Research On Conformal Prediction Method For Positive Response Data Based On Relative Error Criterion

Data with positive response variables(referred as positive response data)are widely exists in the fields of finance,economy,medicine,meteorology,environment and other fields.Modeling and statistical analysis of this kind of data is one of the important issues in statistics and data science and other related fields.It has been shown the reasonability of using the relative error criterion instead of the absolute error criterion when modeling the positive response data.Prediction interval can quantify the uncertainty of prediction,conformal prediction(CP)proposed by Vovk et al.is an interval prediction method that can guarantee coverage probability,the method constructs the prediction interval of the response variable based on the idea of hypothesis testing,with the superiority of loose requirement in base model and easily operation.It has attracted extensive attention by researchers in statistics and machine learning.This thesis focuses on the interval prediction problem of positive response data in regression,constructs three new conformal prediction algorithms based on the relative error criterion that can reach a given confidence level and high computational efficiency,and applies them in the simulated data set and the real data set.The main contents are as follows:Firstly,this thesis studies the interval prediction problem of positive response data under the exchangeable assumption,and constructs a conformal prediction algorithm based on relative error.Considering that the response variable is positive,thesis establishes a nonconformity score function based on the relative error,and establishes a prediction set of the response variable under the given confidence level 1-.In calculation,we propose an binary-research-based search algorithm to construct the prediction interval under the given confidence level,which improves the calculation efficiency.The thesis theoretically proves that the constructed prediction interval is valid.On the other hand,simulation studies show that in the most case of finite samples,the proposed prediction method has shorter prediction interval and the coverage probability is closer to the given confidence level.The results of real data analysis also show that the proposed algorithm has certain advantages in solving the problem of positive response data prediction.Secondly,this thesis studies the interval prediction problem of multiple independent positive response time series in the exchangeable case,and constructs a multi-step conformal prediction algorithm based on relative errors.A multistep nonconformity score function based on relative error is constructed,and a multi-step prediction set of the response variable is established under a given confidence level 1-.In calculation,this thesis divides the sample set into training set and calibration set,combines binary search method and Bonferroni correction to construct a multi-step prediction interval under the given confidence level,which improves the calculation efficiency.The thesis theoretically proves that the algorithm can give effective multi-step prediction intervals.On the other hand,numerical simulation results and real data analysis show that the coverage probability of the multi-step prediction interval by the algorithm is closer to the given confidence level and has higher accuracy.Finally,this thesis studies the interval prediction problem of a single positive response time series in the non-exchangeable situation,and builds a weighted conformal prediction algorithm based on relative errors according to the conformal prediction algorithm by Barber et al.(2023).The algorithm uses weights to restore the correlation of non-exchangeable samples,and constructs a nonconformity score function based on relative error,also it establishes a prediction set which satisfies a given confidence level.In calculation,this thesis divides the sample set into training set and calibration set,and uses the binary search method to construct the prediction interval under a given confidence level,which improves the calculation efficiency and avoids potential overfitting.The thesis theoretically proves that the algorithm can give effective prediction intervals.On the other hand,performances on numerical simulation and real data show that the coverage probability of the prediction interval by the proposed algorithm is closer to the given confidence level and has higher accuracy.