# Information Fusion of Conflicting Input Data

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## Abstract

**:**

## 1. Introduction

**Definition 1**(Direct fusion).

**Definition 2**(Indirect fusion).

## 2. Related Work

#### 2.1. Conflict

**Example 1**(Zadeh’s example).

#### 2.2. Uncertainty

**Definition 3**(Aleatory uncertainty).

**Definition 4**(Epistemic uncertainty).

## 3. Multilayer Attribute-Based Conflict-Reducing Observation

#### 3.1. Architecture

#### 3.2. Balanced Two-Layer Conflict Solving

- adoption of effective human group decision-making principles,
- determination of conflicts between inputs,
- solution of the conflicts, such that their effect on the fusion result is decreased,
- creation of intuitive fusion results, also in high-conflict cases.

**Definition 5**(Basic belief assignment).

**No**

**conflict:**

**Maximum**

**conflict:**

#### 3.2.1. Non-Conflicting Part

**Definition 6**(BalTLCS: non-conflicting part).

#### 3.2.2. Conflicting Part

**Definition 7**(BalTLCS: normalised conflicting coefficient).

**Definition 8**(BalTLCS: conflicting part).

#### 3.2.3. Balanced Group Conflict Redistribution

**Definition 9**(Balanced group conflict redistribution (BalGCR)).

**Lemma 1**(Boundedness of balanced group conflict redistribution).

**Proof.**

#### 3.2.4. Conclusions on Balanced Two-Layer Conflict Solving

- BalTLCS fuses a number of input BBAs by determining intermediate results among the non-conflicting and the conflicting BBAs, and their subsequent additive combination by BalGCR.
- The non-conflicting part of BalTLCS fusion is determined by exhaustive individual combination of pairs of two sensors instead of combination of all sensors at the same time. This is inspired by psychological research findings on human group decision-making.
- In order to derive a decision in all cases, also in high-conflicting cases, the conflicting part is determined by the arithmetic mean amongst all sensors. This is additionally weighed by the normalised conflicting coefficient, such that the conflicting part plays only a subordinate role in the fusion process in case of no or small conflict.
- BalTLCS fusion yields intuitive results. This is evaluated and shown in [17].

#### 3.3. System State Representation

**Definition 10**(System condition).

**normal-condition**.

**abnormal-condition**.

**Definition 11**

**Axiom 1**(Normality). $\underset{\theta \in A}{sup}{\mu}_{A}\left(\theta \right)=1.$**Axiom 2**(Convexity). ${\mu}_{A}(\lambda {\theta}_{1}+(1-\lambda ){\theta}_{2})\ge min({\mu}_{A}\left({\theta}_{1}\right),{\mu}_{A}\left({\theta}_{2}\right)),\forall {\theta}_{1},{\theta}_{2}\in \mathrm{\Theta},\lambda \in [0,1]$.

- ${}^{\mathrm{N}}{\mu}_{s}:\mathbb{R}\to [0,1]$ models the normal condition,
- ${}^{\overline{\mathrm{N}}}{\mu}_{s}:\mathbb{R}\to [0,1]$ models the abnormal condition.

**Definition 12**

**θ**sorted in increasing order, hence: ${\theta}^{\prime}\left[1\right]\le {\theta}^{\prime}\left[2\right]\le \cdots \le {\theta}^{\prime}\left[N\right]$. Then, ${\theta}_{0}$ is determined by:

#### 3.4. Fuzzified Balanced Two-Layer Conflict Solving

**Definition 13**(Constraints on fuzzy basic belief assignment).

- The membership function ${}^{A}\mu $ has finite support on the considered frame of discernment Θ, i.e., the frame of discernment is finite. Hence, ${}^{A}\mu \left(\perp \right)=0$ and ${}^{A}\mu \left(\top \right)=0$, where ⊥ denotes the smallest element in Θ, and ⊤ the largest, respectively.
- Fuzzy set A is a standard fuzzy set, i.e., its membership function ${}^{A}\mu $ is unimodal and normal (cf. Definition 11). This ensures that its α-cuts form nested sets. Only in this case a transfer between fuzzy memberships to basic belief assignments is possible and valid.

