**1. Introduction**

The discipline of information fusion is concerned with the aggregation of uncertain information from several sources. Through the process of fusion, uncertainty is to be reduced, that is, information fusion aims at creating information of higher quality [1].

Uncertainty and ignorance manifest in many forms, such as a lack of confidence, aleatoric uncertainty, or epistemic uncertainty. A comprehensive taxonomy of ignorance is provided by Ayyub and Klir [2]. Uncertain information are modelled in various mathematical frameworks, especially probability theory, Dempster–Shafer theory, fuzzy set theory, and possibility theory [3], and each has strengths and weaknesses with regard to types of uncertainty. Possibilistic information fusion is focused on handling epistemic uncertainty, imprecise information, and incomplete information [4,5], which stem from, e.g., scarce data, repetitive data, or biased data. In possibilistic information fusion, knowledge about the state of affairs is complemented by excluding alternatives, which single information sources deem impossible.

In the following, this paper relies on the nomenclature of information items and information sources adopted from [6].

**Definition 1** (**Information Item**)**.** *Consider an unknown entity v and a non-empty set of possible alternatives X*<sup>A</sup> = {*x*1, ... , *xn*} *with n* ∈ N>0*. An information item models information in the form of plausibilities or probabilities about v regarding X*A*. An information item can be, e.g., a set, an interval, a probability distribution, or a possibility distribution. Consequently, an item may be expressed with certainty (v* = *x or, assuming A* ⊂ *X*A*, v* ∈ *A), may be affected by uncertainty (v is probably x or v is possibly x), or may be expressed imprecisely (x*<sup>1</sup> < *vs*. < *x*2*).*

**Citation:** Holst, C.-A.; Lohweg, V. Designing Possibilistic Information Fusion—The Importance of Associativity, Consistency, and Redundancy. *Metrology* **2022**, *2*, 180–215. https://doi.org/10.3390/ metrology2020012

Academic Editor: Simona Salicone

Received: 17 March 2022 Accepted: 6 April 2022 Published: 11 April 2022

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**Definition 2** (**Information Source**)**.** *An information source S provides information items. It is an ordered concatenation of information items S* = {*I*1, *I*2, ... , *Im*} *with m* ∈ N>0*. Each Ij represents an information item at instance j* ∈ {1, ... , *m*}*. An information source may be, for example, a technical sensor, a variable, a feature, or a human expert.*

Often information fusion benefits from distributing the fusion into a multi-step piecewise process [7–10]. This means, for example, that information items are fused sequentially, in parallel, or hierarchically instead of centralised all at once. The sequence in which items are fused is often referred to as the topology or architecture. While the term architecture is often used in a broader sense to refer to complete fusion frameworks (see [11–14]), the term topology is used in this paper to describe the structure in which the fusion is arranged. Example fusion topologies are shown in Figure 1.

Designing and optimising a fusion topology is one of the main challenges in implementing an information fusion system [15]. An optimal topology reduces communicational and computational loads, increases fusion accuracy [16], and helps to detect defective sources [17]. Fusion topologies are usually designed manually, as e.g., in the dissertation of Mönks [18] or require meta-knowledge about information sources, such as in the work of Fritze et al. [19]. Automated learning processes are rare. Such a learning process is made more difficult by epistemic uncertainty due to, e.g., missing or underrepresented classes in training data or due to having few training data instances to begin with. This calls for approaches of learning topologies based on possibility theory.

**Figure 1.** Three example information fusion topologies. (**a**) Centralised fusion, (**b**) serial fusion, and (**c**) hierarchical fusion.

Key characteristics for designing fusion topologies are the associativity of fusion rules, consistency of information items, and redundancy of information sources. Associativity allows the optimisation of a topology towards, e.g., computational load or other criteria without having to worry about distorting the fusion result. Associativity is especially crucial if a specific topology is necessitated by an application.

Information may not be available at the same time or information sources may be spatially distributed so that a centralised fusion is simply not feasible. Structuring fusion based on consistency or redundancy was proposed quite early [17,20]. The basic idea is to fuse consistent or redundant information in earlier stages and complementary information in later stages. Grouping sources in this way provides the benefits that (i) it is reasonable to conduct fusion conjunctively resulting in maximal certain information [6] and (ii) it is easier to identify defective or malfunctioning sources increasing the robustness of applications [21–24].

In this article, we contribute an approach towards a data-driven automated learning of information fusion topologies. The article focuses on information modelled within the possibility theory. As a foundation, common possibilistic fusion rules are recapitulated and analysed regarding the associativity property. Based on this analysis, design algorithms relying on consistency and redundancy are proposed and discussed. The aim of the design algorithms is to build topologies that result in maximal specific (i.e., minimal uncertain) fusion outcomes and that facilitate source defect detection. The proposed learning algorithm approaches are discussed with regard to their robustness and further improved by exploiting outlier resistant averaging possibilistic fusion rules. As a first step in this article, an overview of the state of the art in fusion topology design is given independent of the mathematical framework.

