Evidence conflict measure based on OWA operator in open world

Dempster-Shafer evidence theory has been extensively used in many information fusion systems since it was proposed by Dempster and extended by Shafer. Many scholars have been conducted on conflict management of Dempster-Shafer evidence theory in past decades. However, how to determine a potent parameter to measure evidence conflict, when the given environment is in an open world, namely the frame of discernment is incomplete, is still an open issue. In this paper, a new method which combines generalized conflict coefficient, generalized evidence distance, and generalized interval correlation coefficient based on ordered weighted averaging (OWA) operator, to measure the conflict of evidence is presented. Through ordered weighted average of these three parameters, the combinatorial coefficient can still measure the conflict effectively when one or two parameters are not valid. Several numerical examples demonstrate the effectiveness of the proposed method.


Introduction
With the rapid development of industry and military, the speed and quantity of information acquisition are growing at a rapid rate, and various demands emerge increasingly based on the collected information, such as opinion exchange [1] and object assessment [2]. At the same time, the obtained information is often uncertain because of its high complexity, susceptibility to interference, and the malicious effects of the enemy in military and radar information [3,4]. Dempster-Shafer evidence theory (D-S theory), proposed by Dempster [5] and extended by Shafer [6], is one of the common methods used to solve the problem of uncertainty. As compared with probability theory, D-S theory uses the confidence interval instead of the specific probability value to describe the possibility and could be able to handle uncertainty issues more effectively; as the generalized Bayesian theory [7][8][9], D-S theory can deal with the evidence when a priori information is not clear. Based on these advantages, D-S theory has been widely applicated in many systems such as risk and reliability analysis [10][11][12][13], decision making [14][15][16][17][18], classifier recognition [19][20][21][22], diagnosis fault [23,24], and uncertain information processing [25][26][27][28][29][30][31][32]. Θ is defined as a sample space, which is called a frame of discernment (FOD). Θ by the state of things might composition, and these states are mutually exclusive and exhaustive. it is also called the basic probability assignment (BPA). BPA must satisfies the following condition X A22 Y mðAÞ ¼ 1; For a BPA, the belief function Bel: 2 Θ ! [0, 1] is defined as the plausibility function Pl: 2 Θ ! [0, 1] is defined as For two BPAs m 1 and m 2 that have been given, they can be combined by Dempster's combination rule.
k is between [0, 1], named as the coefficient of conflict, is a conflict measurement between two pieces of evidence. The larger the value of k, the greater the conflict between the two evidence. When k = 1, Dempster's combination rule will be invalid. The combination rule provides precious properties, as the commutativity and the associativity, thus it is able to dispose multiple pieces of evidence expediently.
growing number of open world problems, that is obtained FOD is incomplete at this time.
Assigning unrecognized mass to m(;) helps handle the open world problems. Suppose that U is a frame of discernment in an open world. Its power set, 2 U G , is composed So, m G is the generalized basic probability assignment (GBPA) of the FOD U. When m G (;) = 0, the GBPA degenerate into a traditional BPA. Given a GBPA m, GBel is defined as GPl is defined as Given two GBPAs m 1 and m 2 , the generalized combination rule (GCR) satisfies Jousselme's evidence distance Jousselme [46] proposed a new distance to measure the difference between two bodies of evidence, which is also called the evidence distance.
Let m 1 and m 2 be two BPAs on the same frame of discernment Θ, which contain N mutually exclusive and exhaustive hypotheses. Let d BPA (m 1 , m 2 ) represent the distance between m 1 and m 2 , defined as where m 1 and m 2 are two BPAs. And D is a 2 N × 2 N matrix whose elements are DðA; BÞ ¼ jA\Bj where A, B 2 P(Θ) are derived from m 1 and m 2 , respectively. And Deng [52] extended it to the generalized evidence distance. The generalized evidence distance dis between m 1 and m 2 is defined as where m 1 and m 2 are two GBPAs. And D is a 2 N × 2 N matrix whose elements are

Interval correlation coefficient
Wu [55] proposed a new coefficient to describe the correlation between the two evidence.
