Stochastic Approaches Systems to Predictive and Modeling Chilean Wildfires

Abstract

Whether due to natural causes or human carelessness, forest fires have the power to cause devastating damage, alter the habitat of animals and endemic species, generate insecurity in the population, and even affect human settlements with significant economic losses. These natural and social disasters are very difficult to control, and despite the multidisciplinary human effort, it has not been possible to create efficient mechanisms to mitigate the effects, and they have become the nightmare of every summer season. This study focuses on forecast models for fire measurements using time-series data from the Chilean Ministry of Agriculture. Specifically, this study proposes a comprehensive methodology of deterministic and stochastic time series to forecast the fire measures required by the programs of the National Forestry Corporation (CONAF). The models used in this research are among those commonly applied for time-series data. For the number of fires series, an Autoregressive Integrated Moving Average (ARIMA) model is selected, while for the affected surface series, a Seasonal Autoregressive Integrated Moving Average (SARIMA) model is selected, in both cases due to the lowest error metrics among the models fitted. The results provide evidence on the forecast for the number of national fires and affected national surface measured by a series of hectares (ha). For the deterministic method, the best model to predict the number of fires and affected surface is double exponential smoothing with damped parameter; for the stochastic approach, the best model for forecasting the number of fires is an ARIMA (2,1,2); and for affected surface, a SARIMA$(1,1,0){(2,0,1)}_4$, forecasting results are determined both with stochastic models due to showing a better performance in terms of error metrics.

1. Introduction

The emergence of forest fires has become one of the greatest problems in global societies [1]. The management of these crises becomes problematic for national and sub-national governments since, at times, they are unable to control the complexity of these events, which results in serious human, environmental, and economic losses [2].

State response systems are seen as vulnerable. In fact, fire brigades, the police, and the army—that is, national realities and available resources—are overcome by the sheer scale of forest fires and their devastating effects, causing tragedy and insecurity in the citizenry. The social consequences of these catastrophes are serious and remain over time, establishing greater tension at the urban/rural interface and affecting the human development of the most deprived countries [3].

In addition, we are in the presence of climate change that raises temperatures and generates prolonged droughts, modifies humidity and the behavior of winds, and increases the combustion of the land and flora that inhabits them [4]. Expectedly, fires become more dangerous and difficult to control, so the development times of these events tend to lengthen, which causes more damage on the surface and destruction of green areas. Even endemic species and endangered animals are put at risk [5], which happened in Australia in 2019–2020.

However, over the recent decades, there have been significant advancements in the theory, modeling, and practical applications of stochastic systems and forecasting tools. Existing literature, research, and advances in methods have introduced new and notable methodologies and approaches for dealing with real problems, leading to the accurate representation of real-world entities through systems that are stochastic, deterministic, or a mixture of both. Additionally, wildfires represent a significant societal challenge and problem, resulting in widespread destruction while substantial resources are allocated for their prevention, containment, and disaster minimization. For this reason, this research wants to advance in ways of predicting fires in a country hit particularly by this type of emergency, such as Chile using wildfire stochastic forecast systems, which endangers the lives of its communities and the local economy and exacerbates the complex national water situation that it has been experiencing in recent years. A contribution in this regard can guide political and administrative decision-making, as well as promote better information and prevention campaigns with citizens, contributing towards communication for risk management that is not giving the expected results.

The main purpose of this study is to provide evidence for the usage of time-series models for forecasting wildfires, either the number of fires or affected national surface, validating classic methods and making comparisons among existing ones in the literature, altogether with provide insights in terms of the behavior of this kind of disaster with a time-series approach.

2. Theoretical Framework

A forest fire is understood as“any uncontrolled fire that affects, at least partially, wooded areas and trees” [6]. Meanwhile, the area affected by a forest fire “corresponds to the area included within the perimeter of a fire, regardless of the degree or level of damage in the affected ecosystem” [7]. It should also be considered what is known as a Great Forest Fire, which is usually linked to the so-called urban/forest interface. In other words, these fires do not stop when they reach an urbanized area and can even devastate it. Examples of this are the case of Los Alamos, New Mexico, in the year 2000, and Valparaíso, Chile, in 2014 [1].

In terrestrial ecosystems, it should not be forgotten that a fire is a natural ecological factor [8,9,10]. It has been present in the environment since long before humans existed [11]. However, human intervention has produced changes in this regard [12]. For example, humans cause approximately 95% of forest fires in Europe [12]. The profile of these claims is not only related to climatic conditions, but also to socioeconomic aspects and becomes a recurring problem [6].

Likewise, it is worrying that, in recent decades, fires have dealt the highest rate of destruction in tropical ecosystems [12]. But even more complex are forest ecosystems, where deforestation rises at an average annual rate of 11.2 million hectares, which is equivalent to 20 hectares per minute. In other words, it is estimated that these forests are heading towards disappearance [6].

Human-caused fires are present in all the world’s plant ecosystems. According to estimates, 10 to 15 million hectares of forest are lost in boreal and temperate regions, while 20 to 40 million are burned in tropical forests. The main causes are attributed to agricultural activity and the reconversion of crops in large areas of surface [12]. Large fires in Australia, California, the Amazon, and Chile have made these incidents a major challenge for governments and civil societies around the world. In California, in 2019, more than 6000 fires occurred, consuming over 80,000 hectares [13], while in South America, only in Bolivia, more than four million hectares were burned in 2019 [14].

In various realities, such as South America, forest fires can be catastrophic due to the erosion and desertification generated, respiratory diseases that affect human populations, or the large amount of carbon dioxide released in a few hours into the atmosphere instead of being stored for decades in trees [11,15,16,17,18]. This reality shows that forest fires are increasing in the world, and their destructive characteristics alarm international public opinion [2], which observes with concern how certain environmental referents like the Amazon Forest suffer damaging episodes.

