时间序列异常值探测的Bayes方法及其在GPS数据处理中的应用
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摘要
时间序列的观测值经常会受到异常扰动的影响,如果忽视这些影响直接进行建模和预测,就会造成虚假的后果,甚至导致错误的结果。因此,在动态测量数据分析和处理中,寻求有效的探测时间序列异常值的策略显得非常重要。
本文在系统地回顾和总结时间序列异常值探测研究现状的基础上,应用现代Bayes统计理论和方法讨论并研究了平稳时间序列异常值的探测问题,在综合利用先验信息与观测信息的基础上,系统地提出了平稳时间序列异常值探测的Bayes方法。进一步,将平稳时间序列异常值探测的Bayes方法应用到GPS数据处理中,优化了GPS钟差序列和电离层VTEC序列的建模和预报方法。
本文的主要工作和创新点如下:
1.总结和分析了现有的时间序列异常值探测方法。概述了线性平稳时间序列的三种模型,指出了时间序列中若出现异常值将会对时间序列建模和预测的影响,回顾和总结了现有的时间序列异常值探测方法,并分析和指出了这些方法的局限性。
2.提出了时间序列异常值定位的Bayes方法。在一定的限制条件下,将时间序列的异常值定位问题转化为线性回归模型的异常值定位问题,结合线性回归模型异常值定位的Bayes方法,提出了时间序列异常值定位的Bayes方法,并进一步给出了无信息先验下和正态—Gamma先验下基于均值漂移模型和方差膨胀模型的后验概率的Bayes公式。
3.提出了时间序列异常值估计的Bayes方法。应用Bayes统计理论,分别在无信息先验分布和正态—Gamma先验分布条件下建立了时间序列异常扰动的Bayes估值公式,进一步完善了时间序列异常值探测的Bayes理论和方法。
4.提出了GPS时间序列建模的Box—Jenkins法和异常值探测的Bayes方法。对GPS卫星钟差序列和电离层VTEC序列进行了Box-Jenkins建模,采用上述Bayes方法探测序列中的异常值并对异常值进行修正,提高了GPS卫星钟差预报和电离层VTEC预报的精度。理论分析和大量的数值试验都表明,本文提出的时间序列异常值探测的Bayes方法具有很好的可靠性和适用性。
The observation of time series may be influenced by outliers. If we forecasts directly, neglecting the influence, will lead to false result. So, in dynamic surveying data analysis and data processing practice, the seeking for approaches to dealing with time series outliers becomes very important.
On the basis of reviewing and summarizing the actual researching state of time series outliers detection systemically, this paper will mainly discuss the approaches to outliers detection in time series utilizing the modern Bayesian theories and methods. And syncretizing prior information with observing information, Bayesian method for outliers detection in stationary time series is put forward. Furthermore, the method is applied in the research on the data processing of GPS; also optimize the modeling and predicting methods of clock error and VTEC series.
The main conclusions are as follows:
1. Firstly, we summarized and analyzed the outliers detecting methods in the present. This paper divided the stationary time series into three classes, then pointed out the influence on modeling and predicting,when there are outliers in time series. Furthermore, reviewed and summarized traditional outliers detecting method. Finally, suggested the limitation of traditional methods.
2. Bayesian method for outliers positioning were given secondly. Given some special restrictions, we can change time series outliers positioning into outliers positioning in linear regression model. Based on the theory of Bayesian Statistical diagnostics, we put forward Bayesian method of outliers positioning. And then, under the condition of both the non-informative priors and normal-gamma prior information, Bayesian method for posterior probability calculating were given respectively.
3. Bayesian methods for outliers estimating were given thirdly. Applying Bayesian statistical estimating theory, under the condition of both the non-informative priors and normal-gamma prior information, Bayesian estimations for outliers are given respectively to perfect outliers estimations as one important aspect of outliers detecting.
4. Box-Jenkins modeling method on GPS time series and Bayesian method for outliers detecting were put forward. Then modeling the GPS clock error series and ionospheric VTEC series with Box-Jenkins method, At last, modify the outliers with the new method, improving the predicting accuracy of the GPS clock error series and ionospheric VTEC series.
Theoretical analysis and mass numerical examples demonstrate that the new method is useful and efficient.
本文在系统地回顾和总结时间序列异常值探测研究现状的基础上,应用现代Bayes统计理论和方法讨论并研究了平稳时间序列异常值的探测问题,在综合利用先验信息与观测信息的基础上,系统地提出了平稳时间序列异常值探测的Bayes方法。进一步,将平稳时间序列异常值探测的Bayes方法应用到GPS数据处理中,优化了GPS钟差序列和电离层VTEC序列的建模和预报方法。
本文的主要工作和创新点如下:
1.总结和分析了现有的时间序列异常值探测方法。概述了线性平稳时间序列的三种模型,指出了时间序列中若出现异常值将会对时间序列建模和预测的影响,回顾和总结了现有的时间序列异常值探测方法,并分析和指出了这些方法的局限性。
2.提出了时间序列异常值定位的Bayes方法。在一定的限制条件下,将时间序列的异常值定位问题转化为线性回归模型的异常值定位问题,结合线性回归模型异常值定位的Bayes方法,提出了时间序列异常值定位的Bayes方法,并进一步给出了无信息先验下和正态—Gamma先验下基于均值漂移模型和方差膨胀模型的后验概率的Bayes公式。
3.提出了时间序列异常值估计的Bayes方法。应用Bayes统计理论,分别在无信息先验分布和正态—Gamma先验分布条件下建立了时间序列异常扰动的Bayes估值公式,进一步完善了时间序列异常值探测的Bayes理论和方法。
4.提出了GPS时间序列建模的Box—Jenkins法和异常值探测的Bayes方法。对GPS卫星钟差序列和电离层VTEC序列进行了Box-Jenkins建模,采用上述Bayes方法探测序列中的异常值并对异常值进行修正,提高了GPS卫星钟差预报和电离层VTEC预报的精度。理论分析和大量的数值试验都表明,本文提出的时间序列异常值探测的Bayes方法具有很好的可靠性和适用性。
The observation of time series may be influenced by outliers. If we forecasts directly, neglecting the influence, will lead to false result. So, in dynamic surveying data analysis and data processing practice, the seeking for approaches to dealing with time series outliers becomes very important.
