Constructing Perturbation Matrices of Prototypes for Enhancing the Performance of Fuzzy Decoding Mechanism

Xu, Kaijie; E, Hanyu; Liu, Junliang; Xiao, Guoyao; Tang, Xiaoan; Xing, Mengdao

doi:https://doi.org/10.1155/2024/5780186

International Journal of Intelligent Systems

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 5780186 | https://doi.org/10.1155/2024/5780186

Constructing Perturbation Matrices of Prototypes for Enhancing the Performance of Fuzzy Decoding Mechanism

Kaijie Xu,^1,2Hanyu E ,³Junliang Liu,⁴Guoyao Xiao,¹Xiaoan Tang,^2,5and Mengdao Xing⁶

Academic Editor: Fabio Caraffini

Received25 Feb 2023

Revised02 Jan 2024

Accepted12 Jan 2024

Published24 Jan 2024

Abstract

Granular computing (GrC) embraces a spectrum of concepts, methodologies, methods, and applications, which dwells upon information granules and their processing. Fuzzy C-means (FCM) based encoding and decoding (granulation-degranulation) mechanism plays a visible role in granular computing. Fuzzy decoding mechanism, also known as the reconstruction (degranulation) problem, has become an intensively studied category in recent years. This study mainly focuses on the improvement of the fuzzy decoding mechanism, and an augmented version achieved through constructing perturbation matrices of prototypes is put forward. Particle swarm optimization is employed to determine a group of optimal perturbation matrices to optimize the prototype matrix and obtain an optimal partition matrix. A series of experiments are carried out to show the enhancement of the proposed method. The experimental results are consistent with the theoretical analysis and demonstrate that the developed method outperforms the traditional FCM-based decoding mechanism.

1. Introduction

As an important technology in artificial intelligence (AI) [1], granular computing (GrC) [2] has become a novel multidisciplinary paradigm. Information granules, such as fuzzy sets and fuzzy relations, have been regarded as the foundation of GrC [3]. Various clustering methods are used to construct information granules so as to discover a structure in the patterns and data [4]. As an important unsupervised learning vehicle, fuzzy C-means (FCM) clustering has been considered as a very powerful technique to build information granules (granulation mechanism) [5]. This fuzzy set-based granulation is also referred to as the fuzzy encoding mechanism. In the encoding progress, fuzzy clustering approaches are used to cluster the numerical data into a collection of fuzzy information granules. The decoding mechanism (degranulation, referred altogether as a data reconstruction problem), as an inverse process of encoding that involves the reconstruction of the original numeric data based on the already-formed information granules, is another problem worth studying [3].

The mechanism of encoding-decoding plays a visible role in GrC, just as the analog-to-digital (A/D) and the digital-to-analog (D/A) conversions in the field of signal processing [3, 6, 7]. However, the topic of the fuzzy encoding-decoding mechanism has not been extensively studied. This lack of a well-established theoretical framework and methodological system opens up some fresh interesting opportunities but also brings more challenges.