The implications of what happened in Brazil in 2019 overcame the domestic reality of the South American giant. The fires in the Amazon jungle had news coverage on the global agenda, as a slow reaction from the Jair Bolsonaro government was noted, with an impact on the image of the country of Brazil and adverse ecological effects for the planet. This fire even spread to areas of the Bolivian Amazon, mobilizing the then-president Evo Morales to officiate as an improvised firefighter.

Whether they are peripheral countries or industrialized powers, these catastrophes become priority aspects for political administrations. Forest fires cause human tragedies and damage to ecosystems in times of global warming. Also, they entail destabilization and economic crises for the affected areas, especially in the most vulnerable regions of the world.

Managing these crises involves political and government decisions of relevance to taxpayers, people’s well-being, and economic stability. Resource mobilization and subsequent aid packages for those affected can be controversial issues in some countries, which must also deal with other types of catastrophes and priorities. Public policies, too, are tested in these situations.

Currently, environmental, scientific, civic, and political sensitivity suggests that fighting fires, especially at the prevention level, is essential to avoid the aspects described above, so they must be increasingly present in national or emergency budgets at the disposal of the governments and the respective institutions. The explosive growth of cities, poor urban planning, and the emergence of irregular settlements of poverty in highly dangerous places make the task of controlling and managing forest fires difficult, and the consequences are dramatic. There is even concern in scientific areas that these fire waves continue to mobilize disease vectors and dangerous viruses [19] due to the loss of natural borders between humans and certain animal species.

2.1. Chilean Context

In recent years, Chile has witnessed multiple forest fires, with serious consequences in various areas of the country. These fires have even entered urban centers, such as Valparaíso, in 2014 [20]. Meanwhile, communities in southern regions of the country, such as the town of Santa Olga (Maule Region), disappeared due to an uncontrolled fire in 2017 [20,21]. In turn, an important international tourist attraction, such as Torres del Paine National Park, in Chilea’s Patagonia region, suffered a loss of more than 17 thousand hectares in a fire declared between December 2010 and March 2011, one of the largest recorded in Chile [22,23].

Chile suffered the worst forest fire season in its history in 2017, with ∼530,000 hectares burned, which is ten times the national average (55,000 ha/year). In fact, 11 people lost their lives, and 3000 houses were destroyed [24]. The peak of fires was reached in 2015, with 8048 events, while in 2019, there were 7219 [25]. In this regard, it is considered that a low percentage of fires, generally less than 1%, lead to large fires. However, large fires explain the significant accumulated damage in some seasons, at least 67 percent [25].

During the last 25 years, there has been a notable increase in the occurrence and incidence of Chilean forest fires. During the 2002–2023 season, the number of fires exceeded over 7500 fires on average per season, in contrast to the pre-1972 period when there were fewer than 1000 fires per season [25]. This increase and upward trend does not seem to reach its maximum point, as there is no discernible decrease in fire occurrences despite the concerted efforts of prevention initiatives. These endeavors are led by both private enterprises and the National Forest Corporation (CONAF), the latter being the state body by law responsible for the prevention and combat of forest fires.

The occurrence of forest fires was historically concentrated in the Mediterranean climate zone, which extends from the Valparaíso Region to the Bío-Bío. However, in recent years, there has been a significant increase in the temperate zones of Chile, which includes the Araucania and Los Lagos regions [26].

Chile has 15.9 million hectares of forest cover, with native forests that represent 85.4% (13.6 million hectares) and forest plantations that reach 14.6%, that is, 2.3 million hectares [27]. In turn, 68% of forest plantations are composed of radiate pine and 23% eucalyptus [28].

Studies on the Chilean reality indicate that forest fires in that country would have the following characteristics [26]:

(a)

High annual occurrence.

(b)

High intensity and spread of fire, generating uncontrollable fires.

(c)

Increase of intentional fires.

(d)

Great occurrence of satellite foci.

(e)

High frequency of fires at the same site in a period of less than 12 years.

(f)

Urban/rural interface with high occurrence of fires.

(g)

Concentration of forest fires from Regions VII to IX.

(h)

Great economic, social, and environmental impact.

Some authors point out that the increase in the number of fires each year allows them to argue that prevention efforts have not had good results due to the low investment and the non-elimination of harvest waste in order to leave the maximum material organic for the benefit of the chemical and physical properties of the soil [26].

It is also argued that there is a great difference in the damage that occurs in Chile between forests and natural forests. In Chilean natural forests, the damage is considerable because it would affect greater plant and fauna diversity, apart from occurring in the High Cordillera, where it is difficult to access [26]. In addition, forests are in private hands and concentrated in the same area of the country, which makes it easier to protect them, receiving an investment of USD 7 per hectare. On the other hand, the natural forests scattered across the country run by CONAF barely receive USD 1 per hectare for their protection [26].

A topic of interest and public discussion in Chile, after the great fires that have affected the country, is the intentionality of these events. For example, one aspect that is often heard in media forums open to the public or on social networks is that the fires would be caused by forest owners in order to circumvent the existing rules of territorial planning [29]. This aspect would occur in other parts of the world [30] because, after a fire, forest areas burned by fire can be transformed into places for urban construction. In Chile, this possibility is only discussed “narratively in public, since accusations can quickly have legal consequences” [31].

Forest fires are perceived as a deliberate strategy of forest expansion [32]. These perceptions arise in sectors where what was planted with the natural forest is replaced by forest after the fires [31] and to avoid this problem in Chile, since 2008 there has been the “Law of recovery of native forests” [33], which prohibits forest owners from converting natural forests into forest plantations. Despite the above, after a fire, it is legal to reforest with endemic species for forest use, so they become accessible for industrial use [34].

Another aspect that is usually in the debate in Chile is the role of insurers or insurance that forest companies hire in the face of fire risks. These insurance companies are generally aware of the problem of the so-called moral hazard [35] because there is a possibility that an owner burns their forest to collect insurance.

To avoid the above, companies design contracts in such a way that the insured always assumes a partial risk that would reduce the incentives to commit any crime [36]. However, there are authors [31] who argue that as long as the contractual conditions negotiated between companies and the insurance sector are not publicly transparent, discursive and ambiguous speculation about the motivations of arsonists will continue.