Let m 1 and m 2 be two BPAs on the same frame of discernment Θ = {θ 1 , θ 2 , . . ., θ N }. It is expressed as a confidence interval of all the single subset of elements on the frame of discernment. That is, m 1 and m 2 is expressed as [Bel 1 (θ i ), Pl 1 (θ i )], i = 1, 2, . . ., N and [Bel 2 (θ i ), Pl 2 (θ i )], i = 1, 2, . . ., N. Let ir(m 1 , m 2 ) represent the interval correlation coefficient between two BPAs m 1 and m 2 , defined as where Ordered weighted averaging (OWA) operator Averaging operator is a tool to implement information fusion [57,58]. Yager [56] proposed an OWA operator that is tantamount to the "or" operation and "and" operation to be extended in fuzzy operation. When the operator is applied, its "and or" degree can be adjusted according to specific requirement, thereby more able to meet the practical application needs. Suppose F: R n ! R, there is an n-dimensional weight vector ω associated with F, ω = (ω 1 , ω 2 , . . ., ω n ), b i is the i-th largest element in array (a 1 , a 2 , . . ., a n ). N-dimensional OWA operator F is defined as which satisfies the following condition For example, when ω = (1, 0, 0, . . ., 0), F(a 1 , a 2 , . . ., a n ) = max(a 1 , a 2 , . . ., a n ) = b 1 , OWA operator is equal to the "or" operator in fuzzy operation. Or when o ¼ 1 n ; 1 n ; Á Á Á ; 1 n À Á , OWA operator is equivalent to arithmetic mean operator now.

Generalized interval correlation coefficient
Interval correlation coefficient, which describe the degree of correlation between evidence, is able to measure evidence conflict. But as the status of problems in the open world becomes more and more important, interval correlation coefficient as a definition of measurement method in the closed world, hindering the application and development of it. In this paper, generalized interval correlation coefficient gir is proposed, which can manage open world problems.
Let m be a GBPA on the frame of discernment Θ = {;, θ 1 , θ 2 , . . ., θ N }. It is expressed as the confidence interval of all the single subset of elements on the frame of discernment. That is, m is expressed as f½GBelð;Þ; GPlð;Þ; ½GBelðy 1 Þ; GPlðy 1 Þ; ½GBelðy 2 Þ; GPlðy 2 Þ; Á Á Á ; ½GBelðy N Þ; GPlðy N Þg: ð20Þ We use the interval information on the belief function which is defined in Eq (10) and the plausibility function which is defined in Eq (11) of propositions in GET. The confidence interval of the single subset θ is expressed as [Bel(θ), Pl(θ)]. Bel(θ) indicates the part of the evidence that explicitly supports θ, and Pl(θ) indicates the part of the evidence that potential supports θ.
The simulation consequence shows that calculating the interval of entire 2 Θ subsets not only resulted in a huge computational quantity, but also caused double counting of information. For instance, both GPl(a) and GPl(a, c) contain the information of m(a), however our purpose is assigning m(a) to only one monad because the information containing in m(a) is definite. Utilizing only the value of the single subset can preserve the vast majority of valid information while having a simpler calculation.
In classic D-S theory, the mass function and its interval representation are equivalent. The mass function can be calculated by its interval representation as follows [59], , and the interrelationships between non-empty sets in GET is the same as above. Therefore GBPA can be obtained from its interval representation by the same method, on the contrary it is clearly that the interval representation can be calculated with GBPA. So we take into account the interval [GBel(θ), GPl(θ)] to reserve the valuable information as much as possible. The following example illustrates the specific calculation method. Example 1. Suppose that there is a frame of discernment of Θ = {a, b, c}, and a GBPA is given as Definition 2. Let m 1 and m 2 be two GBPAs on the same frame of discernment Θ = {θ 1 , θ 2 , . . ., θ N }, the generalized interval correlation coefficient gir is defined as The generalized interval correlation coefficient gir has the following three properties.  (22) is same to Eq (17), and it is clear that Eqs (23) and (16) are also identical. Therefore, the gir degenerates to the interval correlation coefficient when FOD is complete.
