Evaluation of the quality and relevance of a fuzzy partition

1 Introduction

The process of recognition and classification of different stimuli into broader perceptions and concepts is one of the most fundamental activities of human thinking. As a matter of fact, one of the most primitive and common activities of human reasoning consists in generalizing similar objects into concepts or classes. Since the paper presented by Bellman, Kalaba and Zadeh [5] fuzzy techniques stand as one of the fields with major advances and applications in the area of (soft) classification and pattern recognition. In [3], [4], [24], [28] and [14] we can find a variety of applications, perspectives and approaches that have been developed in relation to these two areas.

The concept of a partition is, without any doubt, one of the pillars of unsupervised classification problems. In the fuzzy case, a standard notion for a partition has been that proposed by Ruspini [32], where the total degree of membership is completely distributed among all classes, which was later generalized by Krishnapuram and Keller [22], only requiring that the membership of an object to any class is positive. Therefore, under the notion of a Ruspini’s partition, the total sum of the membership degrees of an object to all classes must be equal to 1, while in the perspective of Krishnapuram and Keller, it is sufficient that for each element, its membership degree is greater than zero. Based on these ideas, and considering that in most cases fuzzy partitions do not verify the Ruspini condition, the notion of fuzzy classification systems have been defined by means of families of aggregation functions, allowing to analyze the classes obtained in a fuzzy partition [1]. Therefore, a classification problem can be evaluated through a fuzzy classification system regardless of the notion of partition used.

Fuzzy classification systems, as proposed in [1] (see also [2]), were conceived as a structure allowing the treatment of complex classification problems following two key ideas. On the one hand, under the theoretical framework introduced by Dombi [15], [16] regarding aggregation operators, fuzzy classification systems were proposed as an alternative approach for non-associative connectives through the use of the concept of recursiveness.

On the other hand, fuzzy classification systems were proposed as a structure allowing the evaluation of the established classification from De Morgan triplets i.e., by using recursive rules (satisfying the De Morgan laws) to evaluate three key characteristics of the family of classes that are obtained in a fuzzy partition: redundancy, coverage and relevance.

Redundancy refers to a certain orthogonality in the family of classes, which is then viewed as a particular representation system of the set of objects [1]. Hence, redundancy suggests the possible existence of an alternative representation to be found by means of an appropriate redefinition of the family of classes or by removing some of the existent classes.

The second characteristic of coverage refers to how different aspects of reality are fully verified by a family of classes. Finally, the third characteristic of relevance is understood as the necessity of including, or not excluding, a class or family of classes from the classification. In this way, decision makers can have a hint on how to improve the classifier performance, e.g., searching for some missing classes, proposing greater or smaller classes, or deleting some classes.

According to the above, fuzzy classification systems propose indexes that allow measuring the degree of redundancy (overlapping) between classes, the degree in which all of the classes accurately cover the aspect(s) of reality under consideration, and the degree in which some classes can be disregarded.

This approach is similar to some standard procedures in statistics and image analysis, where a crisp partition is pursued in such a way that every pixel is member of one and only one category (see e.g. [6]). Any automatic learning procedure should also take into account a number of indices, each one capturing a specific aspect that should be jointly balanced out for learning purposes.

In this line of research, some first approaches led to the development of works such as those presented in [8–10], and [7], and more recently in [29], [30], about a more in-depth study of aggregation functions for evaluating the redundancy and coverage of a particular classification. Such studies allowed the development of what are now known as overlap functions [8], [17] and grouping functions [9], [10]. These new functions do not impose the associativity of the most classic operators, such as t-norms and t-conorms, with which the concepts of redundancy and coverage, respectively, were originally defined (see, e.g., [20]).

An issue that is still open and with a broad field of development, is the study of the relevance property [7], focusing on the optimal number of clusters for a fuzzy partition. Previous works have studied this property from a more statistical perspective [2], or even as a dimensionality reduction problem [12].

Here we propose some first steps towards the characterization of relevance, and with it, a general study of a global quality criterion for a fuzzy partition. Therefore, this approach also allows establishing a stopping criterion for the calculation of the optimal number of classes for a fuzzy partition, based on the relevance of its classes.

The above allows evaluating the quality of a fuzzy partition. In this sense, an important highlight for our approach is that it does not require external tools for its application, such as the comparison with a fuzzy control partition (as suggested in [18]).

