4.3.1. Redundancy-Driven Topology Design

In previous work [39], a redundancy metric was proposed that introduces the notion of *range* of a set of possibility distributions.

**Definition 8** (Range [39])**.** *Given a frame of discernment X* = [*x*a, *x*b]*, the range of a set of possibility distributions* **p** *quantifies how far* **p** *stretches over X. Let* P(**p**) *bet the power set of all possible* **p***, then the range is described by a monotonic increasing function rge* : P(**p**) → [0, 1] *with the following properties:*


The range determines whether a set of possibility distributions covers *X*. Together with the consistency measurement applied in (16), *rge* is adopted into the topology design approach. Consistency and range are balanced against each other, which results in a dual redundancy metric:

**Definition 9** (Possibilistic Redundancy Metric [39])**.** *Let* **S** = {*S*1, *S*2, ... , *Sn*}*, i.e., a set of information sources, and* P(**S**) *be all possible combinations of sources, then a possibilistic redundancy metric ρ is a function that maps* P(**S**) *to the unit interval: ρ* : P(**S**) → [0, 1]*. Information sources are only redundant if their information items both (i) are redundant themselves and (ii) cover the frame of discernment, i.e., have a high range (Definition 8). In accordance*

*with [39], the redundancy of information items is determined via possibilistic similarity measures. Consistency (2) satisfies the requirements to serve as a similarity measure [32].*

*In this context and to qualify as an intuitively meaningful metric, the following requirements have to be met:*


$$
\rho(S\_1, S\_2, \dots, S\_n) = \rho(S\_{p(1)}, S\_{p(2)}, \dots, S\_{p(n)}),
$$

*for any permutation p on* N>0*.*

*The following relations between redundancy of information items and sources hold.*


To capture the idea of a dual metric, *ρ* is designed to be a function of two pieces of evidence. The evidence against redundancy ec : P(**S**) → [0, 1]. As long as information items are redundant, ec(**S**) = 0. Determining the redundancy of information items is both based on the similarity of possibility distributions and related to the notion of possibilistic dependency. An overview of possibilistic redundancy measures for information items is provided by Holst and Lohweg [39]. Dependency measures are reviewed by Dubois et al. [74].

Evidence in favour of redundancy ep : P(**S**) → [0, 1] quantifies the amount of epistemic uncertainty in training data. It incorporates the range of information. It indicates to what degree information is available from the complete frame of discernment. A set of information sources is only redundant if ep(**S**) > 0 and ec(**S**) < 1. The smaller value of ep and (1 − ec) dominates the redundancy metric. In previous work [39], the geometric mean is proposed as an averaging function for ep and ec as follows:

$$\rho(\mathbf{S}) = \rho(\mathbf{e}\_{\mathbf{c}}(\mathbf{S}), \mathbf{e}\_{\mathbf{P}}(\mathbf{S})) \, = \sqrt{\mathbf{e}\_{\mathbf{P}}(\mathbf{S}) \cdot (1 - \mathbf{e}\_{\mathbf{c}}(\mathbf{S}))}. \tag{18}$$

Let the consistency measure h (2) determine the redundancy between information items and let **I***<sup>j</sup>* be the set of information items available at instance *j*, then

$$\text{ie}\_{\mathbb{C}}(\mathbb{S}) = 1 - \mathop{\text{avg}}\_{j = \{1, \dots, m\}} (\mathbf{h}(\mathbb{I}\_{j}))\_{\text{\textquotedblleft}} \tag{19}$$

i.e., ec averages consistencies available from training data with an averaging operator (see Definition 3). Designing MCS-based topologies (16) is based on the notion that the consistency is above a certain *α* for all instances. To keep this notion for the redundancybased design, the minimum operator is used as averaging operator in (19).

