## 1 Introduction  
The main purposes of clustering analyses are to discover the implicit class structure in the data and divide the physical or abstract objects into different classes, where the similarity between a pair of objects in the same class is large and in different classes is small. As a major exploratory data analysis tool, clustering analysis has been widely researched and applied in many fields, such as sociology, biology, medicine, etc. [1, 2, 3]. Most current methods are designed to address single dataset types (numerical or categorical). For example, classical clustering methods, such as the k-means algorithm [4, 5], the EM algorithm [6], etc., are limited to numerical datasets, while some algorithms are also proposed for clustering categorical datasets [7, 8]. However, in the medical and biology fields, many datasets are collected with both numerical and categorical attributes. Hence, many researchers are dedicated to discovering clustering algorithms for mixed types of datasets with categorical and numerical attributes [9, 10]. Many unsupervised clustering algorithms for mixed datasets have been proposed over the years, which can be classified into two types. The first type designs different similarity measurements for numerical and categorical data and then calculates the weighted sum of the two parts. For example, the K-Prototypes algorithm [11] for clustering mixed datasets was put forward simply by combining the k-means algorithm and the K-Modes algorithm, which are used for single types of numerical and categorical datasets, respectively. Additionally, the OCIL algorithm proposed by [PERSON] and [PERSON] [12] is an iterative clustering learning algorithm based on object-cluster similarity metrics. In the second type, the algorithms transform categorical attributes into numerical ones, and then the algorithms apply clustering methods designed for purely numerical datasets to the transformed dataset or vice versa. The most direct method is to map categorical values into numerical vectors. If a categorical attribute contains \(n\) unique values, then each value is mapped into a \(n\)-dimensional vector. This strategy increases the dataset dimensions, resulting in higher computational complexity. It could also transform numerical attributes into categorical ones. For instance, SpectralCAT, proposed by [PERSON] and [PERSON] [13], automatically transforms high-dimensional data into categorical data and then applies spectral clustering [14] to reduce the dimensionality of the transformed datasets through automatic non-linear transformations. When designing clustering algorithms, the similarity or dissimilarity measurement plays an important role. Due to the different nature of numerical attributes and categorical attributes, they should be handled differently. Numerical data use a continuous variable to represent the values of each attribute, and a common distance such as the Euclidean distance usually measures the similarity between numerical objects. However, the values of the categorical data have neither a natural ordering nor a common scale. Due to this distinct nature of these two different data types, methods designed for single-type datasets cannot be applied to other types of datasets. The most direct way is the second of the two types mentioned above. However, this method ignores the similarity information in the categorical attribute values [15]. Therefore, the Hamming distance is used in many dissimilarity measurements. For example, in the K-Prototypes algorithm, the dissimilarity measurement uses the Euclidean distance for the numerical attributes and the Hamming distance for the categorical attributes. This algorithm also controls the contribution of the numerical attributes and the categorical attributes through a user-defined parameter. The K-Prototypes algorithm is simple and easy to implement, so it has been widely used in clustering mixed datasets.