5.1.1. Fusion Rules

In the following, we evaluate whether the computational load of MCS and estimation fusion are decreased by distributing, i.e., whether each fusion node in a distributed topology has a lower load compared to a single centralised node. For MCS fusion, it is assumed that the MCS have already been found, i.e., only (6) is considered.

As (6) consists exclusively of minimum and maximum operations, centralised MCS fusion is O(*n*) with *n* being the number of input information sources. In a distributed twolayer fusion topology, each fusion node has *n*<sup>f</sup> ≤ *n* input sources. First layer nodes operate using renormalised minimum fusion; the final layer node applies maximum fusion. Fusion in each node is therefore O(*n*f). This simple observation shows that computational load of distributed nodes is less than in centralised fusion—for reasonable MCS fusion topologies.

For estimation fusion, the situation is not as simple. Estimation fusion, as defined in (11), (12), and (13), iterates over every *n*-tuple (*x*1, ... , *xn*). Thus, the computational load increases exponentially with its number of inputs *n*.

**Proposition 5.** *Let X*<sup>∗</sup> *be the frame of discernment with the highest cardinality in* {*X*1, ... , *Xn*}*, then the complexity of estimation fusion rule (11) is* O(|*X*∗| *<sup>n</sup>* · <sup>F</sup> +|*X*∗| *<sup>n</sup>* · <sup>G</sup> +|*X*∗| *<sup>n</sup>*)*. If* G *is the minimum operator and* F *is the arithmetic mean operator, then the complexity is* O(|*X*∗| *n*)*.*

**Proof.** Equation (11) is a combination of F, G, and the maximum operator. F and G need to be computed for each *n*-tuple (*x*1, ... , *xn*) for every *xi* ∈ *Xi*, i.e., F and G are computed ∏*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> |*Xi*| times. The maximum operator is computed for each *x* ∈ *X*. Its number of inputs is at worst ∏*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> |*Xi*|. In total, the complexity of (11) is

$$\begin{aligned} &\mathcal{O}\left(\prod\_{i=1}^{n}|X\_{i}|\cdot\mathcal{F} + \prod\_{i=1}^{n}|X\_{i}|\cdot\mathcal{G} + \sum\_{x\in X}\cdot\max\right)\right) \\ &=\mathcal{O}\left(\prod\_{i=1}^{n}|X\_{i}|\cdot\mathcal{F} + \prod\_{i=1}^{n}|X\_{i}|\cdot\mathcal{G} + \sum\_{x\in X}\prod\_{i=1}^{n}|X\_{i}|\right) \\ &=\mathcal{O}\left(\prod\_{i=1}^{n}|X\_{i}|\cdot\mathcal{F} + \prod\_{i=1}^{n}|X\_{i}|\cdot\mathcal{G} + \prod\_{i=1}^{n}|X\_{i}|\right) \end{aligned}$$

Let **<sup>X</sup>** <sup>=</sup> {*X*1,..., *Xn*} and *<sup>X</sup>*<sup>∗</sup> <sup>=</sup> arg max*X*∈**<sup>X</sup>** <sup>|</sup>*<sup>X</sup>* |, then

> =O(|*X*∗| *<sup>n</sup>* · <sup>F</sup> <sup>+</sup>|*X*∗| *<sup>n</sup>* · <sup>G</sup> <sup>+</sup>|*X*∗| *n*).

With G being the minimum operator and F being the arithmetic mean, this becomes

$$\begin{aligned} &\mathcal{O}(|X^\*|^n \cdot n + |X^\*|^n \cdot n + |X^\*|^n) \\ &= \mathcal{O}(|X^\*|^n \cdot n) \\ &= \mathcal{O}(|X^\*|^n). \end{aligned}$$

Therefore, the complexity of (12) relies on the complexities of G and F; however, it is safe to say that the growth |*X*∗| *<sup>n</sup>* leads to issues in practical implementations. Unfortunately, in this case, the lack of scalability cannot be solved by distributing the estimation fusion over several nodes.