Our approach defines relevance as an observable property of a fuzzy partition. Therefore, two processes are necessary to determine the relevance of a class in a fuzzy partition: a comparison process among the classes of the fuzzy partition, and a process allowing to determine the degree of relevance of a class or set of classes. Regarding this comparison process, we identify a commutative group structure on any fuzzy classification system. Such structure is formed by the aggregations rules of any fuzzy classification system and the negation of these rules.

As mentioned above, the rules of a fuzzy classification system determine the degree of coverage and overlap, so their negations represent the degree of non-coverage and non-overlap, respectively. Keeping this in mind, we are interested in comparing such degrees looking for fuzzy partitions with high degrees of coverage and low degrees of overlap. In this sense, similarity relations [23] provide an interesting tool to carry out the mentioned comparison process, as they will allow, under specific properties, defining criteria for evaluating the quality of fuzzy partitions and the relevance of its classes.

The consideration of similarity relations is motivated by two facts: 1) Similarity relations satisfy a “min-transitivity” property that is similar to one of the conditions in the definition of fuzzy subgroups. 2) Similarity relations can be considered to effectively group elements into crips sets whose members are similar to each other to some specified degree, i.e., they generate a set of equivalence relations that can be ordered through inclusion. However, establishing similarity relations may not be a simple task because obtaining operators verifying the property of min-transitivity can be certainly complex. Furthermore, not any similarity relationship is useful for the mentioned comparison process, as we shall see in Section 4.2. In order to examine this problem, proximity relations [21] will be used to subsequently obtain similarity relations through transitive closure. We establish the conditions that are required for both relations to remain invariant.

This paper is organized as follows. In the next section, some preliminary definitions and theorems are presented. In Section 3, we discuss a general framework to address the concept of relevance. Subsequently, we analyze such concept in the perspective of a fuzzy partition. In Section 4, we present the commutative group structure formed from any fuzzy classification system. On this structure, we propose to consider a similarity relation to compare its elements. This similarity relation is in turn constructed through a proximity relation. We prove that under certain conditions, some elements remain invariant. In Section 5, we define a quality criterion and a relevance criterion that allow, through an algorithm, to establish a stopping criterion in the determination of an adequate the number of classes in a fuzzy partition. In the same line, we establish a comparison criterion to select the optimal fuzzy partition as long as the above criteria are satisfied. In Section 6, the proposed method is applied to an image segmentation problem, for which fuzzy partitions are obtained by using the fuzzy c-means algorithm. Finally, Section 7 is devoted to present some conclusions and future work.

2 Preliminaries

In this section some preliminary definitions and necessary results for the development of the proposal are established.

Definition 1. [11] An aggregation operator A : ⋃ n ∈ ℕ [ 0 , 1 ] n → [ 0 , 1 ] is a function that satisfies:

A (0, . . . , 0) = 0 and A (1, . . . , 1) =1,

Note that for each n ∈ ℕ an n-ary aggregation operator A _n is defined and therefore, A is understood as a family of n-ary aggregation operators. For simplicity, in what follows we shall use the term aggregation operator to refer to both a family of n-ary aggregation operators and n-ary operators themselves.

Definition 2. [1] A recursive rule ρ is a family of aggregation operators { ρ n : [ 0 , 1 ] n → [ 0 , 1 ] } n > 1 such that there exist an ordering rule π and two sequences of binary operators {L _n  : [0, 1] ² → [0, 1]} _n>1 and {R _n  : [0, 1] ² → [0, 1]} _n>1 such that for each n and for each (a₁, . . . , a _n ) ∈ [0, 1]  ⁿ , ρ n ( a π ( 1 ) , . . . , a π ( n ) ) = L n ( ρ n - 1 ( a π ( 1 ) , . . . , a π ( n - 1 ) ) , a π ( n ) ) = R n ( a π ( 1 ) , ρ n - 1 ( a π ( 2 ) , . . . , a π ( n ) ) .

A recursive rule is a family of operators allowing a sequential reckoning by means of a successive application of binary operators, once data have been properly ordered: the ordering rule assures that new data do not introduce modifications in the relative position of items already ordered. Recursiveness assures consistency of such a family of operators by assuming that such a rule is operational, in the sense that it can be evaluated both from left and right by means of a sequence of binary operators. A standard recursive rule will be one based upon the identity ordering rule (i.e., the ordering rule that keeps the order of data as they are given to us). Hence, recursiveness indeed allows the generalization of associativity, i.e., in case such a sequence of binary operators is given by a unique binary operator (L _i  = R _j , for all i, j), we should be talking about a rule based upon an associative binary operator.

