# Species Distribution Modeling: Comparison of Fixed and Mixed Effects Models Using INLA

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}and about 780 km of coastline [71,72]. The climate is temperate oceanic with a mean annual temperature of 17 °C at sea level; Relative humidity is high and rainfall ranges from 1500 to more than 3000 mm/m

^{2}per year, increasing with altitude and from east to west [73,74]. Maximum altitude of the islands ranges from 402 m to 2351 m, with several islands reaching more than 1000 m [75].

^{2}and less than 300,000 years, originated by subaerial volcanic activity and is located between the coordinates 38°30′ North and 28°20′ West [81,82]. It is noted for its eponymous volcano, Ponta do Pico, which is the highest mountain in Portugal with 2351 m [83].

^{2}) and the most populous of the Azores, located at 37°50′ North and 25°30′ West [84]. The island is characterized by a large variety of volcanic structures, including three active trachytic composite volcanoes with caldera (i.e., Sete Cidades, Fogo and Furnas; [85]). The highest elevation of the island (Pico da Vara, 1103 m) is localized between Povoação and Nordeste.

#### 2.2. Study Species

#### 2.3. Species Distribution Data

#### 2.3.1. Presence Data

#### 2.3.2. Pseudo-Absences

#### 2.4. Ecogeographical Variables

#### 2.4.1. Topographical Variables

#### 2.4.2. Climatic Variables

#### 2.4.3. Land Use Variables

#### 2.5. Modeling Approach

- (i)
- We started by an intensive evaluation of fixed effects models using both glm() function and INLA, by testing 400 models including different combinations of EGVs.
- (ii)
- After this preliminary evaluation, we tested the bests models including topographic, climatic and land use data, and compared results obtained using glm() and INLA.
- (iii)
- The best models were further selected and we used INLA to derive mixed effects models, including spatial correlation as the random element.

#### 2.6. Model Selection

#### 2.7. Calculation of the Posterior Distribution and of the Random Field

## 3. Results

#### 3.1. Preliminary Selection of Models

#### 3.2. Fixed Effects Models

#### 3.3. Mixed Effects Models

#### 3.3.1. Pico Island

#### 3.3.2. São Miguel Island

#### 3.4. Prediction of the Random Field

#### 3.4.1. Pico Island

#### 3.4.2. São Miguel Island

#### 3.5. Prediction of the Response Grid

#### 3.5.1. Pico Island

#### 3.5.2. São Miguel Island

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Summary of the spatial random effect (Gaussian random field) for the best mixed effects models and their reduction for P. undulatum in Pico Island. Mean (

**a**); Standard deviation (

**b**). PR: Practical range.

**Figure 2.**Summary of the spatial random effect (Gaussian random field) for the best mixed effects models and their reduction for M. faya in Pico Island. Mean (

**a**); Standard deviation (

**b**). PR: Practical range.

**Figure 3.**Summary of the spatial random effect (Gaussian random field) for the best mixed effects models and their reduction for P. undulatum in São Miguel Island. Mean (

**a**); Standard deviation (

**b**). PR: Practical range.

**Figure 4.**Summary of the spatial random effect (Gaussian random field) for the best mixed effects models and their reduction for M. faya in São Miguel Island. Mean (

**a**); Standard deviation (

**b**). PR: Practical range.

**Figure 5.**Prediction of P. undulatum occurrence in Pico Island by posterior mean (

**a**) and standard deviation (

**b**) of the linear predictor. Top: Mixed effects model including a random field; Bottom: Fixed effects model (without a random field). All models calculated using INLA.

**Figure 6.**Prediction of M. faya occurrence in Pico Island by posterior mean (

**a**) and standard deviation (

**b**) of the linear predictor. Top: Mixed effects model including a random field; Bottom: Fixed effects model (without a random field). All models calculated using INLA.

**Figure 7.**Prediction of P. undulatum occurrence in São Miguel Island by posterior mean (

**a**) and standard deviation (

**b**) of the linear predictor. Top: Mixed effects model including a random field; Bottom: Fixed effects model (without a random field). All models calculated using INLA.