In the FCM-based encoding mechanism, the inherent structure within the data is revealed through prototypes (centers) and partition matrices. Subsequently, the data undergo encoding into information granules facilitated by the resultant membership matrices and prototypes [8]. Meanwhile, the FCM-based decoding (degranulation) mechanism refers to the reconstruction of the numeric data based on the prototypes and matrices (constructed information granules). Preliminary research studies have indicated that there exists a relationship between the decoding and the encoding mechanisms. That is to say, the smaller the error of the decoding is, the better performance of the encoding becomes [4]. In the FCM-based decoding, the error of decoding depends on the parameters involved in the granulation process. In addition, the decoding error is commonly used as a criterion to evaluate the quality of the fuzzy clustering [9–11]. One can improve the performance of the fuzzy clustering approaches through reducing the decoding error. In [11], Izakian et al. employ the decoding error as a tool to determine the membership matrix so as to enhance the performance of fuzzy clustering. In [12], a dynamic incremental semisupervised FCM method is designed through introducing the decoding error as an indicator. To optimize the time series, Hesam et al. utilize the decoding error as a criterion in anomaly detection. In [4], the authors construct a novel linear mapping of the partition matrix to achieve the close-linkage of the data to be partitioned, and then establish a novel refinement strategy to modify the prototypes. Ultimately, the decoding error is decreased. Similarly, in [9], Reyes-Galaviz and Pedrycz also develop a refinement mechanism of the prototypes to reduce the decoding error to improve the quality of fuzzy clustering. In [13], Nie et al. utilize neural network technique to improve the degranulation process, which can substantially reduce the reconstruction error. In [14], degranulation mechanism is used to design the fuzzy rule-based models to improve the robustness and the accuracy of prediction. Recently, Pedrycz elucidates the principles of GrC through the granulation-degranulation mechanism and demonstrates the construction of information granules and their subsequent utilization in describing relationships inherent in the data, thereby contributing to the realization of models [15].

Hitherto, the topic of encoding-decoding mechanism has not been intensively studied. At present, this mechanism is mainly used in information granule construction, classification, data compression, anomaly detection in time series data, fuzzy modeling [16–19], etc. The lack of a well-established body of knowledge provides new opportunities but also calls for more investigations.

In brief, the FCM-based decoding mechanism is an important technology in GrC. Enhancing the performance of decoding is an important and well-motivated issue.

The aim of the present research is to design an enhanced scheme for data reconstruction to improve the quality of the decoding process. To facilitate the analysis, in the proposed scheme, we define a prototype matrix and build an integrated encoding-decoding mechanism to expose the transformation relationship between the prototype matrix and the numeric data matrix. A series of subsequent operations are based on the developed mechanism. From the design perspective, the process mainly revolves around the construction and representation of fuzzy information granules, and an enhanced version of decoding is achieved.

Proceeding with more details, we elaborate on an adjustment mechanism (refinement strategy) of the prototypes by introducing a group of perturbation matrices for the defined prototype matrix on the basis of the developed encoding-decoding mechanism (the prototypes are assessed by looking at the decoding error), which increases the flexibility of the encoding-decoding mechanism. The decoding error also serves as a cost function to be employed in the optimization phase, i.e., with the aim of reducing the decoding error, we construct a cost function of the perturbation matrices, a prototype matrix, and a partition matrix.

Subsequently, particle swarm optimization (PSO) [20–23] is employed to determine a group of perturbation matrices of the prototypes through minimizing the decoding error (cost function). Thus, with the resulting perturbation matrices of the prototypes, a modified partition matrix and a refined prototype matrix are obtained. The merit of the developed scheme is that it can produce more reasonable information granules, since both the prototype matrix and partition matrix are refined throughout the optimization process. Ultimately, the quality of the decoding mechanism becomes augmented. To the best of our knowledge, this is the first time that such an approach is explored. In a nutshell, the main contribution of this study is to introduce and develop a novel method which helps minimize the decoding error and improve the quality of numeric data decoding mechanism.

The organization of the paper reflects the key phases of the design issues. A brief introduction of the encoding-decoding mechanism is presented in Section 2. An enhanced method of decoding is detailed in Section 3. Afterward, we offer some experimental studies in Section 4. Section 5 covers some conclusions.

2. FCM-Based Encoding-Decoding Mechanism

In this section, we briefly recall the FCM-based encoding-decoding mechanism.

Consider that a numeric dataset comprising of N samples is partitioned into C clusters. The FCM method minimizes the following cost function to complete the granulation (reveal a structure in the data) [24, 25]:where is the ith datum, is the jth prototype of the cluster, is the degree of membership of object to the prototype j, m (m > 1) is a fuzziness factor, and denotes a certain distance function.

