## 3. Paradoxes (Source of Conflicts) in DS Combination Rule

Dempster–Shafer theory, introduced and developed by Dempster and Shafer [

6,

7,

8], has many merits by contrast to Bayesian probability theory [

9]. However, to use DS sensor fusion algorithm for robust application, we have to overcome the fusion paradoxes. Based on the application in a multi-sensor system, this theory also has its shortcomings [

10]. The different levels of performance of sensors, cluster, and interference of a complex environment may lead to conflicts among evidences. When evidences are highly conflicting, the fusing results obtained by the DS combination method are normally contrary to common sense. When the conflicting factor

K is close to 1, this rule cannot obtain reasonable fusing results as the denominator is approximately 0. These counterintuitive phenomena of the DS theory are called paradoxes. According to Reference [

11], there are mainly three types of paradoxes.

#### 3.1. Completely Conflicting Paradox:

In this situation, there are two sensors and one sensor output completely contradicts the other sensor output. The following example depicts the situation:

**Example** **2.** In the multi-sensor system, assume that there are four evidences in the frame, that $\mathsf{\Theta}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{A,B,C\}$, and that proposition A is true.

Sensor 1: ${m}_{1}$(A) = 1, ${m}_{1}$(B) = 0, ${m}_{1}$(C) = 0,

Sensor 2: ${m}_{2}$(A) = 0, ${m}_{2}$(B) = 1, ${m}_{2}$(C) = 0,

Here, the two sensors are completely conflicting each other. The conflicting factor in Equation (

6) is

K = 1, which reports that evidences from sensor 1 and sensor 2 are completely conflicting. Under such circumstances, the DS combination rule cannot be applied.

#### 3.2. “One Ballot Veto” Paradox

For a multi-sensor system (more than two sensors), one sensor completely contradicts all other sensor outputs. The following example depicts the situation:

**Example** **3.** In the multi-sensor system, assume that there are four evidences in the frame, that $\mathsf{\Theta}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{A,B,C\}$, and that proposition A is true.

Sensor 1: ${m}_{1}$(A) = 0.7, ${m}_{1}$(B) = 0.2, ${m}_{1}$(C) = 0.1,

Sensor 2: ${m}_{2}$(A) = 0, ${m}_{2}$(B) = 0.9, ${m}_{2}$(C) = 0.1,

Sensor 3: ${m}_{3}$(A) = 0.75, ${m}_{3}$(B) = 0.15, ${m}_{3}$(C) = 0.1,

Sensor 4: ${m}_{4}$(A) = 0.8, ${m}_{4}$(B) = 0.1, ${m}_{4}$(C) = 0.1,

Clearly, sensor 2 is faulty and contradicts the results of the other 3 sensors. Applying DS combination rule, we get K = 0.9, ${m}_{1234}\left(A\right)$ = 0/0.1 = 0, ${m}_{1234}\left(B\right)$ = 0.097/.1 = 0.97, and ${m}_{1234}\left(C\right)$ = 0.003/0.1 = 0.03. The fusing results are contrary to the assumed proposition that A is true. A high value of K proposes high contradiction among sensors. This counterintuitive result is caused by the erroneous sensor 2 values. Interestingly, DS combination rule completely omits a proposition even if a single sensor outputs zero evidence.

#### 3.3. “Total Trust” Paradox

Here, one sensor highly contradicts the other sensor but both of them have a common focal element with low evidence. The following example depicts the situation:

**Example** **4.** In the multi-sensor system, assume that there are two evidences in the frame and that $\mathsf{\Theta}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{A,B,C\}$.

Sensor 1: ${m}_{1}$(A) = 0.95, ${m}_{1}$(B) = 0.05, ${m}_{1}$(C) = 0,

Sensor 2: ${m}_{2}$(A) = 0.0, ${m}_{2}$(B) = 0.1, ${m}_{2}$(C) = 0.9,

Applying DS combination rule, we get ${m}_{12}\left(A\right)$ = 0, ${m}_{12}\left(B\right)$ = 1, and ${m}_{12}\left(C\right)$ = 0, K = 0.99. Here, common sense suggests that either $m\left(A\right)$ or $m\left(C\right)$ is correct, but the wrong proposition B is identified to be true with total confidence even though senor 1 and 2 nearly negates this idea.

