#
A Redundancy Metric Set within Possibility Theory for Multi-Sensor Systems^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- 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) [13].
- 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 [14] as well as premature detection of interrelations (such as redundancy) between sensors. The risk of detecting spurious correlations [15] is greatly amplified in intelligent technical or cyber–physical systems. Two examples of premature detection of variable interrelation are shown in Figure 1.

**Definition**

**1**

**Information Item**)

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

**Definition**

**2**

**Information Source**)

**.**An information source S provides information items. It is an ordered concatenation of information items $S=\{{I}_{1},{I}_{2},\dots ,{I}_{m}\}$ with $m\in {\mathbb{N}}_{>0}$. Each ${I}_{j}$ represents an information item at instance $j\in \{1,\dots ,m\}$. In case of multiple information sources, indexing is applied as follows: Let ${S}_{i}$ with $i\in {\mathbb{N}}_{>0}$ be an information source, then its information items are indexed with ${I}_{i,j}$. An information source may be, for example, a technical sensor, a variable, a feature, or a human expert.

## 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. [54] 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 $v\in A$, 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 $0<{\pi}_{v}(x)<1$, it is uncertain if $v=x$ is fully possible. It follows directly that confidence is also represented in the duality measure of $\Pi $ and N as can be seen in the three extreme epistemic situations [50]: (i) if $v\in A$ is certain, ${\Pi}_{v}(A)=1$ and ${N}_{v}(A)=1$, (ii) if $v\notin A$ is certain, ${\Pi}_{v}(A)=0$ and ${N}_{v}(A)=0$, and (iii) in case of ignorance, ${\Pi}_{v}(A)=1$ and ${N}_{v}(A)=0$.

#### 3.3. Fusion within Possibility Theory

- Conjunctive fusion modes implement the principle of minimal specificity most strongly. By applying a triangular norm (t-norm),$${\pi}^{(\mathrm{fu})}=\mathit{fu}({\pi}_{1}(x),\dots ,{\pi}_{n}(x))=t({\pi}_{1}(x),\dots ,{\pi}_{n}(x)),$$$${\pi}^{(\mathrm{fu})}=\frac{t({\pi}_{1}(x),\dots ,{\pi}_{n}(x))}{h({\pi}_{1}(x),\dots ,{\pi}_{n}(x))}$$
- 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:$${\pi}^{(\mathrm{fu})}=s({\pi}_{1}(x),\dots ,{\pi}_{n}(x)).$$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 [69], is$${\pi}^{(\mathrm{fu})}=max(\frac{t({\pi}_{1}(x),\dots ,{\pi}_{n}(x))}{h({\pi}_{1}(x),\dots ,{\pi}_{n}(x))},min(1-h({\pi}_{1}(x),\dots ,{\pi}_{n}(x))),s({\pi}_{1}(x),\dots ,{\pi}_{n}(x))).$$Thus, fusion results in a global level of conflict ($1-h(\xb7)$) 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 [4]. 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

**Definition**

**3**

**Redundancy Type I**)

**.**An information item is type I redundant if the carried information is already included in previously known information. Given an information item I and an unordered set of information items $\{{I}_{1},\dots ,{I}_{n}\}$ with $n\in {\mathbb{N}}_{>0}$, a possibilistic redundancy metric ${r}^{(\mathrm{I})}(I,\{{I}_{1},\dots ,{I}_{n}\})$ quantifies the degree of redundancy of I towards $\{{I}_{1},\dots ,{I}_{n}\}$. A metric for Redundancy Type I satisfies the following properties:

**Boundaries:**Information items can be minimally and maximally redundant. Therefore, ${r}^{(\mathrm{I})}$ is minimally and maximally bounded: ${r}^{(\mathrm{I})}\in [0,1]$.**Inclusion (Upper Bound):**An information item ${I}_{1}$ is fully redundant in relation to ${I}_{2}$ if it encloses (includes) ${I}_{2}$.**Lower Bound:**An information item is non-redundant if it adds new information. Additionally, an item ${I}_{1}$ is fully non-redundant in relation to ${I}_{2}$ if ${I}_{1}$ and ${I}_{2}$ disagree completely on the state of affairs, i.e., in terms of possibility theory $h({\pi}_{1},{\pi}_{2})=0$.**Identity:**Two identical information items are fully redundant, i.e., ${r}^{(\mathrm{I})}(I,I)=1$.

