4.1.1. Pooling Fusion

As can be seen in Table 1, the simple conjunctive and disjunctive fusion rules satisfy associativity. However, depending on the applied t-norm—the renormalisation step already causes nonassociative behaviour. For the product norm (t(*π*1(*x*), *π*2(*x*)) = *π*1(*x*) · *π*2(*x*)), fusion stays associative; however, generally, renormalisation prevents associativity [47].

MCS fusion (6) is based on the idea that consistent information items are to be fused conjunctively first before the results are fused disjunctively. MCS fusion thus specifies a sequence in which information is to be fused. Consequently, MCS fusion is not associative. It is quite easy to see that different sequences result in different outcomes (see Appendix B for an example). Quantified fusion (7) has a similar approach, meaning that it fuses conjunctively and disjunctively in two steps. Quantified fusion is—for the same reasons as MCS fusion—not associative and not quasi-associative.

More sophisticated fusion rules—such as adaptive (8), (9) and progressive (10) rules attempt to make the most of all available information. These fusion rules rely on specific metrics, such as global consistency, consistency between specific subsets, or distances between information items. Many of these metrics are only computable if all information items are available centrally. Since all three rules (8), (9), and (10) are based on the quantified fusion rule, they inherit quantified fusion's nonassociativity.

#### 4.1.2. Estimation and Majority-Guided Fusion

Estimation fusion (11)–(13) as well as the majority-guided MOGPFR (14) relies on Zadeh's extension principle.

**Proposition 3.** *With regard to Zadeh's extension principle, a fusion operator fu*(*π*1, *<sup>π</sup>*2, *<sup>π</sup>*3) <sup>=</sup> \* *<sup>G</sup>*(*π*1(*x*1),*π*2(*x*2),*π*3(*x*3)) *F*(*x*1,*x*2,*x*3) + *satisfies associativity if* G *and* F *are associative functions and* G *is monotonic increasing in all its arguments.*

**Proof.** The operator *fu* is associative if *fu*(*π*1, *π*2, *π*3) = *fu*(*π*1, *fu*(*π*2, *π*3)). With (11), this becomes

$$\max\_{\substack{\mathbf{x}\_{i}\in\mathcal{X}\_{i}:\ \mathbb{F}(\mathbf{x}\_{1},\mathbf{x}\_{2},\mathbf{x}\_{3})=\mathbf{x}}} \mathcal{G}(\pi\_{1}(\mathbf{x}\_{1}),\pi\_{2}(\mathbf{x}\_{2}),\pi\_{3}(\mathbf{x}\_{3})) =$$

$$\max\_{\substack{\mathbf{x}\_{1}\in\mathcal{X}\_{1},\mathbf{x}'\in\mathcal{X}':\\\mathcal{F}(\mathbf{x}\_{1},\mathbf{x}')=\mathbf{x}}} \mathcal{G}\left(\pi\_{1}(\mathbf{x}\_{1})\_{\begin{subarray}{c}\mathbf{x}\_{2}\in\mathcal{X}\_{2},\mathbf{x}\_{3}\in\mathcal{X}\_{3}:\\\mathcal{F}(\mathbf{x}\_{2},\mathbf{x}\_{3})=\mathbf{x}' \end{subarray}} \mathcal{G}(\pi\_{2}(\mathbf{x}\_{2}),\pi\_{3}(\mathbf{x}\_{2})) \right).$$

The frame of discernment *X* contains every unique element given by *F*(*x*2, *x*3) for every 2-tuple (*x*2, *<sup>x</sup>*3) with *<sup>x</sup>*<sup>2</sup> <sup>∈</sup> *<sup>X</sup>*<sup>2</sup> and *<sup>x</sup>*<sup>3</sup> <sup>∈</sup> *<sup>X</sup>*3. In the following, the notation max *<sup>x</sup>*2∈*X*2, *<sup>x</sup>*3∈*X*3: F(*x*2,*x*3)=*x*

is shortened to max *<sup>F</sup>*(*x*2,*x*3)=*<sup>x</sup>* —this also applies to similar notations.