## 5. Entropy in Information Theory under DS Framework

Information is a measure of the compactness of a distribution; logically, if a probability distribution is spread evenly across many states, then its information content is low, and conversely, if a probability distribution is highly peaked on a few states, then its information content is high [

25]. Information is a function of distribution. Entropy measures the compactness of a distribution of information. Entropy is zero when BPA is assigned to a single element, thus creating the most informative distribution. When BPA is uniformly distributed, entropy is at maximum and agrees with the idea of least informative distribution.

In information theory, Shannon entropy [

26] is often used to measure the “amount of information” in a variable.

where

n is the amount of basic states in a state space and

${p}_{i}$ is the probability of state

i. It is clear that the quantity of entropy is always associated with the amount of states in a system. In the framework of DS evidence theory, the uncertain information is represented by both mass functions and the FOD. Deng entropy [

9] considers both.

where |

A| denotes the cardinality of the focal element

A. Other works related to entropy under DS framework can be found in the literature [

27]. Based on Shannon and Deng entropy, we propose a new belief entropy, which considers

Bel and

Pl of mass function, cardinality of focal elements, and number of elements in FOD. The goal of the proposed entropy is to capture the uncertainty of information under DS framework, which are omitted by Shannon and Deng entropy.

where |

X| denotes the cardinality of

X, which represents the number of element in FOD. The exponential factor

${exp}^{(\frac{\left|A\right|-1}{\left|X\right|})}$ in the new belief entropy represents the uncertain information in the number of elements of FOD that has been ignored by Deng entropy. This probability interval considers the lower and upper bounds of evidence that are Bel and Pl, respectively. The new belief entropy which considers Deng entropy and the interval probability can better measure the uncertainty of BPA.

#### Properties of Proposed Entropy Function

**Property** **1.** Mathematically, the value range of the new belief entropy is $(0,+\infty )$. According to DS evidence theory, a focal element A consists of at least one element and the limit of its element number is the scale of FOD. FOD consists of at least one element, and there is no maximum limit; thus, the ranges of $\left|A\right|$ and $\left|X\right|$ are the same, denoted as $[1,+\infty )$. The range of a mass function m(A) is $(0,1]$. Depending on the value of |A|, the believe (Bel) and plausibility (Pl) ranges could be between $(0,+\infty ]$. In the proposed entropy equation, where $\left|A\right|\in [1,+\infty )$ and $\left|X\right|\in [1,+\infty )$, $Bel\left(A\right)\in (0,+\infty ]$ and $Pl\left(A\right)\in (0,+\infty ]$. Thus, the range of the proposed entropy can be denoted $(0,+\infty )$.

**Property** **2.** New belief entropy can degenerate to the Shannon entropy when the mass function is Bayesian. If the mass function m(A) is Bayesian, then BPA is assigned only on single element subset and |A| = 1. In this case, the new belief entropy can degenerate to the following equation, which is exactly equal to Shannon entropy: **Property** **3.** Non-negativity. We know that $0<(Bel\left({m}_{i}\right)+Pl\left({m}_{i}\right))/2<1$. As a result, entropy $\left(m\right)>0$. Only if $m\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1$ and only if A is Bayesian, then Entropy $\left(m\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$. Thus, new entropy satisfies the non-negativity property.

**Property** **4.** Consistency with DS theory framework. The new entropy is consistent with the DS theory framework. Thus, it satisfies the consistency with DS theory framework properties.

**Property** **5.** Probability consistency. If m is Bayesian, then $m\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}Bel\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}Pl\left(A\right)$ for all $A\in X$. Thus, new entropy satisfies the probability consistency property.

The following example shows the properties of proposed entropy and how it is better at capturing uncertainties compared to Shannon and Deng entropy.

**Example** **5.** Given a frame of discernment $\mathsf{\Theta}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{a,b,c\}$ for a mass function $m\left(a\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}m\left(b\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}m\left(c\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1/3$.