**Definition 14**(Fuzzy basic belief assignment (μBBA)).

**Definition 15**(Fuzzified balanced two-layer conflict solving (μBalTLCS)).

**Axiom 3**(Boundary conditions).

**Axiom 4**(Increasing Monotonicity).

**Axiom 5**(Continuity).

**Axiom 6**(Symmetry).

**Proof.**

**Axiom 7.**

#### 3.5. MACRO Attribute Layer Fusion

**Definition 16**

#### 3.6. System Layer Fusion

**Definition 17**

**Definition 18**

- If the attributes are significantly dependent on each other, their information is redundant to a high degree, and external effects affect many or all attributes at the same time. Then the system layer fusion is carried out with a high degree of optimism. This leads to a degradation of the system health only when all attributes determine a deterioration of the system state. Consequently, the system health follows the largest attribute health.
- If the attributes are significantly independent from each other, their information is redundant to a small degree, and external effects affect only some or one attribute. Then the system layer fusion is carried out with a low degree of optimism. This leads to a degradation of the system health when at least one attribute determines a deterioration of the system state. Consequently, the system health follows the smallest attribute health.

## 4. Evaluation

**Example 2**(Printing unit demonstrator).

#### 4.1. Evaluation Setup

**Attribute 1 (Motors):**

**Attribute 2 (Contact Pressure):**

**Attribute 3 (Motor Currents):**

**Naïve Bayes:**

**nB${}_{\mathbf{Gauss}}$:**The nB${}_{\mathrm{Gauss}}$ variant models the prior distribution as normal distribution. It adjusts the distribution’s mean and standard deviation based on the training data.**nB${}_{\mathbf{kern}}$:**No certain probability distribution is assumed for the prior distribution. It is instead estimated based on the training data by kernel density estimation applying Gaussian kernels.- WEKA implements both variants of naïve Bayes in its NaiveBayes classifier. Details on the background of the implementation are found in [99].This classifier (and also all other naïve Bayes implementations found by the authors) is only capable to be applied if data for more than one class is available in the training data. The printing unit demonstrator experiments deliver only data about the demonstrator’s condition, which is per se assumed to represent its normal condition. Thus, the naïve Bayes implementation is applied in combination with the WEKA package OneClassClassifier (WEKA packages are conveniently installed by utilisation of its integrated package manager). This is a meta-classifier, which allows to apply any classifier on one-class problems like the printing unit demonstrator condition monitoring experiments: Based on the training data, artificial data representing its counter-class is generated, facilitating to handle the original one-class problem as two-class problem. The result is obtained by the combination of the prior information derived from the training data with the employed classifier’s output. It utilises Bayes’ theorem for this task. For details on the background of OneClassClassifier see [100].

**Support Vector Machine:**

#### 4.2. PU_{static} Data Set Results

- ${}^{\mathrm{N}}\mu \ge {}^{\mathrm{N}}{\eta}_{\mathrm{warn}}<1$: In this range, the system operation is considered to be normal. Deviations from ${}^{\mathrm{N}}\mu =100$% are intentionally allowed as the behaviour of physical systems is usually not constant (e.g., due to variations in the system’s environment).
- ${}^{\mathrm{N}}{\eta}_{\mathrm{emerg}}\le {}^{\mathrm{N}}\mu <{}^{\mathrm{N}}{\eta}_{\mathrm{warn}}$: If the system health determined during operation is in this range, it is neither considered normal nor in an emergency condition. Instead, it is in a warning condition. This state may, for example, be utilised to increase attention of maintenance personnel. This range is considered as a transient area, in which it is likely that a system defect will follow in the future.
- ${}^{\mathrm{N}}\mu <{}^{\mathrm{N}}{\eta}_{\mathrm{emerg}}$: In this case, the system is considered to be in an emergency condition. It might already bear a defect and appropriate measures, like an emergency stop of the system, have to be taken.