According to the aforementioned background, it was expected that relations with cooperative and collective approaches in risk management would be established in Chile. However, it turned out that forest companies prefer to organize management individually and without much neighborhood involvement [31].

The context in Chile of forest fires is complex, so the study of this reality is important due to the multiple consequences and effects on people’s lives, their assets, the environment, and the national economy. Other studies [37] indicate that the appearance of some fires of an intentional nature would be directed against large forestry companies, which, with their plantations, mark the use of land and dictate the main economic activities of some areas. The authors of [38,39] maintain that forest fires, particularly in the Araucanía region, would have become the symbol of the struggle of sectors of the Mapuche people in their conflict with the Chilean state as a protest against political and/or socioeconomic deficits.

The forest industry contributes 3.1% of the national GDP to the Chilean economy. Chile maintains its economy thanks to exports. The forestry sector exports around 63% of what it produces, highlighting cellulose, pieces for sawing, and pulp and sawn wood, all almost exclusively from radiata pine [27,40]. In addition, the forestry sector contributes a large number of jobs to the country’s economy. It provides a total of 300,000 jobs, considered direct 130,000, and the rest indirect [28]. A forest fire, in addition to affecting the environment, indirectly affects families that depend on this activity to survive [27].

In the Chilean context, CONAF measures the probability of fire occurrence using the Degree of Danger Index (IGP), a mathematical formula based mainly on meteorological conditions (temperature, wind direction, wind speed), the moisture content of the vegetation, and the effect of rainfall in previous days.

2.2. Wildfire Forecasting

Forecasting of wildfires and burned areas can be found in various regions from different countries. Monitoring wildfires can be more effective when it is combined with analytical tools that help decision-makers and organizations focused on controlling this kind of disaster [41].

In Thailand, researchers from the University of Technology Thonburi developed analyses with meteorological factors from northern Thailand in order to forecast conditions linked to wildfires like precipitation, pressure, relative humidity, and others through stochastic and deterministic modeling of time series, specifically ARIMA and Holt–Winters models, whose results showed better results for Holt–Winters in forecasting precipitation, pressure, and relative humidity, while ARIMA showed better results for wind speed and temperature forecasts [42].

In Turkey, researchers from Van Yüzüncü Yil University developed forecasts for burned forest areas with ARIMA models through classic Box and Jenkins methodology with data from 1940 to 2020 from various Turkish regions. An ARIMA (3,1,0) model was selected as suitable for the burned forest area forecast, providing a 3-year projection for the data [43]. An analysis provided by researchers from Hacettepe University in the same country using data from area burned in various Turkish regions from 1937 to 2009 showed a comparative analysis among compound Poisson processes and time-series models, selecting an ARIMA (0,1,1) model for forecasting with better results compared to other methodologies that are not related to time-series analysis themselves [44].

A study led by the National Institute for Space Research in Brazil developed a comparative analysis of time-series models with data from 2003 to 2017 with burned areas in central–west Brazil in South America, east India (Asia), and southwest Mali (Africa) using various forecasting methods, including ARIMA, Exponential Smoothing (ETS), and other approaches derived from Artificial Neural Networks (ANNs). For this study, models based on neural networks showed better results than classical time-series methodologies, but it is worth mentioning that time-series forecasting methods can be used in order to estimate wildfires [45].

In India, the Centre for Development of Advanced Computing (C-DAC) showed how time-series analysis can be used to estimate wildfires in major and various forest types in the country using fire count data from 2003 to 2018. The analysis was developed with models from the ARMA family using Box and Jenkins methodology and ARIMAX with independent variables like temperature and dry days per year [46]. The results showed no significant differences among univariate and ARIMA with regressors (ARIMAX) models in terms of performance. The usage of these models allows including other kinds of variables related to institutional or political factors and decisions, considering that management of these kinds of disasters plays an important role in avoiding spreading and increasing consequences [47,48,49].

Another approach for forecasting wildfires is using spatiotemporal models that combine data mining and machine learning techniques in order to predict wildfires in various regions. In Pakistan, researchers from Arid Agriculture University developed a model considering environmental and socioeconomic factors to detect risks of wildfires using Random Forest and Maxent Models [50]. In Australia, researchers used a physics-informed neural network approach to simulate how the wildfires propagate through the spatiotemporal domain using real and synthetic data for optimizing the model [51]. In Canada, researchers applied deep learning techniques using data from wildfires and weather factors in some regions [52]. The results showed an F1-score greater than 0.95, while regression models showed an r-squared score greater than 0.91.

3. Methodology

This study considers two independent and univariate time series of fire datasets that were used to forecast several performance measures required by CONAF: the number of national fires and the national surface affected measured by hectares (ha).

An exploratory data analysis of the two datasets was performed to detect the attributes and features of each series. Deterministic and stochastic models were applied to forecast the number of fires and affected surfaces series, with the aim of determining the most suitable approach for each one of the datasets. In order to determine the best forecasting method and model, results were compared using measures of goodness-of-fit, namely root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). The model with the lowest MAPE, MAE, and RMSE values was considered to be optimal and selected for forecasting [53,54].

3.1. Deterministic Forecasting

This study employed a range of deterministic Exponential Smoothing (ES) models, including a simple-basic deterministic, a Holt model, a Brown model, and an ES with a damped-trend parameter. The first four models were applied to predict both the number of fires and affected surface series for cases of wildfires across Chile on a national scale, measuring their error metrics and performance in order to compare them with the stochastic models.

