A Redundancy Metric Set within Possibility Theory for Multi-Sensor Systems †
- At the level of single sensor measurements, lack of information, e.g., about the sensor’s detailed characteristics, tolerances, or physical limits, results in imprecise readings. Thus, a sensor is only able to give an approximate measurement. As a result of this, information is often provided in intervals, fuzzy intervals, or uncertainty distributions (either probabilistic or possibilistic) .
- Furthermore, during training, the monitored process may only by observable in specific states. For example, a production machine may create a lot of training data, but these data often originate from the same machine state, that is, data about states of failure are rare. This leads to ambiguous and fuzzy classes  as well as premature detection of interrelations (such as redundancy) between sensors. The risk of detecting spurious correlations  is greatly amplified in intelligent technical or cyber–physical systems. Two examples of premature detection of variable interrelation are shown in Figure 1.
2. Redundancy in Related Work
3. Possibility Theory
3.1. Basics of Possibility Theory
3.2. Possibility Theory in Comparison to Probability Theory
- The application of PosT does not require statistical data to be available. Consequently, it is easier and takes less effort to construct sound possibility distributions than probability distributions (cf.  for methods to construct possibility distributions).
- In contrast to ProbT, both imprecision and confidence can be modelled distinctly within a possibility distribution. Imprecision is modeled by allowing multiple alternatives to be possible, e.g., it may be known that , but not which value v takes within A precisely. Confidence is expressed by the degree of possibility assigned to a value x, i.e., if , it is uncertain if is fully possible. It follows directly that confidence is also represented in the duality measure of and N as can be seen in the three extreme epistemic situations : (i) if is certain, and , (ii) if is certain, and , and (iii) in case of ignorance, and .
3.3. Fusion within Possibility Theory
- Conjunctive fusion modes implement the principle of minimal specificity most strongly. By applying a triangular norm (t-norm),
- In case of fully inconsistent possibility distributions at least one information source must be unreliable. Assuming it is not known which source is unreliable, disjunctive fusion modes apply s-norms so that as much information is kept as possible:Disjunctive fusion is generally not desirable because the fusion does not result in more specific information.
- Adaptive fusion modes combine conjunctive and disjunctive fusion methods. These modes switch from conjunctive to disjunctive aggregation depending on which of the alternatives the sources are inconsistent for. An adaptive fusion mode, proposed in , isThus, fusion results in a global level of conflict () for all alternatives the sources cannot agree on. Otherwise the adaptive fusion reinforces by conjunction.
- A majority-guided fusion searches for the alternatives which are supported by most sources. This is similar to a voting style consensus. Majority-guided fusion requires the identification of a majority subset—usually the subset with highest consistency and maximum number of sources. The possibility distributions of this subset are fused conjunctively. Information outside of the majority subset is discarded which violates the fairness principle postulated in . Applications of majority-guided fusion can be found in previous works of the authors of this contribution [6,7].
4. Quantifying Redundancy within the Possibility Theory
4.1. Redundant Information Items
- Boundaries: Information items can be minimally and maximally redundant. Therefore, is minimally and maximally bounded: .
- Inclusion (Upper Bound): An information item is fully redundant in relation to if it encloses (includes) .
- Lower Bound: An information item is non-redundant if it adds new information. Additionally, an item is fully non-redundant in relation to if and disagree completely on the state of affairs, i.e., in terms of possibility theory .
- Identity: Two identical information items are fully redundant, i.e., .
- Symmetry: A redundancy metric is symmetric in all its arguments, i.e., for any permutation p on .
- Non-Agreement (Lower Bound): Information items are fully non-redundant if they disagree completely on the state of affairs, i.e., they do not agree on at least one alternative in the frame of discernment to be possible, i.e., .
4.1.1. Redundancy Type I
- in case of total ignorance, i.e., .
- iff in case of complete knowledge, i.e., only one unique event is totally possible and all other events are impossible.
- A specificity measure de- and increases with the maximum value of , i.e., let be the kth largest possibility degree in , then .
- , i.e., the specificity decreases as the possibilities of other values approach the maximum value of .
4.1.2. Redundancy Type II
- Boundaries: It is reasonable to assume that possibility distributions can be minimally and maximally similar. The measure is therefore bounded. It is normalized if .
- Identity relation (upper bound): A set of possibility distributions is maximally similar if they are identical, i.e., for any π. The reverse is not necessarily to be true. A set of possibility distributions with does not imply that all are identical.
- Non-agreement (lower bound): The non-agreement property defines that any set of possibility distributions which cannot agree on a common alternative x to be possible are maximal dissimilar, i.e.,
- Least agreement: A set of possibility distributions is at most as similar as the least similar pair :
- Symmetry: A similarity measure is a symmetric function in all its arguments, that is, for any permutation p on .
