Some Studies On Functional Regression And Functional Clustering

With the development of science and technology,data storage and the complexity of data form are increasing.Functional data widely exists in the fields of medicine,meteorology,biology,economics,and other fields.When analyzing this kind of data,it is necessary to consider the infinite-dimensional characteristics of functional data,which brings great difficulties to statistical analysis,and traditional statistical analysis tools are not applicable.In addition,functional data also provide internal characteristics that traditional data cannot bring,more properties and results can be obtained when analyzing this type of data.Therefore,functional data has become a hot research topic at present,and how to use statistical methods to analyze functional data has become a very concerning issue in the field of statistics.We consider functional regression,classification,and clustering problems analysis,the main contents are as follows:(1)Estimation of the mean and covariance function is very important when analyzing multivariate longitudinal and sparse functional data.We define a new covariance function that not only considers the correlation of different observed responses for the same biomarker but different biomarkers.The full quasi-likelihood and the kernel method are used to approximate mean and covariance functions,the covariance decomposition is considered to decompose covariance functions to correlation function and variance function.We use the full quasi-likelihood to solve measurement errors variance and choose the iterative algorithm to update the multivariate mean and covariance functions until convergence.We use leave-oneout cross-validation are used to select bandwidth h.Finally,we give theoretical properties of the unknown functions and prove their convergence.Simulation and application results show the effectiveness of our proposed method.(2)We propose a novel method to analyze multivariate longitudinal data that contains spatial location information.The method has the advantage of analyzing the relationship between curves at neighbor time points and observing the relationship between locations.We offer the spatial covariance function and use functional PCA to estimate unknown parameter functions.A detail solving process and theoretical properties are introduced.Based on the gradient descent method and leave-one-out cross-validation method,we estimate those unknown parameters and select the principal components respectively.Furthermore,compared with the other four methods,the proposed method shows a better category effect on simulation studies and air quality data analysis.(3)Model-based functional data clustering analysis usually has normality assumptions.We consider the non-Gaussian functional data model and cluster analysis and proposes a new non-Gaussian functional mixed-effects model without prior information and cluster categories.Non-Gaussian data is processed using the Box-Cox transformation function,and unknown fixed and random effects are approximated using smooth spline ANOVA and cubic B-splines,respectively.A penalized likelihood function for estimating unknown parameters is constructed and consistent and asymptotically normal properties are demonstrated.Simulations analyses are carried out with different hypothetical distributions of measurement errors,using the air quality of Italian cities as real data.Simulation and real data analysis results show that the proposed method has good performance and a better clustering effect.The different methods proposed in this paper enrich the research on multivariate longitudinal data,which is helpful for modeling and analysis in various fields such as medicine,meteorology,biology,economics,etc.,thereby improving the calculation speed,simplifying the model,and improving the prediction accuracy.