The simple-basic model is suitable for a series with no trends or seasonal patterns. Both Holt’s and Brown’s models are well suited for series with linear trends but lacking seasonality (no periodic movements over time). In Holt’s model, the smoothing parameters are assumed to be equal (for level and trend). Conversely, the damped-trend model is suitable for a series showing a linear trend that decreases over time without seasonal or periodic variations [55]. In a simple moving average model, all the observations are given the same weight. Conversely, in an ES model, more recent data gain greater importance compared to older observations. This concept was initially suggested and introduced by [56] and further development and expansions to the model were introduced by [57]. Brown’s simple ES is expressed in (1)

$$Y_t=\alpha X_t+(1-\alpha )Y_{t-1},$$

(1)

where $Y_t$ corresponds to attenuated average values for the series after observing $X_t$, $X_t$ are the actual values of the series, and $\alpha$ is a attenuation constant, which, for the simple ES model, is considered constant over time.

When there is a discernible trend in the series, the simple ES model may fall, showing poor performance in estimations. To address this issue, double ES models, like Holt’s (2) and (3) and Brown’s (4) and (5) models, have been introduced. These models incorporate an additional component to account for the trend in the forecasted values of the series [55]:

$$Y_t=\alpha X_t+(1-\alpha )(Y_t+T_{t-1}),$$

(2)

$$T_t=\beta (Y_t-Y_{t-1})+(1-\beta )T_{t-1},$$

(3)

$$Y_t=\alpha X_t+(1-\alpha )\left(Y_{t-1}{T}_{t-1}\right),$$

(4)

$$T_t=\beta \left(\frac{Y_t}{Y_{t-1}}\right)+(1-\beta )T_{t-1},$$

(5)

where $Y_t$ corresponds to attenuated average values, $X_t$ corresponds to the actual values of the series, $T_t$ is a trend estimate, and $\alpha$, $\beta$ are attenuation constants that are considered constant over time for the models.

Double ES models assume a constant trend when making forecasts. This can lead to problems related to overestimation and over-forecasting for extended time frames. To address this, Gardner [58] introduced a parameter that decreases the trend over time, ultimately leveling it off for future values, as illustrated in (6) and (7):

$$Y_t=\alpha X_t+(1-\alpha )(Y_{t-1}+\theta T_{t-1}),$$

(6)

$$T_t=\beta (Y_t-Y_{t-1})+(1-\beta )\theta T_{t-1},$$

(7)

where $Y_t$, $X_t$, $T_t$, and $\alpha ,\beta$ correspond to the same variables and parameters of the double ES seen in (2)–(5), and $\theta$ corresponds to the damping parameter for decreasing trend as mentioned before.

If a series shows a seasonal influence over time, suitable models could include a seasonal, a winter-additive, or a winter-multiplicative component depending on the structure of the time series itself. The winter-additive model is shown in (8)–(10):

$$Y_t=\alpha \left(\frac{X_t}{S_{t-c}}\right)+(1-\alpha )(Y_{t-1}+T_{t-1}),$$

(8)

$$T_t=\beta (Y_t-Y_{t-1})+(1-\beta )T_{t-1},$$

(9)

$$S_t=\gamma \left(\frac{X_t}{Y_t}\right)+(1-\gamma )S_{t-c},$$

(10)

where $Y_t$, $X_t$, $T_t$, $\alpha$, and $\beta$ correspond to the parameters and variables shown in (6) and (7), and $S_t$ corresponds to a seasonal parameter added to the model in order to capture periodic movements of the series. In cases where the series does not exhibit a trend, the seasonal model (8) is characterized by the absence of T. On the contrary, when the time series does possess a trend and the smoothing parameters are assumed to be equal, the winter-multiplicative model is a suitable choice for forecasting.

3.2. Stochastic Forecasting

The ARIMA model leverages historical values and previous prediction errors to grasp trends and anticipate values. This model was introduced by Box and Jenkins, and since its origins, various transformations and extensions have been developed in order to improve and adapt the model to various types of problems [59]. When incorporating a seasonal component, an ARIMA (p,d,q) becomes a SARIMA $(p,d,q){(P,D,Q)}_s$ [60], where the seasonal effect denoted by (s) is added to the regression.

In the study, an ARIMA model was employed to predict the national fires and a SARIMA model to predict the national affected surface. Various combinations were experimented and tested for the autoregressive process, applying transformations and differentiations to achieve the stationary condition for each time series. The general equations of the ARIMA (p,d,q) and SARIMA $(p,d,q){(P,D,Q)}_s$ models are presented in (11), (12) and (13), (14), respectively.

$$Y_t=-({\Delta }^d{Y}_t-Y_t)+{\varphi }_0+\sum _{i=1}^p{\varphi }_i{\Delta }^d{Y}_{t-i}-\sum _{i=1}^q{\theta }_i{ϵ}_{t-i}+{ϵ}_t$$

(11)

$$\Delta Y_t=Y_t-Y_{t-i}$$

(12)

where $Y_t$ corresponds to observed values of the series, d is an operator for convergence, $\varphi$ is the autoregressive component parameters, $\theta$ is the moving average component parameters, ${\varphi }_0$ is a constant that is assumed to be different from zero, and ${ϵ}_t$ is the error of the model

$$\varphi \left(B\right)\mathsf{\Phi }\left(B^s\right)(h_t-\mu )=\theta \left(B\right)\mathsf{\Theta }\left(B^s\right){ϵ}_t$$

(13)

$$h_t={(1-B)}^d{(1-B^s)}^D{Y}_t={\Delta }^d{\Delta }_S^D\left(Y_t\right)$$

(14)

where $B^s$ is the seasonal lag operator $B^s\left(Y_t\right)=Y_{t-s}$ for convergence, ${\Delta }^d$: $(1-B)$ is a local difference operator, ${\Delta }_S^D$: $(1-B^s)$ is a seasonal difference operator, $h_t$ is a stationary series, $Y_t$ is the observed values of the series, B is the lag operator (generalized), $\varphi \left(B\right)$ is the autoregressive order p, $\theta \left(B\right)$ is the moving average order q, $\mathsf{\Phi }\left(B\right)$ is an autoregressive order P seasonal, $\mathsf{\Theta }\left(B\right)$ is the moving average order Q seasonal, $\mu$ is the average of the stationary series, ${ϵ}_t$ is the model error, and $D,d$ is the times to apply seasonal and local differences to regulate the original series.

