Theory And Application Of Spatial-Temporal Functional Regression Analysis

With the development of modern technology,a large number of func-tional data have been collected in many scientific fields,such as meteorology,medicine,seismology and economics.To analyze functional data,functional data analysis(FDA)has attracted wide attention in recent years,and many theoretical and applied research results have emerged.Unlike traditional statis-tical analysis,functional data is considered to be generated by a function pro-cess that changes continuously over time or space.This paper aims to explore the relationships among variables including functional variables.The existing regression models have the following shortcomings:First,the information of observation time is ignored.The observation time of much functional data is in-formative.For example,(1)Bidding records of online auctions are a functional data,and many studies(Shmueli&Jank,2004;Borle et al.,2006;Easley&Tenorio,2004;Bapna et al.,2003;Mithas&Jones,2007)have pointed out that bidding time has a significant impact on the final prices;(2)Records of the flow and speed of routes are also functional data,and the recording time of traffic flow is related to traffic conditions.In general,the observation frequency of the fast-changing functional data is higher,and vice versa.This means that the ob-servation or recording time itself is informative.Secondly,most of the existing studies neglect the temporal and spatial correlation information of functional data.In many practical applications,functional data has a temporal and spa-tial correlation.For example,(1)in the analysis of bidding data,the bidding price of earlier similar auctioneers will affect the current product bidding;(2)in the analysis of traffic data,different roads connect with each other,and traffic flow of different roads influences each other;(3)in the analysis of brain image data,the data of magnetic resonance curves of adjacent regions present a similar value.In this paper,we propose the semi-parametric/non-parametric regression models for several types of complex functional data,which can exca-vate the internal relationship between functional variables and other variables and obtain more accurate predictions.Most functional regression analysis restores complete functional data by a pre-smoothing method and uses the integral form to fuse the influence of functional variables on response variables.However,when the observation is sparse and the distribution is irregular,the pre-smoothing method can not be used.Also,the simple integral method erases the observation time information of functional data.In this paper,a dynamic regression model is proposed for functional data with sparse observations and information of observation time.Our method does not need to pre-smooth the observed data,avoids the deviation and variance caused by pre-smooth processing.It has strong data adaptability.At the same time,the prediction accuracy of the model is improved by combining the information of observation time.In particular,our method can predict the transaction price of online auction more accurately.We prove the consistency of the proposed estimators and the minimax risk rate.We also prove the asymptotic normality for statistical inference.Besides,numerical simulation proves the validity of our method and extensive data adaptability.Facing some practical problems,correlated functional data are often col-lected.For example,the online auction data,traffic data and brain image data mentioned above.Therefore,this paper presents a dynamic regression model for correlative functional data.In addition to the advantages of the dynamic regression model mentioned above,the model also considers the cor-relation between functional data and improves the accuracy of estimation.It is worth noting that we do not make specific assumptions about correlation,so we do not bring additional model bias while using more information.Further-more,the binary smooth regression function is used to effectively utilize the correlation information of functional covariates and reduce the computational complexity of the model.We prove the consistency and asymptotic normality of the estimators and give the minimax risk rate.The actual online auction data analysis shows that the use of correlation information improves prediction accuracy.In reality,we also encounter more complex high-dimensional spatial-tempor-al functional data,which not only is high-dimensional but also is temporal and spatial correlated.For example,traffic data,meteorological data,brain image data.Therefore,an additive regression model for high-dimensional spatial cor-relation function data is proposed in this paper.The model makes full use of the information of spatial location and time and fits the spatial and tempo-ral correlation by data-driven smoothing technology.Therefore,the relation-ship information between time and space is utilized without the assumption of the model.This improves the prediction accuracy without additional model bias.Also,the model no longer assumes a linear regression relationship,and flexibly describes the non-linear relationship between response variables and high-dimensional functional covariates through an additive model,so it has extensive data adaptability.We prove the consistency of the estimators and achieve the optimal convergence rate of nonparametric estimation.The validity and robustness of the method are verified by numerical simulation.Compared with other methods,our method has higher prediction accuracy for analyz-ing the traffic speed of a city’s main roads.At the same time,we also reveal the different impact modes of different roads in the morning and evening rush hour.This discovery is helpful to realize the real-time evaluation and analysis of the traffic capacity of main urban corridors and corresponding areas.It is helpful to improve the efficiency of traffic management and the level of urban traffic intelligent management.