Exploring fire incidence in Portugal using generalized additive models for location, scale and shape (GAMLSS)

Abstract

Portuguese wildfires are responsible for large environmental, ecological and socio-economic impacts. This study explores the fire-environment relationships by modeling fire incidence (FI) against vegetation, precipitation and anthropogenic drivers. The mean, dispersion and asymmetry of the FI distribution were modelled on the predictors using the generalized additive models for location, scale and shape. Results show that increasing forests and shrublands increases FI and decreases its dispersion, highlighting high FI regions. Fire absence decreases with all the predictors except human influence, indicating its control on fire hazard. Rain-fed versus irrigated agriculture may have a dual role on FI, pointing the need to explore them separately when modeling FI drivers. Precipitation has a non-linear effect on FI distribution parameters. The role of forests on fire distribution asymmetry needs to be further explored. Modelling the previously unexplored drivers of dispersion and asymmetry of FI gives new insights into fire regime studies and fire-environmental relationships.

Appendices

Appendix 1: Statistical models

According to the proposed modeling approach, first a ZAGA distribution model was fitted to fire data. The probability function of the ZAGA $\left( {\mu ,\sigma ,\nu } \right)$ mixture model is defined by Rigby and Stasinopoulos (2005):

$$f\left( {y~|~\mu ,\sigma ,\nu } \right) = \left\{ {\begin{array}{*{20}c} {\nu ,} & {y = 0} \\ {\left( {1 - \nu } \right)\left[ {\frac{1}{{\left( {\sigma ^{2} \mu } \right)^{{1/\sigma ^{2} }} }}\frac{{y^{{\frac{1}{{\sigma ^{2} }} - 1}} e^{{ - y/\left( {\sigma ^{2} \mu } \right)}} }}{{\Gamma \left( {1/\sigma ^{2} } \right)}}} \right],} & {y> 0} \\ \end{array} } \right.$$

(2)

$$\begin{gathered} {\text{for}}\;0 \leq y\left\langle {\infty ,~\mu } \right\rangle 0,~~\sigma>0~{\text{and~}}0<\nu <1, \hfill \\ {\text{with}}\;E\left( y \right)=\left( {1 - \nu } \right)\mu ~{\text{and}}~Var\left( y \right)=\left( {1 - \nu } \right){\mu ^2}\left( {\nu +{\sigma ^2}} \right). \hfill \\ \end{gathered}$$

(3)

The ZAGA model has three components: the location $\left( \mu \right)$, which is equal to the mean in the GA distribution, the dispersion $\left( \sigma \right)$ of FI, and the probability of fire absence (the zeros), given by the shape parameter $\left( \nu \right)$ of the GA distribution. All distribution parameters are explicitly modeled as a function of the explanatory variables (each distribution parameter can have a different selection of predictors) using the following link functions:

$${\text{log}}\left( \mu \right) = \eta _{1} = x_{1}^{{\prime}} \beta _{1} + \mathop \sum \limits_{{j = 1}}^{h} h_{{j1}} \left( {x_{{j1}} } \right)$$

(4)

$${\text{log}}\left( \sigma \right) = \eta _{2} = x_{2}^{{\prime}} \beta _{2} + \mathop \sum \limits_{{j = 1}}^{h} h_{{j2}} \left( {x_{{j2}} } \right)$$

(5)

$${\text{logit}}\left( \nu \right)=\log \left( {\frac{\nu }{{1 - \nu }}} \right)={\eta _3}=x_{3}^{\prime}{\beta _3}+\mathop \sum \limits_{{j=1}}^{h} {h_{j3}}\left( {{x_{j3}}} \right)$$

(6)

where $x_{k}^{\prime}{\beta _k}$ and ${h_{jk}}\left( {{x_{j3}}} \right)$ are the parametric and non-parametric terms, correspondingly. These last smoothing functions were modeled with penalized B-splines (Eilers and Marx 1996) and they automatically optimize the degree of smoothing based on penalized maximum likelihood estimation.