**Definition**

**4**

**Redundancy Type II**)

**.**Information items are type II redundant if they convey similar information with regard to a given task. This given task can be solved relying on any one of the information items. Let $\mathbf{I}$ be a set of unordered information items and $\mathcal{P}(\mathbf{I})$ all possible combinations of information items, then Redundancy Type II is a function ${r}^{(\mathrm{II})}:\mathcal{P}(\mathbf{I})\to [0,1]$. Similarly to ${r}^{(I)}$, ${r}^{(\mathrm{II})}$ is required to satisfy the properties of boundaries and identity as defined in Definition 3. Additionally, it has the following properties:

**Symmetry:**A redundancy metric ${r}^{(\mathrm{II})}$ is symmetric in all its arguments, i.e., ${r}^{(\mathrm{II})}({I}_{1},{I}_{2},\dots ,{I}_{n})={r}^{(\mathrm{II})}({I}_{p(1)},{I}_{p(2)},\dots ,{I}_{p(n)})$ for any permutation p on ${\mathbb{N}}_{>0}$.**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., $h({\pi}_{1},{\pi}_{2})=0$.

#### 4.1.1. Redundancy Type I

- $\mathit{spec}(\pi )=0$ in case of total ignorance, i.e., $\forall x\in X:\pi (x)=1$.
- $\mathit{spec}(\pi )=1$ 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 $\pi (x)$, i.e., let ${\pi}_{k}$ be the kth largest possibility degree in $\pi (x)$, then $\frac{\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}\mathit{spec}(\pi )}{\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{\pi}_{1}}>0$.
- $\forall k>2:\frac{\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}\mathit{spec}(\pi )}{\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{\pi}_{k}}\le 0$, i.e., the specificity decreases as the possibilities of other values approach the maximum value of $\pi (x)$.

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proposition**

**6.**

**Corollary**

**1.**

**Proposition**

**7.**

#### 4.1.2. Redundancy Type II

**Definition**

**5**

**Possibilistic Similarity Measure**)

**.**Let $\mathbf{p}=\{{\pi}_{1},{\pi}_{2},\dots ,{\pi}_{n}\}$ be an unordered set of possibility distributions defined on the same frame of discernment X. Then a possibilistic similarity measure is a function $\mathit{sim}:\pi {(x)}^{n}\to [0,1]$ satisfying the following properties:

**Boundaries:**It is reasonable to assume that possibility distributions can be minimally and maximally similar. The measure $\mathit{sim}(\xb7)$ is therefore bounded. It is normalized if $\mathit{sim}(\mathbf{p})\in [0,1]$.**Identity relation (upper bound):**A set of possibility distributions is maximally similar if they are identical, i.e., $\mathit{sim}(\pi ,\pi ,\dots ,\pi )=1$ for any π. The reverse is not necessarily to be true. A set of possibility distributions with $\mathit{sim}(\mathbf{p})=1$ does not imply that all $\pi \in \mathbf{p}$ 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.,$$\mathit{sim}(\mathbf{p})=0ifh(\mathbf{p})=0.$$**Least agreement:**A set of possibility distributions $\mathbf{p}$ is at most as similar as the least similar pair $(\pi ,{\pi}^{\prime})\in \mathbf{p}$:$$\mathit{sim}(\mathbf{p})\le \underset{(\pi ,{\pi}^{\prime})\in \mathbf{p}}{min}(\mathit{sim}(\pi ,{\pi}^{\prime})).$$**Symmetry:**A similarity measure is a symmetric function in all its arguments, that is, $\mathit{sim}({\pi}_{1},{\pi}_{2},\dots ,{\pi}_{n})=\mathit{sim}({\pi}_{p(1)},{\pi}_{p(2)},\dots ,{\pi}_{p(n)})$ for any permutation p on ${\mathbb{N}}_{>0}$.**Inclusion:**For any ${\pi}_{1},{\pi}_{2},{\pi}_{3}$, if $\forall x\in X:{\pi}_{1}(x)\le {\pi}_{2}(x)\le {\pi}_{3}(x)$, then $\mathit{sim}({\pi}_{1},{\pi}_{3})\le \mathit{sim}({\pi}_{1},{\pi}_{2})$ and $\mathit{sim}({\pi}_{1},{\pi}_{3})\le \mathit{sim}({\pi}_{2},{\pi}_{3})$.