**Figure 8.**Prediction of M. faya occurrence in São Miguel Island by posterior mean (

**a**) and standard deviation (

**b**) of the linear predictor. Top: Mixed effects model including a random field; Bottom: Fixed effects model (without a random field). All models calculated using INLA.

**Table 1.**Number of species records used in modeling, per island and species. PU: Pittosporum undulatum; MF: Morella faya.

Pico | São Miguel | |||
---|---|---|---|---|

PU | MF | PU | MF | |

Training sample (100%) | 7269 | 3769 | 3188 | 375 |

Sample reduction | ||||

1st step | 5000 | 3000 | 3000 | 300 |

2nd step | 500 | 300 | 300 | 30 |

3th step | 50 | 30 | 30 | --- |

**Table 2.**List of the 27 ecogeographical variables (EGVs) used to model indigenous and non-indigenous woody species.

Variable Category | Variables | Code | Unit |
---|---|---|---|

Topographical | Digital elevation model | DEM | m |

Aspect | ASP | ° | |

Slope | SLP | % | |

Curvature | CRV | ||

Flow accumulation | FLA | ||

Summer hill shade | SHS | ||

Winter hill shade | WHS | ||

Climatic | Annual minimum temperature | TMIN | °C |

Annual mean temperature | TM | ||

Annual maximum temperature | TMAX | ||

Annual temperature range | TRA | ||

Annual mean temperature range | TMRA | ||

Annual minimum relative humidity | RHMIN | % | |

Annual mean relative humidity | RHM | ||

Annual maximum relative humidity | RHMAX | ||

Annual relative humidity range | RHRA | ||

Annual minimum precipitation | PMIN | mm | |

Annual mean precipitation | PM | ||

Annual maximum precipitation | PMAX | ||

Annual precipitation range | PRA | ||

Annual mean precipitation range | PMRA | ||

Land use | Distance to forest | DL 1 | m |

Distance to natural vegetation | DL 2 | m | |

Distance to pastureland | DL 3 | m | |

Distance to agriculture | DL 4 | m | |

Distance to barren/bare areas | DL 5 | m | |

Distance to urban/industrial areas | DL 6 | m |

**Table 3.**Sets of EGV (1–5) that produced models with highest fit in Pico and São Miguel islands using fixed effects GLMs.

Variables | Code | Pico | São Miguel | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

P. undulatum | M. faya | P. undulatum | M. faya | ||||||||||||||||||

EGV Set | EGV Set | EGV Set | EGV Set | ||||||||||||||||||

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | ||

Topographic | |||||||||||||||||||||

Digital elevation model | DEM | + | + | + | + | + | + | + | + | + | |||||||||||

Aspect | ASP | + | + | + | |||||||||||||||||

Slope | SLP | + | + | + | + | + | + | + | + | + | + | ||||||||||

Curvature | CRV | + | + | + | + | ||||||||||||||||

Flow accumulation | FLA | + | + | + | + | + | |||||||||||||||

Summer hill shade | SHS | + | + | + | + | + | |||||||||||||||

Winter hill shade | WHS | + | + | + | + | + | + | + | + | ||||||||||||

Climatic | |||||||||||||||||||||

Temperature | |||||||||||||||||||||

Annual mean temperature | TM | + | + | + | + | + | + | + | + | + | + | ||||||||||

Annual temperature range | TRA | + | + | + | + | + | + | + | |||||||||||||

Annual mean temperature range | TMRA | + | + | + | + | + | + | ||||||||||||||

Humidity | |||||||||||||||||||||

Annual mean relative humidity | RHM | + | + | + | |||||||||||||||||

Annual relative humidity range | RHRA | + | + | + | + | + | + | + | |||||||||||||

Precipitation | |||||||||||||||||||||

Annual mean precipitation | PM | + | + | + | + | + | + | + | + | + | |||||||||||

Annual precipitation range | PRA | + | + | ||||||||||||||||||

Annual mean precipitation range | PMRA | + | + | + | + | + | + | + | |||||||||||||