${E}_{Sh}\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}-(\frac{1}{3}lo{g}_{2}\frac{1}{3}+\frac{1}{3}lo{g}_{2}\frac{1}{3}+\frac{1}{3}lo{g}_{2}\frac{1}{3})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.585$

${E}_{Deng}\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}(\frac{1}{3}lo{g}_{2}\frac{1/3}{{2}^{1}-1}+\frac{1}{3}lo{g}_{2}\frac{1/3}{{2}^{1}-1}+\frac{1}{3}lo{g}_{2}\frac{1/3}{{2}^{1}-1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.585$

${E}_{P}\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}-(\frac{(1/3+1/3)}{2}lo{g}_{2}\frac{(1/3+1/3)}{2.({2}^{1}-1)}.{exp}^{\left(\frac{1-1}{3}\right)}\phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.222222em}{0ex}}\frac{(1/3+1/3)}{2}lo{g}_{2}\frac{(1/3+1/3)}{2.({2}^{1}-1)}.{exp}^{\left(\frac{1-1}{3}\right)}\phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.222222em}{0ex}}$

$\frac{(1/3+1/3)}{2}lo{g}_{2}\frac{(1/3+1/3)}{2.({2}^{1}-1)}.{exp}^{\left(\frac{1-1}{3}\right)})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.585$

This showed that the result of the proposed entropy is identical to Shannon entropy and Deng entropy when the belief is only assigned on single elements (or Bayesian).

**Example** **6.** Given a frame of discernment $\mathsf{\Theta}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{a,b,c\}$, mass function $m\left(a\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}m\left(b\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}m\left(c\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}m(a,b)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}m(a,c)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}m(b,c)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}m(a,b,c)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1/7$. Bel and Pl values can be calculated for all the BPA, which is shown in Table 2. Shannon and Deng entropy is calculated to compare with the values from proposed entropy.

Shannon entropy only considers mass function value and has the lowest entropy. Deng entropy considers both mass function value and cardinality on focal elements. It calculates higher entropy than Shannon. Proposed entropy considers mass function value (central value of probability interval), cardinality of both focal elements, and FOD. It results in the highest entropy value compared to Shannon and Deng. If a FOD consists of 7 elements compared to say 3 elements, intuitively it can be said that the 7-element FOD should have higher entropy because it is less compact. Also, because the proposed entropy considers central value of probability interval $\frac{(Bel+Pl)}{2}$, it is capturing more uncertainty compared to only mass function. As a result, the proposed entropy function abides by the DS framework and is superior in capturing uncertainty compared to Shannon and Deng entropy.

## 6. Proposed Steps to Eliminate Paradoxes

With increasing use of sensors application in real-time decision making, we need an algorithm which can fuse sensor outputs both in the space domain and the time domain. The goal of the proposed method is to eliminate the paradoxes of the original DS combination rule and work as a decision-level sensor fusion algorithm in both the space and time domains. We are adopting “revision of original evidence before combination” because we do not want to lose the associative and commutative properties of the original DS rule. The proposed method is a distance-based method. It calculates the relative distances between the sensor evidences (classification output). Then, based on average distance, it classifies which sensor output is credible and which sensor output is incredible. Then, it penalizes the incredible sensor output using the novel entropy function so that the incredible sensor has less effect on the fused output. It also rewards the credible sensor input so that the credible sensor carries more weight towards the fused output. At the end, modified evidence is fused using the original DS sensor fusion equation. The following example is used to showcase the steps and to compare the final fused results with works from open literature.

**Example** **7.** In a multisensor-based target recognition system, assume there are three types of targets to be recognized: $\{A,B,C\}$. Suppose there are five sensors. They could be any type of sensors. After data acquisition at a specific moment by five sensors, data are processed and classification IDs are generated. Generated IDs from five sensors are listed as BPAs:

Sensor 1: ${m}_{1}:{m}_{1}\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.41,{m}_{1}\left(B\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.29,{m}_{1}\left(C\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.30$

Sensor 2: ${m}_{2}:{m}_{2}\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.00,{m}_{2}\left(B\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.90,{m}_{2}\left(C\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.10$

Sensor 3: ${m}_{3}:{m}_{3}\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.58,{m}_{3}\left(B\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.07,{m}_{3}(A,C)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.35$

Sensor 4: ${m}_{4}:{m}_{4}\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.55,{m}_{4}\left(B\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.10,{m}_{4}(A,C)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.35$

Sensor 5: ${m}_{5}:{m}_{5}\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.60,{m}_{5}\left(B\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.10,{m}_{5}(A,C)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.30$

This is a classic example of the “one ballot veto” paradox. Bel and Pl values can be calculated for all the BPA, which is shown in

Table 3.