The evidence ep is computed based on the range as follows:

$$\text{Re}\_{\mathbf{P}}(\mathbf{S}) = \frac{r\text{g}\mathbf{e}(\mathbf{S}) - \mathbf{x}\_{\mathbf{a}}}{\mathbf{x}\_{\mathbf{b}} - \mathbf{x}\_{\mathbf{a}}}.\tag{20}$$

The range itself is dependent on the position of possibility distributions on the frame of discernment, which is determined by their center of gravity [2]

$$pos(\pi) = \begin{cases} \text{x} & \text{if } \pi(\mathbf{x}) = 1 \text{ and } \forall \mathbf{x}' \in \{X \nmid \mathbf{x}\}: \pi(\mathbf{x}') = 0, \\\frac{\int\_{\frac{\mathbf{x}}{\mathbf{x}\_{\mathbf{a}}} \cdot \pi(\mathbf{x}) \, \mathrm{d}x}{\int\_{\frac{\mathbf{x}}{\mathbf{x}\_{\mathbf{a}}} \cdot \pi(\mathbf{x}) \, \mathrm{d}x} & \text{otherwise.} \end{cases} \tag{21}$$

The position of a set of possibility distributions **p** is obtained by prior disjunctive fusion (5), i.e.,

$$pos(\mathbf{p}) \,:= \, pos(\mathcal{f}u(\mathbf{p})) \,.$$

Given a set of information sources **S** = {*S*1, *S*2, ... , *Sn*} providing information items **I***<sup>j</sup>* = **p***<sup>j</sup>* = {*π*1,*j*, *π*2,*j*,..., *πn*,*j*}, then

$$\text{reg}(\mathbf{S}) = \max\_{j, \nearrow \in \{1, \ldots, m\}} \left( \left| \text{pos}(\mathbf{p}\_{\rangle}) - \text{pos}(\mathbf{p}\_{\neq'}) \right| \right) = \max\_{j \in \{1, \ldots, m\}} \left( \text{pos}(\mathbf{p}\_{\rangle}) \right) - \min\_{j \in \{1, \ldots, m\}} \left( \text{pos}(\mathbf{p}\_{\neq}) \right). \tag{22}$$

At least one pair **p***j*, **p***<sup>j</sup>* of information item sets needs to range over the frame of discernment *X* in order to provide evidence for a redundant behaviour, i.e., ep(**S**) > 0 if ∃*<sup>j</sup>* : *rge*- **p***j* > 0.

The redundancy metric *ρ* (18) is used as a decision criterion to find suitable sets of information sources **S***<sup>ρ</sup>* (*k*) to be fused in fusion nodes *fn*(*k*). Algorithm 2 describes a simple approach that searches all subsets of consistency-based fusion nodes in S<sup>h</sup> (found by Algorithm 1). A set of sources is only assigned to a fusion node if *ρ* ≥ *η*.

**Algorithm 2:** Algorithm that searches for redundancy-based fusion nodes based on S<sup>h</sup> found by Algorithm 1. The algorithm iterates over S<sup>h</sup> and searches all **S** ⊆ **S**, **S** ∈ S<sup>h</sup> for sets meeting the redundancy criterion *η*.

**Input:** Consistency-based fusion topology found by Algorithm 1, i.e., <sup>S</sup><sup>h</sup> <sup>=</sup> {**S**MCS−*<sup>α</sup>* (*k*) }; threshold parameter *η*

**Output:** Redundancy-based fusion toplogy S*<sup>ρ</sup>*

```
Sρ ← {};
S
 ← Sh;
idx ← 1;
while idx ≤ |S

             | do
   S ← S

         [idx];
   if ρ(S) ≥ η or |S| = 1 then
      /* S is added to fusion topology */
      if S  Sρ then
         Sρ.append(S);
      end
   else
      /* create subsets of S to be checked for redundancy */
      foreach S ∈ S do
         S
 ← S \ S;
         if S
 ∈ S / 
 then
            S

              .append(S

                        )
         end
      end
   end
   idx ← idx + 1;
end
```
As motivated previously, the redundancy-based approach of Algorithm 2 results in a more robust MCS-based topology design than Algorithm 1. As (18) includes the range of information items, the effects of incomplete information and epistemic uncertainty in the training data are reduced. This leads to less detections of spurious relations.
