## 3 Uncertainty Modeling Techniques
### Possibilistic Methods
#### 3.2.3 Dempster-Shafer Theory (DST)  
DST offers a framework for combining evidence from different sources to arrive at a degree of belief (represented by a belief function) that takes into account all the available evidence. Unlike traditional probability, it does not require mutually exclusive and exhaustive hypotheses and can operate under less information.  
DST has been effectively applied in power systems, particularly for decision-making and uncertainty analysis. In [40], DST is utilized to address the complex multi-criteria decision-making (MCDM) process involved in selecting optimal offshore wind turbines. This method accommodates the uncertainty inherent in expert judgments, representing them as basic probability assignments and fusing them using DST synthesis rules. By applying DST in conjunction with the stepwise weighted assessment ratio analysis, the study demonstrates how the best turbine can be selected from multiple alternatives under uncertain conditions. This DST-based MCDM framework is particularly useful for projects with high complexity and uncertain data inputs, such as offshore wind farms. In [41], DST is extended to uncertainty and sensitivity analysis for building energy assessments. Since traditional probabilistic methods cannot handle situations where there is insufficient information to define precise probabilities, DST offers an alternative by constructing belief and plausibility measures based on interval-valued probabilities. This allows for more nuanced uncertainty modeling in scenarios where data are incomplete or imprecise. The combination of DST with machine learning techniques in building energy simulations further enhances the robustness and reliability of uncertainty assessments.  
The basic probability assignment, denoted by \(m\), is a function from the power set of the universal set \(X\) to the interval [0,1]. It adheres to the following conditions [42]:  
\[m:P(X) \rightarrow [0,1] \tag{45}\] \[m(\theta) = 0\] (46) \[\sum_{A\in P(X)}m(A) = 1 \tag{47}\]  
where \(P(X)\) represents the power set of \(X\), \(\theta\) is the null set, and \(A\) is a set in the power set (\(A\in P(X)\)).  
In defining the measures for uncertainty, the belief (Bel) of a set \(A\) is computed as the cumulative basic probability assignments (BPAs) for all proper subsets contained within \(A\), expressed mathematically as  
\[Bel(A)=\sum_{B|B\subseteq A}m(B) \tag{48}\]  
Conversely, plausibility (PI) encompasses the sum of BPAs for all sets that have a non-empty intersection with \(A\), formulated as  
\[Pl(A)=\sum_{B|B\cap A\
eq\varnothing}m(B) \tag{49}\]  
Both belief and plausibility are non-additive measures; thus, their sums do not necessarily equate to 1. This non-additivity implies that complete knowledge of one measure does not infer the total of the other.  
Additionally, the relationship between belief and its BPA is inverse, allowing for the derivation of BPA from Bel. Furthermore, plausibility can be derived from belief through the complement rule:  
\[Pl(A)=1-Bel\big{(}\overline{A}\big{)} \tag{50}\]These measures, by bounding the precise classical probability of an event, provide lower and upper limits, respectively, where the precise probability is uniquely determined if \(Bel(A)=Pl(A)\). This relationship emphasizes the theoretical underpinnings of uncertainty quantification in evidential reasoning frameworks.