#### 4.1.3. Reliability and Redundancy Metrics

**Information preservation:**If $\mathit{rel}=1$, then the available information must not be changed but be preserved, i.e., ${\pi}^{\prime}=\pi $.**Specificity interaction:**If $\mathit{rel}=0$, then the information needs to be modified to model total ignorance, i.e., $\forall x\in X:{\pi}^{\prime}(x)=1$. Information must not get more specific by the modification: $\mathit{spec}({\pi}^{\prime})\ge \mathit{spec}(\pi )$ for any $\mathit{rel}\in [0,1]$.

#### 4.2. Redundant Information Sources

**Definition**

**6**

**Possibilistic Redundancy Metric**)

**.**Let S be a possibilistic information source, i.e., the information items ${I}_{j}$ provided by S are possibility distributions: ${I}_{j}={\pi}_{j}$ with $j\in {\mathbb{N}}_{>0}$. Let $\mathbf{S}$ be the set of all available sources and $\mathcal{P}(\mathbf{S})$ be all possible combinations of sources, then a possibilistic redundancy metric ρ is a function which maps $\mathcal{P}(\mathbf{S})$ to the unit interval: $\rho :\mathcal{P}(\mathbf{S})\to [0,1]$.

- If information sources are redundant, then they provide redundant information items. Consequently, $\rho (\mathbf{S})$ increases as the redundancy of information items belonging to the sources in $\mathbf{S}$ 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 $\rho \in [0,1]$.**Symmetry:**The metric ρ is a symmetric function in all its arguments, i.e.,$$\rho ({S}_{1},{S}_{2},\dots ,{S}_{j})=\rho ({S}_{p(1)},{S}_{p(2)},\dots ,{S}_{p(j)})$$for any permutation p on ${\mathbb{N}}_{>0}$.

#### 4.2.1. Evidence Against Redundancy

**Absorbing element:**$avg(r({\mathbf{p}}_{1}),\dots ,r({\mathbf{p}}_{m}),0)=0$ for any $\mathbf{p}$, that is, if information sources in $\mathbf{S}$ produce non-redundant items, then this is evidence that $\mathbf{S}$ are not redundant as well.

#### 4.2.2. Evidence Pro Redundancy

**Definition**

**7**

**Range**)

**.**Given a frame of discernment $X=[{x}_{\mathrm{a}},{x}_{\mathrm{b}}]$, the range of a set of possibility distributions $\mathbf{p}$ quantifies how far $\mathbf{p}$ stretches over X. Let $\mathcal{P}(\mathbf{p})$ bet the power set of al possible $\mathbf{p}$, then the range is described by a function $\mathit{rge}:\mathcal{P}(\mathbf{p})\to [0,1]$ with the following properties:

**Upper bound:**If $\mathit{rge}(\mathbf{p})=1$, then $\exists \pi \in \mathbf{p}:\pi ({x}_{\mathrm{a}})=1$ and $\exists \pi \in \mathbf{p}:\pi ({x}_{\mathrm{b}})=1$.**Lower bound:**$\mathit{rge}(\mathbf{p})=0$ if $\forall \pi ,{\pi}^{\prime}\in \mathbf{p}:\pi ={\pi}^{\prime}$, that is, all possibility distributions $\pi \in \mathbf{p}$ are identical.

