Penalized Estimation Of Logistic Regression Model With Mixed Functional Data

Functional data is an important and popular data type at present,which is widely used in the field of natural sciences,such as meteorology,economics,biology,medicine,etc.Therefore,the analysis of functional data has gradually become a hot issue.With the development of science and technology,the form of data becomes more and more complex,and there may be mixed data mixed with other types of data and functional data.In this paper,Logistic regression model was established for the situation where response variables were dichotomous and covariable was mixed data of functional and numerical types,and it was divided into Logistic partial linear model with mixed function data and Logistic partial single index model with mixed function data.A number of studies have been carried out on the penalty estimation and practical application of these two models.For the Logistic partial linear model with mixed functional data as covariable,we first construct a maximum likelihood function and calculate its log-likelihood function expression.Secondly,we use B-spline basis function to expand the coefficient function,and combine roughness penalty and sparse penalty to get regression and non-parametric penalty likelihood estimation.The effectiveness of the estimation method is verified by numerical simulation.According to the integral average error of empty subregions and non-empty subregions,it is considered that the proposed estimation method has good performance.At the same time for the linear part of the simulation setting,set different dimensions,sample size and different punishment methods,and calculate the linear part of the coefficient variable selection of evaluation and estimation efficiency.Combining the simulation results of the functional part and the linear part,it can be found that the proposed estimation method can identify the empty subregions and zero vector well.Finally,this model was used to fit the spectral data of meat,and it was found that when the wavelength range of spectral data was excluded from [960nm,980nm],the spectral data had a significant impact on the determination of obesity characteristics.Aiming at the Logistic partial single index model with mixed function data,we found that the model could not only avoid the "dimension disaster",but also maintain the flexibility of the non-parametric model.Therefore,we studied the parameter estimation and variable selection of the partial linear single index model.First,the logarithmic likelihood function was constructed,and B-spline was used to estimate the coefficient function and approximate the unknown single index function.The penalty likelihood estimation of the Logistic partial single index model was obtained.Combined with SCAD penalty,the coefficient function and index coefficient estimation were obtained by Newton iterative algorithm.The SCAD penalty method is used to select variables with partial parameters of a single index,and the robustness of the estimation method and good performance of finite samples are demonstrated by numerical simulation.Finally,we demonstrate the rationality and validity of the proposed estimation method by analyzing the meteorological data in 2017.The estimation method proposed in this paper enriches the study of Logistic model with mixed functional data,and the conclusions obtained will help us better analyze the practical problems encountered in the fields of medicine,economics and biology.