In In this example, we should first denote m 1 and m 2 in the form of the confidence interval of all the single subset of elements on the FOD. In order to express more intuitively, we use Table 1 to get the confidence interval. And gir is calculated as follows, It is obvious that the traditional interval correlation coefficient does not apply to the problems in this circumstance, because m 1 and m 2 are not BPAs but GBPAs. Using gir as defined in this paper, the degree of correlation between the two pieces of evidence can be calculated, which is able to determine whether the evidence are in high conflict according to the specific situation and whether it is appropriate to use generalized combination rule.
Example 3. Let a frame of discernment be Θ = {a, b}, six sets of GBPAs and their gir and k are given in Table 2.
As shown in Table 2, gir is decreasing monotone and can be used to measure the degree of conflict between GBPAs. The following Fig 1 graphically show the positive correlation between 1 − gir and k. Moreover, k is unconventionally large because of its defects on dealing with the empty set yet 1 − gir changes smoothly and evenly from 0 to 1. The establishment of above three properties and this example indicate that the generalized interval correlation coefficient  gir is a measure of the consistency between two GBPAs. The larger gir, the greater the correlation between the two evidence, so the smaller the conflict between them, and vice versa.

A combinatorial parameter based on OWA operator
Since Zadeh [40] proposed that Dempster's combination rule in highly conflict evidence would produce a perverse result, many scholars had tried to solve the problem from various angles. Before evidence fusion, the degree of conflict between the evidence must be measured first. How to obtain an effective method to measure the conflict between the evidence, and based on the conflict measurement to determine whether the evidence required for processing prior to combination, has become an essential issue. So far, there have been several methods to measure the evidence conflict in open world. For example, generalized conflict coefficient k mainly indicates the degree of the two evidence contained in each other, generalized evidence distance dis can calculate the difference between the evidence, generalized interval correlation coefficient gir is proposed to measure the correlation between the evidence. The above three coefficients measure the evidence conflict from three different aspects. In order to achieve a comprehensive conflict measure from multiple aspects, a combining parameter based on OWA operator is proposed in this paper. When one or two coefficients are invalid, the combinatorial parameter can still effectively measure the evidence conflict in open world.
Definition 3. The ordered weighted vector [60] ω = (ω 1 , ω 2 , . . ., ω n ) is determined by the following equation: where Q is called fuzzy semantic quantization operator, and satisfies The calculate the process of ω is shown in Table 3. From Table 3 When using OWA operator weighting, the information characterization of each parameter must be unified first. Three parameters are unified into the larger the parameter, the greater the conflict, so calculate with (1 − gir) instead of gir. In this paper, these three parameters k, dis, and (1 − gir) are weighted by the OWA operator, and we adopt the fuzzy semantics of "at least half" (0, 0.5) to retain more information in the variable. When n = 3, α = 0 and β = 0.5, it is easy to get the ordered weighted vector (28) and (29). The combinatorial conflict parameter comb is defined as follows.
Definition 4. Let m 1 and m 2 be two GBPAs on the same frame of discernment Θ, and calculate k, dis, and (1 − gir). The combinatorial conflict parameter comb is defined as where o Ã ¼ 2 3 ; 1 3 ; 0 À Á , and a i is the i-th largest element in array (k, dis, 1 − gir). Deng [52] proposed a conflict model for GET, denoted as The objective of the proposed model is to offer to the decision maker a two-tuples conflict measurement. It also considers two different parameters and has a number of robustness. However, in practical engineering application, we prefer a determinate number rather than a two-tuples model, since it must be analyzed first the respective applicable occasions of each coefficient in the two-dimensional model, which greatly reduces the efficiency. Moreover, as Deng's proposal, when the FOD is incomplete and m(;) 6 ¼ 0, the conflict model should mainly consider K, whereas it had better depend on dis. GET as the extension of D-S theory, its biggest advantage and innovation is capable of managing the problems in open world, yet the processing method of its conflict model appears a recession since many problems in open world are regarded as in the closed world. In the following, some numerical examples are given to demonstrate the OWA operator calculation process and its superiority.