2.1. Encoding Mechanism

The dataset X is expressed as (encoded into) a set of prototypes (j = 1, 2, …, C) and a membership matrix (partition matrix) U through the FCM-based encoding. In this process, the prototypes and the parathion matrix are calculated by minimizing the cost function shown above [26, 27]:

Therefore, the process above involves a double iteration process between the prototypes and the membership matrix. In a word, the dataset is described as (encoded into) a set of prototypes (j = 1, 2, …, C) and a membership matrix , which is the so-called encoding mechanism.

2.2. Decoding Structured

The decoding mechanism primarily involves reconstructing numeric data based on the generated prototypes and the membership matrix, which constitutes the formed information granules. The form of the decoding expression can be obtained by minimizing the following cost function [28–31]:

By using the approach of Lagrange multipliers, the reconstructed numeric data can be solved in the following way:where T stands for the transpose operation, and is the Lagrange multiplier. It can be seen from (4), that each prototype is weighted by the membership grade of the corresponding information granules. On the basis of the underlying operation described above, the decoding mechanism depends on the membership levels and the set of prototypes. Figure 1 describes the process of encoding-decoding.

3. An Augmented Scheme of Decoding

In contrast to the FCM-based decoding mechanism, the proposed scheme in this study is augmented through constructing perturbation matrices of prototypes to refine the prototype matrix and the partition matrix.

In this paper, the error of decoding is evaluated aswhere represents the 2-norm [22], expressed aswhere is the maximum eigenvalue of and H denotes the conjugate transpose [32]. It is clear that the index is directly depended on the membership matrix and the prototypes. Next, we mainly focus on the design of the augmented scheme of the decoding mechanism.

To facilitate the analysis, we define a prototype matrix asand then the FCM-based encoding mechanism is simplified aswhereand δ(t) is a unit pulse function shown as

The structure of reads as follows:

Similarly, the FCM-based decoding mechanism can be written in the following way:where Θ is a matrix in this form:

3.1. Construction of Perturbation Matrices of Prototypes

According to (8), the prototypes generated through the FCM approach are in some sense a mapping of the numeric data matrix and, as such, are obviously incapable of representing the data that are distantly positioned from some prototypes. Possible adjustments of the prototype matrix are considered to augment their dispersion in the data space. In addition, as stated before, the decoding index is directly dependent on the formed partition matrix and the prototype matrix, and there exists an iterative relationship between the two matrices. Relatively speaking, the prototype matrix has a smaller size, which indicates that modifying the prototype matrix directly and indirectly optimizing the partition matrix so as to realize the minimization of the decoding error is a simple and reasonable scheme. Thus, we can consider improving the quality of the decoding mechanism through a modification of the prototype matrix.

To refine the prototype matrix, we construct a group of perturbation matrices (A and B) of prototypes to modify (8) as follows:where A and B are two diagonal matrices:where U(1, δ) stands for the δ-neighborhood of 1, i.e.,

Thus, the prototype matrix can be modified (adjusted) with the perturbation matrices A and B. From the perspective of matrix operation, matrix A is used to adjust the rows of the prototype matrix, and matrix B is used to adjust the columns of the prototype matrix. To obtain a group of sound perturbation matrices A and B, we establish and optimize a cost function in the following form:

The objective of the perturbation matrices of the prototypes is to actualize the modification of the membership matrix and allow for an interaction achieved at the level of information granules so that the decoding error can be reduced.

In the sequel, we will explain how to determine a group of optimal perturbation matrices to modify the prototype matrix and optimize the membership matrix so as to minimize the decoding error (cost function).

3.2. Optimization of the Cost Function with Particle Swarm Optimization (PSO)

The solving of the perturbation matrices is a possible multimodality problem which implies that the traditional mathematical methods [16] are not feasible here. Therefore, an effective alternative is to employ some powerful optimization approaches, such as particle swarm optimization (PSO) [33], which have been successfully applied in multiple fields.