Land use | |||||||||||||||||||||

Distance to forest | DL 1 | + | + | + | + | + | + | + | + | ||||||||||||

Distance to natural vegetation | DL 2 | + | + | ||||||||||||||||||

Distance to pastureland | DL 3 | + | + | ||||||||||||||||||

Distance to agriculture | DL 4 | + | + | + | + | + | |||||||||||||||

Distance to barren/bare areas | DL 5 | + | + |

**Table 4.**Results of fixed and mixed effects models regarding different combinations of variables for P. undulatum and M. faya in Pico and São Miguel islands. BI: Boyce Index; sd: standard deviation; BS: Brier Score.

Fixed Effects Models | Mixed Effects Models | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

GLM | INLA | SPDE | ||||||||||||

AIC | BI | sd (BI) | AUC | 10 k-Fold (AUC) | DIC | DIC | WAIC | Mean BS | Sum LCPO | Mean LCPO | ||||

Model | Study Species | Study Area | Mean | sd | ||||||||||

EGV1 | P. undulatum | Pico Island | 18,629 | 0.988 | 0.002 | 0.740 | 0.752 | 0.007 | 18,629 | |||||

EGV2 | 17,876 | 0.988 | 0.003 | 0.811 | 0.786 | 0.013 | 17,876 | |||||||

EGV3 | 16,687 | 1.000 | 0.000 | 0.809 | 0.824 | 0.012 | 16,687 | |||||||

EGV4 | 17,456 | 0.992 | 0.001 | 0.807 | 0.791 | 0.013 | 17,456 | |||||||

EGV5 | 15,917 | 1.000 | 0.000 | 0.849 | 0.838 | 0.008 | 15,917 | 12,858 | 12,821 | 0.106 | 6415 | 0.371 | ||

EGV1 | M. faya | 11,242 | 0.994 | 0.002 | 0.875 | 0.845 | 0.007 | 11,242 | ||||||

EGV2 | 11,146 | 1.000 | 0.000 | 0.852 | 0.849 | 0.004 | 11,147 | |||||||

EGV3 | 11,226 | 0.999 | 0.001 | 0.878 | 0.848 | 0.011 | 11,226 | |||||||

EGV4 | 10,812 | 1.000 | 0.000 | 0.855 | 0.858 | 0.009 | 10,812 | |||||||

EGV5 | 10,088 | 1.000 | 0.000 | 0.887 | 0.883 | 0.008 | 10,088 | 6685 | 6648 | 0.067 | 3328 | 0.242 | ||

EGV1 | P. undulatum | São Miguel Island | 11,871 | −0.480 | 0.036 | 0.819 | 0.796 | 0.009 | 11,871 | |||||

EGV2 | 13,676 | 0.750 | 0.009 | 0.623 | 0.675 | 0.016 | 13,676 | |||||||

EGV3 | 11,612 | 0.998 | 0.003 | 0.874 | 0.815 | 0.015 | 11,612 | |||||||

EGV4 | 11,640 | 0.425 | 0.017 | 0.820 | 0.802 | 0.018 | 11,640 | |||||||

EGV5 | 9734 | 0.991 | 0.002 | 0.880 | 0.877 | 0.014 | 9734 | 7779 | 7675 | 0.069 | 3862 | 0.293 | ||

EGV1 | M. faya | 2314 | −0.503 | 0.039 | 0.806 | 0.871 | 0.036 | 2314 | ||||||

EGV2 | 3012 | 0.974 | 0.005 | 0.690 | 0.713 | 0.039 | 3012 | |||||||

EGV3 | 2766 | 0.996 | 0.003 | 0.871 | 0.803 | 0.031 | 2766 | |||||||

EGV4 | 2780 | 0.562 | 0.020 | 0.756 | 0.769 | 0.031 | 2780 | |||||||

EGV5 | 2515 | 0.646 | 0.022 | 0.732 | 0.853 | 0.028 | 2515 | INF | 927 | 0.008 | 1211 | 0.117 |

**Table 5.**Results of sampling reduction on fixed and mixed effects models regarding EGV set 5 for P. undulatum and M. faya in Pico Island. BI: Boyce Index; sd: standard deviation; BS: Brier Score; PU: P. undulatum; MF: M. faya; P: Pico. Variables with non-significant values were excluded from the model (code: D) (Supplementary Materials).