With the main goal of modeling the shape of the distribution and the right-skewness of FI, a GG distribution model was fitted to data. The probability function of the flexible GG $\left( {\mu ,\sigma ,\nu } \right)$ distribution is:

$$f\left( {y{\text{~}}|{\text{~}}\mu ,\sigma ,\nu } \right)=\frac{{{\theta ^\theta }{z^\theta }|\nu |{e^{\left( { - \theta z} \right)}}}}{{\left( {\Gamma \left( \theta \right)y} \right)}}$$

(7)

$${\text{where}}\;z={\left( {y/\mu } \right)^\nu },~\theta =1/\left( {{\sigma ^2}{{\left| \nu \right|}^2}} \right)~{\text{for~}}y>0,~\mu>0,~\sigma>0{\text{~and~}} - \infty <\nu <+\infty ,\;{\text{with}}$$

$$E\left( y \right)=\mu \frac{{\Gamma \left( {\theta +\frac{1}{\nu }} \right)}}{{\left[ {{\theta ^{\frac{1}{\nu }}}\Gamma \left( \theta \right)} \right]}},{\text{~~}}Var\left( y \right)=\frac{{\left( {{\mu ^2}} \right)\left\{ {\Gamma \left( \theta \right)\Gamma \left( {\theta +\frac{2}{\nu }} \right) - {{\left[ {\Gamma \left( {\theta +\frac{1}{\nu }} \right)} \right]}^2}} \right\}}}{{\left\{ {{\theta ^{\frac{2}{\nu }}}{{\left[ {\Gamma \left( \theta \right)} \right]}^2}} \right\}}}$$

(8)

and

$${\text{Skew}}\left( {\text{y}} \right)={\text{E}}\left[ {{{\left( {{\text{y}} - {\text{E}}\left( {\text{y}} \right)} \right)}^3}} \right]/\sqrt {{{\left( {{\text{E}}\left[ {{{\left( {{\text{y}} - {\text{E}}\left( {\text{y}} \right)} \right)}^2}} \right]} \right)}^3}} .$$

(9)

The Gamma distribution is obtained when the shape parameter$~\nu ~=1$. For $\nu <0~$ the distribution is asymmetric negative and for $\nu>0$ the distribution is asymmetric positive. The lognormal distribution appears as a limiting distribution of the GG when $\nu =0.$ Thus, the GG family includes other commonly used distributions such as the Exponential, Weibull, Rayleigh, among others (Cox et al. 2007).

Appendix 2: Figures and tables

See Figs. 9, 10, 11, 12, 13 and Table 5.

Fig. 9

Box-and-whisker plots of the bioclimatic variables (Table 1) for classes of Fire Incidence (FI) defined according to its frequency histogram (Fig. 1): (a) mean temperature of the driest quarter (TDq); (b) mean temperature if the warmest quarter (TWaq); (c) annual precipitation (AP); (d) precipitation of the driest month (PDm); (e) precipitation of the wettest quarter (PWq); (f) precipitation of the driest quarter (PDq); (g) precipitation of the warmest quarter (PWaq); (h) precipitation of the coldest quarter (PCq)

Fig. 10

Box-and-whisker plots of the vegetation variables (Table 1) for classes of Fire Incidence (FI) defined according to its frequency histogram (Fig. 1): (a) forest class from Cos2007 (cosFor); (b) shrubland class from Cos2007 (cosShr); (c) cropland class from Cos2007 (cosCrop); (d) forest class from CLC2006 (corFor06); (e) shrubland class from CLC2006 (corShr06); (f) cropland class from CLC2006 (corCrop06); (g) mean forest class area from CLC 1990/2000/2006 (For); (h) mean shrubland class area from CLC 1990/2000/2006 (Shr); and (i) mean shrubland class area from CLC 1990/2000/2006 (Crop)

Fig. 11

Box-and-whisker plots of the human variables (Table 1) for classes of Fire Incidence (FI) defined according to its frequency histogram (Fig. 1): (a) logarithm of population density (logPD); and (b) human influence (HI)

Fig. 12

Mapped weather variables for mainland Portugal: mean temperature of the driest quarter (TDq) (a); mean temperature of the warmest quarter (TWaq) (b); annual precipitation (AP) (c); precipitation of the driest month (PDm) (d); precipitation of the wettest quarter (PWq) (e); precipitation of the driest quarter (PDq) (f); precipitation of the warmest quarter (PWaq) (g); and precipitation of the coldest quarter (PCq) (h). NUTS II level administrative limits are overlaid

Fig. 13

Scatterplot between forest area (For) and estimated shape (nu) parameter of FI from the GG model. Red line represents a linear regression model and R² the model coefficient of determination

Table 5 Univariate and multivariate linear regression models fitted to fire data set to select environmental predictors based on their goodness-of-fit (SBC)