Research On Interval-valued Functional Clustering Method

Based on the theory of interval-valued clustering analysis and functional clustering analysis,interval-valued functional clustering is a analytical method which expands the specific interval-valued similarity measurement into a functional form and applies it to the framework of functional clustering analysis,so as to realize the clustering analysis of interval-valued functional data.It can retain the advantages of functional clustering analysis in exploring the rule of curve development and change.At the same time,the problem of information loss is avoided.By combing the interval-valued functional clustering analysis and functional clustering analysis methods,it is found that the similarity measurement in the existing interval-valued functional clustering analysis only considers the characteristics of the basis functions,which reflect the difference of the curve at the absolute level,ignores the difference of the shape of the curve,and fails to fully mine the trend characteristics of the interval-valued functions.The clustering results often can not fully reflect the data information.What's more,only the research on a single interval function index can not satisfy the need of comprehensively considering the characteristics of the studied object from several aspects in reality.To solve the above problems,this paper extends the interval-valued Euclidean distance to functional data,and further constructs a new similarity measurement which can take into account the numerical distance and curve shape of interval-valued functions--the interval-valued functional Euclidean distance based on the information of original function and derivative function,so as to give the improved K-means clustering process of interval-valued functions.In order to illustrate the characteristics and advantages of the newly constructed similarity measurement,the interval-valued functional clustering method proposed in this paper is further applied to the clustering analysis of the interval-valued functions of air quality index,and the clustering results are compared with the existing interval-valued functional clustering analysis and functional clustering analysis methods.It is found that the interval-valued functional clustering model based on the similarity measurement of interval-valued functional Euclidean distance with the information of original function and derivative function achieves the goal of comprehensively measuring the difference of interval-valued function samples from two aspects of numerical distance and curve shape,and can enrich the original data information without adding variables.In order to satisfy the need of multi-index comprehensive analysis of the characteristics of the studied object,based on the existing interval-valued entropy method and functional entropy method,this paper deduces the calculation steps of interval-valued functional entropy method as an index synthesis method to realize the clustering analysis of multi index interval-valued functional data.The interval functional entropy method is applied to the concentration data of main air pollutants in 31 provincial capitals and municipalities directly as an index synthesis method for multi-index interval-valued functional clustering,which can comprehensively analyze the comprehensive level of air quality and pollutant impact factors in each region without increasing the computational complexity.