#### 4.3. PU_{manip} Data Set Results

## 5. Discussion of the Results

## 6. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

μBBA | fuzzy basic belief assignment |

μBalTLCS | fuzzified balanced two-layer conflict solving |

AAL | ambient assisted living |

BalGCR | balanced group conflict redistribution |

BalTLCS | balanced two-layer conflict solving |

BBA | basic belief assignment |

CMDST | Conflict-Modified-DST |

DRC | Dempster’s rule of combination |

DST | Dempster-Shafer theory of evidence |

FFT | fast Fourier transform |

FST | fuzzy set theory |

GCR | Group-Conflict-Redistribution |

IFU | information fusion |

IIWOWA | implicative importance weighted ordered weighted averaging |

IWOWA | importance weighted ordered weighted averaging |

MACRO | multilayer attribute-based conflict-reducing observation |

MFPC | Modified-Fuzzy-Pattern-Classifier |

OWA | ordered weighted averaging |

probability density function | |

PosT | possibility theory |

ProbT | probability theory |

RMS | root mean square |

RTE | redundant target effect |

SoC | system on chip |

SVM | Support Vector Machine |

TLCS | Two-Layer Conflict Solving |

WAM | weighted arithmetic mean |

WEKA | Waikato Environment for Knowledge Analysis |

## Appendix A. Evaluation Data Set Characteristics

**Printing Unit Demonstrator Condition Monitoring:**The behaviour of the printing unit demonstrator during operation is observed by four analogue sensors. They each output a continuous voltage signal in the range of $[-10,10]$ V, which is proportional to the respective quantity the sensor is observing. Thus, each signal’s unit is irrelevant and abandoned as changes of the original quantity of interest are reflected also in the respective voltage signal. All output time-domain signals are synchronously and equidistantly sampled at a frequency of 20 kHz and quantised with a resolution of 16 bit. The acquired data is then split into non-overlapping batches of $50,000$ samples (corresponding to $2.5$ s of operation), respectively. The length of the time frame was chosen to ensure that 3 revolutions of the plate cylinder are captured in each signal data batch. Each plate cylinder revolution is represented by a data set instance. One of the signals (solid-borne sound) is treated by the fast Fourier transform (FFT) to determine its frequency spectrum per signal batch. Altogether, 5 time- and frequency domain signals are available, from which 5 features per plate cylinder revolution are extracted. That is, every instance in the data set is described by a vector of 5 feature values. This results in 15 feature values per signal data batch. Table A1 and Table A2 summarise which signals were acquired and which features were extracted during the printing unit demonstrator operation.

Symbol | Signal Name | Signal Domain |
---|---|---|

${d}_{1}$ | contact force | time |

${d}_{2}$ | solid-borne sound | time |

${d}_{3}$ | solid-borne sound spectrum | frequency |

${d}_{4}$ | motor current wiping cylinder | time |

${d}_{5}$ | motor current plate cylinder | time |

**Table A2.**Summary of the features extracted from the sensor signals d acquired at the printing unit demonstrator.

Symbol | Feature Name | Feature Description |
---|---|---|

${f}_{1}$ | $\mathrm{mean}\left({d}_{1}\right)$ | arithmetic mean of the contact force |

${f}_{2}$ | $\mathrm{rms}\left({d}_{2}\right)$ | root mean square of the solid-borne sound (sound intensity) |

${f}_{3}$ | $\mathrm{maxAmplFreqInd}\left({d}_{3}\right)$ | index of the frequency component with largest power |

${f}_{4}$ | $\mathrm{mean}\left({d}_{4}\right)$ | arithmetic mean of the wiping cylinder motor current |

${f}_{5}$ | $\mathrm{mean}\left({d}_{5}\right)$ | arithmetic mean of the plate cylinder motor current |

