Designing Possibilistic Information Fusion—The Importance of Associativity, Consistency, and Redundancy
Abstract
:1. Introduction
2. Fusion Topology Design in Related Work
3. Fusion within Possibility Theory
- Possibilistic Pooling Fusion has mainly been advanced by Dubois et al. [4,48]. The aim of possibilistic pooling is to find the possibility degree for each alternative x. Hence, operators work on the grades of possibilities (by applying fuzzy norms). Inside this framework, the choice of fusion rules is most often based on the state of knowledge about the reliability of the information sources involved. Depending on reliability and available knowledge, fusion operators are distinguished into conjunctive, disjunctive, and trade-off modes [32].
- Possibilistic Estimation Fusion was mainly devised and advanced by Yager [54]. In contrast to pooling, estimation operators are based on Zadeh’s extension principle [55], which defines the use of mappings to fuzzy inputs. The goal of estimation concerns itself with finding the result that is the most compatible with all information items. Operators apply averaging functions on the frame of discernment X.
- Majority-guided Fusion identifies majority subsets—often based on consistency measures—and aggregates information from these subsets either exclusively or prioritised—similar to a voting procedure. Majority-guided fusion deliberately violates the fairness principle. It finds application in situations in which it is explicitly known that sources produce consistent readings, e.g., in redundantly engineered technical sensor systems [23]. The operators for majority-guided fusion are often based upon either pooling or fuzzy estimation as is shown in detail in the following.
3.1. Possibilistic Pooling Fusion
3.2. Possibilistic Estimation Fusion
3.3. Majority-Guided Fusion
4. Approach towards Topology Design
- Modularity: A fusion node outputs a fused information item, which qualifies as a possibility distribution π (see Section 3), i.e., π is normal. This property allows self-contained intermediate results in a topology and makes fusion nodes modular. This increases the transparency of the distributed fusion topology.
- Self-Reproducing: Given a single input, a fusion node reproduces this input. It preserves its identity, i.e., .
4.1. Associativity
4.1.1. Pooling Fusion
4.1.2. Estimation and Majority-Guided Fusion
4.2. MCS-Based Topology Design
Algorithm 1: Fast algorithm for finding subsets of information sources, which are consistent at least to degree on every instance of training data. Each subset is assigned to fusion node . The algorithm relies on finding MCS of information items as defined by Dubois et al. [58,61]. |
4.3. Robustness
- First, incomplete information and epistemic uncertainty in the training data may lead to assessing a group of sources as consistent prematurely. Information sources may produce different (in)consistent behaviours depending on the training data’s true value and its position on the frame of discernment. Take, for example, a condition-monitoring scenario of a technical system in which sensors state the condition on a discrete frame of discernment . Two sensors may both detect two of the conditions (e.g., error1, normal); however, only one is able to detect the third condition (error2). If training data does not include data regarding error2, then with Algorithm 1, both sensors are falsely identified as consistent and grouped into a fusion node. If error2 occurs later, then the sensors behave unexpectedly inconsistently. This problem relates to spurious correlations in probability theory [70], which describes that, in large datasets, it is particularly likely that correlations are found between variables incorrectly.
- Second, defective sources are a cause of unexpected inconsistent behaviour. Defective sources are sources that are trustworthy and therefore have a high reliability but nonetheless start to supply incorrect information [71]. Source defects appear in different forms: Information can change suddenly, drift continuously or incrementally, or can be characterised by an increasing number of outliers [72,73]. Countermeasures are majority-guided fusion rules as applied by Ehlenbröker et al. and Holst and Lohweg [21,23]. This requires redundant and reliable sources in a fusion node.
- Redundancy-Driven Topology Design: To counteract non-representative training data, it must be ensured that information sources are not prematurely deemed to be consistent. For this, it must be analysed whether the consistent behaviour between sources extends over the entire frame of discernment. Therefore, instead of the consistency metric used in (16), the redundancy metric originally proposed in previous works [38,39] is adopted, which ensures that the complete frame of discernment is considered.
- Discounting Defective Sources: Grouping the information sources by consistency (or redundancy) eases the detection of defects [23,24]. Items detected as defective are discounted in the fusion node so that they have less influence on the output of the node. This requires an adjustment of the fusion rule (previously minimum or maximum operator) in the nodes. This defect detection step explicitly exploits the distributed topology to its advantage. This deliberately dismisses the associativity of the overall fusion.
- Estimation-fusion-based Nodes: Averaging information is a natural way to favour opinions of the majority. Adopting estimation fusion in nodes results in more robust behaviour against defects—such as outliers—compared to purely conjunctive fusion as applied in (6).
4.3.1. Redundancy-Driven Topology Design
- Upper bound: If , then and .
- Lower bound: if , i.e., all possibility distributions are identical.
- Boundaries: A redundancy metric should be able to model complete redundancy and complete non-redundancy. It follows that ρ is minimally and maximally bounded. It is proposed that .