- if $\forall x\in X:\pi (x)=1$ ($\pi $ models total ignorance), then $\mathit{pos}(\pi )=\frac{1}{2}\xb7({x}_{\mathrm{b}}-{x}_{\mathrm{a}})$,
- $\mathit{pos}(\pi )={x}_{\mathrm{a}}$ if and only if $\pi ({x}_{\mathrm{a}})=1$ and $\forall x\in \{X\backslash {x}_{\mathrm{a}}\}:\pi (x)=0$ ($\pi $ models complete knowledge at ${x}_{\mathrm{a}}$), and
- $\mathit{pos}(\pi )={x}_{\mathrm{b}}$ if and only if $\pi ({x}_{\mathrm{b}})=1$ and $\forall x\in \{X\backslash {x}_{\mathrm{b}}\}:\pi (x)=0$ ($\pi $ models complete knowledge at ${x}_{\mathrm{b}}$).

**Proposition**

**8.**

**Proof.**

**${e}_{\mathrm{p}}$:**The evidence ${e}_{\mathrm{p}}$ (27) is build upon the function $\mathit{rge}$ (26) which in turn is build upon the function $\mathit{pos}$ (24). The position $\mathit{pos}\in [{x}_{\mathrm{a}},{x}_{\mathrm{b}}]$ because it is based on the center of gravity. The range $\mathit{rge}$ takes the difference of maximum and minimum positions and is, therefore, also in $[{x}_{\mathrm{a}},{x}_{\mathrm{b}}]$. The evidence ${e}_{\mathrm{p}}$ normalizes $\mathit{rge}$ to the interval $[0,1]$ in (27).**${e}_{\mathrm{c}}$:**The evidence ${e}_{\mathrm{c}}$ (22) averages the redundancies of information items obtained by ${r}^{(\mathrm{II})}$ which is by definition in $[0,1]$ (see Definition 4).

**Proposition**

**9.**

**Proof.**

**${e}_{\mathrm{c}}$:**The type II redundancy metric ${r}^{(\mathrm{II})}$ is symmetric per definition (Definition 4). The evidence ${e}_{\mathrm{c}}$ (22) averages ${r}^{(\mathrm{II})}$ over all provided information items and is consequently also symmetric.

## 5. Evaluation

#### 5.1. Implementation

- 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 $\mathit{rel}$ and $\beta $ 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 ($\exists x\in X:\pi (x)=1$).**Consistency principle:**What is probable must preliminarily be possible, that is, the possibility of an event A is an upper bound for its probability ($\mathit{Pr}(A)\le \Pi (A)$).**Preference preservation:**Given a probability distribution p, $p(x)<p({x}^{\prime})\to \pi (x)\le \pi ({x}^{\prime})$.

#### 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 ${x}_{i,j}^{(\mathrm{e})}$ be the expected value of the imprecise data provided by source ${S}_{i}$ at instance j and let ${\overline{x}}_{i}^{(\mathrm{e})}$ be the arithmetic mean of the expected values of ${S}_{i}$. Then, the correlation coefficient is computed by$${\rho}_{\mathrm{p}}({S}_{1},{S}_{2})=\left|\frac{{\sum}_{j=1}^{m}\left({x}_{1,j}^{(\mathrm{e})}-{\overline{x}}_{1}^{(\mathrm{e})}\right)\left({x}_{2,j}^{(\mathrm{e})}-{\overline{x}}_{2}^{(\mathrm{e})}\right)}{\sqrt{{\sum}_{j=1}^{m}{\left({x}_{1,j}^{(\mathrm{e})}-{\overline{x}}_{1}^{(\mathrm{e})}\right)}^{2}{\sum}_{i=1}^{m}{\left({x}_{2,j}^{(\mathrm{e})}-{\overline{x}}_{2}^{(\mathrm{e})}\right)}^{2}}}\right|.$$**Inconsistency-based approach:**In [5] the inconsistency $\mathit{inc}$ of a possibility distribution is determined within a set of possibility distributions. The inconsistency is the distance between the distribution’s position $\mathit{pos}(\pi )$ and the position of the majority observation $\mathit{pos}({\pi}_{\mathrm{maj}})$ within the set: $\mathit{inc}=\left|{\mathit{pos}}_{\pi}-\mathit{pos}({\pi}_{\mathrm{maj}})\right|$. 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 [5] is designed for streaming data and the inconsistency of information items is averaged with a moving average filter. Instead of this kind of filter, $\mathit{inc}$ is averaged so that:$$\mathit{inc}({S}_{1},{S}_{2})=\frac{1}{m}\xb7\sum _{j=1}^{m}\left|\mathit{pos}({\pi}_{1,j})-\mathit{pos}({\pi}_{2,j})\right|.$$Similar to our approach, a homogenous frame of discernment between information items is required. Therefore, the inconsistency is computed on the possibility distributions ${\pi}_{\mu}$ obtained by the preprocessing steps detailed previously. The measure $\mathit{inc}$ determines the degree of non-redundancy between information sources.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CPS | Cyber–Physical Systems |