**Static printing unit demonstrator operation (PU${}_{\mathbf{static}}$):**The static experiment observes the printing unit demonstrator during 20:13 (min:s) of operation. The printing unit demonstrator was started immediately before the data acquisition began. No additional manipulations or events occurred during the experiment. Therefore, only data representing the demonstrator’s normal condition ${}^{\mathrm{N}}C$ is contained in the PU${}_{\mathrm{static}}$ data set. It contains $10,000,000$ raw signal samples resulting in 600 instances (plate cylinder revolutions), which are in summary described by 3000 feature values.**Manipulated printing unit demonstrator operation (PU${}_{\mathbf{manip}}$):**The printing unit demonstrator was started ca. 23:00 (min:s) before the data acquisition began. During this 10:31 (min:s) long experiment, the demonstrator application was intentionally manipulated. In addition, the solid-borne sound sensor signal was manipulated through low-pass filtering in order to simulate a defect of this sensor. An unintended incident also occurred during this experiment. Therefore, data representing both the demonstrator’s normal and abnormals conditions ${}^{\mathrm{N}}C$ and ${}^{\overline{\mathrm{N}}}C$ are contained in the PU${}_{\mathrm{manip}}$ data set. The sequence of events along with an objective classification of the demonstrator condition by the human experimenter is summarised in Table A3. The PU${}_{\mathrm{manip}}$ data set contains $5,950,000$ raw signal samples, which are in summary described by 1785 feature values.

**Table A3.**Description of the printing unit demonstrator operation and the events, which occurred during the manipulated printing unit demonstrator operation experiment. These are covered by the PU${}_{\mathrm{manip}}$ data set. The demonstrator condition reflects the objective assessment of the printing unit demonstrator by the human experimenter during operation.

Time $[\mathbf{min}:\mathbf{s}]$ | Instance k | Event and Operation Description | Demonstrator Condition |
---|---|---|---|

00:00–02:55 | 1–100 | training data acquisition | ${}^{\mathrm{N}}C$ |

02:55–03:45 | 101–128 | normal operation without incidents or manipulations | ${}^{\mathrm{N}}C$ |

03:45–03:55 | 129–135 | activation of analogue low-pass signal filter to treat the solid-borne sound signal | ${}^{\mathrm{N}}C$ |

03:55–04:33 | 136–155 | gradual attenuation of the solid-borne sound signal by continuously decreasing the low-pass filter’s cutoff frequency | ${}^{\mathrm{N}}C$ |

04:33–06:36 | 156–224 | normal operation at smallest possible cutoff frequency of the solid-borne sound low-pass filter | ${}^{\mathrm{N}}C$ |

06:36–07:08 | 225–242 | deactivation of the analogue low-pass filter | ${}^{\mathrm{N}}C$ |

07:08–08:33 | 243–290 | uneven turning of the plate cylinder (unintended) | ${}^{\overline{\mathrm{N}}}C$ |

08:33–10:31 | 291–357 | contact pressure decreased until no contact between both cylinders | ${}^{\overline{\mathrm{N}}}C$ |

## Appendix B. MACRO Attribute Evaluation Results

**Figure B1.**Attribute health evaluation over time during normal operation of the printing unit demonstrator (PU${}_{\mathrm{static}}$). No manipulation or fault occurred during the 20:13 (min:s) operation time. Plots (

**a**–

**c**) show attribute healths ${}_{a}^{\mathrm{N}}\mu \left(t\right)$ and their corresponding importances ${I}_{a}\left(t\right)$. Variations in the curves are due to effects of the operation itself. Training data was acquired up to 2:59 (min:s) (100 plate cylinder revolutions, cf. the black dotted line). Attribute composition according to Table 4. (

**a**) Attribute 1: Motors; (

**b**) Attribute 2: Contact Pressure; (

**c**) Attribute 3: Motor Currents.

**Figure B2.**Attribute health evaluation over time during manipulated operation of the printing unit demonstrator (PU${}_{\mathrm{manip}}$). Plots (