3.3. Stationary Time Series

Stationarity plays a fundamental role in the assumptions of some techniques related to forecast and time-series analysis, especially those related to autoregressive methods (AR, MA, ARMA). In many studies and analyses, mainly in statistical ones, it is often useful to compute a covariance matrix, which gives insights about the dependence structure among variables, understanding that there is at least one dependent and one explanatory (independent) variable. For univariate time series $\{Y_t,t\in T\}$, there is only one variable that corresponds to the process $\left\{Y_t\right\}$ itself, so there is a need to extend the concept of the covariance matrix in order to deal with many (or infinite) collections of random variables. For this purpose, the autocovariance function provides a tool for dealing with required extensions. The autocovariance function is presented in (15). Let $\{X_t,t\in T\}$ be a stationary process associated with the process $\left\{Y_t\right\}$

$${\gamma }_X(r,s)=\mathrm{Cov}(X_r,X_s)=E\left[(X_r-E\left[X_r\right])(X_s-E\left[X_s\right])\right]r,s\in T,$$

(15)

Given $\{X_t,t\in T\}$ a process, we say that $X_t$ is stationary if some conditions are met. The first one requires the existence of variance of $\left\{X_t\right\}$ over time T, while the second one requires that the mean of $\left\{X_t\right\}$ exists and is constant over time T. Another assumption is that ${\sigma }_{X_t}$ is constant over time, meaning that ${\sigma }_{X_t}=\sigma$ for any values of $t\in T$. Previous conditions can be stated according to the following $\forall t\in {\mathbb{Z}}_{+}$:

$$E\left[X_t^2\right]<\infty$$

(16)

$$E\left[X_t^2\right]=\sigma$$

(17)

$$E\left[X_t\right]=m$$

(18)

With previous conditions met, it is assured that the following requirement is also met:

$${\gamma }_X(r,s)={\gamma }_X(r+t,s+t),r,s,t\in \mathbb{Z}$$

(19)

This means that the autocovariance function is constant for any subsets $\{X_k:k\in T\}$. Since time-series data deal with the past and future from the process, stationarity requirements are simply a way to ensure that the distribution is the same over time T, avoiding issues related to changes in mean or variance over T, which require that more than one model should be fitted to explain different levels of $E\left[X_t\right]$ and ${\sigma }_{X_t}$ in the process. Most of the time series analyzed in practice do not meet the conditions required for being considered a stationary process. To deal with that issue, differencing techniques can be used to obtain stationary processes, we set a differentiation operator, which is defined in (20):

$$X_t={\Delta }^d{Y}_t=Y_t-Y_{t-d}\forall d<T.$$

(20)

The operator shown in (20) helps to stabilize a time series removing changes in levels, eliminating trends, and making the mean constant over time T, depending on the levels and characteristics of the process sometimes differencing operators are required to be applied more than one time or with different levels of parameter d, such parameter is called local differentiation when d relates to immediate previous values, while if d takes seasons of the series is called seasonal differentiation denoted by D. Moreover, another way for stabilizing a time-series process is applying a transformation to the data, altogether with differentiation, a common practice is to apply a log-transformation, as shown in (21):

$$Y_t^{{}^{\prime }}=\log \left(Y_t\right)$$

(21)

3.4. Evaluation Metrics

For the evaluation and comparison of results MAPE, RMSE, and MAE are calculated for fitted values of each model, either deterministic or stochastic fit. In the case of stochastic modeling, the AIC index is used as a comparison metric with previous ones mentioned before as a way of evaluating the complexity of the models in terms of their variables.

3.4.1. Mean Absolute Percentage Error (MAPE)

MAPE defines the accuracy of a fitted model with respect to current values in the dataset, considering absolute differences between them. The formula for MAPE is given by Equation (22):

$$MAPE=\frac{1}{T}\sum _{i=1}^T\left|\frac{{\mathrm{Actual}}_i-{\mathrm{Fitted}}_i}{{\mathrm{Actual}}_i}\right|$$

(22)

3.4.2. Root Mean Squared Error (RMSE)

RMSE is defined as the square root of the average difference squared between actual and fitted values. The difference between MAPE and RMSE is that the second one keeps the same unit as the target variable, while the first one shown in (22) is a percentage with respect to actual values. The formula for RMSE is given by Equation (23):

$$RMSE=\sqrt{\frac{1}{T}\sum _{i=1}^T{({\mathrm{Actual}}_i-{\mathrm{Fitted}}_i)}^2}$$

(23)

3.4.3. Mean Absolute Error (MAE)

MAE is defined as the absolute error between fitted values and actual values. Similar to RMSE, it keeps the unit of measure of the target variable. The formula for MAE is given in Equation (24):

$$MAE=\frac{1}{T}\sum _{i=1}^T|{\mathrm{Actual}}_i-{\mathrm{Fitted}}_i|$$

(24)

3.4.4. Akaike Information Criterion (AIC)

A criterion for comparing selection models in stochastic forecasting is the information criterion of Akaike, known as AIC. The main purpose of this index is to measure the relative performance of a model involving the complexity of the fitting in terms of its parameters. The formula for AIC is given as follows in (25):

$$AIC=2(p+q)-\ln \left(L\right),$$

(25)

where p represents the number of parameters of the autoregressive section of the model, q is the number of parameters of the moving average section of the model, and L is the likelihood of the fit.

3.5. Series and Models Validations

Validations are tested for times series and models developed. For the series it is a must having a stationary time series, while the model requires some metrics to be evaluated to consider them correct and valid for continuing with forecasting.

3.5.1. Kwiatkowski–Phillips–Schmidt–Shin Test

The test used for assessing stationary processes is the Kwiatkowski–Phillips–Schmidt–Shin test (KPSS test), which evaluates under the null hypothesis if a time series is stationary against an alternative hypothesis of the presence of unit root or non-stationary under a trend.