Fixed Effects Models | Mixed Effects Models | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

GLM | INLA | SPDE | ||||||||||

AIC | BI | sd (BI) | AUC | 10 k-fold (AUC) | DIC | DIC | WAIC | Mean BS | Sum Log CPO | Mean Log CPO | ||

Model | Mean | sd | ||||||||||

P. undulatum—Pico | ||||||||||||

EGV5_PU_P Full | 15,917 | 1.000 | 0.000 | 0.849 | 0.838 | 0.008 | 15,917 | 12,858 | 12,821 | 0.106 | 6415 | 0.371 |

EGV5_PU_P 5000 | 13,836 | 1.000 | 0.000 | 0.816 | 0.818 | 0.015 | 13,836 | 11,917 | 11,892 | 0.120 | 5949 | 0.397 |

EGV5_PU_P 500 | 3445 | 1.000 | 0.000 | 0.887 | 0.791 | 0.017 | 3445 | 3419 | 3418 | 0.042 | 1709 | 0.163 |

EGV5_PU_P_D 500 | 3443 | 1.000 | 0.000 | 0.790 | 0.789 | 0.020 | 3443 | 3416 | 3416 | 0.042 | 1708 | 0.163 |

EGV5_PU_P 50 | 585 | 0.998 | 0.001 | 0.803 | 0.796 | 0.085 | 585 | 579 | 578 | 0.005 | 290 | 0.029 |

EGV5_PU_P_D 50 | 587 | 0.999 | 0.000 | 0.777 | 0.786 | 0.075 | 585 | 578 | 576 | 0.005 | 289 | 0.029 |

EGV5_PU_P_D_D 50 | 582 | 1.000 | 0.000 | 0.644 | 0.777 | 0.061 | 582 | 580 | 578 | 0.005 | 290 | 0.029 |

EGV5_PU_P_D_D_D 50 | 582 | 1.000 | 0.000 | 0.793 | 0.777 | 0.056 | 582 | 578 | 577 | 0.005 | 289 | 0.029 |

EGV5_PU_P_D_D_D_D 50 | 581 | 1.000 | 0.000 | 0.754 | 0.769 | 0.066 | 581 | 579 | 577 | 0.005 | 289 | 0.029 |

Morella faya—Pico | ||||||||||||

EGV5_MF_P Full | 10,088 | 1.000 | 0.000 | 0.887 | 0.883 | 0.008 | 10,088 | 6685 | 6648 | 0.067 | 3328 | 0.242 |

EGV5_MF_P 3000 | 9098 | 1.000 | 0.000 | 0.873 | 0.878 | 0.013 | 9098 | 6481 | 6456 | 0.072 | 3231 | 0.249 |

EGV5_MF_P 300 | 2125 | 1.000 | 0.000 | 0.944 | 0.873 | 0.021 | 2125 | 1982 | 1980 | 0.025 | 990 | 0.096 |

EGV5_MF_P 30 | 364 | 0.997 | 0.001 | 0.843 | 0.884 | 0.047 | 363 | 363 | 363 | 0.003 | 182 | 0.018 |

EGV5_MF_P_D 30 | 358 | 0.989 | 0.002 | 0.855 | 0.872 | 0.052 | 358 | 358 | 357 | 0.003 | 179 | 0.018 |

**Table 6.**Results of sampling reduction for fixed and mixed effects models regarding EGV set 5 for P. undulatum and M. faya in São Miguel Island. BI: Boyce Index; sd: standard deviation; BS: Brier Score; PU: P. undulatum; MF: M. faya; SM: São Miguel. Variables with non-significant values were excluded from the model (code: D) (Supplementary Materials).