Assume G to be monotonic increasing in all its arguments, i.e., for any *ai*, *bi* ∈ [0, 1] with *i* ∈ {1, 2, ... , *n*} and ∀*i* : *ai* ≤ *bi*: *G*(*a*1, *a*2,..., *an*) ≤ *G*(*b*1, *b*2,..., *bn*). If *π*1(*x*1) ≥ G(*π*2(*x*2), *π*3(*x*2)), then G(*π*2(*x*2), *π*3(*x*2)) has no influence on the term max(G(*π*1(*x*1), G(*π*2(*x*2), *π*3(*x*2)))). If, on the other hand, *π*1(*x*1) < G(*π*2(*x*2), *π*3(*x*2)), then G(*π*1(*x*1), G(*π*2(*x*2), *π*3(*x*2))) becomes maximal if *G*(*π*2(*x*2), *π*3(*x*3)) is maximal. Consequently,

$$\begin{split} f\mu(\pi\_1, f\mu(\pi\_2, \pi\_3)) &= \max\_{F(\mathbf{x}\_1, \mathbf{x'}) = \mathbf{x}} G\left(\pi\_1(\mathbf{x}\_1)\_{\prime} \max\_{F(\mathbf{x}\_2, \mathbf{x}\_3) = \mathbf{x}'} G(\pi\_2(\mathbf{x}\_2), \pi\_3(\mathbf{x}\_2))\right) \\ &= \max\_{F(\mathbf{x}\_1, \mathbf{x'}) = \mathbf{x}} \max\_{F(\mathbf{x}\_2, \mathbf{x}\_3) = \mathbf{x}'} G(\pi\_1(\mathbf{x}\_1), G(\pi\_2(\mathbf{x}\_2), \pi\_3(\mathbf{x}\_2))) .\end{split}$$

If *G* is also associative, then

$$f\mathfrak{u}(\pi\_1, f\mathfrak{u}(\pi\_2, \pi\_3)) := \max\_{\substack{F(\mathbf{x}\_1, \mathbf{x'}) = \mathbf{x} \ \ F(\mathbf{x}\_2, \mathbf{x}\_3) = \mathbf{x'}}} \max\_{F(\pi\_1(\mathbf{x}\_1), \pi\_2(\mathbf{x}\_2), \pi\_3(\mathbf{x}\_2)) .} G(\pi\_1(\mathbf{x}\_1), \pi\_2(\mathbf{x}\_2), \pi\_3(\mathbf{x}\_3)) .$$

If *F* is associative, then

$$f\_{\mathfrak{u}}(\pi\_1, \mathfrak{f}\mathfrak{u}(\pi\_2, \pi\_3)) \, = \max\_{\mathfrak{F}(\mathfrak{x}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3) = \mathfrak{x}} G(\pi\_1(\mathfrak{x}\_1), \pi\_2(\mathfrak{x}\_2), \pi\_3(\mathfrak{x}\_2)) .$$

In contrast to the estimation fusion rules, the MOGPFR (14) uses the IIWOWA operator for the functions F and G and is, therefore, not associative. The IIWOWA operator is an extension of the OWA operator. The OWA operator sorts the inputs (*π*1(*x*1), ... , *πn*(*xn*)) in descending order. It then weights the inputs with a predefined weighting vector (*w*1, ... *wn*) with *wi* ∈ [0, 1]. For (1, 0, ... , 0) the OWA operator becomes the maximum operator, and for (0, ... , 0, 1), the minimum operator. In these cases, the OWA operator is associative. In all other cases, sorting the input values prevents associativity and quasi-associativity. Consequently, the IIWOWA operator and the MOGPFR are nonassociative as well.

