| 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. |
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