MI | Mutual Information |

OWA | Ordered Weighted Averaging |

Probability Density Function | |

PosT | Possibility Theory |

ProbT | Probability Theory |

TTPPT | Truncated Triangular Probability-Possibility Transform |

UPF | Unimodal Potential Function |

## Appendix A. Additional Proofs

#### Appendix A.1. Proofs of Section 4.1.1

**Proof**

**of**

**Proposition**

**5.**

**Proof**

**of**

**Proposition**

**6.**

**Proof**

**of**

**Corollary**

**1.**

**Proof**

**of**

**Proposition**

**7.**

#### Appendix A.2. Proofs of Section 4.1.2

**Proposition**

**A1.**

**Proof**

**of**

**Boundaries**

**Property.**

**Proof of Identity Relation**

**(Upper**

**Bound)**

**Property.**

**Proof of Non-Agreement**

**(Lower**

**Bound)**

**Property.**

**Proof**

**of**

**Least**

**Agreement**

**Property.**

**Proof**

**of**

**Symmetry**

**Property.**

**Proof**

**of**

**Inclusion**

**Property.**

- Part (i)
- ${\mathit{sim}}_{\mathrm{r}}({\pi}_{1},{\pi}_{3})\le {\mathit{sim}}_{\mathrm{r}}({\pi}_{1},{\pi}_{2})$ if ${r}^{(\mathrm{I})}({\pi}_{1},{\pi}_{3})\le {r}^{(\mathrm{I})}({\pi}_{1},{\pi}_{2})$. It follows that$$\begin{array}{cc}\hfill {r}^{(\mathrm{I})}({\pi}_{1},{\pi}_{3})& \le {r}^{(\mathrm{I})}({\pi}_{1},{\pi}_{2}),\hfill \\ \hfill \left(1-|g(\mathit{fu}({\pi}_{1},{\pi}_{3}),{\pi}_{3})|\right)\xb7h({\pi}_{1},{\pi}_{3})& \le \left(1-|g(\mathit{fu}({\pi}_{1},{\pi}_{2}),{\pi}_{2})|\right)\xb7h({\pi}_{1},{\pi}_{2}),\hfill \\ \hfill 1-|g(\mathit{fu}({\pi}_{1},{\pi}_{3}),{\pi}_{3})|& \le 1-|g(\mathit{fu}({\pi}_{1},{\pi}_{2}),{\pi}_{2})|,\hfill \\ \hfill 1-|g({\pi}_{1},{\pi}_{3})|& \le 1-|g({\pi}_{1},{\pi}_{2})|,\hfill \\ \hfill -|g({\pi}_{1},{\pi}_{3})|& \le -|g({\pi}_{1},{\pi}_{2})|,\hfill \\ \hfill -|\underset{<0}{\underbrace{\mathit{spec}({\pi}_{1}(-\mathit{spec}({\pi}_{3})}}|& \le -|\underset{<0}{\underbrace{\mathit{spec}({\pi}_{1})-\mathit{spec}({\pi}_{2})}}|,\hfill \\ \hfill \mathit{spec}({\pi}_{1})-\mathit{spec}({\pi}_{3})& \le \mathit{spec}({\pi}_{1})-\mathit{spec}({\pi}_{2}),\hfill \\ \hfill \mathit{spec}({\pi}_{3})& \ge \mathit{spec}({\pi}_{2}).\hfill \end{array}$$
- Part (ii)
- ${\mathit{sim}}_{\mathrm{r}}({\pi}_{1},{\pi}_{3})\le {\mathit{sim}}_{\mathrm{r}}({\pi}_{2},{\pi}_{3})$ if ${r}^{(\mathrm{I})}({\pi}_{1},{\pi}_{3})\le {r}^{(\mathrm{I})}({\pi}_{2},{\pi}_{3})$. The same steps as carried out in part (i) can be applied. This leads to$$\begin{array}{cc}\hfill -|\underset{<0}{\underbrace{\mathit{spec}({\pi}_{1})-\mathit{spec}({\pi}_{3})}}|& \le -|\underset{<0}{\underbrace{\mathit{spec}({\pi}_{2})-\mathit{spec}({\pi}_{3})}}|,\hfill \\ \hfill \mathit{spec}({\pi}_{1})-\mathit{spec}({\pi}_{3})& \le \mathit{spec}({\pi}_{2})-\mathit{spec}({\pi}_{3}),\hfill \\ \hfill \mathit{spec}({\pi}_{1})& \le \mathit{spec}({\pi}_{2}).\hfill \end{array}$$