Fixed Effects Models | Mixed Effects Models | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

GLM | INLA | SPDE | ||||||||||

AIC | BI | sd (BI) | AUC | 10 k-fold (AUC) | DIC | DIC | WAIC | Mean BS | Sum Log CPO | Mean Log CPO | ||

Model | Mean | sd | ||||||||||

Pittosporum undulatum—São Miguel | ||||||||||||

EGV5_PU_SM Full | 9734 | 0.991 | 0.002 | 0.880 | 0.877 | 0.014 | 9734 | 7779 | 7675 | 0.069 | 3862 | 0.293 |

EGV5_ PU_SM 3000 | 9864 | 0.993 | 0.002 | 0.880 | 0.864 | 0.009 | 8849 | 7591 | 7503 | 0.069 | 3773 | 0.290 |

EGV5_ PU_SM 300 | 2169 | 0.996 | 0.004 | 0.887 | 0.866 | 0.020 | 2168 | 2098 | 2103 | 0.024 | 1052 | 0.102 |

EGV5_ PU_SM _D 300 | 2169 | 0.997 | 0.002 | 0.849 | 0.865 | 0.016 | 2168 | 2099 | 2103 | 0.024 | 1052 | 0.102 |

EGV5_ PU_SM 30 | 379 | 0.999 | 0.001 | 0.592 | 0.851 | 0.087 | 378 | 378 | 378 | 0.003 | 189 | 0.019 |

EGV5_ PU_SM _D 30 | 375 | 0.999 | 0.002 | 0.964 | 0.807 | 0.139 | 375 | 375 | 374 | 0.003 | 187 | 0.019 |

Morella faya—São Miguel | ||||||||||||

EGV5_ MF_SM Full | 2515 | 0.646 | 0.022 | 0.732 | 0.853 | 0.028 | 2515 | INF | 927 | 0.008 | 1211 | 0.117 |

EGV5_ MF_SM 300 | 2150 | 0.717 | 0.022 | 0.841 | 0.765 | 0.030 | 2149 | 919 | 895 | 0.009 | 462 | 0.045 |

EGV5_ MF_SM 30 | 359 | 0.325 | 0.031 | 0.988 | 0.827 | 0.203 | 359 | INF | 243 | 0.002 | 124 | 0.012 |

EGV5_ MF_SM _D 30 | 360 | 0.003 | 0.049 | 0.753 | 0.884 | 0.115 | 359 | 273 | 293 | 0.002 | 145 | 0.014 |

**Table 7.**Numerical summary of the posterior distributions of the parameters for the best mixed effects model for the two species studied and islands. This summary contains the mean, the standard deviation (sd), and 95% credibility interval. SM: São Miguel.

Island | Species | Predictor | Mean | sd | Q0.025 | Q0.975 |
---|---|---|---|---|---|---|

