In the area of **fuzzy** **measures** and capacities some research has been done to prove a Radon-Nikodym type theorem, as Graf [6] puts it. That is, re- searchers try to solve the question of when a given **fuzzy** measure can be expressed as the (Choquet) in- tegral [1] of a function with respect to another given measure. When such relationship is found, we can say that this function is the Radon-Nikodym deriva- tive. Graf [6] was one of the first authors to deal with this problem. He focuses on sub-additive **fuzzy** **measures** and gives (Theorem 4.3) necessary and sufficient conditions for this to happen. Sugeno [21] deals with the same problem but considering dis- torted probabilities. Rébillé [17] deals with the case of almost subadditive set functions of bounded sum. In this paper we consider the definition of the **Hellinger** **distance** for **fuzzy** **measures**. To do so, we use the concept of derivative as used in [21], and the Choquet integral as an alternative to the Lebesgue integral. We will illustrate the definition with some examples using distorted probabilities, and prove some properties.

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A simulation study is conducted to compare the es- timation performances of the considered **distance** mea- sures. Considering the overall statement of the simu- lation results, we reached minimum M AE values with taking into account the **distance** measure described by Kaufmann and Gupta [13] and Heilpern-2 [12] for Case- II. Besides, the **distance** measure described by Chen and Hsieh [4] gives minimum M AE values for Case- III. It is demonstrated that the **distance** measure used by Abdalla and Buckley [1][2] is not convenient to esti- mate **fuzzy** linear regression model parameters with MC methods. Since all maximum values of M AE are calcu- lated with the **distance** measure that is considered by Abdalla and Buckley [1][2] for both Case-II and Case- III.

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We discuss a new notion of **distance** on the space of finite and nonnegative **measures** on Ω ⊂ R d , which we call **Hellinger**–Kantorovich **distance**. It can be seen as an inf- convolution of the well-known Kantorovich–Wasserstein **distance** and the **Hellinger**- Kakutani **distance**. The new **distance** is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass.

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based on dynamic plans are proved. We will briefly recall these results, however, since the cone structure did not play a role in [Man07] we will reinterpret the results in our setting. In the following we understand weak convergence in the space of **measures** as convergence against bounded and continuous functions. Moreover, a curve s 7→ µ(s) ∈ M(Ω) is called weakly continuous if and only if µ(s) weakly converges to µ(t) in M(Ω) for s → t.

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studied on the **distance** between set order **fuzzy** **Measures**. Chan et al. (Bhattacharjee, 1991) have extensively studied set ordering among functions with a application to reliability. In the light of these journals we apply the notion to the case of **Fuzzy** **Measures** on the dispersive nature of **Fuzzy** **Measures**. *Corresponding author: Beulah, A.

Sets f The comparative measure indicates with a high degree of certainty that the FSs of f are nearly identical. A possible application of the comparative measure is to the problem of ranking. Comparing the comparative measure against the DM, which may also be used for ranking [4], the ordering of the FSs differs. By observ- ing the absolute values of the **measures**, according to the DM the most distant pair is a and the second most distant is e, however, it is the other way round according to the comparative measure. This is because the SM indicates there is some similarity between the FSs within Fig. 1(a), which causes the comparative measure to decrease in dissimilarity/**distance**. However, the FSs

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This paper presents new axiomatic definitions of entropy measure using concept of probability and **distance** for interval-valued intuitionistic **fuzzy** sets (IvIFSs) by considering degree of hesitancy which is consistent with the definition of entropy given by De Luca and Termini. Thereafter, we propose some entropy **measures** and also derived relation between **distance**, entropy and similarity **measures** for IvIFSs. Further, we checked the performance of proposed entropy and similarity **measures** on the basis of intuition and compared with the existing entropy and similarity **measures** using numerical examples. Lastly, proposed similarity **measures** are used to solve problems in the field of pattern recognition and medical diagnoses.

