Introduction

Fire is an important disturbance factor in forest ecosystems. It has significant impacts on the balance of carbon and energy, regeneration, and forest succession (Johnstone et al. 2010; Kasischke et al. 2010). The area burned annually by forest fire causes enormous costs and losses, and strongly influences decision-making and land-use planning of the public agencies of natural resources management in China and in the World. On the other hand, climate changes and variables have been a well-known driver of forest fires under various spatial and temporal scales (Flannigan and van Wagner 1991; Johnson and Wowchuk 1993; Duff et al. 2005). Thus, a number of studies regarding the area burned by forest fires and climate variables have been conducted in the past decades (e.g., Flannigan and van Wagner 1991; Skinner et al. 2002; Duff et al. 2005; Flannigan et al. 2005; McCoy and Burn 2005).

To date, linear regression models are the most common approach for modeling the relationships between the area burned by forest fires and climate changes/variables (e.g., Balling et al. 1992; Flannigan et al. 2005; Tymstra et al. 2005). However, some researchers indicated that there were no significant linear relationships between the fire burned area and climate variables such as fuel moisture, precipitation, and temperature. Instead, thresholds occurred when the burned area varied with time based on the changes of temperature and precipitation. The relationships between the burned area and temperature and precipitation contrasted with what they were when their thresholds were exceeded (Schoenberg et al. 2003). Alternatively, researchers applied logarithmic transformation to fire data, namely log-linear models, in order to deal with the situations where non-linear relationships existed between the fire burned area and climate variables (e.g., Littell et al. 2009). However, a log-linear model has limited flexibility to linearize non-linear relationships between variables, because (1) it assumes that the frequency distribution of the response variable follows a log-normal distribution, and (2) the relationships between variables are either exponential (when applying log-transformation on the response variable only) or power (when applying log-transformation on both response and predictor variables). It may not be the best choice for modeling the highly skewed frequency distributions of the response variables like the area burned by forest fire. Thus, a more flexible regression model is desirable to investigate the relationships between the area burned by forest fires and climate variables.

In recent years, generalized linear models (GLIM) have proved to be a better and more effective approach dealing with the frequency distributions of response variables that are grossly departed from normal. GLIM is designed to model response variables that may follow a general distribution called the exponential family, which includes normal, lognormal, binomial, Poisson, beta, gamma, etc. (Myers et al. 2002). GLIM is viewed as a unification of linear and nonlinear regression models and has three components: a response variable distribution, a linear predictor that involves a number of independent or regressor variables, and a link function that connects the linear predictor to the natural mean of the response variable. In the situations where the distributions of response variables are assumed to follow a beta or gamma distribution, a natural log link function is commonly used to link the mean of the response variable to the linear predictors. However, this log transformation of the response mean in GLIM is fundamentally different from a log transformation on the observed response variable in a log-linear model. This is because that log-transforming the mean of the response variable in GLIM does not alter the error distribution of the model, whereas log-transforming the values of the response variable in log-linear models does (Myers et al. 2002). The two methods of log transformation can lead to quite different results. In general, log-transforming the response mean (GLIM) often allows the results to be more easily interpreted, especially in that mean parameters remain on the same scale as the measured response variables. It is well known that gamma distribution is suitable to dealing with heteroskedasticity in non-negative, continuous data, in a way that a log-linear model cannot do without weighted least squares.

However, gamma modeling remained quite difficult to conduct until fairly recently when powerful statistical computing packages became available. The gamma distribution has been applied in many study fields; for example, it is commonly used in meteorology and climatology to represent variations in precipitation amount (Wilks 1990). Littell et al. (2009) analyzed the fire burned area of western US using gamma-generalized linear regression model. Their results revealed that both goodness-of-fit and significance level of the parameters of gamma model were superior to those of multiple linear regressions.

The Tahe forest region is located in the boreal forest in northeast China. This area has experienced high frequencies of forest fires, resulting in large burned areas annually. To date, some studies were conducted relating the burned area in Tahe to climate changes/variables, but all utilized linear regression models (e.g., Qu and Hu 2007; Zhao et al. 2009; Yang et al. 2010; Wang et al. 2013). Other regression models such as GLIM have not been thoroughly compared and discussed for modeling the fire burned area in the region.