The KPSS test decomposes the series into the sum of a deterministic trend, a random walk, and an error that is considered stationary [61] The decomposition is shown in (26):

$$X_t=\alpha \times t+u_t+{ϵ}_t,$$

(26)

where $u_t=u_{t-1}+{\alpha }_t$, and ${\alpha }_t\sim (0,{\sigma }^2)$, the hypothesis for the test is $H_0:{\sigma }^2=0$ against $H_1:{\sigma }^2\ne 0$. In case the null hypothesis is not rejected, we can assume that $\left\{X_t\right\}$ is a stationary process.

3.6. Unit Root

An ARMA model with p and q parameters is defined in terms of pass or lagged values of the process $\left\{X_t\right\}$ and innovations ${ϵ}_t$. This can be written as follows in (27) and (28):

$$X_t=\sum _{i=1}^p{\varphi }_i{X}_{t-i}+\sum _{i=1}^q{\theta }_i{ϵ}_{t-i}+{ϵ}_t$$

(27)

$$X_t-\sum _{i=1}^p{\varphi }_i{X}_{t-i}=\sum _{i=1}^q{\theta }_i{ϵ}_{t-i}+{ϵ}_t$$

(28)

Applying back-shift operators Equation (28) can be written as follows:

$$\varphi \left(L\right)X_t=\theta \left(L\right){ϵ}_t$$

(29)

In (29), $\varphi \left(L\right)$ is known as the characteristic polynomial of the AR(p) process of the form

$$\varphi \left(z\right)=1-\varphi \left(1\right)z-\varphi \left(2\right)z-...-\varphi \left(p\right)z^p$$

(30)

Characteristic polynomial allows for testing if $\left\{X_t\right\}$ is stationary or not. That is, if the roots of $\varphi \left(L\right)=0$ lie outside the unit circle, the process can be considered as stationary [62,63]. For addressing this feature, we consider the inverse AR and MA roots that must lie within the unit circle for considering $\left\{X_t\right\}$ stationary. This is considering the AR components of the process. The same can be applied over MA components with $\theta \left(L\right)$ in (29) by letting $\theta \left(L\right)=0$ and repeating the procedure. Time series with unit roots are non-stationary, which means that the moments of the process depend on t, particularly the variance is non-constant:

$$\mathrm{Var}\left(X_t\right)=\sum _{i=1}^t{\sigma }^2=t{\sigma }^2$$

(31)

As illustrated in (31), the variance of $\left\{X_t\right\}$ depends on t, as $t\to +\infty$ the variance diverges.

The step-by-step methodology is shown in Figure 1. For the data pre-processing stage, the transformations and differentiations over time-series data are considered. This step is strictly necessary for stochastic models that require a stationary process to bring stable and valid results [59]. The modeling and forecasting stage considers the identification, estimation, validation, and selection of the forecasting model. The proposed methodology of Figure 1 considers the KPSS test and transformations shown in Section 3.3 and Section 3.5.1. For the pre-processing stage, model validation is performed through unit roots of models, as shown in Section 3.6, and model selection with metric errors shown in Section 3.4.1, Section 3.4.2 and Section 3.4.3. It is important to note that the methodology in Figure 1 applies entirely to stochastic times series, as shown in Section 3.2, due to the condition of the stationary process imposed by these kinds of models. Deterministic forecasting considers up to series transformation due to not assuming the process to be stationary, but the transformation and adjustment for the entire series [59,64] are necessary.

4. Results

This section is divided into results for stochastic models and later for deterministic models. Stochastic models consider transformations shown in Section 3.3, applying transformations and adjustments presented in (20) and (21) to obtain a stationary process and a valid ARMA model. Conversely, for deterministic models, there is only application of (21) to time-series data; previous operators are applied for both number of fires and affected surface (ha) series.

4.1. Data Pre-Processing

For the number of national fires and national affected surface (measured in hectares (ha)), 59 data points were collected corresponding to the period from the years 1964 to 2022. Figure 2 shows the series of number of fires for the period previously mentioned, Figure 2a shows the series without any adjustment of transformation, showing how the number of fires has increased over the years, going from about 2000 to 8000 fires in the national territory in the last years. Figure 2b shows the series after applying log-transformation. It can be seen how the dispersion in the series has decreased, maintaining the trend over the years. Figure 2c shows series after local differentiation ($d=1$) with the operator shown in (20). Values after applying local differentiation move around zero between −0.3 and 0.6 for the years up to 1980, and between −0.3 and 0.3 for 1981 onward. Also, the trend has been eliminated from the series, so the mean value is constant of about zero.

Series for affected surface are shown in Figure 3 from the years 1964 to 2022. Figure 3a shows the original series without any adjustment. It can be seen how there is an outlier value for the year 2017, which corresponds to a wildfire that affected three regions from central and southern Chilean territory [65]. Figure 3b shows series after log-transformation and treatment of outliers. There can be seen a slightly positive trend over the years. Figure 3c shows series after local differentiation ($d=1$), showing a constant mean of about zero with values between −2 and 2 after transformation.

Differentiated series for the number of fires and affected surface shown in Figure 2c and Figure 3c are tested with the KPSS test for stationarity. Both of them show a significance level higher than 0.10 as shown in

Table 1. Results of KPSS test for stationarity.

Variable	Statistic	Significance p (≥|t|)
Number of fires	0.077	>0.10
Affected surface (ha)	0.042	>0.10

, not rejecting the null hypothesis, and according to (26) series can be considered as stationary. With previous results derived from series adjustments and transformations, prior conditions on assumptions for applying ARMA models in the process are met according to Section 3.3.

4.2. Stochastic Model Identification

Figure 4 shows autocorrelations (a) and partial autocorrelations (b) for the number of fires series allowing for identifying the orders of time-series models. According to the behavior of ACF and PACF in Figure 4, they suggest ARIMA models with values of $p\in \{0,1,2\}$ and $q\in \{0,1,2\}$ with a local differentiation of the series $(d=1)$.

