基于众数的函数型数据聚类方法

In recent decades, functional data has received increasing attention and functional data clustering is successfully and widely used in economics, meteorology and many other fields. However, most existing clustering methods for functional data are extensions of traditional K-means clustering and model-based clustering. Their drawbacks, such as requiring a pre-given number of clusters and strict assumptions of parameters, limit their applications. Nonparametric modal clustering method to be studied in the project not only overcomes the drawbacks of the above methods, but also performs well when different clusters have very different variances, which can provide new ideas for complicated practical problems. Previous functional data clustering methods usually reduce the data dimension by projection first, then use the projected data for cluster analysis. In this project, cluster analysis and dimension reduction are performed simultaneously to find a procedure that constructs a low-dimensional representation of functional data such that the cluster structure in the data is maximally revealed, which greatly improves the accuracy of clustering. This project will also extend the nonparametric modal clustering method to multi-dimensional functional data, functional data with spatial information and so on, and consider applying modal regression to clustering, hoping to find new research directions of functional data clustering.

近几十年来，函数型数据得到了越来越多的关注,函数型数据的聚类分析也被广泛应用于经济学、气象学等领域，并取得了一定的成绩。然而，当前的文献大都是对传统K均值聚类和基于模型的聚类等方法的推广和改进，这些方法的缺点如需要猜测聚类数目、参数假设过于严格等,限制了它们在一些实际问题中的应用。本项目拟研究的基于众数的函数型数据聚类方法将克服上述方法的缺点，且对簇与簇之间方差差异很大的数据表现良好，为解决复杂多变的实际问题和不断扩展的应用提供新的思路。以往的函数型数据聚类方法一般先将数据降维，再利用降维后的数据做聚类分析。本项目则将函数型数据降维与聚类分析相结合，不仅可以找出最大化函数型数据聚类特征的最优降维空间，还提高了聚类的精确性。本项目还将基于众数的聚类方法推广到多维函数型数据、带有空间相关性的函数型数据等，并考虑将基于众数的回归模型应用到函数型数据聚类，进一步开拓函数型数据聚类分析新方向。

近几十年来，函数型数据得到了越来越多的关注，函数型数据的聚类分析也被广泛应用于经济学、气象学、社会学等领域。然而，当前的研究大都是对传统K均值聚类和基于模型的聚类等方法的推广和改进，这些方法的缺点如需要猜测聚类数目、参数假设过于严格等，限制了它们在一些实际问题中的应用。本项目拓展了多维函数型数据的系统聚类法和K均值聚类法，并建立了基于众数的函数型数据聚类方法。本项目提出的基于众数的聚类法克服了系统聚类法和K均值聚类法的很多缺点，且对簇与簇之间方差差异很大的数据表现良好。以往的函数型数据聚类方法一般先将数据降维，再利用降维后的数据做聚类分析。本项目则将函数型数据降维与聚类分析相结合，不仅可以找出最大化函数型数据聚类特征的最优降维空间，还提高了聚类的精确性。最后，本项目将所建立的聚类方法应用到我国33个城市可持续发展潜力的评估和我国70家电子设备制造业上市企业创新能力的评价。