In this study, we applied three regression models to determine the relationships between climate variables and the burned area caused by lightning-caused fire, human-caused fire, and the combination of both. The lighting-caused fire is induced by lightning strikes. The previous study showed that the lightning-caused fire of Daxing’an Mountains mainly occurred in June and distributed in the elevation of 200–1300 m (Du et al. 2010). The human-caused fire is that the fire ignitions are directly or indirectly related to human activities such as smoking, hunting, fireworks, escaped fire from locomotives and residents’ homes. The three models used in this study were a multiple linear regression model, a log-linear regression model, and a gamma-generalized linear model. The model fitting was evaluated and compared using model statistics such as R2, AIC and parameter analysis.

Materials and methods

Study area

The study area is the Tahe forest region (52°09′–53°23′N, 125°19′–125°48′E) located in northeast China with a total forest area of 920,000 ha (Fig. 1). The area belongs to cold temperate continental monsoon climate. Winter is long, cold, and dry because of the cold air from Siberia and Mongolia. The annual sunshine is 2560 h, the annual precipitation is 428 mm, and the frost-free period is 100 days. The dominant tree species includes Dahurian larch (Larix gmelinii Rupr.), accompanied with white birch (Betula platyphylla Suk.) and Mongolian pine (Pinus sylvestris L. var. mongolica Litv.). The Tahe forest region has been suffering high fire disturbances. Thus, the fire prevention is one of the main responsibilities of the local bureau of forest management.

Fig. 1
figure 1

Map of the location of the study area, the Tahe forest region, in northeast China

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Data collection

The Fire Prevention Office of Tahe County (FPOT) is responsible for collecting and recording wildfire information, including fire location, burned area, forest type, causes, and the dates of forest fires. In this study, the annual fire burned area of the Tahe region from 1974 to 2009 was provided by FPOT, which included the area burned by lightning-caused fire (L-fire), the area burned by human-caused fire (H-fire), and the total burned area of both causes (T-fire). We used these three variables as the response variables in this study.

The corresponding historical climate data of the study area during the period of 1974–2009 were obtained from the China Meteorological Data and Sharing Network (http://cdc.cma.gov.cn/). There is a national weather station located in the center of the Tahe forest region (Station ID is 50246). The weather data have been recorded continuously and completely since 1952, except for some missing data that were caused by accidents such as equipment failures and natural disasters. In this study, we used the average meteorological data during the fire season from April to November. There were about twenty meteorological or climate variables recorded by the station, and we chose nine of them in this study, including average precipitation (PD), average wind speed (WD), average relative humidity (RH), average temperature (MTE), average maximum temperature (MAT), days that rainfall exceeded 0.1 mm (DA), annual hours of sunshine (SH), average maximum wind speed (MAW), and average minimum relative humidity (MIRH). The descriptive statistics of the three response variables and nine independent or predictor variables were provided in Table 1.

Table 1 Descriptive statistics of dependent and independent variables
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Regression models

We briefly describe the three regression models used in this study as follows:

  1. (1)

    Multiple linear regression model Given a set of n observations on p independent or predictor variables (X1, X2, …, Xp), and a dependent or response variable Y, the relationship between Y and Xs can be regressed as follows:

    $$ Y = X\beta = \beta_{0} + \beta_{1} \cdot X_{1} + \beta_{2} \cdot X_{2} + \cdots + \beta_{p} \cdot X_{p} + \varepsilon $$
    (1)

    where Y is a n × 1 vector of the observed response variable, X is a n × (p + 1) known matrix including a column of 1 (for intercept) and p predictor variables, β is a p × 1 vector of unknown model parameters (including β0, β1, …, βp) that are estimated from data, and ε is a random error term with assumed distribution N(0, σ2I), where I denotes an identity matrix and σ2 represents the common error variance. The ordinary least squares (OLS) estimate of β is obtained by

    $$ \widehat{\beta } = ({\text{X}}^{\text{T}} {\text{X}})^{ - 1} {\text{X}}^{\text{T}} {\text{Y}} $$
    (2)

    where superscript T denotes the transpose of a matrix. The relationship represented by equation [1] is assumed to be universal or constant across the geographic area (Zhang and Gove 2005). Multiple linear regression models are the traditional approach to investigating the relationships between fire burned area and climate variables.