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While **distance** **measures** traditionally use a single real value to express **distance**, representing the **distance** as a FS would give a richer, more accurate comparison, reflecting the uncertainty inherent in FSs. This work follows and draws on work by [5], which describes a real-valued directional **distance** measure, and presents a **distance** measure which describes **distance** as a FS. In [5], alpha-cuts (α-cuts) are used to measure **distance** by comparing each α-cut of one FS with the same α-cut of another FS. This, however, introduces difficulties for non-normal FSs where an α-cut results in the empty set. Though the problem was addressed, the method taken is limited by using a substituted value of **distance** for α-cuts where one of the **fuzzy** sets is not present. The method introduced in this paper removes this problem by comparing every α-cut (or mass assignment) of one

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72 state of students knowing the results of their performance. The problem description uses the concept of IFS that makes it possible to render two important facts. First, Values of each subject performance changes for each careers. Second, values of each student’s for subject performance. We use the the normalized Euclidean **distance** measure method given in [5, 7, 10] to measure the **distance** between each student and each career.

So, in parallel we tried to develop the dynamic approach, which was not too successful at the early stages. Only after realizing and exploiting the connection to the cone **distance** in Summer and Autumn of 2013 we were able to connect LET systematically with the dynamic approach. The crucial and surprising observation was that optimal plans for E and lifts of **measures** µ ∈ M (X) to **measures** λ on the cone C could be identified by exploiting the optimality conditions systematically. Corresponding results were presented in workshops on Optimal Transport in Banff (June 2014) and Pisa (November 2014).

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Note during final preparation. The earliest parts of the work developed here were first presented at the ERC Workshop on Optimal Transportation and Applications in Pisa in 2012. Since then the authors developed the theory continuously further and presented results at different workshops and seminars, see Appendix A for some remarks concerning the chronological development of our theory. In June 2015 they became aware of the parallel work [24], which mainly concerns the dynamical approach to the **Hellinger**- Kantorovich **distance** discussed in Section 8.5 and the metric-topological properties of Section 7.5 in the Euclidean case. Moreover, in mid August 2015 we became aware of the work [11, 12], which starts from the dynamical formulation of the **Hellinger**-Kantorovich **distance** in the Euclidean case, prove existence of geodesics and sufficient optimality and uniqueness conditions (which we state in a stronger form in Section 8.6) with a precise characterization in the case of a couple of Dirac masses, provide a detailed discussion of curvature properties following Otto’s formalism [33], and study more general dynamic costs on the cone space with their equivalent primal and dual static formulation (leading to characterizations analogous to (7.1) and (6.14) in the **Hellinger**-Kantorovich case).

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Part II is devoted to Logarithmic Entropy-Transport (LET) problems (Sect. 6) and to their applications to the **Hellinger**–Kantorovich **distance** HK on M( X ) . The **Hellinger**–Kantorovich **distance** is introduced by the lifting technique in the cone space in Sect. 7, where we try to follow a presentation modeled on the standard one for the Kantorovich–Wasserstein **distance**, independently from the results on the LET-problems. After a brief review of the cone geom- etry (Sect. 7.1) we discuss in some detail the crucial notion of homogeneous marginals in Sect. 7.2 and the useful tightness conditions (Lemma 7.3) for plans with prescribed homogeneous marginals. Section 7.3 introduces the definition of the HK **distance** and its basic properties. The crucial rescaling and gluing techniques are discussed in Sect. 7.4: they lie at the core of the main metric properties of HK, leading to the proof of the triangle inequality and to the characterizations of various metric and topological properties in Sect. 7.5. The equivalence with the LET formulation is the main achievement of Sect. 7.6 (Theorem 7.20), with applications to the duality formula (Theo- rem 7.21), to the comparisons with the classical **Hellinger** and Kantorovich distances (Sect. 7.7) and with the Gaussian **Hellinger**–Kantorovich **distance** (Sect. 7.8).

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Many different **measures** quantifying the degree of dis- crimination between two probability distributions have been studied in the past. They are frequently called dis- tance **measures**, although some of them are not strictly metrics. They have been applied to different areas, such as medical image registration (Josien P.W. Pluim [21], classification and retrieval, among others. This class of distances is referred, in the literature, as the class of φ, f or g-divergences (Csisza’r [11]; Vajda [22]; Morales et al. [23]; the class of disparities (Lindsay [16]). The di- vergence **measures** play an important role in statistical theory, especially in large theories of estimation and testing.