Recursive rules constitute an elegant formal mechanism to deal with aggregations of any arbitrary number of elements. Moreover, the recursive approach allows this mechanism to be robust under changes in the cardinality of the data, as it guarantees that all the operators in a family of aggregation functions are tightly related to a unique aggregation procedure: the binary operators that build up the recursive rule (see also [31]). This is useful in the context of unsupervised classification, where the number of classes in which the data has to be segmented is typically unknown a priori. In this context, recursive rules provide a robust mechanism to deal with the aggregation of diverse class information for different number of classes [1], automatically assessing the classification performance. We recall next the definition of fuzzy classification systems.

Definition 3. [1] Let us assume a finite set of objects X. A fuzzy classification system is a finite family C of n fuzzy classes, where each c ∈ C has an associated membership function μ _c  : X → [0, 1] , together with a recursive triplet (φ, φ, N) such that

φ is a standard recursive rule, where φ₂ (0, 1) = φ₂ (1, 0) =0;

According to Definition 3, a fuzzy classification system can be denoted by (C, φ, φ, N). Notice that in a fuzzy classification system each x ∈ X has a membership degree μ _c  (x) associated with each class c ∈ C. As our purpose is to analyze the fuzzy classes, from now on we consider each standard recursive rule to act on such membership degrees, that is, for any given object x ∈ X we are interested in the sets φ {μ _c  (x)/ c ∈ C} or φ {μ _c  (x)/ c ∈ C} . In other words, the elements a _i in the previous definition will be given by the membership degrees μ _i  (x), where μ _i  (x) denotes the membership degree of the object x to the i-th class of C, i = 1, . . . , n.

Thus, for instance, we have that φ n ( μ 1 ( x ) , μ 2 ( x ) , … , μ n ( x ) ) = N - 1 [ φ n ( N ( μ 1 ( x ) ) , … , N ( μ n ( x ) ) ) ] .

In the same line, notice that φ is a conjunctive recursive rule, in the sense that φ _n  (μ₁ (x) , …, μ _n  (x)) =0, whenever μ _j  (x) =0 for certain j ∈ {1, . . . , n}. As a direct consequence, φ is a disjunctive recursive rule, in the sense that φ _n  (μ₁ (x) , …, μ _n  (x)) =1, whenever there is μ _j  (x) such that μ _j  (x) =1.

Then, given a fuzzy partition C the coverage of an object x ∈ X is analyzed by means of the disjunctive rule φ {μ _c  (x)/ c ∈ C}, and its redundancy is analyzed by means of the conjunctive rule φ {μ _c  (x)/ c ∈ C}. Without loss of generality, we refer to φ _n  (μ₁ (x) , μ₂ (x) , …, μ _n  (x)) as φ _n  (c₁, c₂, …, c _n ).

Example 1. We can consider a recursive triplet (φ, φ, N) such that,

φ n ( μ 1 ( x ) , … , μ n ( x ) ) = 3 ∏ k = 1 n μ k ( x ) 1 + 2 ∏ k = 1 n μ k ( x )

According to each selected rule, an aggregated value is obtained, to be understood as the degree up to which the family of fuzzy classes C satisfies a particular property or characteristic with respect to an object x ∈ X (in this case, coverage and redundancy). In this sense, it is also desirable to analyze the aggregated information of such characteristic for all objects x. To this aim, we propose the following definition.

Definition 4. Let X = {x₁, . . . , x _m } be a finite set of objects, C = {c₁, . . . , c _n } a family of fuzzy classes over this set and ρ _n an aggregation operator representing the degree up to which the family of fuzzy classes satisfies a specific property for an object x ∈ X. Then the global degree of ρ _n on X, denoted ρ n T , is defined as the aggregation of the values of ρ _n for all objects x ∈ X by means of an aggregation operator A. This is, ρ n T = A ( ρ 1 n , . . . , ρ mn ) . Where ρ _jn denotes the degree up to which the n fuzzy classes satisfy a specific property for object x _j ,  j = 1, …, m.