#### *4.2. MCS-Based Topology Design*

In addition to relying on associative and quasi-associative rules, there is the third option to design a fusion topology and its fusion process based on the characteristics of the information items themselves. In this case, the possibility distributions of sources are analysed, which guides the design towards desired effects. In a sense, the information provided by the multi-source system dictates the topology.

One approach to do so is to build upon the MCS fusion rule (6). It itself is not quasiassociative, and thus information items cannot be freely assigned to fusion nodes. However, by carefully searching for all the most consistent subsets, fusion can be distributed in a way that each fusion node produces the most specific intermediate result from agreeing sources, thus, emphasizing the consensus of this agreeing subset. In such a two-layer topology, all *<sup>I</sup>* <sup>∈</sup> **<sup>I</sup>***MCS-<sup>α</sup>* (*k*) are fused in separate fusion nodes *fn*(*k*) using, at the first level, a mix of renormalised conjunctive minimum fusion and maximum fusion:

$$\pi\_{(k)}(\mu) = \begin{cases} \frac{\min\_{i} \pi\_{i}(\mu)}{\mathbf{h}\_{i}(\pi\_{i}(\mu))} & \text{if } \mathbf{h}\_{i}(\pi\_{i}(\mu)) > 0\\ \max\_{i} \pi\_{i}(\mu) & \text{if } \mathbf{h}\_{i}(\pi\_{i}(\mu)) = 0 \end{cases} \tag{15}$$

with *i* indexing *Ii* ∈ **I** MCS−*α* (*k*) . At the second level, all intermediate results are fused disjunctively using the maximum operator. An exemplary fusion topology based on the MCS fusion rule is shown in Figure 4.

**Figure 4.** An example of a MCS-based fusion topology. Depicted are seven information sources fused in a two-layer topology. On the left side (**a**), the topology itself is shown with minimum fusion on the first layer and maximum fusion on the second layer. The right side (**b**) illustrates the associated possibility distributions from which the topology is constructed.

As MCS fusion analyses the consistency of information items, the inferred topology needs to be adapted for each new set of items. This is, particularly in a technical system, often not practical or feasible. Think, for example, of a technical multi-sensor system in which sensors give updated measurements in periodic time increments. In this case, the advantages of distributed fusion—such as the distribution of computing load into local nodes or lower communication loads by condensing information—are negated by the reorganisation with each measurement. Finding the MCS requires having all information items at hand in one central node rendering the distribution of the fusion process pointless. Therefore, topology design based on MCS fusion is only beneficial, if knowledge about the sources' expected behaviour regarding consistency exists a priori. In other words, if it

is known that sources produce consistent items continually, then they are assigned to a fusion node without the need for an update with each new instance or measurement. This knowledge can be derived or learned from representative training data. Conclusions about the sources' consistency in the training data are used to build up the MCS fusion topology.

Let **S**MCS−*<sup>α</sup>* (*k*) be a set of information sources that are assigned to fusion node *fn*(*k*). Furthermore, let *j* = {1, ... , *m*} be indices of training data, *I*(*k*),*<sup>j</sup>* be an information item produced by source *<sup>S</sup>*(*k*) at instance *j*, and **<sup>I</sup>**(*k*),*<sup>j</sup>* be all information items of **<sup>S</sup>**MCS−*<sup>α</sup>* (*k*) at instance *j*, then

$$\mathcal{S}\_{\{k\}} \in \mathbf{S}\_{\{k\}}^{\text{MCS}-a} \text{ if } \begin{cases} \forall j = \{1, \dots, m\}: \text{h}\Big(I\_{\{k\},j'} \mathbf{I}\_{\{k\},j} \big) \ge a \quad \text{and if } a \in (0, 1],\\ \forall j = \{1, \dots, m\}: \text{h}\Big(I\_{\{k\},j'} \mathbf{I}\_{\{k\},j} \big) > 0 \quad \text{and if } a = 0, \end{cases} \tag{16}$$

i.e., a source *<sup>S</sup>*(*k*) belongs to **<sup>S</sup>**MCS−*<sup>α</sup>* (*k*) if all its information items are consistent with the items of **S**MCS−*<sup>α</sup>* (*k*) at least to a degree of *α*.