The proposed scheme includes two phases: an unsupervised granulation phase, and a supervised prototype matrix and partition matrix refinement phase (supervised by the decoding error). During the first phase, a dataset X is encoded into a collection of prototypes and a membership matrix with the fuzzy clustering approach. At the second phase, the decoding error is employed to supervise and modify the prototype matrix to generate a more reasonable membership matrix. Ultimately, the quality of decoding becomes augmented. To this end, we need to determine a group of perturbation matrices.

First, we need to initialize the variables (perturbation matrices, the prototype matrix, or membership matrix) involved in the cost function to facilitate the implementation of the algorithm (cost function). Theoretically, the initial elements in these matrices can be random or certain values. To accelerate the convergence of the algorithm, we can consider employing the prototype matrix and the membership matrix produced by the FCM as the initial matrices. For the perturbation matrices, we can set them as two unit matrices. This initialization implies that we start the method from the generic FCM-based granulation mechanism. This granulation minimizes a certain cost function, and the performance is not directly linked with the decoding error. However, the FCM-produced prototype matrix and the membership matrix can always keep the decoding error at the local minimum. At this point, under the action of the perturbation matrices, the cost function runs in a smaller direction (smaller than the above local minimum). In this sense, another role of the perturbation matrices is to accelerate the convergence of the cost function.

The above cost function is minimized by iteratively updating the prototype matrix, membership matrix, and perturbation matrices. In the iterative process, the prototype matrix is updated according to (14), i.e.,where t denotes the index of the iteration loop of the algorithm. The perturbation matrices A and B are captured through the PSO optimizing in the (C + n)-dimensional space. Afterwards, the prototype matrix is modified and the membership matrix can be refined so that the quality of the decoding mechanism becomes enhanced. Figure 2 shows the principle of the proposed scheme. Ideally, we wish the cost function can be decreased to the minimum value (say, zero). If the cost function (error of decoding) is reduced to zero, then we realize a perfect decoding mechanism (reconstruction of the numerical data).

4. Experimental Studies

In this section, we report on a variety of experiments on synthetic data and numerous publicly available datasets [34]. All these datasets are normalized to (0, 1). Our ultimate goal is to examine the decoding quality of the proposed scheme compared with the FCM method.

In the experiments, the fuzzification factor is set from 1.1 to 5.1 with a step size of 0.5, and the number of information granules (prototypes) is taken from 2 to 6. Normalized Euclidean distance [3] is used. The decoding error is taken as the evaluation index. For the proposed scheme, to accelerate the convergence of the PSO method, we set the initial perturbation matrices as follows:where stands for the S order unit matrix. Without loss of generality, we set the neighbor parameter as 0.25. Furthermore, the parameters of the PSO [35] method are listed in Table 1. A 10-fold cross-validation is used, which is commonly used to validate the classification performance [36]. For each set of parameters, fuzzy-based algorithms are repeated 10 times, and the means and standard deviations of the decoding error are recorded.

4.1. Synthetic Data

The first experiment is performed on a 2-D synthetic dataset comprising 420 samples, each having six features, and the synthetic dataset is visualized in Figure 3. In Figures 4 and 5, we plot the optimization of the decoding and the corresponding perturbation matrices with PSO. It is evident that both the error of reconstruction and the cost function decrease with the iteration progresses (Figure 4). Specifically, the values of the starting point of the two group curves are produced with the FCM method (Figure 5), since the initial perturbation matrices are set as unit matrices when using the PSO optimization. In particular, the cost functions and the error of decoding have the roughly the same changing trends. To visualize and contrast the performance of the two methods, in Figure 6, we plot the original synthetic data and the reconstructed data. It can be seen that the shape of the reconstructed data obtained through the developed scheme is much closer to the original data than that of the FCM method.