Such aggregation operator can be of very different nature (conjunctive, disjunctive or averaging). Definition 4 is a generalization of the definition given in [13] (about the degree of global coverage) to any property analyzed by a recursive rule ρ _n  . The Example 2 illustrates the aim of this definition.

Example 2. Let C = {c₁, c₂, c₃} be a partition with three classes of the set X = {x₁, x₂, x₃, x₄}. We select the disjunctive recursive rule φ _n from Example 1 to evaluate the coverage (see Table 1).

If we select the aggregation operator A ( φ 13 , φ 23 , φ 33 , φ 43 ) = 1 4 ( φ 13 + φ 23 + φ 33 + φ 43 ) (remember that φ _ij denotes the degree of coverage of j classes with respect to the object i), then a global degree of coverage of the class set, i.e., φ 3 T = A (φ₁₃, φ₂₃, φ₃₃, φ₄₃) =0.5445, is obtained.

Next we will remember the definitions corresponding to similarity relations, proximity relations and fuzzy subgroups.

Definition 5. Let X be a set and w be a fuzzy subset of X × X. Then w is called a proximity relation (also, tolerance relation) on X if the following properties hold, ∀x, y ∈ X:

Reflexivity: w (x, x) =1

Definition 6. [23] Let X be a set and v be a fuzzy subset of X × X. Then v is called a similarity relation on X if the following properties hold, ∀x, y, z ∈ X:

Reflexivity: v (x, x) =1

Definition 7. [27] The set of all fuzzy subsets of X is called the fuzzy power set of X and is denoted by FP (X). Let G = (G, ·) be an arbitrary group and let μ ∈ FP (G). Then μ is called a fuzzy subgroup of G if

μ (x · y) ≥ min {μ (x) , μ (y)}, ∀x, y ∈ G and

Example 3. Let G = {e, a, b, ab} be the Klein four-group. The fuzzy subset of G defined by μ (e) = μ (ab) =0.7 and μ (a) = μ (b) =0.2 is a fuzzy subgroup of G.

Definition 8. [27] Let G be a group and let μ be a fuzzy subgroup of G. Then μ is called a normal fuzzy subgroup of G if it is an Abelian fuzzy subset of G, i.e., μ (x · y) = μ (y · x), ∀x, y ∈ G.

According to [27] it is well-known that if v is a similarity relation on a set Ω, then v _t  = {(x, y) |v (x, y) ≥ t} is an equivalence relation on Ω for all t ∈ [0, 1]. This leads to the following theorem.

Theorem 1. [23] Let v be a similarity relation on a finite set G. Then there is a binary operation · on G such that G = (G, ·) is a group and a fuzzy subgroup μ of G such that v (x, y) = μ (x · y^-1) or v (x, y) = μ (x^-1 · y), ∀x, y, z ∈ G if and only if the equivalence classes determined by the crisp equivalence relation v _t  = {(x, y) |v (x, y) ≥ t} have the same size for all t ∈ [0, 1].

Definition 9. [23] Let G be a group and let v be a similarity relation on G. Then v is said to be right-invariant if (∀ x, y, z ∈ G) v (x, y) = v (x · z, y · z). In a similar way, v is said to be left-invariant if (∀ x, y, z ∈ G) v (x, y) = v (z · x, z · y). v is translation invariant if it is both left and right-invariant.

According to Definition 8, the next theorem establishes the connection between a translation invariant similarity relation and a normal fuzzy subgroup.

Theorem 2. [23] Suppose v is a translation invariant similarity relation on G. Then v is a fuzzy subgroup of G × G if, and only if, μ is a fuzzy subgroup of G such that μ (x) = v (e, x) and μ is commutative, where e is the identity element of G.

In this section we address the concept of relevance, establishing the elements to determine the relevance of an object. We use such elements to evaluate the relevance of a class in a fuzzy partition from a fuzzy classification system.

In this section we present the commutative group structure formed from any fuzzy classification system. On such a structure we propose to consider similarity relations as a way of comparing the elements of such a structure.

Based on the algebraic structure of commutative group and a translation invariant similarity relation on such group, in this section we define a quality criterion for a fuzzy partition and a relevance criterion for classes. An algorithm that describes the comparison process is presented. Such algorithm allows establishing the minimum number of classes in a fuzzy partition in such a way that a high-quality fuzzy partition is obtained. Additionally, we present a criterion that allows comparing two fuzzy partitions that meet the quality criterion and the relevance criterion.