**Proposition 6.** *Let* G *be the minimum operator and* F *be an averaging operator as defined in (13). Assume a topology of fusion nodes using estimation fusion (13) exclusively, then fusion at the final fusion node in the last layer still grows exponentially, that is, has* <sup>O</sup>(∏*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> |*Xi*|) *or* O(|*X*∗| *<sup>n</sup>*)*, respectively.*

**Proof.** Looking at a single fusion node with *nk* inputs, F maps in worst case each tuple (*x*1, ... , *xnk* ) to a unique point *x*. Then, the size of the output's frame of discernment is ∏*nk <sup>i</sup>*=<sup>1</sup> |*Xi*|. Let *fn*(*k*,*l*) be fusion nodes arranged in a topology so that the fusion topology outputs a single information item, i.e., there is a final fusion node *fn*(1,*L*), *L* ∈ N+. Assume all *n* available information items are input into a fusion node exactly once. Then, the final node has to process 2 ≤ *n*final ≤ *n* input information items. The number of tuples to iterate is then ∏*n*final *<sup>k</sup>*=<sup>1</sup> <sup>|</sup>*Xk*,*L*−1|. In a two-layer topology, <sup>∏</sup>*n*final *<sup>k</sup>*=<sup>1</sup> <sup>|</sup>*Xk*,*L*−1<sup>|</sup> <sup>=</sup> <sup>∏</sup>*n*final *<sup>k</sup>*=<sup>1</sup> <sup>∏</sup>*nk <sup>i</sup>*=<sup>1</sup> |*Xk*|. As *<sup>n</sup>*final <sup>=</sup> <sup>∑</sup>max(*k*) *<sup>k</sup>*=<sup>1</sup> *nk*, this is <sup>∏</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> |*Xi*|≤|*X*∗| *<sup>n</sup>*. Thus, fusion at the final node has O(|*X*∗| *n*).

For estimation fusion, the number of elements in the frame of discernment grows with each fusion node. The final fusion node has to process in worst case |*X*∗| *<sup>n</sup>* tuples, which is the same for centralised fusion.

Yager demonstrated [65] that, if all *π<sup>i</sup>* are convex and if *X* contains only real-valued ordered elements, then (13) (that is G = min and F is an averaging operator) can also be computed via the crisp-set *α*-cuts

$$A\_i^a = \{ \mathbf{x} \in X\_i : \pi\_i(\mathbf{x}) \ge a \} \text{ with } a \in (0, 1]. \tag{28}$$

**Definition 11.** *A possibility distribution π is said to be convex iff (1) each of its α-cuts A<sup>α</sup> are a single closed interval, i.e., A<sup>α</sup>* <sup>=</sup> [*a*, *<sup>b</sup>*] *and (2) all A<sup>α</sup> are nested, i.e.,* <sup>∀</sup>*α*<sup>1</sup> <sup>&</sup>gt; *<sup>α</sup>*<sup>2</sup> : *<sup>A</sup>α*<sup>1</sup> <sup>⊆</sup> *<sup>A</sup>α*<sup>2</sup> *.*

For each *α*-level the crisp sets *A<sup>α</sup> <sup>i</sup>* are fused using the averaging operator F, which results in

$$\begin{aligned} A^{\mathbf{f}\mathbf{u}-\mathbf{a}} &= \, \mathrm{F}(A\_1^a, \dots, A\_n^a) \\ &= \left[ \mathrm{F} \left( \min\_{x \in A\_1^a} \mathbf{x}, \dots, \min\_{x \in A\_n^a} x \right), \mathrm{F} \left( \max\_{x \in A\_1^a} \mathbf{x}, \dots, \max\_{x \in A\_n^a} x \right) \right]. \end{aligned} \tag{29}$$

The fused possibility distribution is then obtained by taking the maximum *α*-level as follows:

$$\pi^{(\text{fu})}(\mathbf{x}) = \max\_{\mathbf{x}} \begin{cases} \mathfrak{a} & \text{if } \mathbf{x} \in A^{\text{fu}-\mathfrak{a}}, \\ 0 & \text{if } \mathbf{x} \notin A^{\text{fu}-\mathfrak{a}}. \end{cases} \tag{30}$$

**Proposition 7.** *The computational load of (28)–(30) grows linearly in number of input possibility distributions n, number of elements in X*∗*, and number of α-levels nα, i.e., (28)–(30) have in total* O(*n* · |*X*∗| · *nα*)*.*

**Proof.** Equation (28) grows linearly in |*Xi*|. It has to be for each *α*-level and each input possibility distribution, i.e., (28) is O(*n* · |*X*∗| · *nα*).

