马尔可夫链预测方法及其在水文序列中的应用研究
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摘要
本文首先讨论了随机变量序列的“马尔可夫性”检验,接着讨论了基于绝对分布的马尔可夫链预测方法、叠加马尔可夫链预测方法和加权马氏链预测方法。特别是研究和完善了加权马氏链预测理论。本文的亮点是基于统计试验的各种马尔可夫链预测方法的比较分析以及研究各种因素对马尔可夫链预测精度的影响。通过随机模拟,得到了如下一批成果:
(1) 加权马尔可夫链预测方法的预测精度最高;叠加马尔可夫链预测方法的预测精度次之;基于绝对分布的马尔可夫链预测方法的预测精度最低。
(2) 统计试验的次数c对预测结果有一定的影响,当c≥200时,预测精度趋向稳定。
(3) 各种马尔可夫链预测方法的预测精度具有对不同分布的适应性。
(4) 序列的变差系数C_V和偏态系数C_S对预测精度的影响显著。随着C_V的增大,预测精度稍有下降;随着C_S的增大,预测精度呈现较大幅度的下降趋势,WMCP法的预测结果最为稳健。
(5) 样本容量的增大会提高预测的精度,这与任何一种统计预测方法的性质是一致的;三种马尔可夫链预测方法的优劣不随样本容量的变化而变化,即对样本容量的变化具有自适应性。
(6) 一步自相关系数ρ_X(1)的变化对马尔可夫链预测方法的预测精度有一定的影响,但并不显著。
(7) 不管是对于服从P-Ⅲ型分布的马氏序列,还是对于服从对数正态分布的马氏序列,指标值分级的不同都将对预测精度产生显著的影响。
另外,本文还研究了加权马尔可夫链预测理论在水文水资源科学中的应用;并以实例的形式分别讨论了加权马尔可夫链在河流丰枯状态预测中的应用、降水丰枯状态预测中的应用和长江中下游地区梅雨强度预测中的应用。三个实例均表明加权马尔可夫链预测方法是一种行之有效的预测方法。
Firstly, the "Markov property" test of random variable series was discussed, then the Markov chain prediction method based on absolute distribution (ADMCP), the Markov chain prediction method based on probability summation (SPMCP) and weighted Markov chain prediction method (WMCP) were discussed. Especially the weighted Markov chain prediction method was studied and completed in the dissertation. The bright point of this dissertation was that various Markov chain prediction methods were studied and compared, and various factors influenced on accuracy of Markov chain predictions based on statistical experiments. Through stochastic simulation, a series results were obtained as follows:(1) The prediction accuracy of WMCP method is the highest, the prediction accuracy of ADMCP method is the lowest, and the prediction accuracy of SPMCP method is in the middle.(2) The time of statistical experiments c has a kind of influence on prediction results, and the prediction accuracy approached stable when c ≥200.(3) The prediction accuracy of various Markov chain prediction methods had a suitability to difference distributions.(4) The coefficients Cv and Cg had a heavy influence on prediction accuracy. When Cv increased, the prediction biases a little increased, and when Cg increased, the prediction biases decreased. But the prediction results of WMCP method were the most robust.(5) It would increase prediction accuracy when sample size enlarged, which was similar to any efficient statistical prediction method. The goodness of these three kinds Markov chain prediction methods were not changed with sample size, that is that various Markov chain prediction methods ail have a auto consistent property.(6) The change of a step self-correlation coefficient px(l)had a kind of influence on various Markov chain predictions accuracy, but the influence was not so significant.(7) The classification of observed values from a Markov series had a strong influence on prediction accuracy, whether or not that the series had a P-III distribution or three parameters LN distribution.Moreover, the application of WMCP method on hydrology and water resources was studied in the dissertation. The following three real instances were discussed: The application of WMCP method to the prediction of river runoff state, the application of WMCP method to the prediction of precipitation state, and the application of WMCP method to the prediction of plum rains intensity. Three real instances all indicated that WMCP method was a valuable prediction method.
(1) 加权马尔可夫链预测方法的预测精度最高;叠加马尔可夫链预测方法的预测精度次之;基于绝对分布的马尔可夫链预测方法的预测精度最低。
(2) 统计试验的次数c对预测结果有一定的影响,当c≥200时,预测精度趋向稳定。
(3) 各种马尔可夫链预测方法的预测精度具有对不同分布的适应性。
(4) 序列的变差系数C_V和偏态系数C_S对预测精度的影响显著。随着C_V的增大,预测精度稍有下降;随着C_S的增大,预测精度呈现较大幅度的下降趋势,WMCP法的预测结果最为稳健。
(5) 样本容量的增大会提高预测的精度,这与任何一种统计预测方法的性质是一致的;三种马尔可夫链预测方法的优劣不随样本容量的变化而变化,即对样本容量的变化具有自适应性。
(6) 一步自相关系数ρ_X(1)的变化对马尔可夫链预测方法的预测精度有一定的影响,但并不显著。
(7) 不管是对于服从P-Ⅲ型分布的马氏序列,还是对于服从对数正态分布的马氏序列,指标值分级的不同都将对预测精度产生显著的影响。
另外,本文还研究了加权马尔可夫链预测理论在水文水资源科学中的应用;并以实例的形式分别讨论了加权马尔可夫链在河流丰枯状态预测中的应用、降水丰枯状态预测中的应用和长江中下游地区梅雨强度预测中的应用。三个实例均表明加权马尔可夫链预测方法是一种行之有效的预测方法。
Firstly, the "Markov property" test of random variable series was discussed, then the Markov chain prediction method based on absolute distribution (ADMCP), the Markov chain prediction method based on probability summation (SPMCP) and weighted Markov chain prediction method (WMCP) were discussed. Especially the weighted Markov chain prediction method was studied and completed in the dissertation. The bright point of this dissertation was that various Markov chain prediction methods were studied and compared, and various factors influenced on accuracy of Markov chain predictions based on statistical experiments. Through stochastic simulation, a series results were obtained as follows:(1) The prediction accuracy of WMCP method is the highest, the prediction accuracy of ADMCP method is the lowest, and the prediction accuracy of SPMCP method is in the middle.(2) The time of statistical experiments c has a kind of influence on prediction results, and the prediction accuracy approached stable when c ≥200.(3) The prediction accuracy of various Markov chain prediction methods had a suitability to difference distributions.(4) The coefficients Cv and Cg had a heavy influence on prediction accuracy. When Cv increased, the prediction biases a little increased, and when Cg increased, the prediction biases decreased. But the prediction results of WMCP method were the most robust.(5) It would increase prediction accuracy when sample size enlarged, which was similar to any efficient statistical prediction method. The goodness of these three kinds Markov chain prediction methods were not changed with sample size, that is that various Markov chain prediction methods ail have a auto consistent property.(6) The change of a step self-correlation coefficient px(l)had a kind of influence on various Markov chain predictions accuracy, but the influence was not so significant.(7) The classification of observed values from a Markov series had a strong influence on prediction accuracy, whether or not that the series had a P-III distribution or three parameters LN distribution.Moreover, the application of WMCP method on hydrology and water resources was studied in the dissertation. The following three real instances were discussed: The application of WMCP method to the prediction of river runoff state, the application of WMCP method to the prediction of precipitation state, and the application of WMCP method to the prediction of plum rains intensity. Three real instances all indicated that WMCP method was a valuable prediction method.
引文
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