4.3.2. Discounting Defective Sources

Information items that deviate from the expected level of consistency *α*<sup>r</sup> (17) are seen as unreliable and, consequently, are discounted in each fusion node. Therefore, the degree of reliability *rel* ∈ [0, 1] is determined with regard to *<sup>α</sup>*<sup>r</sup> . Let **I** be information items fused in a node and **<sup>I</sup>**<sup>∗</sup> be the largest subset in **<sup>I</sup>**, which has (i) h(**I**∗) ≥ *<sup>α</sup>*<sup>r</sup> and (ii) |**I**∗| > 1; then,

$$rel(I) = \begin{cases} 1 & \text{if } \operatorname{h}(I, \mathbf{I}^\*) > a^\mathbf{r}, \\ \frac{\operatorname{h}(I, \mathbf{I}^\*)}{a^\mathbf{r}} & \text{if } \operatorname{h}(I, \mathbf{I}^\*) \le a^\mathbf{r}. \end{cases} \tag{23}$$

In the case that there is no unique **<sup>I</sup>**<sup>∗</sup> with <sup>h</sup>(**I**∗) ≥ *<sup>α</sup>*<sup>r</sup> and at least two elements, then all items are seen as fully reliable, and fusion needs to switch to disjunctive fusion.

Information items' possibility distributions are modified prior to fusion so that they have a lesser effect on the fusion results [4,75]. A modification function for discounting information items has to satisfy the following requirements (extended from previous work [39]).

**Definition 10** (Requirements for Information Item Modification)**.** *As modification aims at changing fusion outputs, the requirements interact with fusion rules to be applied on π:*


Modification functions were proposed by Yager and Kelman [75]

$$
\pi'(\mathfrak{x}) = \operatorname{rel} \cdot \pi(\mathfrak{x}) + 1 - \operatorname{rel}\_r
$$

and Dubois and Prade [4]

$$\pi'(\mathfrak{x}) = \max\_{\mathfrak{x} \in \mathfrak{X}} (\pi(\mathfrak{x}), 1 - n!).$$

Both satisfy the requirements for modification only for conjunctive fusion. A general modification function for the use with OWA operators was proposed by Larsen [68]. It is defined based on the *andness* degree *and* ∈ [0, 1] of OWA fusion:

$$
\pi'(\mathbf{x}) = \textit{and} + \textit{rel} \cdot (\pi(\mathbf{x}) - \textit{and}).\tag{24}
$$

The OWA operator results in the minimum fusion for *and* = 1 and in maximum fusion for *and* = 0. The OWA modification (24) introduces a global possibility level of *and* to the distribution *π* . As of this, the modification satisfies the requirement of *neutral element* only if *and* = 1 or *and* = 0 but not for 0 < *and* < 1.

All three modification functions raise the overall possibility level globally. As argued in previous work [39], this kind of approach towards modification functions is counterintuitive if it is considered that defective or unreliable sources may err in their estimation of the unknown value *v*. An unreliable source may be slightly incorrect. Raising the possibility level globally cannot model such a situation. A modification function that widens or shrinks the possibility distribution is proposed as (adapted from previous work [39]):

$$\begin{aligned} \pi'(\mathbf{x}) &= \begin{cases} \max\_{\mathbf{x}' \in \mathcal{C}} (\pi(\mathbf{x}')) & \text{if minimum fusion,} \\ \min\_{\mathbf{x}' \in \mathcal{C}} (\pi(\mathbf{x}')) & \text{if maximum fusion,} \end{cases} \\ \text{C} &= \left[ \mathbf{x} - (1 - rel)^{\beta} \cdot (\mathbf{x}\_{\mathbf{b}} - \mathbf{x}\_{\mathbf{a}}), \mathbf{x} + (1 - rel)^{\beta} \cdot (\mathbf{x}\_{\mathbf{b}} - \mathbf{x}\_{\mathbf{a}}) \right], \text{and} \\ &X = \left[ \mathbf{x}\_{\mathbf{b}}, \mathbf{x}\_{\mathbf{b}} \right]. \end{aligned} \tag{25}$$

This modification considers both minimum and maximum fusion as they occur in the MCS-based fusion topology but does not approach a global modification. The reliability *rel* and the control parameter *β* ∈ R≥<sup>1</sup> define a vicinity around *x*. The new possibility *π* (*x*) is taken from this vicinity. This creates a widening or shrinking effect, respectively. The parameter *β* allows to control the size of the vicinity and, thus, the extent to which *rel* alters *π*(*x*). The larger *β* is, the less effect *rel* has on *π*(*x*). If *rel* > 0 and *β* → ∞, then (25) has no widening or shrinking effect.

#### 4.3.3. Estimation-Based Fusion Nodes

The third adaptation to increase the robustness of the proposed MCS-based fusion topology is to replace fusion in the first layer (15) with estimation fusion (13). In this way, defective sources have a lesser impact on the fusion result of a node.

Associativity needs to hold for first layer fusion nodes (see Figure 4) if multi-level fusion is to be achieved (splitting fusion nodes into smaller ones). Estimation fusion is only associative if G is associative and monotonic increasing and F is associative. In the proposed estimation-based fusion nodes, G is the minimum operator that satisfies associativity and monotonicity. The function F is defined to be an averaging operator, which is rarely associative, e.g., the arithmetic mean. Multi-level distributed fusion can still be achieved by using a fusion node's ability to output auxiliary information (see Definition 4).