  2. (2)

    Log-linear regression model An alternative approach to modeling relationships among variables is to take natural logarithmic transformation on both response variable and predictor variables such that

    $$ {\text{logY}} = \beta_{0} + \beta_{1} \cdot {\text{log (X}}_{ 1} ) + \beta_{2} \cdot \log (X_{2} ) + \cdots + \beta_{p} \cdot \log (X_{p} ) + \varepsilon $$
    (3)

    where log is natural logarithm. This log–log model is called a log-linear model because the relationships between the log-transformed variables are linear. Log-linear models are commonly used (1) to handle the situations where the relationships between variables are nonlinear, (2) to transform a skewed frequency distribution of the response variable into one that is more approximately normal (in fact, there is a distribution called log-normal distribution defined as a distribution whose logarithm is normally distributed – but whose untransformed scale is skewed), and (3) to stabilize the heterogeneous variance in data. In this case, the interpretation of model coefficients is given as an expected percentage change in Y when X increases by some percentage. Such relationships, where both Y and X are log-transformed, are commonly referred to as elastic in econometrics, and the coefficient of logX is referred to as an elasticity.

    However, the three response variables (L-fire, H-fire and T-fire) had some zeros (e.g., no area burned by forest fire in some years). In order to take logarithmic transformation we used 0.01 to replace 0 for the three response variables. The effect of these changes on modeling was trivial.

  3. (3)

    Gamma-generalized linear model The gamma distribution belongs to exponential family and is one of the commonly used distributions in generalized linear models (GLIM). The probability density function of gamma distribution can be expressed as:

    $$ {\text{f}}\left( {{\text{Y;}}\;\gamma ,\lambda } \right) = \frac{1}{{\varGamma \left( \gamma \right)\lambda^{\gamma } }}Y^{\gamma - 1} \exp \left( { - \frac{Y}{\lambda }} \right)\quad{\text{Y }} \ge \, 0; \, \gamma , \, \lambda \, > \, 0 $$
    (4)

    where γ and λ are called shape and scale parameters, respectively. It can take on a wide range of shapes, and provides the link between the mean and the variance through its two parameters such that μ = E(Y) = γ·λ and σ2 = Var(Y) = γ·λ2.

    One of the link functions for gamma-generalized linear model is natural log as follows:

    $$ g\left( \mu \right) = \log \left( \mu \right) = \log \left( {E\left( Y \right)} \right) = \beta_{0} + \beta_{1} \cdot X_{1} + \beta_{2} \cdot X_{2} + \cdots + \beta_{p} \cdot X_{p} $$
    (5)

The model coefficients of the gamma-generalized linear model can be estimated by the maximum likelihood method (McCullagh and Nelder 1989). It is known that using a log-link function with gamma-generalized linear models is different from fitting a log-linear model to log-transformed data because on the log scale the gamma is left skew to varying degrees, while the lognormal is symmetric. This makes the gamma-generalized linear model useful in a variety of situations (McCullagh and Nelder 1989; Myers et al. 2002).

Model assessment

There are several issues that we had to consider in the modeling process:

  1. (1)

    Multicollinearity among independent variables may result in inaccurate estimation of model coefficients because a high degree of multicollinearity can prevent the computation of the matrix inversion in Eq. (3), which is required for solving for the estimates of regression coefficients. In this study the variance inflation factor (VIF) was used as a diagnostic measure to detect the multicollinearity among the nine predictor s. Generally, VIF = 10 is a threshold or red-flag on possible multicollinearity in a multiple linear regression model (Rawlings et al. 1998; Hanna 2002).

  2. (2)

    Temporal autocorrelation may exist in our fire data because the response variables (the area burned by L-fire, H-fire and T-fire) and independent variables (climate variables) were collected over years (from 1974 to 2009). The temporal autocorrelation in model errors may result in underestimated standard errors for the model coefficients, consequently causing inaccurate hypothesis testing on the model coefficients. In this study the Durbin–Watson (DW) test was performed on the residuals of the models to test on the temporal autocorrelation in model residuals (Rawlings et al. 1998; Hanna 2002).