#### Appendix A.3. Proofs of Section 4.1.3

**Proposition**

**A2.**

**Proof.**

**Proposition**

**A3.**

**Proof.**

#### Appendix A.4. Proofs of Section 4.2.2

**Proof.**

**Proof.**

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**Figure 1.**Examples of variables ${x}_{1},{x}_{2}\in \mathbb{R}$ showing (

**a**) similar behaviour which is not apparent in the sample data and showing (

**b**) non-similar behaviour although sample data indicate otherwise (which is an example of spurious correlation). These kinds of biased or skewed sample data commonly occur, for example, in production systems. Production systems execute tasks repetitively in a normal (as in functioning properly) condition. In this case, data are not sampled randomly and do not match the population distribution.

**Figure 2.**A possibility distribution ${\pi}_{v}$. For any element $x\in B$, $v=x$ is fully plausible; for $x\in (A\cap {B}^{\mathsf{c}})$, $v=x$ is only partially plausible; and for $x\in {A}^{\mathsf{c}}$, $v=x$ is impossible. The accompanying possibility and necessity measures for $A,B$ are: $\Pi (A)=1$, $N(A)=1$ and $\Pi (B)=1$, $N(B)=0.5$.

**Figure 3.**Different fusion approaches in possibility theory. Part (

**a**) shows conjunctive fusion (4) using the minimum operator as t-norm, (

**b**) illustrates disjunctive fusion (5) using the maximum operator as s-norm, and (

**c**) shows the adaptive fusion rule (6) presented in [69] (also relying on minimum and maximum operators).

**Figure 4.**Possibility distributions and their fusion results as examples for the proposed type I redundancy metric. In (

**a**), ${r}^{(\mathrm{I})}({\pi}_{1},{\pi}_{2})=1$ and $0<{r}^{(\mathrm{I})}({\pi}_{2},{\pi}_{1})<1$. Subfigure (

**b**) shows a case in which both possibility distributions are not redundant, i.e., $0<{r}^{(\mathrm{I})}({\pi}_{1},{\pi}_{2})<1$ and $0<{r}^{(\mathrm{I})}({\pi}_{2},{\pi}_{1})<1$. Although the fusion result is less specific (more uncertain) in (

**c**) due to renormalisation, both ${\pi}_{1}$ and ${\pi}_{2}$ are not redundant (similar to (

**b**)).

**Figure 5.**An incorrect (${\pi}_{1}$), a partially erroneous (${\pi}_{2}$), and a correct possibility distribution (${\pi}_{3}$). The degree of error is dependent on the level of possibility $\pi v$, v being the unknown ground truth. Note that it is difficult to determine the error of a possibility distribution since v is unknown and it is precisely the task of $\pi $ to give an imprecise estimation of v.