For (29), both minimum and maximum have to be computed *n* times, F has to be computed two times. This has to be performed for each *α*-level. This results in

$$\begin{aligned} &\mathcal{O}(n\_{\boldsymbol{\alpha}} \cdot 2 \cdot (\boldsymbol{F} + \boldsymbol{n} \cdot \min + \boldsymbol{n} \cdot \max)) \\ &= \mathcal{O}(n\_{\boldsymbol{\alpha}} \cdot (\boldsymbol{n} + \boldsymbol{n} \cdot |\boldsymbol{X} \ast| + \boldsymbol{n} \cdot |\boldsymbol{X}^\*|)) \\ &= \mathcal{O}(n\_{\boldsymbol{\alpha}} \cdot \boldsymbol{n} \cdot |\boldsymbol{X}^\*|). \end{aligned}$$

Equation (28) is a single maximum with *n<sup>α</sup>* inputs, i.e., it is O(*nα*). In total, (28)–(30) is O(*n<sup>α</sup>* · *n* · |*X*∗|).

In contrast to (13), the computational load is distributed over fusion nodes if (28)–(30) are distributed. Using *α*-cuts, neither |*X*| nor *n<sup>α</sup>* grow with each fusion node. Rather, they stay constant. Consequently, increasing the number of fusion nodes in a topology—which decreases the number of inputs per fusion node—reduces the computational load per node. In conclusion, both estimation fusion as well as MCS fusion profit from reduced computational load per node if fusion is distributed.

## 5.1.2. Fusion Topology Algorithms

Using (16) naively to search all possible subsets of a set of information sources **S** for fusion nodes is computationally demanding. Such an approach grows exponentially in number of sources *n*. The proposed Algorithm 1 presents a computational faster approach.

**Proposition 8.** *The Algorithm 1 for finding consistency-based fusion nodes has complexity* O- *<sup>m</sup>* · *<sup>n</sup>*<sup>2</sup> *with n* = |**S**| *and m being the number of training data instances.*

**Proof.** Algorithm 1 iterates over all training data instances *j*. For *j* = 1, it searches **S** for all MCS. As the algorithm of [58,61] grows linearly in *n*, this step is O(*n*). For each subsequent iteration with *j* > 1, it searches all previously MCS found at *j* − 1 again for MCS. The maximum number of found MCS is *n*. The maximum number of sources belonging to an MCS is also *n*, i.e., each iteration at *j* > 1 grows with *n*2. Consequently, Algorithm 1 is O- *<sup>m</sup>* · *<sup>n</sup>*<sup>2</sup> .

The redundancy-based Algorithm 2 takes the fusion nodes found by Algorithm 1 as input. If an MCS does not meet the redundancy criterion, then Algorithm 2 searches within each MCS for largest subsets with *ρ* ≥ *η*.

**Proposition 9.** *The Algorithm <sup>2</sup> for finding redundancy-based fusion nodes has complexity* O(2*n*) *with n* = |**S**|*.*

**Proof.** Algorithm 2 searches the power set of each MCS **S**MCS−*<sup>α</sup>* (*k*) . As the maximum number of sources in **S**MCS−*<sup>α</sup>* (*k*) is *<sup>n</sup>*, Algorithm <sup>2</sup> is <sup>O</sup>(2*n*).

In contrast to the consistency-based algorithm, the redundancy-based version scales in its current implementation poorly with number of sources. For reasons of practical implementation, this needs to be addressed in future works. In this regard, plausibility checks are promising as to whether subsets of **S**MCS−*<sup>α</sup>* (*k*) can actually exhibit the required range. In such cases, it would not make sense to search these subsets at all, saving computational time.