Example 5. Suppose that there is a frame of discernment of Θ = {a, b, c}, and two GBPAs are given as This example is Zadeh's classic paradox, the direct combination of these two GBPAs through the Dempster's combination rule will produce results contrary to common sense. The new combinatorial parameter comb in this example is very close to 1, which indicates there is a quite high degree of conflict between two evidence, and not suitable for Dempster's rule in this case. Example 6. Assume a frame of discernment Θ = {a, b, c}, and that two GBPAs are given as We first calculate such that In addition, Lefèvre [47] proposed an interesting method to measure the conflict of evidence. He ingeniously combined the information of conflict coefficient and evidence distance by weighted average, and regarded the mass assigned to conflict, that is m(;) as an alarm signal announcing disagreement. However, the essence of Lefèvre's method is an adaptive weighting between two combination methods which have the completely opposite allocation rule about m(;), namely m(;) = 0 in Dempster's combination rule while it is approximately 1 in conjunctive rule. The evidence distance plays a weight role to command the result on the basis of the conflict coefficient k. Therefore, as long as the m(;) in conjunctive rule is small, videlicet k is close to 0, the conflict management results will be incorrect just like this example which shows the evidences are totally identical. Based on the above analysis, our method comb is obviously more reasonable.
The evidence in this case is relatively easy to appear in the application. For instance, the frame of discernment of the multi-sensor contains two very similar units b and c in military reconnaissance, the information obtained by sensors has a high probability of confusing b and c with malicious interference from the enemy. If the conflict between the evidence is measured by conflict coefficient k, the result is 0. And at this time using Dempster's combination rule will get the fallacious result m(a) = 1, which indicates the information may be unable to identify b and c. From the results, m 1 and m 2 are conflicting evidence of each other, and must be carefully combined. In this case directly using the combination rule to deal with the GBPAs may lead to an erroneous result, and should try to modify the evidence model first. According to the results of Example 2, gir between m 1 and m 2 is 0.298, and we can calculate that k = 0.78, 1 − gir = 0.702, so Obviously, comb is able to effectively measure the conflict between two GBPAs in open world.  Table 4 and Fig 2.  Table 4 and Fig 2 show that, regardless of how the subset A changes, the value of k remains constant, which is obviously contrary to common sense. Although dis and (1 − gir) vary with the subset A, due to lopsided processing of the empty set, the value of dis and (1 − gir) is too small when A = {1, 2, 3} or in its vicinity. The new combinatorial parameter comb correctly reflects that the conflict between m 1 and m 2 is minimal when A = {1, 2, 3}, and the greater the deviation of the subset A from {1, 2, 3}, the greater the conflict of evidence. Furthermore, the value of comb is also intuitive.
In summary, three existing measurements k, dis and 1 − gir sometimes produce counterintuitive and inaccurate results. This does not explain that they are worthless, but is actually quite the opposite. Three coefficient measure the conflict from different aspects, that is, k mainly calculates the degree of the two evidence contained in each other, dis can indicate the difference between the evidence, and gir is proposed to measure the correlation between the evidence. comb is a synthesis of the three coefficients, it blends various effective information and avoids the errors caused by the one-sidedness of single factor, accordingly is highly accurate.

Conclusion
In the application of information fusion, GET has been extensively used due to its ability to deal with evidence conflict when the frame of discernment is incomplete. But how to determine a valid parameter to measure the degree of conflict between the evidence in open world is still an open issue. We first extend the interval correlation coefficient to gir in this paper, which is capable of calculating the correlation between the evidence in incomplete FOD. D-S theory and GET are the methods of processing information fusion, the parameters describing the evidence conflict are also a kind of information. So we use the "information fusion" idea to combine the three coefficients. By extracting their respective valid information, we obtain a combinatorial parameter based on OWA operator to measure the evidence conflict. Several numerical examples demonstrate that compared with using a single parameter, the proposed method is more reliable and effective.