Evidence, theory

Evidence theory also known as the Dempster-Shafer theory, has been first introduced by Dempster [5], then formalized by [25] and finally axiomized later into the framework of the Transferable Belief Model (Evidence theory) by Smets [28]. The Evidence theory can be understood as an alternative to probability theory for the representation of uncertainty It allows one to manipulate non-necessarily exclusive events and thus to represent explicitly information uncertainty... 

This theory assumes the definition of (i) the frame of discernment Q consisting of the exhaustive and exclusive h3rpothesis and (ii) the reference set 2 of all the disjunctions of the elements of fi. The Evidence theory defines the basic belief assignment (bba) function as an elementary mass function m 2 —> [0,1] verif3dng for all elements A of 2 ... 

Finally, the last step is the decision making process which is supported by the results provided by the combination rules. Indeed, as previously highlighted, the combination of the available sources of information provides us with a new belief function which represents the most reliable and complete information. However, if the choice of the most likely h3rpothesis is straightforward in the probabilistic framework, it can become quite complex in the Evidence theory. 

Because it offers a framework to manage uncertain and conflicting information, the Evidence theory can be relevant to combine and to cross check fault signals. In the context of fault diagnosis, the frame of discernment 17 will be the set of all possible states of the system, i.e. all the faults that can occur on the supervised process. In other terms, we have ... 

In the Evidence theory framework, the bba corresponding to residuals ri, T2 and rs will have respectively the cores ... 

The value applied to each mass is derived from the value of the associated residual. In a boolean framework, a threshold would be defined such that some values are declared as fault-free i.e., when the residual is lower than the threshold) and other ones as faulty i.e., when the residual is higher than the threshold). This method produces for each situation a vector of 0 and 1 that can be compared to the known signatures to isolate the fault. As already pointed out one main drawback is that if the value of the residual oscillates around the threshold, the state associated to the residual oscillates too. On the contrary, in the Evidence theory framework, an infinity of values are possible for each focal element. It is then possible to use smooth functions to produce a bba from the residual. For example, the following function has been proposed in [23] ... 

Fig. 6. Comparison of booiean (b) and Evidence theory (c) based approaches with coherent residuals (a). Abscissa dimension is hours.

The beliefs are derived from the residuals with the relation (15) with the same threshold than for boolean evolution, i.e., t = 0.4 for a = 0.5. Clearly, the boolean combination leads to an unstable isolation of the faults because the value of ri is often oscillating around 0.4. On the contrary, the approach based on the Evidence theory isolates perfectly the faults and does not induce any oscillation. In another situation where the value of residual rs is not affected as it should be when the fault /s occurs, the boolean combination does not succeed again to isolate the faults (see in particular /s in Figure 7.b) whereas the Evidence theory combination methods correctly the faults most of the time Cf. Figure 7.c). 

These two simple examples highlight how an uncertainty managing framework like Evidence theory can improve fault isolation performances. 

Section 3.2 has shown that Evidence theory could improve the performances of a FDI system by supporting a non boolean i.e., not crisp) evaluation of the residuals. In the following, it will be shown how the method performs information crosschecking and exhibits uncertainty. 

Table 3. Fault isolation performances of the Evidence theory based FDI system...

The core of the Evidence theory lies in the combination of belief assignments. As already described, several belief combination rules exist, each of them corresponding to an interpretation of the conflict between bba. As a consequence, a rule should be first chosen according to the interpretation of conflict appropriate to the final objective. In addition, some mathematical considerations should be also accounted for. Indeed, only the Dempster s and Smets rules are associative. This means that for other combination rules e.g., Yager s or Dubois and Trade s rules), we have (i) either to perform a simultaneous combination of all available bba or (ii) to determine the sequence of combinations appropriate to the final objective. The first choice is satisfactory since it does not need to perform any assumption on the combination order. However, it implies to compute a great number of intersections of focal elements and is thus difficult to apply for more than seven bba (due to the computation time required). 