  3. (3)

    If a log-linear model (Eq. (3)) is used to fit the log-transformed fire data, the fitted log-model yields the prediction of log(Y). To obtain the desired prediction of Y, anti-log transformation is needed to convert log(Y) to Y, i.e., \( \hat{Y} = \exp \left( {\log Y} \right) \). It is well known that this back-transformation process introduces bias into the estimation of Y. Consequently, a correction factor is typically applied to remove or reduce the bias (Baskerville 1972). However, Madgwick and Satoo (1975) finds that anti-log transformation tends to overestimate Y by applying the corrections factor, and suggests that the correction factor may be ignored if the bias from anti-log is relatively small compared to the overall variation in the estimate of Y.

  4. (4)

    Model fitting is commonly evaluated by model R2 (Rawlings et al. 1998; Hanna 2002). However, using R2 to compare two models is meaningful and appropriate only if the response variables are the same for both models. In this study, the response variables were in different scales: Y for the multiple linear regression model (Eq. (1)) and logY for the log-linear model (Eq. (3)). Therefore, it was inappropriate to compare these two models using the model R2 calculated by any statistical software. On the other hand, the model R2 can be written as (Nakagawa and Schielzeth 2013):

    $$ R^{2} = 1 - \frac{SSE}{SST} = 1 - \frac{{\sum\nolimits_{i = 1}^{n} {\left( {Y_{i} - \hat{Y}_{i} } \right)^{2} } }}{{\sum\nolimits_{i = 1}^{n} {\left( {Y_{i} - \bar{Y}} \right)^{2} } }} = 1 - \frac{{Var\left( {e_{i} } \right)}}{{Var\left( {Y_{i} } \right)}} $$
    (6)

    where SST is the total sum of squares of the response variable Y, SSE is the residual sum of squares, \( Var\left( {e_{i} } \right) \) is the variance of the model residuals, \( Var\left( {Y_{i} } \right) \) is the variance of the observed Y, and Yi, \( \hat{Y} \) and \( \bar{Y} \) represent the observed, predicted, and the mean value of Y, respectively. We used the model R2 of Eq. (6) as the assessment for model fitting to the data in this study. For the log-linear model (Eq. (3)), the predicted response variable, \( \hat{Y} \), was computed by taking the exponential to the predicted logY by Eq. (3), i.e., \( \hat{Y} = \exp \left( {\log Y} \right) \).

  5. (5)

    Akaike Information Criterion (AIC). In recent years AIC is also popular to measure the goodness-of-fit of a regression model by incorporating both the likelihood of the model and a penalty for extra model parameters. The following rule is commonly accepted: the smaller AIC is, the better the model fits the data. Again, using AIC to compare two models is meaningful and appropriate only if the response variables are the same for both models. To compare AIC for the MLP models against the log-linear models, the probability density function for the transformed data must be adjusted (Xiao et al. 2011). The likelihood that the data are generated from a log-normal distribution can be calculated based on the following formula:

    $$ \log L = \sum\limits_{i = 1}^{n} {\left[ { - \log Y_{i} - \frac{1}{2}\log \left( {2\pi \sigma_{{}}^{2} } \right) - \frac{1}{{2\sigma_{{}}^{2} }}\left( {\log Y_{i} - \log \hat{Y}_{i} } \right)^{2} } \right]} $$
    (7)

    Thus, the AIC of the log-linear model can be computed as:

    $$ AIC = - 2\log L + 2p $$
    (8)

    where p is the number of model coefficients in the model.

Results

Our fire data showed that the area burned by three types of fire-causes had a dramatic variance, which was mainly due to the extreme fire events in the past. For example, the human-caused fires (H-fire) in 1987 resulted in nearly 360,000 hectares of burned area. Figure 2 demonstrates that the lightning-caused fire (L-fire) mostly resulted in small burned areas (1–10 ha) such that its frequency distribution is strongly skewed to the right, while the area burned by human activities (H-fire) was mainly distributed in 1–10 and 11–100 ha. On the other hand, the frequency distribution of the total burned area (T-fire) followed a normal distribution (Fig. 2).