If a node outputs the number of information items that contributed to its fusion result as a weight *w*, then a weighted arithmetic mean operator of the form

$$F^{\mathsf{WAM}}(\mathbf{x}\_1, \dots, \mathbf{x}\_n) = \frac{\sum\_{i=1}^n w\_i \cdot \mathbf{x}\_i}{\sum\_{i=1}^n w\_i}$$

results in associative fusion. In the following, we refer back to the notation of fusion nodes as defined in Definition 4, i.e., **I***k*,*<sup>l</sup>* denotes the set of information items that serve as input to fusion node *fn*(*k*,*l*). To achieve associativity, a weight *w*(*k*,*l*) is assigned to the output of *fn*(*k*,*l*), which is defined as

$$\begin{aligned} w\_{(k,l)} &= \sum\_{I\_{(o,p)} \in \mathbf{I}\_{(k,l)}} w\_{(o,p)} \text{ with},\\ w\_{(k,1)} &= |\mathbf{I}\_{(k,1)}|.\end{aligned}$$

The distributed weighted average function

$$F\_{\left(k,l\right)}^{\text{WAM}}\left(\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}\right) = \sum\_{\substack{I\_{\left(o,p\right)} \in \mathbf{I}\_{\left(k,l\right)} \atop \left(o,p\right)}} \frac{1}{w\_{\left(o,p\right)}} \sum\_{\substack{I\_{\left(o,p\right)} \in \mathbf{I}\_{\left(k,l\right)}}} w\_{\left(o,p\right)} \cdot F\_{\left(o,p\right)}^{\text{WAM}}\left(\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}\right)$$
 
$$\text{with } F\_{\left(k,1\right)}^{\text{WAM}}\left(\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}\right) = \frac{1}{w\_{\left(k,1\right)}} \sum\_{i=1}^{\left|I\_{\left(k,1\right)}\right|} \mathbf{x}\_{i}$$

allows splitting nodes without changing the fusion result. An overview of a distributed fusion topology based on estimation fusion rules is given in Figure 6.

**Figure 6.** Example of an MCS-based fusion topology adapted with weighted estimation fusion. Previously conjunctive fusion nodes (first level fusion; see also Figure 4) are replaced with the estimation fusion rule described by (13). To preserve the associativity of the first level fusion nodes, the weighted averaging operator *F*WAM described by (26) is applied as function F. If the estimationbased nodes are split into a multi-level topology, then *F*WAM requires fusion nodes to communicate the number of input information items.

To keep the option of discounting defective sources, weights *w*(*k*,*l*) are modified in the case a defect is detected via (23) as follows:

$$w\_{k,l}' = \; w\_{k,l} \cdot rel \left( I\_{(k),l} \right) . \tag{27}$$

If *rel I*(*k*),*<sup>l</sup>* <sup>=</sup> 1, then information is preserved. Otherwise, if *rel I*(*k*),*<sup>l</sup>* = 0 the information item is completely discounted.

#### *4.4. Remark on Multi-Level Fusion by Splitting Nodes*

The MCS-based design approach describes a two-layer fusion topology by first fusing consistent or redundant information items conjunctively and then fusing the intermediate results disjunctively. In this context, multi-layer fusion can be achieved by splitting a single fusion node into multiple smaller ones. This may be beneficial if, e.g., communication or computational loads per node need to be optimised. While this approach of splitting is feasible due to the associativity of applied fusion rules, the ability of the fusion topology to detect and discount defective sources is reduced by doing so.

Discounting information items requires finding the unique largest subset of items whose consistency is greater than *α*<sup>r</sup> . If multiple sources are defective simultaneously, then—depending on the fusion node size—the largest subset may be made up by defective sources. In the worst case, the maximum number of defective sources a fusion node can handle is *<sup>n</sup>*−<sup>1</sup> <sup>2</sup> [24], with *n* being the number of sources contributing to a fusion node. As the proposed discounting approach is node-specific, the ability of a node to discount defective sources is hindered by splitting nodes. The smaller *n* is, the smaller is the maximum number of detectable defective sources. This hast to be kept in mind in designing an MCS-based fusion topology.

#### **5. Evaluation**

The evaluation is structured into three parts in which the computational complexity, topology design approaches, and the robustness of distributed MCS fusion are focused. Distributing information fusion is motivated—as outlined in Section 1—by the assumption that computational load per distributed node is less than the load for a single centralised node. First, this assumption is examined for MCS and estimation fusion.

Subsequently, the computational complexity of design Algorithms 1 and 2 are discussed. Their performance and the effectiveness of the MCS-fusion adaptations (see Section 4.3) are then evaluated on selected real-world datasets.

## *5.1. Computational Complexity*

The following evaluation of computational time complexity relies on the Bachmann–Landau notation *f*(*n*) = O(*g*(*n*)), which states that a function *f*(*n*) does not grow faster for *n* → ∞ than *g*(*n*). *f*(*n*) is therefore asymptotically upper bounded by *g*(*n*). O(*g*(*n*)) denotes the set of all *f*(*n*) such that there exist positive constants *c* and *n*0: *f*(*n*) ≤ *c* · *g*(*n*), ∀*n* ≥ *n*<sup>0</sup> [76].