- Identity relation: An information source is fully redundant with identical copies of itself: . Note that sources can be redundant without necessarily being identical.
- Symmetry: The metric ρ is a symmetric function in all its arguments, i.e.,for any permutation p on .
- If information sources are redundant, then they provide redundant information items. Consequently, increases as the redundancy of information items increase.
- Redundant information items do no necessitate that their information sources are also redundant. Due to cases of incomplete information, redundant information items may be a case of spurious redundancy (similar to spurious correlation).
Algorithm 2: Algorithm that searches for redundancy-based fusion nodes based on found by Algorithm 1. The algorithm iterates over and searches all , for sets meeting the redundancy criterion . |
4.3.2. Discounting Defective Sources
- Information preservation: If , then the information must not be changed but instead preserved. Let be a modified possibility distribution based on π. If , then .
- Neutral element: If , then I needs to have no effect on the fusion. The item I needs to act as a neutral element on fusion operator , i.e., .
- Monotonicity: For increasing , I needs to have a monotonic increasing effect on .
4.3.3. Estimation-Based Fusion Nodes
4.4. Remark on Multi-Level Fusion by Splitting Nodes
5. Evaluation
5.1. Computational Complexity
5.1.1. Fusion Rules
5.1.2. Fusion Topology Algorithms
5.2. Robustness
5.2.1. Data Preprocessing
- If data are singular values or probability distributions, they are transferred into possibility distributions first. For this step, singular values x are interpreted as probability distributions with and . Transformation is conducted by the truncated triangular probability-possibility transformation [49,77,78] resulting in .
- Second, sources providing noisy data are regarded as partially unreliable. Their possibility distribution are modified using (25) accordingly. Unreliability values for information sources are determined heuristically.
- Third, modified possibility distributions are mapped to a common, shared frame of discernment. This X is based on fuzzy memberships , i.e., . This requires a fuzzy class to be defined to which indicates the degree of membership of x. The class membership function can either be provided by an expert or trained automatically [18,38,39]. Here, is trained by a parametric unimodal potential function proposed proposed by Lohweg et al. [79]:The possibility distribution is then mapped to via the extension principle as follows:
5.2.2. Nonrepresentative Training Data
- For the consistency-based approach, fusion nodes trained on complete data are expected to be smaller or of equal size compared with nodes trained on reduced data. More specifically, because (16) requires consistencies of all data instances to be above the threshold .
- Sources grouped by the redundancy-based approach are expected to always be a subset of at least one consistency-based found group , i.e., because the redundancy metric (18) is more restrictive than pure consistency. The additional range information (22) prevents sources being added to a fusion node when it is not known that they behave consistently over the complete frame of discernment.
5.2.3. Defective Sources
- with ,
- with , and
- with .
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. Specificity as a Measure of Information Content
- 1.
- in the case of total ignorance, i.e., .
- 2.
- in the case of complete knowledge, i.e., only one unique event is totally possible and all other events are impossible.
- 3.
- A specificity measure de- and increases with the maximum value of , i.e., let be the k-th largest possibility degree in , then .
- 4.
- , i.e., the specificity decreases as the possibilities of other values approach the maximum value of .
Appendix B. Proofs of (Non-)Associativity of Fusion Rules
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Fusion Rule | Equation(s) | Associative | Proof of Associativity | Quasi-Associative | Proof of Quasi-Associativity |
---|---|---|---|---|---|
Conjunctive | (3) | yes | Inherited from t-norm | yes | See Proposition 2 |
Renormalised Conjunctive | (4) | Dependent on t-norm | Proof for nonassociativity in the case of minimum-norm and associativity in the case of product-norm given by Dubois and Prade [47] | yes | and |
Disjunctive | (5) | yes | Inherited from s-norm | yes | See Proposition 2 |
MCS fusion | (6) | no | [61] | no | [61] |
Quantified | (7) | no | Proof given in Appendix B | no | Similar to MCS fusion |
Adaptive | (8), (9) | no | [69] | no | [69] |
Progressive | (9), (10) | no | Inherited from adaptive fusion | no | Inherited from adaptive fusion |
Estimation | (13) | yes (with restrictions) | See Proposition 3 | yes (with restrictions) | See Propositions 2 and 3 |
MOGPFR | (14) | no | Proof given in Appendix B | no | OWA operator prevents quasi-associativity |
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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
Holst C-A, Lohweg V. Designing Possibilistic Information Fusion—The Importance of Associativity, Consistency, and Redundancy. Metrology. 2022; 2(2):180-215. https://doi.org/10.3390/metrology2020012
Chicago/Turabian StyleHolst, Christoph-Alexander, and Volker Lohweg. 2022. "Designing Possibilistic Information Fusion—The Importance of Associativity, Consistency, and Redundancy" Metrology 2, no. 2: 180-215. https://doi.org/10.3390/metrology2020012