Pico | P. undulatum | Intercept (A0) | 1.2717 | 2.4951 | −3.6659 | 6.1333 |

Digital elevation model | −0.0021 | 0.0013 | −0.0047 | 0.0005 | ||

Slope | 0.0519 | 0.0091 | 0.0340 | 0.0699 | ||

Annual mean temperature | −0.1498 | 0.0830 | −0.3126 | 0.0131 | ||

Annual temperature range | −0.0145 | 0.1458 | −0.2998 | 0.2725 | ||

Annual relative humidity range | 0.0988 | 0.0458 | 0.0094 | 0.1892 | ||

Annual mean precipitation | −0.0013 | 0.0003 | −0.0019 | −0.0007 | ||

Annual mean precipitation range | 26.9892 | 6.2672 | 14.5772 | 39.2030 | ||

Distance to forest | −0.0053 | 0.0004 | −0.0060 | −0.0046 | ||

Distance to agriculture | −0.0004 | 0.0002 | −0.0008 | 0.0001 | ||

M. faya | Intercept (A0) | 10.8081 | 3.8892 | 3.1496 | 18.4210 | |

Digital elevation model | −0.0067 | 0.0024 | −0.0113 | −0.0020 | ||

Slope | 0.0486 | 0.0144 | 0.0204 | 0.0770 | ||

Winter hill shade | 0.0009 | 0.0037 | −0.0063 | 0.0081 | ||

Annual mean temperature | −0.1502 | 0.1294 | −0.4043 | 0.1038 | ||

Annual temperature range | −0.5037 | 0.2117 | −0.9201 | −0.0889 | ||

Annual mean precipitation | −0.0022 | 0.0007 | −0.0036 | −0.0009 | ||

Distance to forest | −0.0047 | 0.0005 | −0.0057 | −0.0037 | ||

Distance to agriculture | −0.0009 | 0.0004 | −0.0016 | −0.0001 | ||

SM | P. undulatum | Intercept (A0) | 29.7124 | 6.3398 | 17.2904 | 42.1730 |

Slope | 0.0638 | 0.0055 | 0.0530 | 0.0746 | ||

Flow accumulation | 0.0086 | 0.0012 | 0.0063 | 0.0109 | ||

Winter hill shade | 0.0022 | 0.0010 | 0.0003 | 0.0042 | ||

Annual mean temperature | −0.4494 | 0.1673 | −0.7784 | −0.1217 | ||

Annual mean temperature range | −9.3326 | 2.2925 | −13.8510 | −4.8493 | ||

Annual mean relative humidity | −0.1601 | 0.0374 | −0.2337 | −0.0870 | ||

Annual mean precipitation | −0.0010 | 0.0003 | −0.0015 | −0.0005 | ||

Distance to forest | −0.0072 | 0.0004 | −0.0080 | −0.0064 | ||

Distance to natural vegetation | −0.0007 | 0.0001 | −0.0009 | −0.0004 | ||

M. faya | Intercept (A0) | 12.8597 | 4.8016 | 3.3689 | 22.2520 | |

Summer hill shade | −0.0228 | 0.0049 | −0.0328 | −0.0133 | ||

Annual mean temperature range | −12.0140 | 4.1734 | −20.3010 | −3.8892 | ||

Annual relative humidity range | −0.4172 | 0.1079 | −0.6343 | −0.2101 | ||

Annual mean precipitation range | 0.4804 | 22.0600 | −42.7840 | 43.8530 | ||

Distance to forest | −0.0082 | 0.0015 | −0.0114 | −0.0054 | ||

Distance to barren/bare areas | −0.0049 | 0.0010 | −0.0070 | −0.0033 |

Study Species | Study Area | Hyperparameters | Mean | sd | Q0.025 | Q0.05 | Q0.975 |
---|---|---|---|---|---|---|---|

P. undulatum | Pico Island | ${\theta}_{1}$ | 4.499 | 0.062 | 4.370 | 4.503 | 4.611 |

${\theta}_{2}$ | −6.508 | 0.097 | −6.681 | −6.514 | −6.303 | ||

M. faya | ${\theta}_{1}$ | 4.385 | 0.097 | 4.171 | 4.396 | 4.544 | |

${\theta}_{2}$ | −6.750 | 0.293 | −7.180 | −6.797 | −6.077 | ||

P. undulatum | São Miguel Island | ${\theta}_{1}$ | 3.789 | 0.100 | 3.622 | 3.779 | 4.010 |

${\theta}_{2}$ | −5.889 | 0.099 | −6.112 | −5.877 | −5.731 | ||

M. faya | ${\theta}_{1}$ | 3.348 | 0.176 | 2.963 | 3.365 | 3.647 | |

${\theta}_{2}$ | −6.434 | 0.157 | −6.716 | −6.444 | −6.103 |

**Table 9.**Numerical summary of the posterior distributions of the best mixed effects models and their reduction for the two species studied in Pico and São Miguel islands. This summary contains the DIC, the posterior marginal distribution of log ($k$) and 1/$k$, the posterior marginal distribution of nominal variance of random field$({\sigma}_{W}^{2}$) and posterior marginal distribution of the practical range (meters). PU: P. undulatum; MF: M. faya; P: Pico; SM: São Miguel. Variables with non-significant values were excluded from the model (code: D) (Supplementary Materials).