Fig. 2
figure 2

The frequency distribution of the area burned by different types of forest fire in the period of 1974–2009. The X-axis represents the different scales of burned area. The Y-axis represents the frequency of the burned area at a certain scale. L-Fire is lightning-caused fire, H-fire is human-caused fire, and T-Fire is total amount of fires

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Pearson correlations were computed between the area burned by three types of fire-causes and nine independent variables (climate variables). The results showed that only average maximum temperature was significantly correlated with the area burned by human-caused fire and the total burned area. The scatter plots were drawn to show the relationships between the area burned by three types of fire-causes and each climate variable (Figs. 3, 4, 5), which revealed that there was no obvious linear relationships between three response variables and independent variables.

Fig. 3
figure 3

Scatter plots of the area burned by lightning-caused fire versus each climate variable. L-fire is lightning-caused fire. The abbreviation of the climate variables on the X-axis is the same as in Table 1

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Fig. 4
figure 4

Scatter plots of the area burned by human-caused fire versus each climate variable. H-fire is human-caused fire. The abbreviation of the climate variables on the X-axis is the same as in Table 1. Because there was a huge inter-annual variation in the fire burned area, mainly due to an outlier of H-fire in 1987, we reduced the area burned by H-fire by 100 times, just for plotting purpose (i.e., not did so in modeling processes). The outlier was represented by triangles

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Fig. 5
figure 5

Scatter plots of the total area burned versus each climate variable. T-fire is the total burned area. The abbreviation of the climate variables on the X-axis is the same as in Table 1. Because there was a huge inter-annual variation in the fire burned area, mainly due to an outlier of H-fire in 1987, we reduced the area burned by T-fire by 100 times, just for plotting purpose (i.e., not did so in modeling processes). The outlier was represented by squares

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We attempted to model the area burned by forest fire using nine climate variables, and used the variance inflate factor (VIF) as the diagnostic measure for multicollinearity among the nine predictors (Rawlings et al. 1998; Hanna 2002). The resulted showed that VIF of each climate variable in the models was less than 5, indicating there was basically no serious multicollinearity among them.

Given that our fire data were collected from 1974 to 2009, we conducted the Durbin-Watson (DW) test on the temporal autocorrelations in the model residuals. The result revealed that the p-values of the DW tests were >0.2 (definitely larger than the significance level α = 0.05) for all models of the three response variables. Thus, there was no temporal autocorrelation in the data. Although our fire data were collected over 36 years, the area burned by forest fire seemed more random, rather than dependent from year to year.

We regressed the three response variables (L-fire, H-fire and T-fire) to the nine climate variables (i.e., full models) by each of the multiple linear regression model (Eq. (1)), log-linear model (Eq. (3)), and gamma generalized linear model (Eq. (5)). The model fitting statistics, R2, AIC and RMSE, of the three models were listed in Table 2. The result indicated that, among the three response variables, L-fire was the worst one fitting to the fire data. The R2 of the three models was smaller than 0.1. The AICs of the gamma-generalized linear models were significantly smaller than those of MLR and log-linear models for all three response variables, indicating that the gamma model fitted each fire data much better than the other two models when all nine predictor variables were included (full model). On the other hand, log-linear model did not show any obvious advantages compared to MLR in the three model fitting statistics (Table 2). In addition, most predictor variables were statistically insignificant at the significance level α = 0.05 in both MLR and log-linear models.

Table 2 Comparison of full models and best models of multiple linear regression model (MLR), log-linear model and gamma-generalized linear model
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Further, we selected the best models for each response variable by removing the non-significant predictor variables one by one at the significance level α = 0.05. For L-fire there was no best model selected for either MLR or log-linear models (i.e., none of the predictor variables was statistically significant). Among the best models the AICs of the gamma-generalized linear models were also significantly smaller than those of MLR and log-linear models for H-fire and T-fire (Table 2).

The details of the gamma-generalized linear best models for the three response variables (L-fire, H-fire and T-fire) were examined by the parameter analysis (Littell et al. 2009). Table 3 provided the sign and significance of the predictor variables in the gamma full models. Table 4 listed the model coefficient estimate and p-values of the predictor variables for the gamma best models. The indicated that precipitation (PA) and maximum wind speed (MAW) were negatively related to the area burned area by lightning-caused fire (L-fire), while the days that rainfall greater than 0.1 mm (DA) was positively impacted the L-fire. The mean temperature (MTE) was positively associated with the area burned caused by human activities (H-fire), whereas the minimum relative humidity (MIRH) and days that rainfall greater than 0.1 mm (DA) negatively impacted the human-caused burned area. For the total area burned by forest fire, the days that rainfall greater than 0.1 mm (DA) and minimum relative humidity (MIRH) were the important climate variables for the fire burned area (Table 4).

