Research And Application Of Functional Logistic Regression Model Based On Gibbs Sampling Algorithm

With the increasing demand for data,the demand for data quality is getting higher and higher in many fields,and the data analysis is also leaping from low-frequency data analysis to high-frequency data analysis.However,in many cases,the data we get are discrete data,which cannot fully capture the information of the data.Based on this,Ramsay proposed functional data analysis(FDA)^[1]in 1982.Compared with traditional data analysis,FDA has more advantages.It can dig out more important information by analyzing the curve properties of data.And Logistic regression as a widely used classification model,also has an important application in functional data.Therefore,in this paper,the Logistic regression model whose response variables were binary data and multi-classification ordered data was explored under the functional framework.Firstly,in view of the functional Logistic regression model whose response variables are binary data,the regression coefficient function and the regression function-type independent variable are expanded by the selected data-driven principal component basis function.By using the orthogonality of the principal component basis function,the high-dimensional data are expressed in a low-dimensional manner.The appropriate prior information was set for the expansion coefficient of the regression function and the posterior distribution of the expansion coefficient of the regression function was obtained by using the Polya-Gamma transform,in addition the Gibbs sampling algorithm was established.Monte Carlo simulation results show that this method has good classification performance.Finally,this paper applies this method to Tecator actual data and ECG actual data,finds that its classification effect is better than other methods.For the functional cumulative Logistic regression model whose response variables are ordered multi-classification data,the relationship between the ordered response variable and the function covariable is related by the latent variable.The regression coefficient function and the independent variable of regression function were expanded by selected principal component basis,the error term is set to follow the standard logistic regression distribution.The complexity of the model likelihood function was solved by Polya-Gamma transform,and the posterior distribution of the expansion coefficient is obtained to construct Gibbs sampling algorithm.In addition applied the method to the simulated data and actual AQI data,the results show that it can classify them well.