## 1 Introduction  
Symbolic interval data analysis (SIDA) is a statistical method that allows for more efficient and intuitive analysis. In general, SIDA allows for the precise analysis of interval data by converting the data into symbolic representations. This makes computations easier and more accurate. Regarding the advantages and characteristics of SIDA, (1) SIDA can incorporate uncertainty and imprecision in data, which are often present in real-world scenarios. This makes it a useful tool for clustering in uncertain situations. (2) The symbolic representation of interval data can significantly reduce the size of the dataset, making it easier and faster to analyze. (3) SIDA can be applied to a wide range of data types and can be used for a variety of analyses, including clustering, classification, and regression. (4) SIDA is computationally efficient, making it a valuable tool for analyzing large datasets. Hence, SIDA is a useful method for analyzing interval data in a precise, intuitive, and efficient manner. Its flexibility and ability to incorporate uncertainty make it a valuable tool for fuzzy clustering in a wide range of fields [1, 2]. [PERSON] et al. [3] introduced an interval optimization method for applying SIDA in finance. [PERSON] et al. [4] presented a trajectory-recovery algorithm based on interval analysis, applied to SIDA in engineering. [PERSON] et al. [5]suggested a convex combination approach for artificial neural networks with interval data to forecast fluctuations or uncertainty in economic, financial, and environmental data. [PERSON] et al. [6] proposed a maximum entropy fuzzy clustering approach for interval-valued data in cancer detection within the realm of SIDA in medicine. [PERSON] et al. [7] introduced a novel autocorrelation index with applications in COVID-19 and rent prices, using interval-valued data within the framework of SIDA in medicine. These studies explored various applications of SIDA across diverse fields, such as finance, engineering, environmental science, and medicine. The publication of these works underscores the significance of SIDA in addressing challenges under different domains. As information technology rapidly advances, collecting extensive datasets has become more accessible. When dealing with substantial data, the goal is to uncover the latent information within it. Clustering analysis is a valuable tool for extracting meaningful insights from data [8; 9]. In general, clustering analysis can be categorized into two types based on their modeling assumptions: probabilistic models and non-probabilistic models [10]. Methods relying on probabilistic models typically assume that the data follows a specific probability distribution and are often solved using the maximum likelihood method, with the expectation-maximization (EM) algorithm being one of the most prevalent techniques [11]. Non-probabilistic methods, on the other hand, are grounded in objective functions that aim to either maximize similarity or minimize dissimilarity to achieve optimization. These methods include hierarchical clustering [12; 13; 14] and partitioning clustering [15; 16; 17]. An important development was [PERSON]'s introduction of an improved K-means clustering algorithm, known as the fuzzy C-means (FCM) clustering algorithm [18]. FCM incorporates fuzzy theory concepts, allowing data points to have membership in multiple groups rather than belonging to a single, specific cluster. Papers [19; 20] proposed a modified fuzzy C-means clustering algorithm under single-valued data on image process. Papers [21; 22; 23] proposed a modified fuzzy C-means clustering algorithm under single-valued data on data analysis. [PERSON] et al. [24] proposed guided filter-based fuzzy clustering under single-valued data for general data analysis. Papers [25; 26; 27; 28] proposed hybrid algorithms under the fuzzy C-means algorithm with single-valued data on different applications.