## 1 Introduction
However, when implementing similarity measurements for categorical attributes, the Hamming distance is rough, and the clustering result is very sensitive to this parameter in the K-Prototypes algorithm. Subsequently, some improved similarity measurements for categorical attributes are proposed, which are based on the frequency of categorical values, the co-occurrence, and the conditional probability estimate [7; 16; 17]. Based on these improved similarity measurements for categorical attributes, some combined similarity measurements for both categorical and numerical datasets have been developed. For instance, the OCIL algorithm [12] uses the frequency of categorical object values that occur in the cluster for categorical attributes and the numerical distance for numerical attributes when measuring similarity. It can be found that each attribute often contributes differently to the desired clustering results in many practical applications, which should be considered when measuring the similarities. For example, we want to cluster a mammographic mass dataset into two groups, corresponding to benign types and malignant types. In this task, the age attribute may play a more important role than the mass density attribute. Therefore, it is very important to identify different attribute contributions to improve the quality of the clustering results. Actually, some researchers have realized this problem and proposed several strategies. However, most research focuses on single-type datasets; e.g., for categorical attributes, the weights could be assigned based on the overall distribution of attribute values [18] or based on the frequency the class center appearances and the average distance between objects and the clustering center [19]. When handling mixed datasets, current algorithms only assign weights for single-type attributes. For example, when the OCIL algorithm [12] measuring the similarity, which only assigns weight for each categorical attribute based on information entropy, and it lets each numerical attribute take the same weight. The result is to weakens the importance of the numerical attributes. [PERSON] and [PERSON] proposed an algorithm for mixed datasets by adding weights to only numerical attributes [20]. Actually, both the numerical attributes and the categorical attributes should be evaluated when designing the similarity, and the weight strategy should be applied to both types of attributes in order to simplify the computational complexity. In this paper, we propose a similarity measurement with entropy-based weighting for mixed datasets with both categorical and numerical attributes. First, a similarity metric for the categorical attributes is designed by assigning a different weight to each attribute based on information entropy theory. Second, we present an automatic categorization technique that transforms numerical data into categorical data, which is achieved by automatically discovering the optimal number of categorizations for each attribute based on the Calinski-Harabasz index. Then, the similarity metric for categorical data can be used to measure the similarity for transformed data. In this way, this similarity measurement can be applied to the mixed dataset containing both numerical and categorical attributes. Subsequently, this similarity measurement with entropy-based weighting is applied to the k-means framework. We accessed several datasets from UCI and compared the proposed algorithm with the OCIL and K-Prototype methods on mixed datasets as well as with the k-means algorithm on numerical datasets. The experimental results show that the iterative clustering algorithm based on the proposed similarity measurement is superior to these three algorithms. The remainder of this paper is organized as follows. Section 2 introduces the problem formulation and then proposes a similarity measurement with entropy-based weighting for mixed datasets and applies this similarity measurement to the k-means algorithm framework.