Table 3 Parameter analysis of gamma full model by different response variables
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Table 4 Parameter estimates of the gamma best models by different response variables
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Discussion

Our study revealed that the gamma-generalized regression models were more suitable for analyzing the area burned by forest fires, especially the human-caused and total burned area, in the Tahe forest region. Our results were partly in accordance with the finding by Littell et al. (2009), which indicated that the gamma-generalized linear models were generally superior in the southwestern ecoprovinces of USA, whereas log-linear models fitted data better in the cooler ecoprovinces. However, we did not find the advantage of log-linear model for fitting our fire data. However, we did not investigate the applicability of different models on different ecozones in this study, due to the relatively small forest region and unique forest type of the study area.

According to other relevant studies in the literature, the relative humidity is a good indicator of the area burned by forest fire (Skvarenina et al. 2004; Holsten et al. 2013). In this study, the similar result was found in the gamma model on the total burned area and the area burned by human-caused fire, but not by lightning-caused fire. We also identified that the average temperature (MTE) of fire seasons was significantly correlated with the area burned by human-caused fire, which was supported by Larsen (1996). Some fire-climate researches in the boreal forest ecosystems around the World indicated that the precipitation (PA) was a significant impact factor on the area burned by forest fire (e.g., Larsen 1996; Bergeron et al. 2001; Carcaillet et al. 2001; Anderson et al. 2006). Our results showed that the days that rainfall greater than 0.1 mm (DA) was an important factor for the total burn area and the area burned by human-caused fire, which provided a collateral proof for the influence of the precipitation. It is worth noting that the gamma best model showed an opposite influence of DA on the area burned by lightning-caused fire and human-caused fire. One possible explanation is that the DA may be a factor causing more lightning strikes (thus more lightning-caused fire), but DA may decrease human activities (thus less human-caused fire occurrence).

In this study, the R2 of all models, including full models and best models, were less than 0.4, meaning that these models can only explain limited percent of the total variations in the three response variables. In other words, these nine climate variables may not be sufficient to predict the area burned by forest fire. Other factors such as forest type, topography, infrastructures or human-activities, may also significantly influence the fire burned area. On the other hand, the major tasks of local forest fire management are actively eliminating forest fires; hence the fire burned area may vary dramatically along with the fire suppression efficiency.

Besides, there are considerable studies on the association between climate change and forest fire in the past decade. The results of these studies almost all revealed the significant influence of Atlantic Multidecadal Oscillation (AMO), Pacific Decadal Oscillation (PDO)/El Niño (EI Nino Southern Oscillation, ENSO) and Palmer’s Drought Severity Index (PDSI) on the fire frequency and burned area (e.g., Fauria and Johnson 2008; Collins et al. 2006; Littell et al. 2009; Gillett et al. 2004; Hess et al. 2001; Hessl et al. 2004). We did not take the above climate indices into account in this study because our study area was relatively small compared to other studies which were either conducted in North American (Fauria and Johnson 2008) or western US (Littell et al. 2009). In our case, the local climate/weather factors can impact local fire regime more significantly and directly in the study area.

Conclusions

We applied multiple linear regression, log-linear regression and gamma-generalized linear regression models to investigate the relationships between the area burned by forest fire and local climate variables. The results showed that gamma-generalized linear model was superior to both multiple linear regression and log-linear regression models on different causes of forest fires according to the model fitting statistic tests such as AIC.

Moreover, the best models of the gamma-generalized linear regression revealed that the maximum wind speed (MAW), precipitation (PA) and days that rainfall greater than 0.1 mm (DA) were significantly impacting the area burned by the lightning-caused fire. The mean temperature (MTE), minimum relative humidity (MIRH) and days that rainfall greater than 0.1 mm (DA) were the main drivers of the area burned by human-caused fire. Overall, the total burned area by forest fire was significantly influenced by the days that rainfall greater than 0.1 mm and minimum relative humidity, meaning the moisture condition of forest stand will determine the burned area.