**a**–

**c**) show attribute healths ${}_{a}^{\mathrm{N}}\mu \left(t\right)$ and their corresponding importances ${I}_{a}\left(t\right)$. Training data was acquired up to 2:55 (min:s) (100 plate cylinder revolutions, cf. the black dotted line). Variations in the curves are due to effects of the operation itself and result from manipulations or faults, marked by vertical black lines (cf. Table A3). Attribute composition according to Table 4. (

**a**) Attribute 1: Motors; (

**b**) Attribute 2: Contact Pressure; (

**c**) Attribute 3: Motor Currents.

## Appendix C. Proofs

**Proof of Lemma 1.**

**Proof of Axiom 3**. First ${}^{A}{\mu}_{s}=0$ for all s is considered, i.e., no belief is assigned to proposition A by any sensor. As no information is provided about further propositions, the remaining belief is assigned to the frame of discernment to satisfy Definition 5, i.e., ${}^{\mathrm{\Theta}}{\mu}_{s}=1-{}^{A}{\mu}_{s}=1$ for all s. Then

**Proof of Axiom 4**. Let ${}^{A}{\mu}_{s}=\theta $ for all $s\in {\mathbb{N}}_{n}$ and ${}^{A}{\mu}_{s}^{\prime}=\theta +\epsilon \le 1$ for all s without loss of generality, where $\theta \in [0,1]$ and $\epsilon \in [0,1-\theta ]$. Hence ${}^{A}{\mu}_{s}\le {}^{A}{\mu}_{s}^{\prime}$ for all s and ${}^{A}{\mu}_{s},{}^{A}{\mu}_{s}^{\prime}\in [0,1]$.

**Roots.**

**Extrema.**

- Necessary criterion $\left(\frac{\mathrm{d}f}{\mathrm{d}\theta}f\left(\theta \right)=0\right)$:$$0=-12\epsilon \theta +6\left(\epsilon -{\epsilon}^{2}\right)\iff \theta =\frac{1}{2}\left(1-\epsilon \right).$$
- Sufficient criterion $\left(\frac{{\mathrm{d}}^{2}f}{\mathrm{d}{\theta}^{2}}f\left(\theta \right)\ne 0\right)$:$$\frac{{\mathrm{d}}^{2}f}{\mathrm{d}{\theta}^{2}}f\left(\theta \right)=-12\epsilon >0\mathrm{for}\epsilon 0.$$

- $\epsilon =0:{\theta}_{1/2}=\frac{1}{2}\left(1\pm 1\right)\Rightarrow {\theta}_{1}=0,{\theta}_{2}=1$.
- $\epsilon =1-\theta :{\theta}_{1/2}=\frac{1}{2}\left(1-(1-\theta )\pm \sqrt{1-\frac{{\theta}^{2}}{3}}\right)$. From $\theta +\epsilon \le 1$ follows that θ is further constrained by $\theta \le 0.5$, hence ${\theta}_{1}=-0.229,{\theta}_{2}\approx 0.738$.

**Proof of Axiom 5**.

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**Figure 1.**Architecture of the multilayer attribute-based conflict-reducing observation system MACRO [20].

**Figure 2.**Fuzzy membership functions ${}^{\mathrm{N}}{\mu}_{s}\left({\theta}_{s}\right)$ and ${}^{\overline{\mathrm{N}}}{\mu}_{s}\left({\theta}_{s}\right)$ for two exemplary sensor measurements representing their normal and abnormal conditions, respectively. (

**a**) Normal and abnormal condition for sensor ${S}_{1}$; (

**b**) Normal and abnormal condition for sensor ${S}_{2}$.

**Figure 3.**Effect of wiping errors in the intaglio printing process [92]. (

**a**) Error-free intaglio print result; (

**b**) Print errors caused by wiping errors.

**Figure 4.**Structural design of the printing unit simulator along with the applied sensors (printed in italic) [92].

**Figure 5.**Evaluations of the system health ${}^{\mathrm{N}}\mu $ over time during static operation of the printing unit demonstrator (PU${}_{\mathrm{static}}$). The result depicted here is based on the attribute healths shown in Figure B1.

