Integration Estimation And XGBoost Algorithm Based On Quasi-Monte

The Monte Carlo method is a statistical computing method based on probability theory,aimed at solving complex mathematical problems through random sampling.In order to further improve the sampling efficiency of the Monte Carlo method,scholars have proposed variance reduction techniques,such as control variates and importance sampling.These methods can effectively reduce the variance of the estimator,improve computational accuracy and efficiency.However,these methods do not change the convergence rate of the Monte Carlo method,which is still O(N-1/2).To overcome this limitation,scholars have proposed quasi-Monte Carlo methods.This method uses low-discrepancy sequences instead of traditional random sequences,thereby improving the convergence rate of the Monte Carlo method to O(N-1).Quasi-M onte Carlo method is an efficient numerical computation method,which has great potential for applications in the two fields of integration estimation and machine learning focused on in this paper.In the study of integration estimation,this paper explores the randomized quasi-Monte Carlo method with combinable variance reduction techniques.It elaborates on the relevant theory of the variance reduction effect of the(0,m,s)-scrambled net sampling method from the perspective of variance reduction,and compares its performance on various functions in different dimensions through experiments.The research results show that with the increase of sample size,the randomized quasi-Monte Carlo method can reduce the variance faster,and has higher efficiency compared to the control variates method.Moreover,the randomized quasi-Monte Carlo method combined with control variates performs better.In addition,this paper has developed corresponding Python packages for the randomized scrambled method and the randomized quasi-Monte Carlo method combined with control variates,which can be used by other scholars for learning and research.In the field of machine learning,XGBoost algorithm is an efficient ensemble learning algorithm.In this paper,we use the low-discrepancy property of quasi-Monte Carlo sequences to improve the sampling technique in such algorithms,and propose an XGBoost algorithm based on quasi-Monte Carlo sequences.Through numerical simulations and applications in practical cases such as credit loans for diabetic patients,we found that the proposed algorithm has significant advantages in accuracy,AUC,and other evaluation metrics.