**Step 1**: Build a multi-sensor information matrix. Assume, for a multi-sensor system, there are N evidences (sensors) in the frame

$\mathsf{\Theta}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{{H}_{1},{H}_{2},\dots ..,{H}_{M}\}$ (objects to be detected).

**Step 2**: Measure the relative distance between evidences. Several distance function can be used to measure the relative distance. They all have their own advantages and disadvantages regarding runtime and accuracy. We have used Jousselme’s distance [

28] function. Jousselme’s distance function uses cardinality in measuring distance which is an important metric when multiple elements are present in one BPA under DS framework. The effect of different distance functions (Euclidean, Jousselme, Minkowsky, Manhttan, Jffreys, and Camberra distance function) on simulation time and information fusion can be found in the literature [

29]. Assuming that there are two mass functions indicated by

${m}_{i}$ and

${m}_{j}$ on the discriminant frame

$\mathsf{\Theta}$, the Jousselme distance between

${m}_{i}$ and

${m}_{j}$ is defined as follows:

where

$D\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{|A\cap B|}{|A\cup B|}$ and |.| represents cardinality.

**Step 3**: Calculate sum of evidence distance for each sensor.

**Step 4**: Calculate global average of evidence distance.

**Step 5**: Calculate belief entropy for each sensor by using Equation (

11), and normalize.

It is interesting to note that, although

${m}_{3},{m}_{4},$ and

${m}_{5}$ have zero

m(C) values, it has nonzero

Pl values. As a result, it will consider nonzero

Pl values of

m(C) when calculating entropy.

${E}_{P}\left({m}_{1}\right)=1.5664,\phantom{\rule{0.222222em}{0ex}}{E}_{P}\left({m}_{2}\right)=0.469,\phantom{\rule{0.222222em}{0ex}}{E}_{P}\left({m}_{3}\right)=1.3861,\phantom{\rule{0.222222em}{0ex}}{E}_{P}\left({m}_{4}\right)=1.513,\phantom{\rule{0.222222em}{0ex}}{E}_{P}\left({m}_{5}\right)=1.483$. Normalize the entropy:

**Step 6**: The evidence set is divided into two parts: the credible evidence and the incredible evidence. From Equations (14) and (15):

The intuition is that, if an evidence has higher distance than average distance (which is calculated using all the evidences), then probably that evidence is faulty and should be penalized (incredible evidence). If an evidence distance is lower than average, then that evidence is in harmony with other evidence and should be rewarded (credible evidence). Lower entropy means lower uncertainty, and that evidence should be rewarded more for credible evidence. The opposite is true for incredible evidence. Therefore, we needed a function which has large slope as it goes near to zero. Natural log function fits the bill. As a result, the following reward and penalty function is proposed:

Using Equations (16)–(18), calculate reward and penalty value for each evidence. $Rewar{d}_{1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.4103,\phantom{\rule{0.222222em}{0ex}}Penalt{y}_{2}=0.0759,\phantom{\rule{0.222222em}{0ex}}Rewar{d}_{3}=1.5326,\phantom{\rule{0.222222em}{0ex}}Rewar{d}_{4}=1.445,\phantom{\rule{0.222222em}{0ex}}Rewar{d}_{5}=1.4647$.