- Inclusion: For any , if , then and .
4.1.3. Reliability and Redundancy Metrics
- Information preservation: If , then the available information must not be changed but be preserved, i.e., .
- Specificity interaction: If , then the information needs to be modified to model total ignorance, i.e., . Information must not get more specific by the modification: for any .
4.2. Redundant Information Sources
- If information sources are redundant, then they provide redundant information items. Consequently, increases as the redundancy of information items belonging to the sources in increase.
- The reverse is not necessarily true. Redundant information items do no necessitate that their information sources are also redundant. Due to cases of incomplete information, redundant information items may support spurious redundancy (similar to spurious correlation which is depicted in Figure 1).
- 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 .
- Symmetry: The metric ρ is a symmetric function in all its arguments, i.e.,for any permutation p on .
4.2.1. Evidence Against Redundancy
- Absorbing element: for any , that is, if information sources in produce non-redundant items, then this is evidence that are not redundant as well.
4.2.2. Evidence Pro Redundancy
- Upper bound: If , then and .
- Lower bound: if , that is, all possibility distributions are identical.
- if ( models total ignorance), then ,
- if and only if and ( models complete knowledge at ), and
- if and only if and ( models complete knowledge at ).
- : The evidence (22) averages the redundancies of information items obtained by which is by definition in (see Definition 4).
- : The type II redundancy metric is symmetric per definition (Definition 4). The evidence (22) averages over all provided information items and is consequently also symmetric.
- Imprecision is modelled with probability distributions or not at all rather than with possibility distributions. Precise information items given as singletons are often only allegedly so—modelling the imprecision is often neglected.
- Information comes from unreliable sources.
- Information comes from heterogeneous sensors meaning that information is provided regarding different frame of discernments.
- If information are provided as singletons or probability distributions, they are transformed into possibility distributions.
- The unreliability of information sources is taken into account by modifying (widening) the possibility distribution using (20) with parameters and selected appropriately for each dataset.
- All information are mapped to a common frame of discernment.
5.1.1. Probability Possibility Transform
- Normalization condition: The resulting possibility distribution is required to be normal ().
- Consistency principle: What is probable must preliminarily be possible, that is, the possibility of an event A is an upper bound for its probability ().
- Preference preservation: Given a probability distribution p, .
5.1.2. Unifying Heterogeneous Information
5.2. Results and Discussion
- Pearson’s correlation coefficient: Correlation coefficients are computed on the expected value of the original data because sources from the TSD dataset provide information associated with an imprecision interval modeled by a uniform PDF. Let be the expected value of the imprecise data provided by source at instance j and let be the arithmetic mean of the expected values of . Then, the correlation coefficient is computed by
- Inconsistency-based approach: In  the inconsistency of a possibility distribution is determined within a set of possibility distributions. The inconsistency is the distance between the distribution’s position and the position of the majority observation within the set: . The position is determined by (24). Since we compare only pairs of information sources, no majority observation can be found and the distance between the positions of both information items is taken. The approach in  is designed for streaming data and the inconsistency of information items is averaged with a moving average filter. Instead of this kind of filter, is averaged so that:Similar to our approach, a homogenous frame of discernment between information items is required. Therefore, the inconsistency is computed on the possibility distributions obtained by the preprocessing steps detailed previously. The measure determines the degree of non-redundancy between information sources.
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
|OWA||Ordered Weighted Averaging|
|Probability Density Function|
|TTPPT||Truncated Triangular Probability-Possibility Transform|
|UPF||Unimodal Potential Function|
Appendix A. Additional Proofs
Appendix A.1. Proofs of Section 4.1.1
Appendix A.2. Proofs of Section 4.1.2
- Part (i)
- if . It follows that
- Part (ii)
- if . The same steps as carried out in part (i) can be applied. This leads to
Appendix A.3. Proofs of Section 4.1.3
Appendix A.4. Proofs of Section 4.2.2
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|Dataset||Information Sources (Columns)||Information Items (Rows)||Format||Imprecision||Noteworthy Characteristics|
|highly linearly correlated|
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Holst, C.-A.; Lohweg, V. A Redundancy Metric Set within Possibility Theory for Multi-Sensor Systems. Sensors 2021, 21, 2508. https://doi.org/10.3390/s21072508
Holst C-A, Lohweg V. A Redundancy Metric Set within Possibility Theory for Multi-Sensor Systems. Sensors. 2021; 21(7):2508. https://doi.org/10.3390/s21072508Chicago/Turabian Style
Holst, Christoph-Alexander, and Volker Lohweg. 2021. "A Redundancy Metric Set within Possibility Theory for Multi-Sensor Systems" Sensors 21, no. 7: 2508. https://doi.org/10.3390/s21072508