Some Topics On Regularized Regression With Complex Data And Its Applications

Much progress has been made in the field of data collection,resulting in numerous types of data,such as panel data,genomic data,functional data,and matrix data,among others.Complex data involves a large number of variables,observations,or both,and as technology has improved,numerous types of covariates have emerged.This thesis focuses on regularized regression models for complex data,particularly functional and matrix data,which are relatively novel in the field of statistics.Regularized regression is a powerful and extensively used technique in various domains,including statistics,machine learning,and data science,for generating prediction models.Regularized regression modifies the traditional loss function of linear regression by incorporating a penalty term,which helps to decrease overfitting and improve the model’s generalization performance throughout.In our dissertation,the main objective is to determine the parameter function for the generalized functional additive model(G-FAM)in the reproducing kernel Hilbert space(RKHS)framework by utilizing the penalized negative log-likelihood function.In addition,we aim to analyze the prediction convergence rate while examining the G-FAM’s potential for investigating the relationship between a functional predictor and a scalar response.The G-FAM is an innovative regression model that offers increased flexibility while maintaining ease of interpretation and estimation.Our proposed model has been tested on simulated and real-world datasets,and the results demonstrate superior performance compared to existing models.Additionally,the dissertation employs the penalized function-on-scalar FOSR regression model and the Bayesian function-on-scalar model(Bayesian FOSR)model to assess the relationship between COVID-19 reproduction rates and vaccination rates across 46 African nations from January to November 2021.The study reveals a noteworthy correlation between vaccine rates and the virus’s reproductive rate,with a statistically significant association being observed.Furthermore,the dissertation explores the impact of temperature on the monthly average air quality in the Cairo region of Egypt through the application of the penalized function-on-function(penalized FF)model and function-on-function principal component analysis(FF-FPCA).The findings indicate that air quality substantially influences mitigating particulate matter in Egypt more than temperature.Also,the results show that penalized FF is more efficient than FF-FPCA.The dissertation expands the sparse trace regression model to accommodate exponential β-mixing errors in the matrix data section.This expansion establishes the estimator’s convergence rate and asymptotic properties.Our proposed method is shown to provide remarkable prediction results through both simulation and analysis of real data.Finally,the dissertation establishes concentration inequalities under exponential β-mixing conditions and explores the low rank of the expectile trace regression.The efficiency of our proposed methods has been demonstrated in numerical simulations and real-world data applications.In conclusion,the thesis presents regularized regression models for complex data.The simulation and real-world data analyses demonstrate their effectiveness in predicting and providing new insights into the data.