Method For Modeling Functional Data And Its Application

Recent years have witnessed lots of functional data with the rapid development of information technology,these data widely exist in economic,finance,bioinformatics,medicine,meteorology,kinesiology,speech recognition,and many other domains,the analytical methods for functional data have been more widely studied.In the traditional data mining methods,one-dimensional functional data is regarded as a discrete and limited observation sequence,thus the continuity and highdimensionality of functional data are ignored,and the knowledge discovery for functional data is limited.Considering the limitation of using the traditional data mining methods to deal with functional data,based on the basis representation for functional data,modeling theory and modeling method are explored in the problems of classification?clustering and regression,and concrete examples are used to test the validity of these methods.Specifically,the main results and innovations are summarized as follows:(1)For the representation problem from functional data,we explore the modeling principle of functional principal component representation.We have obtained the model that functional principal component basis driven by data satisfy using variational theory,this provides the method for acquiring functional principal component basis.We also verify that functional principal component basis are the optimal orthonormal basis theoretically in sense of the mean-square error criterion,this provides a theoretical basis for representing functional data by means of functional principal component basis.(2)For the classification problem from functional data,we explore the classification performance difference between different basis representations.We have proved that L2 distances between functional samples are equivalent to Euclidean distances between their basis coefficient vectors with the orthonormal representation for functional data,this lays a solid foundation for using two-stage classification method to classify functional samples.Based on two-stage classification algorithm,from the perspective of classification performance,the data type that every kind of basis function is fit for is analyzed among Fourier basis,wavelet basis and functional principal component basis.At the same time,we compare classification performance difference between non-orthogonal representation for functional data and orthogonal representation for functional data.(3)For the clustering problem from functional data,we explore the representation of cluster centroids in functional k-means clustering algorithm.We have proved a similarity measure between multidimensional functional samples is a distance,which includes the derivative information of functional samples,this lays a foundation for functional k-means clustering algorithm.With respect to the given distance,we have given the representations of cluster centroids(for each cluster,the mean of functional samples within cluster is regarded as cluster centroid)and proved that the cluster centroids can minimize the sum of squared distances within cluster.In addition,the temperature data of 26 capital cities in China are used to test the effectiveness of functional k-means clustering algorithm under given cluster centroids.(4)For the regression problem from functional data,we explore the modeling method of partial functional linear model which is used to deal with mixed data.In order to improve the prediction accuracy of the model,we have transformed the semi-parametric model into a parameter model by means of basis representation of functional coefficient in SobolevHilbert space;at the same time,in order to increase the stability of the model,we have utilized the least-squares method with much looser penalty to learn the model.Experimental results on the artificial data and the real data show that the proposed method is effective.Aiming at the limitations of traditional data mining methods in process of dealing with functional data,based on the basis representation strategy of functional data,we provide the theory and method for functional data modeling,the research results are of great theoretical and practical significance in functional data mining domain.