**Figure 6.**System health evaluation over time during static operation of the printing unit demonstrator (PU${}_{\mathrm{static}}$) by one-class naïve Bayes applying Gaussian (nB${}_{\mathrm{Gauss}}$) and kernel-density estimated (nB${}_{\mathrm{kern}}$) priors.

**Figure 7.**System health evaluation over time during static operation of the printing unit demonstrator (PU${}_{\mathrm{static}}$) by one-class SVM.

**Figure 8.**Evaluation of the system health ${}^{\mathrm{N}}\mu $ over time during manipulated operation of the printing unit demonstrator (PU${}_{\mathrm{manip}}$). The result depicted here is based on the attribute healths shown in Figure B2.

**Figure 9.**System health evaluation over time during manipulated operation of the printing unit demonstrator (PU${}_{\mathrm{manip}}$) by one-class naïve Bayes applying Gaussian (nB${}_{\mathrm{Gauss}}$) and kernel-density estimated (nB${}_{\mathrm{kern}}$) priors.

**Figure 10.**System health evaluation over time during manipulated operation of the printing unit demonstrator (PU${}_{\mathrm{manip}}$) by one-class SVM.

**Table 1.**Physicians’ beliefs about a patient’s disease (according to [65]).

Disease | Meningitis | Brain Tumor | Concussion |
---|---|---|---|

Doctor A | $0.99$ | $0.01$ | $0.00$ |

Doctor B | $0.00$ | $0.01$ | $0.99$ |

**Table 2.**Fusion result of Dempster’s rule of combination (DRC) given the individual beliefs presented in Table 1.

Disease | Meningitis | Brain Tumor | Concussion |
---|---|---|---|

DRC | $0.00$ | $1.00$ | $0.00$ |

**Table 3.**Membership function parameters of the features with respect to the printing unit demonstrator condition monitoring data sets.

RD${}_{\mathbf{static}}$ | RD${}_{\mathbf{manip}}$ | ||||
---|---|---|---|---|---|

Feature | ${D}_{l}$ | ${D}_{r}$ | ${D}_{l}$ | ${D}_{r}$ | |

${f}_{1}$: arithmetic mean of the contact force | 16 | 8 | 8 | 20 | |

${f}_{2}$: root mean square of the solid-borne sound (sound intensity) | 16 | 8 | 20 | 8 | |

${f}_{3}$: index of the frequency component with largest amplitude | 16 | 16 | 8 | 16 | |

${f}_{4}$: arithmetic mean of the wiping cylinder motor current | 16 | 16 | 16 | 8 | |

${f}_{5}$: arithmetic mean of the plate cylinder motor current | 8 | 16 | 16 | 8 |

Attribute | Attribute Description | Number of Features | Features |
---|---|---|---|

1 | motors | 3 | ${f}_{3}$, ${f}_{4}$, ${f}_{5}$ |

2 | contact pressure | 3 | ${f}_{1}$, ${f}_{2}$, ${f}_{4}$ |

3 | motor currents | 2 | ${f}_{4}$, ${f}_{5}$ |

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Mönks, U.; Dörksen, H.; Lohweg, V.; Hübner, M. Information Fusion of Conflicting Input Data. *Sensors* **2016**, *16*, 1798.
https://doi.org/10.3390/s16111798

**AMA Style**

Mönks U, Dörksen H, Lohweg V, Hübner M. Information Fusion of Conflicting Input Data. *Sensors*. 2016; 16(11):1798.
https://doi.org/10.3390/s16111798

**Chicago/Turabian Style**

Mönks, Uwe, Helene Dörksen, Volker Lohweg, and Michael Hübner. 2016. "Information Fusion of Conflicting Input Data" *Sensors* 16, no. 11: 1798.
https://doi.org/10.3390/s16111798