Mathematical Problems in Engineering

Volume 2015, Article ID 495829, 11 pages

http://dx.doi.org/10.1155/2015/495829

## Fuzzy Collaborative Clustering-Based Ranking Approach for Complex Objects

^{1}School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650500, China^{2}Key Laboratory of IOT Application Technology of Universities in Yunnan Province, Yunnan Minzu University, Kunming 650031, China^{3}Department of Mathematics and Statistics, Allama Iqbal Open University, Islamabad 44000, Pakistan^{4}School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received 14 March 2015; Revised 4 July 2015; Accepted 6 July 2015

Academic Editor: Laura Gardini

Copyright © 2015 Shihu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper makes a discussion on the ranking problem of complex objects where each object is composed of some patterns described by individual attribute information as well as the relational information between patterns. This paper presents a fuzzy collaborative clustering-based ranking approach for this kind of ranking problem. In this approach, a referential object is employed to guide the ranking process. To achieve the final ranking result, fuzzy collaborative clustering is carried on the patterns in the referential object by using the collaborative information obtained from each ranked object. Since the collaborative information of ranking objects is represented by cluster centers and/or partition matrices, we give two forms of the proposed approach. With the aid of fuzzy collaborative clustering, the ranking results can be obtained by comparing the difference of the referential object before and after collaboration with respect to ranking objects. One can find that this proposed ranking approach is totally different from the previous ranking methods because of its completely collaborative clustering mechanism. Moreover, some synthetic examples show that our proposed ranking algorithm is valid.

#### 1. Introduction

For decision makers, the goal of ranking is to discover a mechanism that induces an increasing or decreasing order over the data set of given objects to be ranked. Ranking problem appears in many fields such as information retrieval [1], selection and evaluation [2, 3], image similarity measure [4], and income distribution [5]. It can be easily found that, during the ranking process, only the attribute information is usually considered. Thereinto, the attribute information may be given by fuzzy numbers [6, 7], intuitionistic fuzzy numbers [8], or interval-valued intuitionistic fuzzy numbers [9]. Generally speaking, in these cases, the ranking problem can be regarded as a multiple criteria decision making problem [10, 11], and the detailed decision making process can be executed by many approaches [11–13].

Most ranking algorithms receive attribute information and output a real-valued sequence as the ranking results. Sometimes, besides the attribute information, the relational information of the patterns is also provided for the ranking, especially in the fields of machine learning [14–16]. Up to now, there have been several developments in theory and algorithms for learning over relational information. For example, Agarwal [17] developed an algorithmic framework for learning ranking functions on graph data; Mihalcea [18] presented an innovative unsupervised method for automatic sentence extraction using graph-based ranking algorithms; Lee et al. [19] focused on the flexibility of recommendation and proposed a graph-based multidimensional recommendation method; Agarwal [20] considered the graph learning problems through ranking objects relative to one another, and so on [21, 22]. No matter what problem to investigate or which algorithm to construct, in general, the patterns that need to be sorted are described by only one kind of information: attribute information or relational information.

Obviously, the above-mentioned methods are concentrated on attribute information or relational information separately. In practical terms, it is not enough when the attribute information and relational information of patterns are available at the same time. In fact, there have been many ranking problems that need to take both of these types of information into consideration, for the purpose of improving knowledge recognition degree.

Bearing this in mind, in this paper, we make a discussion on the ranking problem of complex objects. Each complex object consists of some simple patterns which are described by attribute information as well as the relational information between patterns. Because of the complexity of the objects, inspired by the famous TOPSIS method [23], in our proposed ranking approach, a referential object is proposed in advance. Hereinto, the number of patterns of the referential object is the same as that of each complex object. To fully fulfill the ranking problem, the fuzzy collaborative clustering mechanism is proposed and it is carried on the patterns in the referential object by using the collaborative information. The collaborative information refers to the partition matrices and/or cluster centers with respect to the complex objects, and it can be employed freely during the ranking process. During the process of ranking, the essence of the ranking algorithm is to compare the difference of the referential object before and after collaboration with respect to the ranking objects. Throughout the ranking process, only the collaborative information participates in the sequence calculation.

The remainder of this paper is organized as follows. In Section 2, we recall some basic concepts, such as the mathematical description of complex object and fuzzy collaborative clustering. In Section 3, we discuss the ranking problem about complex objects when the collaborative information is cluster centers that derived from the corresponding complex object by some clustering approaches. In Section 4, the partition matrices are regarded as the collaborative information and corresponding ranking approach is constructed. In Section 5, some synthetic examples are simulated to illustrate the validity of the proposed ranking approaches. Finally, Section 6 concludes this paper.

#### 2. Preliminaries

In this section, at first we introduce the concept of complex objects and then make a brief review of the fuzzy collaborative clustering algorithm. For simplicity, here we ignore the presentation of fuzzy -means clustering algorithm [24–26], not to mention other associated research topics [27–29].

##### 2.1. Mathematical Description of Complex Objects

Mathematically, the complex object can be expressed as a tuple , where is a nonempty finite set and for are patterns, is a relational information set, and for is the possible relational information between the patterns and .

Formally, if is nothing but a symbolic representation of the th pattern, then the relational information set can be only used during the process of data analysis. If not, the pattern can be expressed as , where represents the value of with respect to the attribution . In general, the value may not be a real number. Sometimes it may be a set [30], an interval [31], or a fuzzy number [32, 33], and so forth.