4.2. Publicly Available Data

To further demonstrate the effectiveness of the proposed scheme, we also use six publicly available datasets [37]: Iris, Glass Identification, Pima Indians Diabetes, Wine, User Knowledge Modeling, and BuddyMove. The iris dataset contains three classes with four features and 50 instances each, viz. 1-setosa, 2-versicolor, and 3-virginica. The Glass Identification dataset contains nine features and 214 instances. The Pima Indians Diabetes dataset consists of several medical predictor variables and one target variable. The Wine dataset contains 13 features and 178 instances. These data are the results of a chemical analysis of wines grown in the same region in Italy but derived from three different cultivars. The analysis determined the quantities of 13 constituents found in each of the three types of wines. The User Knowledge Modeling dataset has five features and 403 instances. The BuddyMove dataset contains seven features and 249 instances, and it was populated from destination reviews published by 249 reviewers of holidayiq.com till October 2014. The further detailed introduction of these datasets can be found from https://archive.ics.uci.edu/ml. The errors of decoding for all these datasets are plotted in Figure 7. Obviously, the proposed method achieves better reconstruction performance than the FCM algorithm for all the publicly available datasets. In a nutshell, through introducing a group of perturbation matrices to optimizing the prototypes and the partition matrix, the performance of the decoding mechanism achieves significant improvement, which is consistent with theoretical analysis. The improvement is about 20% on average and varies in-between a minimal improvement of 5% and 40% in the case of the most visible improvement. That is, the proposed scheme also provides a powerful way for optimizing the structure of the information granules, which has a certain driving effect on the research and development of granular computing.

5. Conclusions

In this paper, an augmented scheme for the decoding mechanism is put forward, where the idea is to relocate the locations of prototypes initially generated through implementing the FCM approach on a data matrix. In the design process, we define a prototype matrix to build an integrated encoding-decoding mechanism. By introducing a group of perturbation matrices, a modification model of the prototype matrix is developed on the basis of the already established encoding-decoding mechanism. In the developed scheme, the perturbation matrices offer a direct adjustment mechanism for the prototype matrix and increase the flexibility of the method.

With the supervised learning mode of the encoding-decoding mechanism, we construct a cost function of the perturbation matrices, a prototype matrix, and a partition matrix, and use the decoding error as a performance index to capture the optimization of the perturbation matrices to determine an acceptable prototype matrix. Then the PSO method is employed to carry out the optimization so as to determine a refined prototype matrix and a reasonable membership matrix.

Finally, the quality of the decoding mechanism becomes augmented with the optimization-produced results. We carry out a comprehensive theoretical analysis and present a number of numerical experiments. Combining theoretical analysis and numerical experiments, we analyze and demonstrate the performance of proposed scheme.

In conclusion, the developed scheme is particularly beneficial to slow down the deterioration of the decoding results and augment the quality of the overall encoding-decoding mechanism, which is significant for transforming data between numerical form and granular format. The model proposed in this study opens a specific way for improving the performance of the encoding-decoding mechanism and also poses a more general problem: How to get a closed form solution for the perturbation matrices? This could be an interesting issue. In addition, future work could also consider the research of the deep relationship between the degranulation cost function and the cost function of the FCM clustering.

Data Availability

The data used to support the findings of this study are available from Kaijie Xu (kjxu@xidian.edu.cn).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Kaijie Xu conceptualized the study, proposed the methodology, and wrote the original draft. Hanyu E contributed to data curation and provided software. Junliang Liu reviewed the article. Guoyao Xiao edited the article. Xiaoan Tang performed validation. Mengdao Xing performed supervision and validation.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (nos. 62101400, 72101075, 72171069, and 92367206), in part by the China Postdoctoral Science Foundation under Grant 2023M732743, in part by the Shaanxi Fundamental Science Research Project for Mathematics and Physics under Grant 22JSQ032, and in part by the Fundamental Research Funds for the Central Universities under Grant PA2023IISL0104.

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Copyright © 2024 Kaijie Xu 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.

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