MCS-based fusion nodes are then created by Algorithm 1, which is based on the algorithm provided for finding MCS [58,61]. Algorithm 1 starts with **S** and searches all MCS for the first data instance (*j* = 1). The found MCS are stored and themselves searched for new MCS for the next data instance and so forth.

**Algorithm 1:** Fast algorithm for finding subsets of information sources, which are consistent at least to degree *α* on every instance of training data. Each subset **S**MCS−*<sup>α</sup>* (*k*) is assigned to fusion node *fn*(*k*). The algorithm relies on finding MCS of information items as defined by Dubois et al. [58,61].

**Input:** A set of information sources **S**, alpha-cut-level *α* **Output:** Set of sets <sup>S</sup><sup>h</sup> with fusion node set **<sup>S</sup>**MCS−*<sup>α</sup>* (*k*) ∈ S<sup>h</sup>

```
m ← number of training data instances;
if j = 1 then
  Sh ← {S};
end
for j ← 1 to m do
  S ← {};
  foreach S
 ∈ Sh do
     S
 = findMCS(Ij, α);
     ; /* findMCS() as defined by Dubois et al. [58,61] */
     ; /* Ij provided by S
 */
     S = {S ∪ S

                };
  end
  Sh ← S;
end
```
For the following computations, the minimum consistency in each group is stored as a reference value: 

$$\mathfrak{a}\_{(k)}^{\mathfrak{r}} = \min\_{\mathfrak{j}} \mathbf{h} \Big( \mathbf{I}\_{(k), \mathfrak{j}} \Big). \tag{17}$$

In an MCS fusion topology, which is learned from training data rather than updated each *j*, it is not guaranteed that, for new data instances, intermediate results *I*fu (*k*,1) are disjoint. As of this, the maximum fusion rule of the final layer as described previously is replaced with (15). This means that, in the case that the topology is learned using Algorithm 1, all fusion nodes use the same fusion rule.

Regarding parameter *α*, the following observation leads to maximal specific fusion results at the first layer. If ∀*j* cores of the possibility distributions are disjoint, then fusion with MCS-1 is equal to maximum fusion [6]. Therefore, MCS-1 fusion demands continuous mutual consistency. In contrast, MCS-0 results in minimum fusion if ∀*j* the supports overlap and is less restrictive.

**Proposition 4.** *MCS fusion as outlined in (15) results in the maximal specific information items if Algorithm 1 is executed with α* = 0*.*

**Proof.** With decreasing *α*, the condition for grouping items into fusion nodes becomes less strict—as can be seen in (16). Thus, fusion node sizes increase with decreasing *α*. It follows that the maximum node sizes are achieved if *α* = 0. The more information items belong to a node, the more alternatives for the unknown true value are eliminated by the minimum operator in (6). Consequently, the integral . *<sup>x</sup>*<sup>b</sup> *<sup>x</sup>*<sup>a</sup> *π*(*x*) d*x* inside the specificity measure (A2) becomes minimal if *α* = 0, and therefore specificity (A2) itself becomes maximal.

Consequently, we propose the design of MCS fusion by using *α* = 0 to achieve maximal node sizes and maximal specific fusion results.

The approach presented in (16) and Alg. 1 allows the transfer of the MCS-fusion rule (6) to distributed fusion topologies. This is an alternative to designing topologies based on (quasi-)associative fusion rules, which are rare in a possibilistic setting. An MCS-based topology is aimed at producing maximal specific and precise fusion subresults. However, distributed MCS-fusion lacks robustness in the case of nonrepresentative training data or defective sources, which is detailed in the next section.