Additional information can be withdrawn from these curves. Indeed, the evidence theory is one of the very few diagnosis approaches that allows one to detect explicitly conflicts in the measurements signals. As a consequence, one can easily imagine that, in case of persisting doubt, the supervisory software should call an expert (if present on site). Also, such an analysis of the sensors network can invalidate the assumptions made on the influent composition (see the beliefs on CODin and between days 8 and 13 in Figure 4). The person in charge of the process has here an indication on the needed manual analysis to perform for better characterization of the wastewater to be treated. These two aspects are key requirements for telemonitoring issues as detailed in [18]. 

The previous section concentrated on the management of a hard and soft sensors network. This is an important step since the information sources must be carefully checked before being further used. This section will be devoted to the diagnosis of the overall biological state of the process. In particular, it will illustrate that the use of the Evidence theory approach improves the fault diagnosis system in terms of modularity and d3mamical adaptation. 

For all the reasons detailed throughout the chapter i.e., modularity, robustness, simplicity) it is our strong belief that Evidence theory will play a major role in the next future for advanced diagnosis of bioprocesses. 

L. Lardon and J.P. Steyer. Using evidence theory for diagnosis of sensors networks application to a wastewater treatment process. In Int. Joint Conf. Artificial Intell. (IJCAI), pages 29-36, Acapulco, Mexico, 2003. 

Another important advantage of Evidence Theory is that the Dempster-Shafer law of combination (the orthogonal sum) allows us to combine data from different independent sources. Thus, if we have the same frame of discernment for two mass functions which have been derived independently from different data, we may obtain a unified mass assignment. 

As a result, in evidence theory there are two measures of likelihood, behef and plausibility the belief about events and propositions is represented as intervals, bounded by two values, beUef and plausibility. The belief in a proposition (set) A is quantified as the sum of the probability masses assigned to aU sets enclosed by it ... 

Irrespective of the taxonomy used, epistemic and aleatory uncertainty, or uncertainty and variation, alternatives to probability have been suggested for the representation of the epistemic concept. These include interval or imprecise probabihty (Coolen 2004, Coolen Utkin 2007, Utkin Coolen 2007, Weichselberger 2000), fuzzy set theory and the associated theory of possibility (Zadeh 1965, Zadeh 1978, Unwin 1986), and the theory of behef functions (Shafer 1976), also known as evidence theory or the Dempster-Shafer theory of evidence. 

Ker J, Bradley B. Simulation in medical education. In Swanwick T, editor. Understanding medical education evidence, theory, and practice. Oxford Wiley-Blackwell 2010. p. 164-80. 

Brickhouse, N. W., Dagher, Z. R., Letts, W. J., Shipman, H. L. (2000). Diversity of students views about evidence, theory, and the interface between science and religion in an Astronomy course. Journal of Research in Science Teaching, 37,4 340-362. 

ABSTRACT Dependencies or failure dependencies in probabilistic risk assessments may lead to significant errors if not properly analyzed. In order to overcome the limitations of tradition methods, a modified Bayesian Network (BN), which is called Evidential Network (EN), was proposed with evidence theory to handle dependencies in Probabilistic Risk Assessment (PRA). Fault Trees (IT s) and Event Trees (ETs) were transformed into an EN which constructs a uniform framework to represent accident scenarios. Dependencies can be processed through the corresponding evidential networks where uncertainties are characterized by basic belief mass. A case study was discussed to demonstrate the proposed approach. Frequencies of end states were obtained and expressed by belief and plausibility measures. The proposed approach can be easily applied to probabilistic risk assessments that involve dependencies while addresses the uncertainties in experts knowledge. 

The remainder of the article is organized as follows In Section 2, the factor model, which is a typical CCF factor model, is briefly introduced. Uncertainties brought by P factors are discussed and serve as an inspiration of using modified models to deal with dependencies in PR A. The D-S evidence theory and evidential networks are briefly introduced in Section 3. The EN based approach is then discussed in detail in Section 4. In Section 5, a part of a practical probabilistic risk assessment is analyzed using the proposed approach. The conclusion is expressed in the end. 

A BRIEF INTRODUCTION TO D-S EVIDENCE THEORY AND EVIDENTIAL NETWORKS... 

---