Following the arguments and remarks previous to Theorem 3, we propose the first quality criterion. Consider a fuzzy classification system (C, φ, φ, N) with C = {c₁, . . . , c _n }, where C is obtained from a fuzzy iterative, non-hierarchical classification algorithm (e.g., fuzzy c-means or possibilistic c-means among others). Also, consider the groups G _o  = {φ _n , φ _n , σ _n , δ _n } and G _p  = {e, a, b, c} together with a translation invariant similarity relation v on G _p (according to Eq. 4). Then, it is possible to compute the equivalence relations v _t  = {(x, y) |v (x, y) ≥ t} for all t ∈ [0, 1]. In this way, it holds that if 0 ≤ t₁ < ⋯ < t _p  = 1, then v _{t p}  ⊆ ⋯ ⊆ v _{t 1}  = G _o  × G _o .

Let 1 = w ( φ n T , φ n T ) , k = w ( φ n T , φ n T ) , m = w ( φ n T , δ n T ) and s = w ( φ n T , σ n T ) and let x, y ∈ G _p  = {e, a, b, c}. Then, let us define the equivalence relations

K = v_t=k (x, y) ={ (x, y) |v (x, y) ≥ k },

M = v_t=m (x, y) = {(x, y) |v (x, y) ≥ m},

S = v_t=s (x, y) = {(x, y) |v (x, y) ≥ s}.

Taking into account the above considerations and as well as Theorem 3 and Theorem 4, the following quality criterion is defined:

Quality criterion. A fuzzy partition Cwith nclasses is a quality partition if m > s > k,or equivalently, if it holds that S ⊂ M. Otherwise, we say that it is of low quality. Hence, the value of mdetermines the degree of quality of the fuzzy partition, and the higher, the better.

Remember that m provides the degree of coverage without considering the redundancy of the classes. Furthermore, we can extend the same argument for a second relevance criterion.

Relevance criterion. Given a fuzzy partition Cwith nclasses, let C _i  ⊆ C = {c₁, …, c _n } be a fixed nonempty family of classes, such that C _i is the subset of n - 1classes obtained by deleting class c _i  ∈ C, with i = {1, . . . , n}. Also, let k _i , m _i , and s _i be the corresponding values for won C _i . If k _i  > s _i , or equivalently if it holds that S _i  ⊂ K _i ,then c _i is a highly relevant class in partition C. Otherwise, we say that c _i is of low relevance for partition C.

Notice that this criterion proceeds by removing a class from the partition under consideration, one at a time, in order to study its relevance. In this sense, the removed class will be relevant if the aggregation of the remaining classes indicates that the degree of non-coverage is greater than the degree of overlap. In other words, a class is considered relevant if its elimination decreases the degree of coverage in a greater proportion than the degree of overlap.

Therefore, we can state on more formal terms the basic intuition supporting both criteria, by the following definition.

Definition 11. A fuzzy quality partition is called a high-quality fuzzy partition if all classes are relevant.

According to the criteria established above, it is possible to develop a procedure to assess the quality and relevance of a given fuzzy partition. This procedure is summarized in the Algorithm 1.

This Algorithm 1 proceeds as follows: In the first step, a partition C = {c₁, c₂} is computed through an iterative, non-hierarchical classification algorithm. For practical purposes we will use the fuzzy c-means. From a proximity relation, Algorithm 1 determines if the partition C is of quality. In the case n = 2 (two classes), the relevance of the classes is not evaluated because, under our framework, the aggregation of a single element or fuzzy partitions with a single class are not considered. Therefore, if the partition with two classes is of quality, then the algorithm ends. Otherwise, the algorithm increases the number of classes by one, returning to step 1, until a high-quality partition is found, that is, a quality partition in which all its classes are relevant.

Algorithm 1 Evaluation of quality of a fuzzy partition

Input: X finite set of objects, |X| = r, recursive rules φ and φ and the negation N (corresponding to a fuzzy classification system), proximity relation w and n = 2. Let z be an integer denoting the maximum number of classes to be considered.

1: C = output partition of the fuzzy c-means with n classes

2: Compute φ _n , φ _n , σ _n and δ _n . Consider G _o  = {φ _n , φ _n , σ _n , δ _n }

3: Compute w on G _o . Let  k = w (φ _n , φ _n ), m =. w (φ _n , δ _n ), s = w (φ _n , σ _n )

4: If m < s ≤ k then

5:  While n < z do

6:   n = n + 1, return to 1.