Normalize reward and penalty values to get evidence weights. ${w}_{1}=0.2379,\phantom{\rule{0.222222em}{0ex}}{w}_{2}=0.0128,\phantom{\rule{0.277778em}{0ex}}{w}_{3}=0.2585,\phantom{\rule{0.222222em}{0ex}}{w}_{4}=0.2437,\phantom{\rule{0.222222em}{0ex}}{w}_{5}=0.2471$. Obviously, we can observe that there is a high conflict between the evidence ${m}_{2}$ and other evidences. Therefore, ${m}_{2}$ is defined as an incredible evidence and has very low weight. Other evidences are supported by each other, so their weights are higher than ${m}_{2}$.

**Step 7**: Modify the original evidences.

The resulting modified evidences are m(A) = 0.5298, m(B) = 0.1477, m(C) = 0.0726 and m(A,C) = 0.2499.

**Step 8**: Combine modified evidence for ($n-1$) times (for this example, 4 times) with DS combination rule by using Equations (5) and (6). How to apply the fusion rule is important. For this example, if evidences ${m}_{1}$ and ${m}_{2}$ are fused with modified evidence, then ${m}_{12}\left(A\right)=0.8125,\phantom{\rule{0.222222em}{0ex}}{m}_{12}\left(B\right)=0.0325,\phantom{\rule{0.222222em}{0ex}}{m}_{12}\left(C\right)=0.062\phantom{\rule{3.33333pt}{0ex}}and\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.222222em}{0ex}}{m}_{12}(A,C)=0.093$. Now, to get ${m}_{123}$, if ${m}_{12}$ values are fused with ${m}_{12}$ values using Equations (5) and (6), that would be wrong. To get ${m}_{123}$, ${m}_{12}$ values should be fused with the original modified evidence from step 7. It is also evident that, for single elements, if that element has higher value after step 7, it will have highest value after fusing ($n-1$) times. The higher the value after step 7, the higher the value after fusion.

Table 4 compares the results of the proposed algorithm with other combination methods from open literature for Example 7.

As seen from

Table 4, when evidences are in high conflict, classical Dempster’s combination rule produces counterintuitive results that are not correct. With increases in number of sensors, Murphy’s simple averaging, Deng’s weighted averaging, and Han’s novel weight averaging, Wang’s weighted evidence and Jiang’s uncertainty measure give reasonable results, although their final combination results are slightly inferior to the outcomes of our proposed approach. Wang et al. [

32] showed in his paper that the modified evidences before the fusion steps are

$m\left(A\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5048,m\left(B\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.184,m\left(C\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.068,$ and

$m\left(AC\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.243$. Now, the modified evidence for

m(A) is lower than our proposed method as stated in step 7. Also as explained in step 8, it is unlikely that, after fusing these evidences (

$n-1$) times (4 for this example) using original DS combination rule, the fused

${m}_{12345}\left(A\right)$ will be higher than our proposed method. Using the evidences presented in Wang’s work, the recalculated fused evidences are presented in

Table 4. The proposed method also has the highest convergence rate (rate of

$m\left(A\right)$ value goes towards 1) after sensor 3. It is reasonable to say that the proposed method overcomes the paradoxes of classical DS rule and produces competitive fusion results compared to that of combination rule available in open literature.

Figure 2 shows how fused evidence of

$m\left(A\right)$ changes with the addition of new sensors and compares multiple methods from the literature. As

$m\left(A\right)$ is the correct evidence, how it is changing with the inclusion of new sensor evidence is important for justification of the fused result. The proposed method penalizes

$m\left(A\right)$ when only two sensors are used. As a result,

$m\left(A\right)$ starts with lower evidence for the proposed method compared to other methods (number of sensors = 2). However, with the inclusion of correct evidences from sensors 3 and 4,

m(A) converges towards 1 quickly for the proposed method compared to other methods. As

m(A) evidence converges towards 1, the convergence rate becomes slow for all the methods. A zoomed-in view shows that the proposed method has higher

m(A) evidence after fusing 5 sensor evidences compared to other methods from the literature.

It can be seen from Example 7 that this method is applicable for any multi-sensor system because fusion occurs after classification ID is created from sensor output. As an example, let us assume multiple cameras are used for object classification. Camera output (video/image) will go through a classifier (example: neural network) for object ID classification. After classification, the output may have similar syntax to Example 7. Then, the proposed method can be applied to find out which sensor is providing erroneous data and to fuse them accordingly.