 |

elev.poisson.gp | 78.4 | 0.1 | 0.19 | |

dist.poisson.gp | 78.5 | 0.3 | 0.17 | |

fulle.poisson.gp | 78.7 | 0.4 | 0.16 | |

full.poisson.gp | 78.9 | 0.7 | 0.14 | |

diste.poisson.gp | 79.1 | 0.8 | 0.13 |

Archipelago | Model | WAIC | ΔWAIC | Weight |
---|---|---|---|---|

Azores | elev.poisson.gp | 85.7 | 0.0 | 0.18 |

area.poisson.gp | 85.8 | 0.1 | 0.17 | |

fulle.poisson.gp | 85.8 | 0.1 | 0.17 | |

full.poisson.gp | 85.8 | 0.2 | 0.16 | |

dist.poisson.gp | 85.9 | 0.2 | 0.16 | |

diste.poisson.gp | 85.9 | 0.2 | 0.16 | |

Canary Islands | fulle.poisson.gp | 62.1 | 0.0 | 0.20 |

elev.poisson.gp | 62.2 | 0.1 | 0.19 | |

area.poisson.gp | 62.4 | 0.3 | 0.17 | |

full.poisson.gp | 62.4 | 0.3 | 0.17 | |

dist.poisson.gp | 62.8 | 0.8 | 0.14 | |

diste.poisson.gp | 62.9 | 0.8 | 0.13 |

Archipelago | Model | WAIC | ΔWAIC | Weight |
---|---|---|---|---|

Azores | area.poisson.gp | 70.7 | 0.0 | 0.38 |

fulle.poisson.gp | 72.2 | 1.5 | 0.18 | |

full.poisson.gp | 72.8 | 2.1 | 0.13 | |

dist.poisson.gp | 73.2 | 2.5 | 0.11 | |

diste.poisson.gp | 73.3 | 2.6 | 0.10 | |

Canary Islands | fulle.poisson.gp | 70.8 | 0.0 | 0.20 |

area.poisson.gp | 70.9 | 0.1 | 0.18 | |

elev.poisson.gp | 71.0 | 0.2 | 0.17 | |

full.poisson.gp | 71.2 | 0.4 | 0.16 | |

dist.poisson.gp | 71.3 | 0.5 | 0.15 | |

diste.poisson.gp | 71.6 | 0.8 | 0.13 |

Archipelago | Model | WAIC | ΔWAIC | Weight |
---|---|---|---|---|

Azores | fulle.poisson.gp | 57.8 | 0.0 | 0.45 |

diste.poisson.gp | 59.7 | 2.0 | 0.17 | |

full.poisson.gp | 60.3 | 2.6 | 0.13 | |

Canary Islands | dist.poisson.gp | 61.3 | 0.0 | 0.19 |

elev.poisson.gp | 61.4 | 0.1 | 0.18 | |

diste.poisson.gp | 61.4 | 0.1 | 0.18 | |

area.poisson.gp | 61.6 | 0.3 | 0.17 | |

full.poisson.gp | 61.6 | 0.3 | 0.16 | |

fulle.poisson.gp | 62.2 | 0.9 | 0.12 |

Archipelago | Model | WAIC | ΔWAIC | Weight |
---|---|---|---|---|

Azores | area.poisson.gp | 69.6 | 0.0 | 0.4.0 |

fulle.poisson.gp | 70.2 | 0.6 | 0.29 | |

dist.poisson.gp | 72.1 | 2.6 | 0.11 | |

elev.poisson.gp | 72.4 | 2.8 | 0.10 | |

diste.poisson.gp | 72.4 | 2.9 | 0.10 | |

Canary Islands | elev.poisson.gp | 68.7 | 0.0 | 0.23 |

dist.poisson.gp | 68.8 | 0.1 | 0.22 | |

fulle.poisson.gp | 68.9 | 0.2 | 0.20 | |

area.poisson.gp | 68.9 | 0.3 | 0.20 | |

diste.poisson.gp | 69.5 | 0.9 | 0.15 |

**Table 11.**Ratio of the correlation matrices for the best models for MIE and SIE species of the Azores. The abbreviations mean: C—Corvo, Fl—Flores, Fa—Faial, P—Pico, G—Graciosa, SJ—São Jorge, T—Terceira, SM—São Miguel, SMa—Santa Maria.

C | Fl | Fa | P | G | SJ | T | SM | SMa | |

C | 1.00 | 1.14 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Fl | 1.14 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Fa | 0.00 | 0.00 | 1.00 | 1.20 | 2.74 | 1.72 | 14.85 | 0.00 | 0.00 |

P | 0.00 | 0.00 | 1.20 | 1.00 | 2.20 | 1.17 | 5.13 | 0.00 | 0.00 |

G | 0.00 | 0.00 | 2.74 | 2.20 | 1.00 | 1.40 | 2.63 | 0.00 | 0.00 |

SJ | 0.00 | 0.00 | 1.72 | 1.17 | 1.40 | 1.00 | 2.25 | 0.00 | 0.00 |

T | 0.00 | 0.00 | 14.85 | 5.13 | 2.63 | 2.25 | 1.00 | 204.25 | 0.00 |

SM | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 204.25 | 1.00 | 4.54 |

SMa | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 4.54 | 1.00 |

**Table 12.**Ratio of the correlation matrices for the best models for MIE and SIE species of the Canary Islands. The abbreviations mean: F—Fuerteventura, L—Lanzarote, GC—Gran Canaria, T—Tenerife, LG—La Gomera, EH—El Hierro, LP—La Palma.

F | L | GC | T | LG | EH | LP | |

F | 1.00 | 1.44 | 5.08 | 47.14 | 0.00 | 0.00 | 0.00 |

L | 1.44 | 1.00 | 24.88 | 0.00 | 0.00 | 0.00 | 0.00 |

GC | 5.08 | 24.88 | 1.00 | 1.88 | 5.22 | 37.23 | 35.4 |

T | 47.14 | 0.00 | 1.88 | 1.00 | 1.31 | 4.51 | 3.11 |

LG | 0.00 | 0.00 | 5.22 | 1.31 | 1.00 | 1.56 | 1.60 |

EH | 0.00 | 0.00 | 37.23 | 4.51 | 1.56 | 1.00 | 1.99 |

LP | 0.00 | 0.00 | 35.4 | 3.11 | 1.6 | 1.99 | 1.00 |

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## Share and Cite

**MDPI and ACS Style**

Barros, D.D.; Mathias, M.d.L.; Borges, P.A.V.; Borda-de-Água, L.
The Importance of Including Spatial Autocorrelation When Modelling Species Richness in Archipelagos: A Bayesian Approach. *Diversity* **2023**, *15*, 127.
https://doi.org/10.3390/d15020127

**AMA Style**

Barros DD, Mathias MdL, Borges PAV, Borda-de-Água L.
The Importance of Including Spatial Autocorrelation When Modelling Species Richness in Archipelagos: A Bayesian Approach. *Diversity*. 2023; 15(2):127.
https://doi.org/10.3390/d15020127

**Chicago/Turabian Style**

Barros, Diogo Duarte, Maria da Luz Mathias, Paulo A. V. Borges, and Luís Borda-de-Água.
2023. "The Importance of Including Spatial Autocorrelation When Modelling Species Richness in Archipelagos: A Bayesian Approach" *Diversity* 15, no. 2: 127.
https://doi.org/10.3390/d15020127