7:  end while

8:  RETURN “No quality partition found” and END.

9: else if m > k > s and c = 2 then return C and END.

10: else if m > k > s and c > 2 then go to 12

11: end if

12: Compute C _i for all i = 1, …, n

13: Compute φ _i , φ _i , σ _i and δ _i for each C _i and consider G _i  = {φ _i , φ _i , σ _i , δ _i }

14: for each C _i do compute w on G _i

15:  if s _i  ≤ k _i and n < z then n = n + 1, return to 1.

16:  else if k _i  > s _i then RETURN C and END.

17:  else if n = z then RETURN “No high-quality partition found” and END.

18:  end if

19: end for

Notice that, in particular, the process ends when a high-quality fuzzy partition has been found, i.e, a partition fullfiling the quality and relevance criteria and with the smallest number of classes. However, it can happen that there is no optimal number of classes. In this case, it is suggested to restart the fuzzy c-means algorithm with a different seed or with a different fuzziness parameter. Similarly, it can happen that there are two partitions fulfilling all the proposed criteria. In principle, the partition with the smallest number of classes is desirable, but this does not guarantee that relevant information is not lost. Therefore, we propose an additional criterion to compare two high-quality fuzzy partitions when we use Algorithm 1 several times under different configurations (different seeds or fuzziness parameter values). Let us stress that this criterion is not used in Algorithm 1.

Comparison criterion. Let C _{n 1} and C _{n 2} be two high-quality fuzzy partition on X. Let m₁ = and m₂ = respectively be the values for w for each partition. Then C _{n 1} is better that C _{n 2} if m₁ > m₂.

6 Application

In order to apply the defined criteria and Algorithm 1, we have selected the image presented in Fig 1., considering the unsupervised classification problem of obtaining classes of similar pixels. We consider the recursive triplet of Example 1 and the translation invariant similarity relation of Example 5.

OPEN IN VIEWER

Fig.1

Aurora borealis.

According to Algorithm 1, we start by obtaining a partition with n=2 classes through the fuzzy c-means algorithm, leading to obtain m = 1, s = 0.44, k = 0.44. As s = k, then the partition is of low quality.The corresponding classes of the fuzzy partition are presented in Fig. 2.

OPEN IN VIEWER

Fig.2

Classes obtained by applying the fuzzy 2-means algorithm (left: class 1, right: class 2). The gray scale represents the membership degree of each pixel to each class, where black = 0 and white = 1.

The low quality of the partition obtained from the image can be observed in some regions where objects with different intensity of color have been classified in the same class. For instance, there is almost no distinction between trees and much of the sky.

According to Algorithm 1, we again apply the fuzzy c-means obtain a new partition with n = 3 classes. Then, we compute the proximity relation w, which is shown in Table 5.

As w (φ₃, δ₃) =0.7 > w (φ _n , σ _n ) =0.61 this 3-class partition is a quality partition. After computing the transitive closure of w, we compute the (translation invariant) similarity relation v, which is shown in Table 6.

Thus, we have that v_t=0.7 (c, b) ⊂ v_t=0.61 (c, a). Equivalently, by Theorem 2 and Theorem 3, we have that v (c, b) =0.7 ⊂ v (c, a) =0, 61 (being v a fuzzy subgroup).

As the partition is a quality partition, following Algorithm 1 and according to Step 12, we evaluate the relevance of the classes considering the 2-class sub-partitions  C₁, C₂, C₃ and computing the corresponding values for w.

For C₃ it is k₃ = w (φ₂, φ₂) =0.67 > w (φ₂, σ₂) =0.46 = s₃. For C₂ it is k₂ = w (φ₂, φ₂) =0.59 > w (φ₂, σ₂) =0.4 = s ₂ and for C₁ it is k₃ = w (φ₂, φ₂) =0.69 > w (φ₂, σ₂) =0.48 = s₃.

According to the Relevance Criterion and Step 16, all the classes are then highly relevant. The corresponding classes of the fuzzy partition are presented in Fig. 3.

OPEN IN VIEWER

Fig.3

Classes obtained after applying the fuzzy 3-means algorithm (left: class 1, center: class 2 and right: class 3). The gray scale represents the membership degree of each pixel to each class, where black = 0 and white = 1.