#### **2. Fusion Topology Design in Related Work**

Information fusion systems are composed of various interacting parts and methodologies, such as information sources, information pre-processing, fusion nodes, mathematical frameworks, or fusion algorithms. This results in in high-dimensional design spaces, i.e., a large amount of hyperparameters. Deciding on and designing the topology is an important subtask in fusion system design as identified by Raz et al. [16]. The authors explored the design space of a relatively simple fusion task (still > <sup>2</sup> × 105 design combinations) with the help of machine-learning algorithms. Their goal was to estimate the impact of design choices on the performance of the fusion system. Among other design parameters, the topology and allocation of sources to fusion nodes were identified to be crucial to the performance. This motivated ongoing work on topology design.

A widely used approach towards designing topologies and allocating information sources is to rely on meta-knowledge about the information sources. Mönks et al. [18,25] grouped information sources (here: technical sensors) into a two-level fusion topology based on the sensor's observed objects, measured physical property, or spatial location. Semantically close (e.g., observing the same object) or spatially close sensors are assumed to be at least partly-redundant and are allocated to the same fusion node. This manual approach has been partly automated by Fritze et al. [19,26,27] who equipped sensors with a self-description containing information about the sensor's characteristics, its contextual environment, and observed objects. A rule-based system then matches and groups sensors based on their self-description. Other ontology-based approaches have been proposed by Boury-Brisset [28] and Martí et al. [29]. Both do not focus on topology design specifically but rather on designing or facilitating a fusion system. Boury-Brisset [28] discussed ontological methods for the integration in the Joint Directors of Laboratories (JDL) fusion architecture [30] including the semantic integration of information. Martí et al. [29] proposed an ontology-based adaptive sensor fusion architecture, and this architecture organises sensors and external sources into preprocessing nodes and fusion nodes depending on the task at hand. A recent application of ontology-based design of information fusion systems can be found in the field of assisted living [31]. Ontological approaches reduce the manual effort needed for structuring fusion topologies; however, they still require profound expert knowledge about the information sources and their context. Building the ontology requires manual engineering and is time-consuming [28].

Designing information fusion topologies is closely related to the *data association* step predominately but not exclusively used in the JDL fusion architecture. Solaiman and Bossé [32] refer to the task of data association as the identification of any relation between information elements and monitored objects. Waltz and Llinas [33] defined the data association problem with regard to fusion systems more specifically as the "Cross correlation of measurements and *m*-ary decisions to partition all measurements into sets of common origin. One can distinguish between associating a set of measurements (partitioning) and associating a measurement (or a set of measurements) to a given object. [. . . ]".

In this definition, the partitioning of measurements refers to preparing a fusion task in which each partition represents the input to a fusion node; hence, the relation to designing fusion topologies. Data-driven approaches for data association are given by Grabisch and Prade [34] and Ayoun and Smets [35]. Both approaches cluster sensor measurements based on quantifications of the measurements' proximities. Grabisch and Prade [34] modelled information within the possibility theory and computed the proximity based on the degree of intersection of possibility distributions. Ayoun and Smets [35] used Dempster–Shafer theory instead and clustered based on the degree of conflict between measurements. A similar approach was taken by Schubert [36,37]—although not explicitly labelled as data association—who clustered basic belief functions (evidential masses) based on their conflict and attraction with each other. All of these works ([34–37]) partition information sources based on single instances of measurements (the current measurement) and not on historical data. More sophisticated interdependencies and interrelations between information sources can only be detected robustly in historical data. For example, for the identification and quantification of redundancies between sources, meaningful data are necessary, which spans over the sources' frame of discernment as shown by Holst and Lohweg [38,39].

Regarding the problem of data association, it has to be mentioned that more recent publications focus solely on the specific application task of visual target tracking (see for example the works of Kamal et al. and Yoon et al. [40,41]). This focus comes with a shift in interpretation of the data association problem as shown by the definition given by Khaleghi et al. [42]: "[. . . ] the data association problem, which may come in two forms: measurement-to-track and track-to-track association. The former refers to the problem of identifying from which target, if any, each measurement is originated, while the latter deals with distinguishing and combining tracks, [. . . ]". Publications with this shifted focus are less related to the problem of designing fusion topologies.

In summary, in related works, the task of structuring fusion topologies has been approached based on expert knowledge, ontologies, or based on current measurements. Approaches that consequently analyse historical data or information in order to derive a fusion topology are missing. While this section considered topology design independently from the mathematical fusion framework, the remainder of this paper focuses on possibility theory.