Results for models tested are shown in

Table 2. AIC and error metrics on models tested.

p	q	AIC	MAPE	MAE	RMSE
2	2	955.98	17.1%	629.67	832.03
2	1	955.37	17.2%	641.44	842.32
0	1	956.02	17.3%	659.87	878.18
2	0	953.73	17.5%	646.25	845.40
1	2	954.82	17.5%	644.12	838.75
1	1	957.15	17.7%	666.52	871.77
1	0	959.71	18.4%	706.32	907.79
0	2	955.63	18.4%	675.31	806.15
0	0	960.61	19.6%	734.85	931.10

, showing that ARIMA(2,1,2) performs better than other models in terms of all error metrics while having the lowest AIC value. MAPE values seem to be equal in almost all models except for the last three ones. MAE increases while moving through models, and RMSE shows no clear behavior. Model ARIMA(2,1,2) also shows all unit roots for AR and MA components within the unit circle. This can be seen in Figure 5, where each one of the dots corresponds to a component of AR and MA operators, so the model does not have issues regarding the presence of unit roots in its operators.

Figure 6 shows autocorrelations (a) and partial autocorrelations (b) for affected surface series. Identification is similar to the previous one shown in Figure 4. According to the behavior of ACF and PACF in Figure 6, they suggest ARIMA models with values of $p\in \{0,1,2\}$, $q\in \{0,1,2,3\}$. There are are tested values of $P\in \{0,1,2\}$, and $Q\in \{0,1,2\}$ The last ones are related to seasonal components involving SARIMA models.

Results for the model tested are shown in

Table 3. AIC and error metrics on models tested for seasonal components.

p	q	P	Q	AIC	MAPE	MAE	RMSE
1	0	2	1	140.19	67.9%	26,377	5039
1	0	0	1	136.27	68.2%	26,372	4934
0	0	0	1	142.01	70.46%	27,272	1746
0	0	2	1	145.82	70.66%	27,192	1523

. Despite having various models according to Figure 6, some of them are not considered due to having problems with unit roots. Model SARIMA$(1,1,0){(2,0,1)}_4$ performs better than other models, the second one having the lowest AIC value, the lowest MAPE, and the second lowest MAE while having the highest RMSE. Model SARIMA$(1,1,0){(2,0,1)}_4$ also shows all unit roots for AR and MA components within the unit circle shown in Figure 7, so the model, similar to the one related to the number of fires, does not have issues regarding presence of unit roots in its operators either local or seasonal.

Previous results deliver two valid models applying ARMA models, for the number of fires series best model is ARIMA(2,1,2) which has the lowest error metrics and shows no issues in terms of unit roots, while for affected surface (ha), the SARIMA$(1,1,0){(2,0,1)}_4$ model shows the best performance in terms of error metrics and has no issues related to unit roots.

4.3. Results for Deterministic Forecasting Models

Deterministic models are tested for the series of number of fires and affected surface (ha). It is important to note that deterministic models do not make assumptions about the processes, so the series applied over deterministic models are those shown in Figure 2b and Figure 3b that considers only log-transformation, as shown in (21). Results of models on the number of fires are shown in

Table 4. Error metrics for deterministic models tested over number of fires series.

Model	MAPE	MAE	RMSE
Exponential smoothing	18.8%	750.4	954.5
Double exponential smoothing (without damped)	18.3%	747.4	973.5
Double exponential smoothing (with damped)	17.4%	658.8	857.8
Holt–Winters	23.7%	808.1	1052.1
Brown	18.2%	693.1	891.5

; it can be seen how MAPE values are slightly similar to those shown by stochastic models in Table 2. RMSE and MAE values are all above the same metrics for stochastic models, except by double exponential smoothing with damped parameter, whose error metrics are the lowest of deterministic models, and the three of them are similar to the best model on Table 2 corresponding to ARIMA(2,1,2).

Table 5. Error metrics for deterministic models tested over affected surface (ha) series.

Model	MAPE	MAE	RMSE
Exponential smoothing	51.7%	23,277.9	31,103.8
Double exponential smoothing (without damped)	55.4%	22,452.2	28,539.6
Double exponential smoothing (with damped)	53.1%	21,908.8	28,639.1
Holt–Winters	62.2%	24,849.7	32,023.5
Brown	51.9%	23,231.8	30,995.3

shows the results of deterministic models for the affected surface (ha) series. It can be seen how MAPE and MAE values are similar to those shown in Table 3 in terms of MAPE and MAE. RMSE shows values over six times higher than those shown by stochastic models because this metric considers quadratic errors, so greater distances between fitted and actual values increase while being squared in the calculation.

From Table 4 and Table 5, the differences in error metrics compared to Table 2 and Table 3 can be seen. These different results are clear when looking at affected surface models. Deterministic models for this series suffer with RMSE metric indicators due to the way of calculation the metric, behaving differently from MAPE and MAE, which seem slightly similar and for some models yield even better results. Previous results show how a stochastic approach gives better results in terms of performance over series for number of fires and affected surface with lower error metrics values, this difference being more noticeable in the errors shown by stochastic and deterministic models over affected surface series.

4.4. Forecasting

Table 6. Forecasted values for number of fires series with ARIMA(2,1,2) with six years ahead.

Year	Forecast	90% Lower Interval	90% Upper Interval
2023	7305	4967	10,745
2024	6971	4251	11,430
2025	7115	4098	12,351
2026	7155	3707	13,807
2027	7028	3446	14,335
2028	7139	3299	15,450

shows predicted values for the number of fires with the ARIMA(2,1,2) model, which is selected due to better performance compared to other stochastic and deterministic models fitted for the series. Forecasting is calculated with a window ahead of six years up to the year 2028. It can be seen how values seem to be a straight line of about 7000 fires per year, maintaining the level of the last 20 years. Lower and upper confidence intervals are calculated at a 90% level, showing increasing values for the upper interval as years increase in the forecasting window, fitted values, forecasting, and confidence intervals for predictions can be seen in Figure 8. The model fitted follows the behavior of actual values in terms of trend and variability over the years.