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Abstract—The **fuzzy** integral (FI) with respect to a **fuzzy** measure (FM) is a powerful means of aggregating information. The most popular FIs are the Choquet and Sugeno and most research focuses on these two variants. The arena of the FM is much more populated, including numerically-derived FMs such as the Sugeno λ-measure and decomposable measure, expert-defined FMs, and data-informed FMs. The drawback of numerically-derived and expert-defined FMs is that one must know something about the relative values of the input sources. However, there are many problems where this information is unavailable, such as crowd-sourcing. This paper focuses on data- informed FMs, or those FMs that are computed by an algorithm that analyzes some property of the input data itself, gleaning the importance of each input source by the data they provide. The original instantiation of a data-informed FM is the agreement FM, which assigns high confidence to combinations of sources that numerically agree with one another. This paper extends upon our previous work in data-informed FMs by proposing the uniqueness measure and additive measure of agreement for interval-valued evidence. We then extend data-informed FMs to **fuzzy** number (FN)-valued inputs. We demonstrate the proposed FMs by aggregating interval and FN evidence with the Choquet and Sugeno FIs for both synthetic and real-world data.

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they are characterized by an independence axiom [73, 43]. This property implies some limitations in the way the weighted sum can model typi- cal decision behaviours. To make this more precise, let us consider the example of two criteria having the same importance, an example which we will consider in more details in Section 3.5. We are interested in the following four alternatives: x is bad in both criteria, y is bad in the first criterion but good at the second one, z is good in the first criterion but bad in the second one, and t is good in both. Clearly x ≺ t and the DM is equally satisfied by y and z since the two criteria have the same importance. However, the comparison of y, z with x and t leads to several cases. First, the DM may say that x ∼ y ∼ z ≺ t, where ∼ means indifference. This depicts a DM who is intolerant, since both criteria have to be satisfied in order to get a satisfactory alternative. In the opposite way, the DM may think that x ≺ y ∼ z ∼ t, which depicts a tolerant DM, since only one criterion has to be satisfactory in order to get a satisfactory alternative. Finally, we may have all intermediate cases, where x ≺ y ∼ z ≺ t. An important fact is that, due to addi- tivity, the weighted sum is unable to distinguish among all these cases, in particular, all decision behaviours related to tolerance or intolerance are missed. These phenomena are called interaction between criteria. They encompass also other phenomena such as veto. We will show in this chapter that the notions of capacity and **fuzzy** integrals enable to model previous phenomena.

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The compression algorithms builds a dictionary or a model using training text documents set. These generated models are used to the train classifiers. Test document can be assigned to a particular author by compressing this test document for each author specific model or dictionary which is generated during training phase. The test document is attributed to an author which is produced the highest compression rate [2]. In order to measure the compressed **distance** similarity many metrics were proposed in the literature [5].

The above equation DM characterized by distance measures can be considered by characterization of a symmetrically compositive sum-form information measurable functional equation.. The fu[r]

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From the Comparison of three methods **Fuzzy** **Distance**, Trident **Distance** and Sub-Trident **Distance** it is observed that, the optimal path for Method: 1 is 1→3→6→9→11, the optimal path for Method: 2 is 1→2→6→9→11 and the optimal path for Method: 3 is 1→5→8→10→11.Among all these three methods, the average value using Sub-Trident **Distance** is very small i.e., 0.002.So the path 1→5→8→10→11 obtained from Sub-Trident **Distance** is the best optimal path.

Abstract—The **fuzzy** integral (FI) with respect to a **fuzzy** measure (FM) is a powerful means of aggregating information. The most popular FIs are the Choquet and Sugeno and most research focuses on these two variants. The arena of the FM is much more populated, including numerically-derived FMs such as the Sugeno λ-measure and decomposable measure, expert-defined FMs, and data-informed FMs. The drawback of numerically-derived and expert-defined FMs is that one must know something about the relative values of the input sources. However, there are many problems where this information is unavailable, such as crowd-sourcing. This paper focuses on data- informed FMs, or those FMs that are computed by an algorithm that analyzes some property of the input data itself, gleaning the importance of each input source by the data they provide. The original instantiation of a data-informed FM is the agreement FM, which assigns high confidence to combinations of sources that numerically agree with one another. This paper extends upon our previous work in data-informed FMs by proposing the uniqueness measure and additive measure of agreement for interval-valued evidence. We then extend data-informed FMs to **fuzzy** number (FN)-valued inputs. We demonstrate the proposed FMs by aggregating interval and FN evidence with the Choquet and Sugeno FIs for both synthetic and real-world data.

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