**Figure 6.**Modifying possibility distributions depending on the reliability of their information source S. Subfigure (

**a**) shows the approach of Yager and Kelman (18), (

**b**) shows the method of Dubois et al. (19), and (

**c**) shows the proposed method (20). Only the method in (

**c**) has a widening effect, both methods in (

**a**,

**b**) raise the level of possibility along the complete frame of discernment. All methods result in total ignorance for $\mathit{rel}(S)=0$ and ${\pi}^{\prime}=\pi $ for $\mathit{rel}(S)=1$. For these plots, parameter $\beta =2$.

**Figure 7.**Information items in the form of triangular possibility distributions provided by two information sources. Available (e.g., measured) information is scattered throughout the frame of discernment $X=[{x}_{\mathrm{a}},{x}_{\mathrm{b}}]$. The left side shows a two-dimensional scatter plot in which each marker represents the maximum of each possibility distribution. The right side depicts the possibility distributions of three exemplary selected datapoints (marked by an encompassing circle). Each cluster considered in isolation represents a case of incomplete information because only parts of the frame of discernment are covered. For example, cluster 1 (marked by ×) suggests redundancy (as long as information items are similar). This may not hold when new information from both sources become available. Clusters 1 (×) and 2 (✶) together suggest redundancy more strongly. Any data containing cluster 3 (+) evidences no redundancy. Relying esclusively on ${e}_{\mathrm{c}}$ (22) may result in detecting redundancy prematurely. A second evidence measure is needed to put ${e}_{\mathrm{c}}$ into context. This second measure—denoted as evidence pro redundancy ${e}_{\mathrm{p}}$—is presented in the following.

**Figure 8.**Preprocessing steps (i)–(v) carried out on three information items provided as probability distributions $p(x)$—as (

**a**) singular value, (

**b**) uniform probability density function, and (

**c**) Gaussian probability density function. Each item gives information regarding an unknown measurand in its own frame of discernment (${X}_{1}=[{x}_{\mathrm{a},1},{x}_{\mathrm{b},1}]$, ${X}_{2}=[{x}_{\mathrm{a},2},{x}_{\mathrm{b},2}]$, ${X}_{3}=[{x}_{\mathrm{a},3},{x}_{\mathrm{b},3}]$). As a result of this, preprocessing is necessary to be able to derive conclusions about potential redundancy. First, in step (ii) the probability distributions are transformed into possibility distributions via the truncated triangular probability-possibility transformation [53,65,66]. Step (iii) takes account of potential unreliability of information sources by widening $\pi (x)$ using (20) (here with $\mathit{rel}=0.95$ and $\beta =1$). Steps (iv), (v) transform the frame of discernment into fuzzy memberships ${X}_{\mu}=[{\mu}_{\mathrm{a}},{\mu}_{\mathrm{b}}]=[0,1]$. Assuming a binary fuzzy classification task, one fuzzy class (e.g., the normal condition in condition monitoring) is represented by a unimodal potential function (UPF) (29) either learned from training data or provided by an expert (iv) (here: arbitrary selected UPFs are shown as an example). Whereas $\pi (x)$ in (iii) represents the imprecision of a single information item, $\mu (x)$ represents the fuzzy set of the given class. In the final step (v), $\pi (x)$ is transformed into $\pi (\mu )$ (30). Note that $\pi (x)$ aligns with $\mu (x)$ in such a way that (

**a**) $\pi (\mu )$ is close to 0 and (

**b**), (

**c**) $\pi (\mu )$ is close to 1.

**Figure 9.**Information items of the selected information sources. Each row, consisting of a scatter and linear plot, belongs to sources from the datasets Sensorless Drive Diagnosis (SDD) (

