Dempster-Shafer theory of evidence: A new approach to spatially model wildfire risk potential in central Chile

Highlights

• Wildfires can be modeled through evidence theory.

• There are spatial patterns that repeat, associated to risk areas.

• The evidence theory is a tool used to determine wildfire risk areas.

• Metropolitan Region shows risk concentration patterns.

• This method can be replicated in other Mediterranean climates.

Abstract

A spatial modeling was applied to Chilean wildfire occurrence, through the Dempster-Shafer's evidence theory and considering the 2006–2010 period for the Valparaiso Region (central Chile), a representative area for this experiment. Results indicate strong spatial correlation between documented wildfires and cumulative evidence maps, resulting in a powerful tool for future wildfire risk prevention programs.

Introduction

Forest fires have been widely studied in terms of cause spatial behavior, and severity when linked to events unrelated to landscape dynamics. However, there are very few scientific references focusing on the risk for fire ignition, based on spatial modeling founded on uncertainty theories and statistical evidence (Castillo, 2012a, Castillo, 2012b).

The analysis of evidence-based modeling for recurrent wildfire events is becoming increasingly important in countries with high fire recurrence, especially when there are recurring references for their ignition (Preisler et al., 2004, Finney et al., 2011). In Mediterranean climates, for example, the study of fire occurrence (ignition) is based on historical fire risk modeling (Castillo, 2012a, Castillo, 2012b, Miller and Ager, 2013) and the fire potential conditions present when the wildfire began (Perestrello de Vasconcelos et al., 2001). Maps of areas susceptible to the presence of new wildfires are usually based on historical data based on the location and size of the event, allowing for areas of greater impact to be determined, which are then used to develop fire-prevention strategies (Finney et al., 2011).

Each wildfire event is preceded by a factor of uncertainty, associated with a probability of occurrence, due to either natural or human causes. This degree of uncertainty is then moved to fire behavior's characteristics, if ignition occurred. This procedure is part of the spreading danger's analysis, a function of weather, topography, and vegetation (Castillo et al., 2013). Thus, fire risk is evaluated through the development of predictive models based on historic wildfire occurrence. However, there is a clear lack of studies focusing on models based on uncertainty. Available theories in this field include interval analysis (Moore, 1966), theory of confidence (Kanal and Lemmer, 1986), theory of possibility (Dubois and Prade, 1988), theory of evidence (Schafer, 1976), and fuzzy set theory (Zimmermann, 1985). The above theories were developed as an alternative to Bayesian theories for decisions based on human opinions and judgments (Zimmermann, 1985, Dubois and Prade, 1988).

Fire risk is generally attributed to very specific aspects: the occurrence of fires (number and location), the effects of roads and populated centers, and also human activities. These factors contribute to the definition and evaluation of historic and potential wildfire risk (Castillo et al., 2013). Even though fire risk is a phenomenon that can be evaluated in case studies, authors such as Finney et al. (2011) introduced risk-related variables, including them when developing simulation models to estimate probability of occurrence. Similarly, Preisler et al. (2004) studied this phenomenon via logit models, considering vegetation and climate parameters for an area in Oregon, United States.

Considering the above, this study addresses a methodological process to produce fire risk maps, based on the theory of evidence developed by Dempster (1967) and later complemented by Schafer (1976), all together known as the “Dempster-Shafer theory”, as a new tool in this type of study, to provide a foundation for current methods to calculate wildfire risk. To facilitate the reader's understanding of this complicated topic, a review of the Dempster-Shafer theory is included below.

The Dempster-Shafer theory is an extension of the probability theory used to describe uncertainty of evidence. It focuses on the belief that an event may happen (or has happened) in accordance with the experience of decision-makers, as opposed to classic probability, which implies the existence of probability values associated with events determined independently of what the observer may know about the actual probability of occurrence. Furthermore, it enables evidence obtained through observation or through experiments to support various mutually-exclusive conclusions at the same time, or no conclusion in particular. The Dempster-Shafer theory aims to reallocate the probability of belief in hypotheses when evidence changes. This supposes that there is an exhaustive set of mutually-exclusive hypotheses Θ = {θ₁, θ₂… θ_n}, known as the Frame of Discernment (FD), around which subsequent evidence may be considered (Zadeh, 1965, Zimmermann, 2000). Unlike other approaches, this theory states that one must consider the impact of evidence not only on the original individual hypotheses, but also on the groups of these hypotheses, which are subsets of Θ and the considered hypotheses. Thus, the new hypotheses are possible options or alternatives to the original hypotheses. The set of Θ parts, represented by P(Θ), is made up of all subsets of Θ, including the empty set (Ø) and Θ itself. The P(Θ) set, and not Θ, is thus the set of the hypothesis considered. The theory of evidence uses a function μ, called Basic Probability Assignment (BPA), in order to assign to each element of P(Θ) an indicative value of the belief that is allocated to it when evidence is added. The function μ fulfills the following properties:

$$\mathrm{\mu }\left(\mathrm{Ø}\right)=0$$

$$\left.\forall \mathrm{{\rm A}}\right|\mathrm{{\rm A}}\in \mathrm{P}\left(\mathrm{\Theta }\right):0\le \mathrm{\mu }\left(\mathrm{{\rm A}}\right)\le 1$$

$${\sum }_{\mathrm{A}\in \mathrm{P}\left(\mathrm{\Theta }\right)}\mathrm{\mu }\left(\mathrm{A}\right)=1$$

Expression (1) indicates that the belief with evidence placed in the empty set is always 0. Expression (2) refers to a value of actual belief between 0 and 1 is assigned to all subsets of Θ. Finally, Expression (3) shows that the sum of all assigned values must be 1. Function μ is similar to the Probability Density Function (PDF) of the probability theory, but in which the Bayesian restriction that the sum of beliefs assigned to the original hypotheses must be 1, does not apply. Therefore, confirmation of a determined belief for a subset does not mean that the remaining belief is confirmed by its rejection; only when μ has values different from zero for the unitary subsets of Θ, would μ behave as a PDF. On the other hand, if evidence supports a subset of Θ hypotheses that is not unitary, this means that the evidence confirms belief in the Θ hypotheses forming the subset, but without determining the impact on the belief of each of these. The theory of evidence with BPA provides a way of representing the impact of evidence on the FD. A series of measuring instruments are established to try and determine the Degree of Belief that may be placed in each hypothesis using available evidence. This involves degrees of Belief, Verisimilitude, and Belief Interval, which will be defined below.

• The Degree of Belief in an element A of P(Θ), is written as Bel(A) and represents the minimum belief in hypothesis A, as a result of evidence. Therefore, it is defined as the sum of basic probability assignments undertaken for all subsets of A:

$$\forall \mathrm{A}\mid \mathrm{A}\in \mathrm{P}\left(\mathrm{\Theta }\right):\mathrm{Bel}\left(\mathrm{A}\right)={\sum }_{\mathrm{X}\subseteq \mathrm{A}}\mathrm{\mu }\left(\mathrm{X}\right)$$

• The Degree of Verisimilitude (or Plausibility) of an element A of P(Θ) is written as Pl(A) and represents the maximum belief in hypothesis A, as a result of evidence. It may also be seen as the sum of basic probability assignments undertaken for all elements X of P(Θ) whose intersection with A is not empty:

$$\forall \mathrm{A}\mid \mathrm{A}\in \mathrm{P}\left(\mathrm{\Theta }\right):\mathrm{Pl}\left(\mathrm{A}\right)={\sum }_{\mathrm{X}\cap \mathrm{A}}\mathrm{\mu }\left(\mathrm{X}\right)$$

It follows that the Degree of Belief is always less than the Degree of Verisimilitude.

• The interval between the Degree of Belief and that of Verisimilitude, of an element A of P(Θ), is the Interval of Belief in A. It is written as a pair [Bel(A), Pl(A)] and represents a level of uncertainty about hypothesis A, as a result of evidence. According to the Theory of Evidence, the difference between Bel(A) and Pl(A) is a measure of this uncertainty. When Bel(A) and Pl(A) are equal, there is total certainty about the impact of evidence on hypothesis A. When Bel(A) is 0 and Pl(A) is 1, the difference between both measures is maximum, and nothing is known about the effect of the evidence on A. When the values of Bel(A) and Pl(A) are different, the greater the difference between them, the greater the uncertainty about the impact of evidence on hypothesis A. For the case of wildfires, the phenomena of fire occurrence responds to a casuistic that is possible to model from the spatial and temporal perspective, in which Dempster-Shafer contributes to the construction of an evidence framework, just like it has been done traditionally through the analysis of time series, or simply through descriptive statistics (Castillo and Rodríguez y Silva, 2015). This spatial modeling method is an excellent analysis complement for the understanding of wildfire risk and also for danger control, since one can know well the areas susceptible for fire and, as a consequence, their local characteristics from the potential fire spreading perspective (Castillo et al., 2013).