Certainly, the foregoing proposed data representation formats are also suitable for the entries in relational information set . Particularly, the relationship between any two patterns may be described by more than one relation; that is, with , where describes the th relationship between patterns. Notice that, in this paper, we adhere to the hypothesis that for and is a real number, for belongs to unit interval , and for complex object .

##### 2.2. Fuzzy Collaborative Clustering

It is well known that the nature of clustering analysis is to find out the potential structure of the data. If one wants to search for the common structure of some different data sets while the information of these data sets cannot be shared freely, the collaborative clustering algorithm [34, 35] plays an important role.

One has that the collaborative clustering can be divided into horizontal collaborative clustering, vertical collaborative clustering, and hybrid collaborative clustering. Because we have assumed that the number of patterns for each ranking object is equal, in what follows, we will use the horizontal collaborative clustering algorithm; thus we give a brief introduction of it.

Given that are data sets, and there are patterns in each for , the collaborative mechanism based objective function can be constructed aswhere is the distance between the pattern and the cluster center in and the nonnegative parameter represents the collaborative degree of to .

The concrete optimization process of objective function in (1) is the same as that of fuzzy -means clustering algorithm [24–26] or partial supervised fuzzy clustering algorithm [36, 37]: applying Lagrange multiplier method. Just as [35] did, the partition matrix can be expressed aswhere and . Similarly, the cluster centers can be expressed aswhere

In fact, the process of collaborative clustering can be partitioned into two phases: the first stage is to execute fuzzy -means clustering algorithm for each data set separately, in which case we can obtain partition matrix and cluster centers . The second stage is to compute the collaborated partition matrix by (2) and the cluster centers by (3).

#### 3. Cluster Center-Based Ranking Approach for Complex Objects

Here, we pay our attention to the construction of ranking algorithm when the collaborative information of each complex object is provided in terms of cluster centers. For all ranking objects , we suppose that they have the same number of patterns, as well as the referential object .

In the approach to be given in this section, there are three kinds of collaborative information, producing corresponding cluster center matrices: if it is derived from the attribute information of complex object , then we apply to denote the collaborative information; if it is derived from the relational information of complex object , then we apply to denote the collaborative information; if it is derived from both of the attribute information and relational information of the complex object , then we apply to denote the collaborative information. It should be noted that if the collaborative information is composed of two irrelative parts (one is derived from attribute information and another is derived from relational information), then comes naturally. Certainly, the information contained in these matrices can be combined to be embedded in collaborative fuzzy -means clustering.

With the above description, one knows that the collaborative information has at least three possible representation types, and different type represents different meaning. No matter which types the collaborative information belongs to, once the collaborative mechanism is introduced to rank the objects for , the objective function of collaborated by can be constructed as follows:where is the distance between pattern and center and is a label function with respect to the collaborative information. Moreover, is the distance between and the collaborative information :

Obviously, in (5) describes the collaboration influence of on , and parameter can be viewed as the collaborative coefficient that is relevant to the types of collaborative information. Meanwhile, (6) represents the distances between the cluster center of referential object and the cluster center of complex object .

*Remark 1. *The collaborative coefficient can be determined from the types of the collaborative information:(i)If the collaborative information is , then and .(ii)If the collaborative information is , then and .(iii)If the collaborative information is , then and .(iv)If the collaborative information is , then , but and .

As can be seen from Remark 1, the types of collaborative information can determine which component of is equal to 0, but the value of other nonzero components is not provided. No matter which types of collaborative information, that is, the collaborative information is , , , and/or , the concrete value of depends entirely on the requirement of practical ranking problem, and it can be calculated by many methods.

So far, one can regard the objective function in (5) as an optimization problem with the constraint . In what follows, we still apply the Lagrange multiplier method to solve it. At first, the Lagrange function of (5) can be rewritten as

Computing the derivative of with respect to and making it equal to 0, we haveOn account of the constraint , we have that the Lagrange multiplier can be determined asInserting the expression of the above into (9) yields

The computations of the cluster centers are straightforward, as no constraints are imposed on them. By executing the derivation of (5) with respect to variant , and letting the result be equal to 0, then we have thatFor the solving of , by (6), we have thatTherefore, taking (10) into (9) yieldswhere and .

Up to now, the partition matrix in (11) and the cluster centers in (14) can be used to describe the clustering results of referential object under the collaboration of complex object . Notice that, throughout this section, the notation “” denotes the cluster centers or collaborative information of the ranking object . To avoid unnecessary ambiguity, in the rest of this subsection, we apply to replace the computing results of (14).

By the collaboration of , one can obtain the new partition matrix and cluster centers of the referential data . Certainty, if the difference of and is smaller, then we can say that and have a bigger similarity in aspects of partition matrix. Similarly, if the difference of and is smaller, then we can say that and have a bigger similarity in aspect of cluster center.

Bearing this in mind, the equation can be used to compute the similarity of and about the referential object , where represents the similarity of and , and can be calculated by many methods [38]. Similarly, the equationcan be used to calculate the similarity of and , where . By (15) and (16), the similarity of clustering results of referential object before and after collaboration of can be determined by the following equation:

Obviously, if the similarity between and is greater than that of and , then is nearer to than the object in aspects of collaborative information. Therefore, complex object should be sorted in front of complex object . So much for this, the collaborative clustering-based ranking approach for complex objects can be summarized in Algorithm 1.