Under the procedure above, we may conclude that this 3-class partition obtained through the fuzzy c-means is a high-quality partition. In this partition, for example, there is a much better distinction than before between the trees and the sky. In general, with a fuzzy partition with three classes, the image is segmented into regions that allow understanding it in terms of the intensity of the color of the objects.

As a way of checking these results, we later applied the fuzzy c-means to obtain a partition with n = 4 classes, for which the different quality indicators were computed. Particularly, for this partition we obtained m = w (φ₄, δ₄) =0.62 < w (φ₄, σ₄) =0.73 = s. Therefore, this 4-class partition was assessed as of low quality, which somehow provides a confirmation of the previous result.

Furthermore, in order to illustrate the robustness of the obtained results, the procedure has been performed with two different De Morgan triplets (given in Table 7). Table 8 summarizes the results.

Under triplets 1 and 2 the same result was obtained, allowing us to assert that the optimal number of classes for the image worked are three classes.

Through this paper, some basic elements are examined for characterizing the property of relevance regarding the evaluation of a fuzzy partition from a classification system. In particular, three aspects are considered for the study of relevance: 1) the comparison process between classes and the way they cover the objects under consideration; 2) the estimation of degrees of intensity in the changes generated by the elements in the valuation space; and 3) the specification of a stopping criterion for inclusion of classes in a fuzzy partition.

We explore the algebraic group structure G _o  = {φ _n , φ _n , σ _n , δ _n }, established from a fuzzy classification system. On such a structure, a translation invariant similarity relation v is applied in such a way that the equivalence relations v _t form a lattice under inclusion. Such equivalence relations acquire practical meaning because they allow comparing the coverage of a family of classes and the overlap degree, non-covering degree and non-overlap degree. However, our proposal establishes similarity relations based on proximity relations. In this way, some values of both relations remain invariant. This is, we prove that under certain conditions the degrees of similarity and the degrees of proximity for certain elements of the group are the same. Such conditions allow establishing a criterion of relevance for the classes and a criterion of quality for the partitions.

Degrees of similarity have been presented by examining pairs of properties, such as the similarity between the coverage degree and the overlap degree, or the similarity between the degree of grouping and non-overlap. Based on this, it is desirable that there is a greater similarity between the degrees of coverage and non-overlap, than between the degree of coverage and overlap. Hence, we proposed a couple of criteria that allow determining the quality and relevance of a fuzzy partition, showing that relevance is a local property and is responsible for determining the size of the partition, i.e., the optimal number of classes of a fuzzy partition.

We established a process that helps the decision maker to choose the best family of classes. Such process, that we have described algorithmically, is a iterative process because such an algorithm requires obtaining fuzzy partitions with different number of classes, and possibly also different initialization, as many times as necessary, until a high-quality partition is found. In this sense, the process ends when it has found the high-quality fuzzy partition with the lowest number of classes (if possible).

In general, a fuzzy partition is said to be of quality if it has high degrees of coverage and low degrees of overlap. However, this is a feature that several partitions may have. Therefore, we established an extra criterion that allows comparing two or more high-quality fuzzy partitions. In such a process, the evaluation of the relevance of the classes, as expected, is a determining aspect.

As future work, we propose to compare the established criteria with traditional indexes for fuzzy partitions, in such a way that two scenarios can be presented: on the one hand, complementing the existing indexes with the criteria we have proposed or, on the other hand, finding other advantages in the use of the criteria.

In the same line, it is necessary to carry out a study on the performance of the criteria with different families of De Morgan triplets and different proximity relations. In general, we established a process that does not depend on these two elements, however, we may carry out processes that are computationally inefficient. A key aspect in our proposal is to establish a proximity relation that effectively reflects the idea of closeness.

Another future work that may be addressed in this context is the introduction of paired fuzzy sets (see [25]). In this way we could extend the classification model not only introducing general aggregation tools like overlap and grouping degrees, but incorporating opposite arguments, as for instance, the degree of class separation, perhaps understanding separate as the opposite concept of overlap. Despite the apparent increase in computation difficulties due to the extra information being included, the existence of opposites should help in collecting more evidence for properly assessing the quality of a partition, and hopefully improve the confidence in the model. Similarly, our proposal may be extended to the alternative approach proposed in [26], where the concept of an aggregation rule should be defined from a computational point of view, focusing on the computational properties of such an aggregation, i.e., on the manner in which the aggregation values are computed.