**a**–

**c**), HAR (

**d**–

**f**), and Typical Sensor Defects (TSD) (

**g**–

**i**). Each point in the scatter plots represents the center of gravity (24) of an information item, i.e., of ${\pi}_{\mu}(\mu (x))$. To get an intuition about the imprecision in the information, the possibility distributions of a single pair of information items are plotted below each scatter plot. The selected cases show linear relations, non-linear relations, non-redundancy, and aleatoric noise. In some only part of the frame of discernment is perceived. Note that plots (

**g**–

**i**) are zoomed in for better visibility.

Dataset | Information Sources (Columns) | Information Items (Rows) | Format | Imprecision | Noteworthy Characteristics |
---|---|---|---|---|---|

SDD | 48 | 58509 | real-valued, $x\in \mathbb{R}$ | precise, $px=v=1$, $px\ne v=0$ | highly linearly correlated |

HAR | 561 | 5744 | real-valued, $x\in \mathbb{R}$ | precise, $px=v=1$, $px\ne v=0$ | noisy |

TSD | 22 | 72500 | real-valued, $x\in \mathbb{R}$ binary-valued, $x\in \{0,1\}$ | imprecise, uniform PDF | incomplete information |

**Table 2.**Parameters ${x}_{\mathrm{n}}$, ${x}_{\u03f5}$, and $\u03f5$ for the truncated triangular probability-possibility transform of different probability density functions (PDF).

${\mathit{x}}_{\mathbf{n}}$ | ${\mathit{x}}_{\mathit{\u03f5}}$ | $\mathit{\u03f5}$ | |
---|---|---|---|

Gaussian | $2.58\xb7\sigma $ | $1.54\xb7\sigma $ | $0.12$ |

Laplace | $3.20\xb7\sigma $ | $1.46\xb7\sigma $ | $0.13$ |

Triangular | $2.45\xb7\sigma $ | $1.63\xb7\sigma $ | $0.11$ |

Uniform | $1.73\xb7\sigma $ | $1.73\xb7\sigma $ | 0 |

**Table 3.**Results of the possibilistic redundancy metric $\rho $ (21) along with the evidences ${e}_{\mathrm{p}}$ (27) and ${e}_{\mathrm{c}}$ (22). The metric is compared to (i) the Pearson’s correlation coefficient ${\rho}_{\mathrm{p}}$ computed on the expected values of the original data and an inconsistency-based approach (measure $\mathit{inc}$). The cases (a)–(i) refer to the selected information sources as plotted in Figure 9.

Case | Dataset | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{e}}_{\mathbf{p}}$ | ${\mathit{e}}_{\mathbf{c}}$ | $\mathit{\rho}$ | ${\mathit{\rho}}_{\mathbf{p}}$ | $\mathit{inc}$ |
---|---|---|---|---|---|---|---|---|

(a) | SDD | 7 | 8 | $0.92$ | 0 | $0.96$ | 1 | 0 |

(b) | SDD | 2 | 46 | $0.85$ | 1 | 0 | $0.09$ | $0.06$ |

(c) | SDD | 20 | 36 | $0.76$ | 1 | 0 | $0.07$ | $0.19$ |

(d) | HAR | 89 | 102 | $0.47$ | 0 | $0.69$ | $0.98$ | $0.02$ |

(e) | HAR | 86 | 99 | $0.47$ | 0 | $0.69$ | $0.94$ | $0.09$ |

(f) | HAR | 12 | 50 | $0.47$ | 1 | 0 | $0.27$ | $0.16$ |

(g) | TSD | 9 | 15 | $0.04$ | 0 | $0.20$ | $0.90$ | $0.16$ |

(h) | TSD | 9 | 18 | $0.04$ | $0.10$ | $0.19$ | $0.89$ | $0.02$ |

(i) | TSD | 14 | 18 | $0.05$ | 0 | $0.23$ | $0.99$ | $0.02$ |

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Holst, C.-A.; Lohweg, V. A Redundancy Metric Set within Possibility Theory for Multi-Sensor Systems. *Sensors* **2021**, *21*, 2508.
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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