On the basis of reviewing and summarizing the actual researching state of time series outliers detection systemically, this paper will mainly discuss the approaches to outliers detection in time series utilizing the modern Bayesian theories and methods. And syncretizing prior information with observing information, Bayesian method for outliers detection in stationary time series is put forward. Furthermore, the method is applied in the research on the data processing of GPS; also optimize the modeling and predicting methods of clock error and VTEC series.
The main conclusions are as follows:
1. Firstly, we summarized and analyzed the outliers detecting methods in the present. This paper divided the stationary time series into three classes, then pointed out the influence on modeling and predicting,when there are outliers in time series. Furthermore, reviewed and summarized traditional outliers detecting method. Finally, suggested the limitation of traditional methods.
2. Bayesian method for outliers positioning were given secondly. Given some special restrictions, we can change time series outliers positioning into outliers positioning in linear regression model. Based on the theory of Bayesian Statistical diagnostics, we put forward Bayesian method of outliers positioning. And then, under the condition of both the non-informative priors and normal-gamma prior information, Bayesian method for posterior probability calculating were given respectively.
3. Bayesian methods for outliers estimating were given thirdly. Applying Bayesian statistical estimating theory, under the condition of both the non-informative priors and normal-gamma prior information, Bayesian estimations for outliers are given respectively to perfect outliers estimations as one important aspect of outliers detecting.
4. Box-Jenkins modeling method on GPS time series and Bayesian method for outliers detecting were put forward. Then modeling the GPS clock error series and ionospheric VTEC series with Box-Jenkins method, At last, modify the outliers with the new method, improving the predicting accuracy of the GPS clock error series and ionospheric VTEC series.
Theoretical analysis and mass numerical examples demonstrate that the new method is useful and efficient.
引文
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[2] Decoste, D. Mining multivariate time-series sensor data to discover behavior envelopes[R]. Proc of the 3rd Conference on Knowledge Discovery and Data-Mining, [S.l.]:AAAI Press, 1997, 151-154.
[3] Dasgupt, A.D., Forrest, S. Novelty detection in time series data-using ideas from immunology[R]. Proc of the 5th International Conference on Intelligent Systems, 1996, 82-87.
[4] Ma, J., Perkins, S. Time-series novelty detection using one-class support vector machines[R]. Proc of International Joint Conference on Neural Networks, 2003.
[5] Ma, J., Perkins, S. Online novelty detection on temporal sequences[M]. ACM Press, New York, 2003.
[6] Borisyuk, R., Denham, M., Hoppenstead, T.F., et al. An oscillatory neural network model of sparse distributed memory and novelty detection[J]. Bio-Systems, 2000, 58 (1):265-272.
[7] Shahab, I.C., Tian, X., Zhao, W. A wavelet-based approach to improve the efficiency of multi-level surprise and trend queries[J]. Proc of the 12th International Conference on Scientific and Statistical Data-base Management, IEEE Computer Society, Washington DC, 2000, 55-68.
[8] Chakrabarti, S., Saraqwagi, S., Dom, B. Mining surprising patterns using temporal description length[M]. Morgan Kaufmann Publishers, San Francisco, 1998, 606-617.
[9] Yamanish, I.K., Takeuch, I.J. A unifying framework for detecting outliers and change points from non-stationary time series data[M]. ACM Press, New York, 2002, 676-681.
[10] Jagadish, H.V., Koudas, N., Muthukrishnan, S. Mining deviants in a time series data base[M]. S Morgan Kaufmann Publishers, San Francisco, 1999, 102-113.
[11] Deutsch, D., Richards, Swain, J.J. Effects of a signal outlier on ARMA identification[J]. Communications in Statistics, Theory and methods, 1990, 19:2207-2227.
[12] Kleiner, B., Martin, R.D., Thomson, D.J. Robust estimation of power spectra[J]. Journal of the Royal Statistical Society, Series B, 1979, 41:313-351.
[13] Ledolter, J. The effect of additive outliers on the forecasts from ARIMA models[J]. International Journal of Forecasting, 1989, 5:231-240.
[14] Hotta, L.K. The effect of additive outliers on the estimates from aggregated and disaggregated ARIMA models[J]. International Journal of Forecasting, 1993, 9:85-93.
[15]茆诗松.贝叶斯统计学[M].北京:中国统计出版社,1999.
[16]吴喜之.现代贝叶斯统计学[M].北京:中国统计出版社,2000.
[17] Fox, A.J. Outliers in time series[J]. Journal of the Royal Statistical Society, Series B, 1972, 34:350-363.
[18] Ledolter, J. Outlier diagnostics in time series analysis[J]. Journal of Time Series Analysis, 1990, 11:317-324.
[19] Ljung, G.M. On outlier detection in time series[J]. Journal of the Royal Statistical Society, Series B, 1993, 55:559-567.
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