【作者】 王红；

【导师】 类淑河；

【作者基本信息】 中国海洋大学 ， 统计学， 2015， 硕士

【摘要】 带有调和项与有色噪声(加性噪声)的混合谱的模型在电力、气象、海洋、经济等方面有广泛应用。由于受时间、人力物力等因素的影响,我们可能得到较短的混合谱时间序列。经典的谱估计方法周期图法、Welch法,谱估计精度较低,谱密度函数已不能很好体现出短时序列混合谱调和项波动的特性,反映在经济上可能会造成波动周期的误判。近几年处理短时序列的MCMC方法主要针对混合谱中加性噪声线性模型的参数估计,对短时序列中调和项的参数估计并未给出明确说明,且方法的实现还需要模型参数分布的先验信息,验证模型也较简单,未用于实证分析。本文利用混合谱具有的波动特性和时序特征,将混合谱过程看作调和项与自回归模型AR模型的叠加。由于两部分叠加的相互影响,本文又采用了交替迭代估计,即在零噪声假设下先进行调和项估计,然后在原数据上减掉已估计出的调和项模拟数据,对剩余数据进行噪声项的估计；将估计出的噪声项模拟数据从原数据中减掉,对剩余数据进行调和项的估计,反复迭代直到参数估计值稳定为止。其中,对调和项的估计提出一种改进的方法,将MUSIC算法的频率识别代替扩展Prony法中频率的识别,然后利用Prony算法对周期振幅进行估计。改进的方法结合了MUSIC算法中高精度频率测量、不过度依赖数据长度的优越性的特点,同时弥补Prony法在噪声较强时对调和项估计准确度不高的缺陷。而加性噪声的估计则采用了适用于AR模型的最大熵法。在混合谱数值实验模拟中,通过观察混合谱主要参数的均方误差随样本容量变化(以1000样本容量为基础,以50为步长,逐渐减小样本容量直至50)得出,均方误差随样本容量波动不大,且在样本容量约为200后变化率较小。在样本容量为200时,比较改进方法与其他谱估计方法在同等阶数判别下频率识别度,比较结果得出改进方法具有明显的优势,说明本文提出的改进方法在处理短时序列混合谱方面是有效、可靠的。利用改进方法对近期上证指数日成交线进行了混合谱模型的整体数据处理和分段数据处理。分段数据处理是以200个数据作为一组,对12组数据进行了分析。通过短时序列的分析结果得出12组数据大致具有相同的四个显著周期频率0.007813 Hz、0.02344 Hz、0.0625 Hz、0.08594 Hz,与上证指数的未分段数据分析得到的周期相一致,与以往经济周期分析也较为一致。由于估计方法误差及分段数据所处时期的不同经济策略等影响因素,上证指数的混合谱分析分段数据的加性噪声估计会存在一些差异。更多还原

【Abstract】 Mixed spectrum model with harmonic items and colored noise (or additive noise) has a wide range of application in power, meteorology, oceanography, economics, etc. Due to the impact of time, manpower and other factors, we may get a short-term time series. Classical spectrum estimation methods, like periodogram method and Welch method, their spectral estimation accuracy is low, the spectral density is not good enough to reflect characteristics of fluctuations in the short-term sequence of mixed spectral with harmonics, which might lead to misjudgment fluctuation cycle on the economy. And MCMC method, emerged in recent years, is only applied to the additive noise term parameters’ estimation. Whether there is a suitable method to estimate the parameters of the harmonic items in short-term mixed spectrum sequence is unknown. Implementation also requires a priori information of model parameters’distribution, besides, validation is also relatively simple, and empirical analysis is not used.In this paper, combined the characteristics of timing and fluctuation, mixed spectrum can be regard as a mixture of harmonic items and autoregressive model AR model. Due to the mutual influence of the superposition of two parts, alternating iterative estimation is applied. That is, under the assumption of zero additive noise, do the estimation of harmonic items, and then remove harmonic items that have been estimated from the original data, do the estimation of additive noise terms; to remove the estimated noise term from the original data, and then estimate the harmonic items, as go on, till the parameter estimates are stable. Besides, an improved method is provided in this paper. That is, MUSIC algorithm and Prony method are combined as one method, which means that MUSIC algorithm frequency identification takes the part of Prony algorithm frequency identification, and then use Prony algorithm other parts to estimate the amplitudes of harmonic items. Improved method has the characteristics of precision frequency measurements and the degree of dependence of the superiority of the data length, which come from MUSIC algorithm, while improving the Prony algorithm accuracy defect in parameter estimation at low signal to noise ratio. And the additive noise is estimated by maximum entropy method, a model applicable to AR process.In the mixed spectral numerical simulation experiments, by observing the mean square error of the mixed spectral main parameters with the sample size (based on the sample size of 1000,50 in steps, gradually reduce the sample size up to 50) we conclude that the mean square error with the sample size is small and smaller after the sample size of 200. In the sample size of 200, we compare the improved method with other methods in the frequency spectrum estimation by the degree discrimination in the same order. The result obtained improved that the improved method has obvious advantages, indicating that the proposed method is effective and reliable in dealing with short sequences of mixed spectrum.The improved method is used in the Shanghai index for mixed spectrum analysis of the overall data processing and segmentation data processing. There are 12 sets of data being analyzed, and each set has 200 numbers. The results obtained 12 sets roughly have four distinct frequencies 0.007813 Hz,0.02344 Hz,0.0625 Hz and 0.08594 Hz, which are the same as the results of the overall data processing, and keep in consistence with previous economic cycle analysis. Due to estimation method error and the impact of factors of economic policy in segmented data, there will be some differences in the Shanghai Composite Index additive noise mixed spectrum analysis.更多还原