Model | DIC | Log($\mathit{k}$) | 1/$\mathit{k}$ | $\left({\mathit{\sigma}}_{\mathit{W}}^{2}\right)$ | Practical Range |
---|---|---|---|---|---|

Pittosporum undulatum—Pico | |||||

EGV5_PU_P Full | 12,858 | −6.504 | 667.838 | 4.472 | 1904 |

EGV5_PU_P 5000 | 11,917 | −6.486 | 655.872 | 2.665 | 1864 |

EGV5_PU_P 500 | 3419 | −6.743 | 848.099 | 0.282 | 2718 |

EGV5_PU_P_D 500 | 3416 | −6.683 | 798.443 | 0.267 | 2536 |

EGV5_PU_P 50 | 579 | −8.067 | 3187.146 | 1.131 | 11,936 |

EGV5_PU_P_D 50 | 578 | −8.382 | 4373.173 | 1.290 | 19,805 |

EGV5_PU_P_D_D 50 | 580 | −7.982 | 2928.797 | 0.671 | 17,047 |

EGV5_PU_P_D_D_D 50 | 578 | −7.871 | 2620.682 | 0.641 | 12,842 |

EGV5_PU_P_D_D_D_D 50 | 579 | −7.286 | 1459.476 | 0.535 | 7370 |

Morella faya—Pico | |||||

EGV5_MF_P Full | 6685 | −6.725 | 833.101 | 124.597 | 2511 |

EGV5_MF_P 3000 | 6481 | −7.104 | 1217.079 | 11.668 | 3531 |

EGV5_MF_P 300 | 1982 | −7.944 | 2818.350 | 3.522 | 8626 |

EGV5_MF_P 30 | 363 | −6.328 | 559.804 | 0.051 | 6522 |

EGV5_MF_P_D 30 | 358 | −6.730 | 920.246 | 0.060 | 13,930 |

Pittosporum undulatum—São Miguel | |||||

EGV5_PU_SM Full | 7779 | −5.867 | 353.313 | 5.339 | 1026 |

EGV5_ PU_SM 3000 | 7591 | −5.993 | 400.794 | 5.688 | 1141 |

EGV5_ PU_SM 300 | 2098 | −6.379 | 589.498 | 0.795 | 1767 |

EGV5_ PU_SM _D 300 | 2099 | −6.395 | 599.029 | 0.793 | 1791 |

EGV5_ PU_SM 30 | 378 | −6.988 | 1083.141 | 0.123 | 6546 |

EGV5_ PU_SM _D 30 | 375 | −9.422 | 12,351.790 | 1.437 | 11,2128 |

Morella faya—São Miguel | |||||

EGV5_ MF_SM Full | INF | −6.428 | 618.996 | 43.046 | 1780 |

EGV5_ MF_SM 300 | 919 | −6.714 | 823.773 | 31.754 | 2333 |

EGV5_ MF_SM 30 | INF | −6.339 | 566.232 | 8.557 | 1972 |

EGV5_ MF_SM _D 30 | 273 | −6.762 | 864.418 | 5.053 | 3126 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dutra Silva, L.; Brito de Azevedo, E.; Bento Elias, R.; Silva, L. Species Distribution Modeling: Comparison of Fixed and Mixed Effects Models Using INLA. *ISPRS Int. J. Geo-Inf.* **2017**, *6*, 391.
https://doi.org/10.3390/ijgi6120391

**AMA Style**

Dutra Silva L, Brito de Azevedo E, Bento Elias R, Silva L. Species Distribution Modeling: Comparison of Fixed and Mixed Effects Models Using INLA. *ISPRS International Journal of Geo-Information*. 2017; 6(12):391.
https://doi.org/10.3390/ijgi6120391

**Chicago/Turabian Style**

Dutra Silva, Lara, Eduardo Brito de Azevedo, Rui Bento Elias, and Luís Silva. 2017. "Species Distribution Modeling: Comparison of Fixed and Mixed Effects Models Using INLA" *ISPRS International Journal of Geo-Information* 6, no. 12: 391.
https://doi.org/10.3390/ijgi6120391