混合回归模型的变点检测

发布时间：2019-01-14 11:04

【摘要】：大数据时代的到来产生了丰富海量的数据,但如何有效地检测出数据中异常突变点备受人们关注.变点检测技术无疑是目前解决这一问题的最有效手段之一,该检测手段已广泛应用到经济学、气候模拟、生物医药、反恐安检等各个领域,具有重要的研究意义.但对带有时间序列自回归影响,又有协变量线性回归影响的数据模型,即本文提出的混合回归模型已有文献鲜有涉及.本文计划就混合回归模型中的变点检测问题进行研究.本文首先对一些经典文献做简要的回顾,并对简单的均值模型、自回归模型、回归模型进行了分析总结,构造出新的模型,即混合回归模型.然后,采用带有惩罚的最小二乘法对模型参数进行估计.考虑到变点的稀疏性,我们借助了Group Lasso(Least absolute shrinkage and selection operator)技术将变点估计问题转化变量选择问题,同时给出了寻找变点的坐标下降算法.接着,给出了混合回归模型变点检测的方法,即提出检测统计量,并在一定条件下给出相应的统计性质和理论证明.最后,利用数据模拟进一步分析说明.本文主要结论:在给定条件下,变点个数的估计不小于真实变点个数是依概率趋于1的;变点位置的估计与真实位置、模型参数的估计与真实参数都具有一致性(指在给定条件下,变点位置的估计与真实位置或者模型参数的估计与真实参数差的绝对值能被限定在某个范围内);若变点个数的估计不小于真实变点个数时,变点估计集与真实变点集具有一致性;在给定的信息准则下,仅与模型调节参数有关的变点数量的估计依概率1趋于真实变点数量.模拟结果显示,混合回归模型与已有三种模型相比,可以更加准确的检测出变点位置及数量,且稳定性高,优势明显.
[Abstract]:The arrival of big data era has produced a wealth of data, but how to effectively detect the abnormal mutation in the data has attracted people's attention. Change-point detection technology is undoubtedly one of the most effective methods to solve this problem. It has been widely used in economics, climate simulation, biomedicine, anti-terrorist security inspection and other fields, which has important research significance. However, the data model with time series autoregressive effect and covariable linear regression effect, that is, the mixed regression model proposed in this paper, has rarely been involved in the literature. This paper plans to study the problem of change point detection in mixed regression model. In this paper, some classical literatures are reviewed briefly, and the simple mean model, autoregressive model and regression model are analyzed and summarized, and a new model, that is, mixed regression model, is constructed. Then, the least square method with penalty is used to estimate the model parameters. Considering the sparsity of the change point, we transform the variable selection problem into the variable selection problem with the help of Group Lasso (Least absolute shrinkage and selection operator) technique. At the same time, we give a coordinate descent algorithm for finding the change point. Then, the method of change point detection in mixed regression model is given, that is, the detection statistics are proposed, and the corresponding statistical properties and theoretical proof are given under certain conditions. Finally, the data simulation is used to further analyze and explain. The main conclusions of this paper are as follows: under given conditions, the number of change points is estimated to be not less than the number of real change points. The estimation of the position of the change point is consistent with the real position, the estimation of the model parameter is consistent with the real parameter (that is, under given conditions, The absolute value of the difference between the estimation of the position of the change point and the real position or the estimation of the model parameters and the real parameter can be limited to a certain range. If the number of change points is not less than the number of real change points, the set of change point estimators is consistent with the set of real change points, and under the given information criterion, the estimation of the number of change points related only to the adjustment parameters of the model tends to the number of real change points according to probability 1. The simulation results show that the hybrid regression model can detect the position and number of change points more accurately than the existing three models, and the stability is high and the advantages are obvious.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：C81

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