## 3 Materials and Methods
### Methodology
#### 3.3.2 Dempster-Shafer Theory (DST)  
The concept of DST is based on a frame of discernment and is known as a belief function (Bel) that is derived from the Bayesian probability theory (BPT), which identifies this algorithm as a soft classifier. However, the difference between BPT and DST is that the priors and conditionals need not be specified and the basic assumption of DST is that ignorance exists instead of supplying prior probabilities [56]. The name DST is obtained from the extensions and clarifications presented by [PERSON]. The DST is considered as a useful approach to integrating spatial data and can provide a mathematical framework for the description of incomplete knowledge [57]. Moreover, the DST is well known as evidence theory, which defines basic probability assignment (bpa) functions to provide a combination of evidence [58]. It defines hypotheses in a hierarchical structure developed from a basic set of hypotheses that form the framework of discernment [59]. The DST is an effective method for modelling imprecision and uncertainty assessment, and it can provide additional flexibility for the specification of uncertainty in probabilistic models and hypothesis testing. DST has mostly been applied with a focus on uncertain reasoning in artificial intelligence and expert systems. In this theory, displaying information requires two essential functions, namely the Bel and plausibility function (PI), from which the lower bound value for a (known) probability function and the upper bound for a probability function (an unknown probability function) are derived, respectively. The differentiation between Bel and PI illustrates the uncertainty of the knowledge about the objective proposition [60]. The DST provides an extension of the probability framework for assessing the uncertainty of any imprecision event of the probability P(MI) that the alternative method MI, I = 1,..., n is correct. The lower bound indicates the degree of knowledge or belief that supports MI and represented by Bel (MI), whereas the upper bound contributes to MI and is called plausibility PI (MI) [61] (see Equations (1) and (2)  
\[Bel\;(A)=\sum_{B\subseteq A}m(B) \tag{1}\]  
\[Pl\;(A)=\sum_{B\subseteq A\
eq 0}m(B) \tag{2}\]  
where the summation is taken over all sets \(B\subseteq 2\emptyset\) with \(B\subseteq A\) in the definition of Bel and the summation in that of PI is taken over all sets \(B\in 2\emptyset\) with \(B\cap A\
eq 0\), for which the set of \({}^{\,\Theta}\) is a mutually exclusive and collectively exhaustive hypothesis, and the power set of \({}^{\,\Theta}\) is denoted by \(2\emptyset\). The Bel is the summation of all masses directly assigned to that by a set of hypothesis A, while the plausibility sums all masses not assigned to the complement of hypothesis A. An uncertainty interval is defined as \(Bel(A),Pl(A)\) and has the following properties:  
\[Bel\;(A)\leq Pl(A)\]  
\[Pl(A)=1-Bel(\bar{A})\]  
where \(\bar{A}\) is the negation of \(A\) and Bel (\(\bar{A}\)) is called the disbelief function. These properties indicate that the unknown true probability lies somewhere between the belief and plausibility functions. The difference between these two functions, which is called the belief interval or ignorance, represents the ignorance of one's belief of the target proposition \(A\) and can be regarded as a measure of the imprecision of knowledge about the uncertainty of set \(A\)[62]. This belief interval or ignorance is the main distinguishing feature of the DST of evidence compared to the traditional probability theory. From the general point of view, unlike a probabilistic theory that allocates a mass to the individual elementary events, the theory of evidence, or the bpa, makes \(m\;(A)\) on set \(A\) of the P(z), the power sets of space \(Z\) event (i.e., on a set of results rather than a single elementary event). In more detail, \(m\) (\(A\)) expresses the degree of belief that a specific element x belongs to set \(A\) only, and not to any subset of \(A\).