Analysis And Research On The Dimension Reduction Method For Functional Data

With the development of The Times and the progress of science andtechnology, people are more curious about the more complex laws after they canexplain many of the law on the objective laws. But people will find that the morecomplex the objective laws of contain more large amount of data. Objective lawscontained in the factors of concern, the more the greater the amount of datadimension, and the increase of the dimension is to analyze the data associated withthe difficulty of exponential growth. With the development of science andtechnology, data acquisition technology constantly improving, the accuracy of dataacquisition and frequency increases. Nowadays, every data of time intervalbetween two collections can become very short, although in fact the data collectedis discrete, but almost can be seen as a continuous curve or continuous function.The data contains function characteristics called functional data. Dimensionreduction method is used to overcome the "dimension disaster" or thehigh-dimensional data model of a typical data processing technology. In general,we need to reduce the dimensionality of high-dimensional data to a reasonable size,at the same time also to retain the original information as much as possible, thepurpose is to eliminate redundant information, the representation of a morereasonable way. Said to achieve a more reasonable way, and then will put the datainto the processing system, thereby reducing processing data of time and effort, toachieve the goal of improve the efficiency of data processing.The purpose of this paper is to study and review the functional data analysis ofseveral common dimension reduction method and the results of other researchers,to lay the foundation for myself for functional data analysis work in the financialindustry in the future. The first chapter firstly introduces functional data and thenecessity of dimension reduction related concept and the functional data analysis are summarized. Including the origin of the "dimension disaster", the word"dimension disaster" is by Bellman, Richard Emest[1]. For the first time in1961, thedifficulty of the multivariate function is estimated with the linear increase of thenumber of sampling points growing exponentially; What is the functional data: datafunction is the most basic ideas as a functional data is smooth, and then to extractinformation from functional data for statistical inference. The classic functional datamethod is assuming that the functional data curve was observed complete anderror-free. Because of this assumption is too perfect, mainly used in themeteorological data; The definition of "dimension reduction": when the datadimension linear increase processing these data are facing difficulty in exponentialtrend growth, so when the data dimension need to be taken when it reaches acertain degree of certain means to make the data processing system caneffectively deal with these data, this approach is now referred to a growing numberof dimension reduction method; The basic theory of functional data analysis:J.O.Ramsay addicted himself to the study of functional data analysis for a long time.When he and C.J.Dalzell publish the book Some of the Tools for Functional DataAnalysis and put forward many practical time used to study the functional the infinitedimensional Data on methods and Tools, and the Functional Data of principalcomponent Analysis method is applied to the empirical study of the relationshipbetween temperature and precipitation in Canada. After that, J.O.Ramsay in1997with B.W.Silcerman cooperation summarizes the theories and methods ofFunctional Data Analysis, and published the Functional Data Analysis. The book verycomprehensively expounds the basic characteristics of the functional data andstatistical analysis of ideas, greatly promoted the development of the functionaldata analysis and academic understanding of the functional data.The second chapter reviewed the development of dimension reduction and theformer research achievements of the functional data analysis method, including thefunctional data analysis methods including functional data principal componentanalysis, canonical correlation analysis and functional data sliced inverseregression analysis. Due to the functional data analysis in the field of generaldimension reduction method is by multiple case analysis of the data dimensionreduction method has evolved, so in the second chapter first introducesrespectively the three dimension reduction in diverse circumstances of dimension reduction method is proposed, by data analysis and function in the three ways ofconnection and comparison. Principal component analysis method[4]requires pcomponents can represent the variability in the system, but most of the variabilityoften can be specified only by k principal components. The k principal componentscontained the information and the original p variables contain almost the same. Soby the time of a p-dimension variable measured values of the original data iscompressed into the measurements of the k principal component values of datasets. Canonical correlation analysis[7]is the earliest named by Hotelling (H.Hotelling)proposed in1936. The purpose of multiple canonical correlation analysis is toidentify and quantify the connection between the two groups of variables. Thepurpose is: a set of linear combination of the relationship between variables[8].Sliced inverse regression[10]was first named by Li (1991) on Sliced inverseregression for dimension reduction. Without any parameters or nonparametricmodeling process reduce the dimensions of the input variables. Sliced inverseregression method to estimate based on inverse regression. Different from positiveregression used to do regression analysis, inverse regression is used to doregression analysis. The benefit of the most direct is exchanged and the location ofthe translation can be dimension problems. In essence, we will forward regressionhigh-dimensional problem into one dimension to the one-dimensional regressionproblems. In order to challenge the dimension disaster, we hope to take advantageof low dimensional projection to compensate for high-dimensional data we areinterested in.The third chapter summarizes and analysis the advantages and disadvantagesof these three dimension reduction methods for functional data, and hope that inthe near future people can develop some functional data analysis software to meetthe needs of various industries for the functional data analysis.