Table 7. Forecasted values for affected surface (ha) series with $SARIMA(1,1,0){(2,0,1)}_4$ model with six years ahead.

Year	Forecast	90% Lower Interval	90% Upper Interval
2023	73,788	21,767	250,134
2024	93,843	22,362	393,307
2025	88,837	16,058	491,456
2026	89,538	13,178	608,376
2027	86,375	10,513	709,647
2028	85,859	8793	838,812

shows forecasted values for affected surface (ha) with the SARIMA$(1,1,0){(2,0,1)}_4$ model, which was selected due to performing better in almost all metric errors compared to other stochastic and deterministic models fitted over the series. Forecasting is developed with a time window of six years ahead up to the year 2028. In a similar way to a number of fires, results of forecasting show a prediction level about of 90,000 hectares affected due to wildfires for years from 2023 up to 2028, maintaining levels of the last 10 years, which move around 80,000 hectares affected per year. Figure 9 shows actual, fitted, and forecasted values; it can be seen how the model captures the behavior and movements of actual series, reaching higher and lower values over the years. It is important to note that the series shown in Figure 9 are values after cleaning series and do not consider extreme value shown in Figure 3a, which is an atypical value that was not considered due to being a value whose behavior could bias or influence over model selection and parameters estimation [66,67].

5. Discussion

Applied methodology for estimating a model and forecasting with it can be described as a process with various steps. The first one is related to obtaining a stationary process for applying autoregressive models, which require some conditions to be met. For these requirements, the KPSS test showed that the process can be considered as stationary after applying transformations and differentiation operators. Later on, model identification and selection are conducted for stochastic and deterministic models. Stochastic models showed a better performance in terms of error metrics, while deterministic models tended to adjust to the mean of values in the series [68]. It is worth noting that error metrics are calculated with the same values in the fitting stage, so there is no out-of-sample during the process, which is an approach used widely while fitting machine learning models with time-series data to avoid overfitting [69]. Finally, forecasted values are obtained through stochastic models due to performing better in error metrics.

Forecasting was developed for six years up to 2028, with annual data for 59 years from 1964 to 2022. Predictions for number of fires are about 7000 fires in the national territory with an ARIMA(2,1,2) model, while for affected surface (ha) predictions are about 86,000 hectares burned due to wildfires with SARIMA$(1,1,0){(2,0,1)}_4$. Efforts for forecasting this kind of disaster and managing its consequences can be found in [70,71,72,73], with different approaches but the same objective: forecast and managing risks of wildfire, either using complex systems such as image and mapping processing through computational algorithms, mathematical models, or controlling key meteorological indicators that give insight about climate conditions and risk of wildfires.

The usage of stochastic models such as ARMA models and their variants can be found in different analyses for various regions and countries. Results on applying seasonal models are similar to [43,74,75] where seasonal components are included in autoregressive models in order to forecast burned area and wildfires. Other methods for forecasting and prediction of wildfires can be found by applying different approaches such as deep learning and machine learning tools that expand and complexify the analysis, adding other variables such as meteorological measures, images, or geodata to improve the performance of models [76,77,78,79]. These kinds of techniques allow for the usage of ARIMAX models, which are a special case of ARIMA models with external regressors, including different kinds of variables like meteorological data, weather indexes, and observations such as temperature, humidity, or wind speed that can improve performance of stochastic models that follow classical techniques of model estimation [80,81].

6. Conclusions

Forest wildfires are and will be an issue to consider in risk management for every country that faces risks related to this kind of disaster. No single factor produces wildfires. They occur according to conditions in the environment that trigger the disaster, without considering the ones initiated by human activity, either intentional or not [82]. Risk management must focus on controlling these conditions in order to anticipate and foresee potential disasters, trying to mitigate as much as possible the impact of disasters in terms of human and economic losses. Results for the number of fires and affected surface suggest that the models performed well in describing the series and capturing the variability of them as much as possible, specifically in affected surfaces. This type of analysis should be complementary to other tools related to helping wildfire management, not only in forecasting for years ahead, but also in understanding the behavior of processes over the years in the past.

Results of the model fitted are similar to the models and studies developed in China [83] where a SARIMA model is estimated for forecasting fire frequency applying optimization techniques for data pre-processing, showing better performance in terms of error metrics compared to traditional stochastic methods. Another study related to wildfire forecasting shows an ARIMA model compared to neural networks, support vector machines, and integrated spatiotemporal forecasting frameworks for the predictive modeling of wildfires [41], the ARIMA model being one of the better performing models compared to more complex models. Researchers from the United States used ARIMA models for forecasting area burned (ha), proportion of areas burned, and area burned at different levels of severity in Western US through different autoregressive and moving average parameters [84]. In northeastern Chinam researchers compared different methods and approaches for forecasting fire danger. Models included long short-term memory, ANN, ARIMA, gradient boosting, and support vector regressor for the estimation method [85]. Other approaches involve more complex models using satellite images with deep generative models [86] and a combination of naïve Bayes with autoregressive approaches for forecasting burned area [87]. In all the previous studies mentioned, the pre-processing of time-series data plays a crucial role in estimating a model with valid results, considering not only the requirements of stationarity, but also the transformation or development of new variables from the initial available ones.

Forecasted values for the number of fires follow the trend of the previous series. Predictions are about 7000 fires per year, maintaining the levels of the last ten years. The same occurs with affected surfaces (ha), whose forecasted values are about 90,000 hectares burned due to wildfires on average for 2024 to 2028, meaning that, according to results and levels of forecasted values, there must be a concern about wildfires in years to come, both considering results from ARIMA (for number of fires) and SARIMA (for affected surfaces). In September 2023, the Chilean government announced an increase in budget to prevent and mitigate the consequences of wildfires in the country. The previous announcement was due to the repercussions previous wildfires have had on the country over the last ten years [88] where, annually, there has been at least one disaster involving wildfires affecting more than one region at the time [89,90].