Research And Application Of Functional Principal Components Analysis And Functionl Linear Regression Model

With the development of stochastic science and technology,a large number of complex data sets have emerged.For this kind of data,scholars have explored a new data processing method,namely functional data analysis.Functional data analysis considers the observed data as a whole rather than as a series of data,which is the main difference between traditional data analysis and functional data analysis.Functional principal component analysis and functional linear regression model are two common methods of functional data analysis.However,in the analysis and research of practical problems,most of the data we collect are recorded in the form of time series,and we often encounter the phenomenon of autocorrelation.Therefore,this paper studies the estimation of slope function of a functional linear regression model with an autoregressive error term,and obtains the estimation of slope function in the model by using the method of functional principal component analysis.In the deep understanding of the functional principal component analysis method,the discrete high temperature data of 29 major cities were collected and synthesized into a smooth curve based on the Fourier basis function.The results show that the cumulative contribution rate of the first four principal components is more than 95%.The principal component function shows the data characteristics of continuous changes over time.The temperature variation characteristics are interpreted in combination with the actual situation.In the deep understanding of the linear regression model with independent identical distribution,a functional linear regression model was established by taking the temperature curve as a functional covariable and the unit yield increment of maize as a scalar response variable.By using the function principal component analysis,the score vector of the infinite dimensional function data is transformed into the finite dimensional score vector,and then the slope function is estimated to obtain the estimated value of maize yield increment,and the average relative error is calculated to be 0.02%.According to the change of slope function over time,the effect of temperature on increment of maize yield per unit was analyzed.Based on the understanding of the method of functional principal component analysis and functional linear regression model,the estimation of slope function of functional linear regression model with autoregressive error term is further discussed,and the convergence rate is obtained.