| Functional data analysis is a prominent research domain within the field of statistics,boast-ing significant applicability in medicine,genetics,and various other disciplines.Traditional functional data analysis methodologies based on Gaussian processes,frequently encounter chal-lenges associated with outliers,particularly in scenarios involving large and intricate datasets.This paper proposed a robust inferential approach based on Student-t process for functional principal component analysis and functional regression.This paper encompasses three distinct aspects:The first part of this thesis introduces a robust inference method for functional principal component analysis(FPCA)to minimize the influence of anomalous data on the outcomes.Our approach utilizes the Student-t process,enabling robust estimation of functional PCA methods.The variational EM algorithm is employed for parameter estimation and prediction.Our method delivers more precise and robust data analysis,enhancing the reliability of the results.The second part of this thesis presents a robust inference method for a function-based linear regression model of scalar pair functions.Our model adopts three distinct Student-t processes to model the distribution of the response variable,the random effect term of the covariate function,and the random error term,respectively.We perform data dimensionality reduction by utilizing functional principal components analysis.The model parameters are estimated via a variational EM approach,resulting in more precise and robust analysis outcomes.The third part of this thesis introduces a robust inference method for functional generalised linear models utilizing principal component analysis.Our model assumes that the response vari-ables follow other exponential family distributions,with a functional logistic regression model serving as an example.We use different Student- t processes to model the random effect terms and the random error terms of the covariate functions.In terms of estimation,the variational EM method is used to estimate the model parameters,and the likelihood function of the binomial distribution is approximated by local variation,which achieves robust data analysis results. |
