平稳随机信号的可预测性分析与熵值估计

Predictability Analysis and Entropy Estimating for Stationary Random Signal

自回归模型中的预测目标信号通常被视为其线性组合的部分输出以及从外部引入的谱白化噪声之和,并以其预测误差即外部噪声的均方差最小作为模型自回归系数计算的约束条件,但信号的不确定性使得其中的随机变化部分不可能被有效预测,所引入的外部噪声无法对其预测误差进行任何有效补偿,模型只能根据以往已知数据部分地预测到当前信号,故此应将模型的组合输出直接作为其预测结果而无需引入无益的外部噪声,其中组合输出在目标信号上的投影即为信号中的可预测部分,而与目标信号相正交的偏差部分则是对信号中不可预测部分的功率补偿,且同时还应要求模型的自回归系数为谱白化形式,以维持预测过程前后信号功率谱分布的一致性,此时信号可预测部分的最大权重比即等于其功率谱值的平方和,其余值即信号不可预测部分的权重比即可被作为随机信号的预测熵值以衡量其不确定性,信号频点状态分量的熵值函数则等于该频点处的可预测权重比与除此频点之外的其余各点信号功率谱累加值之间的乘积,模型预测及谱估计的目标都应是期望获得最小的信号预测熵值。

The predicted target signal of the Auto-Regressive model was usually regarded as the sum of partial linear combination of the model and external white noise.The set of AR coefficients of the model was solved under the constraint on the minimum mean square error of the external noise as its prediction error.However,owing to the uncertainty of the random signal,the variable random portion comprised in the signal is unable to be predicted reliably,the introduced external noise is unable to compensate for its actual prediction error any more.As the current signal can only be predicted incompletely by the AR model through its previous known data,it needs to treat the linear combination of the model directly as its predicted output without any useless external noise accordingly.Then the projection of the linear combination onto its target signal is exactly the predictable portion of the signal,and the orthogonal deviation from the linear combination and its target signal is considered as the power compensation for the unpredictable random variable portion of the signal.Meanwhile a set of spectral whitening prediction coefficients of the model as its filtering coefficients is essential to keep up the invariance of the power spectrum of the signal during the procedure of its in-out of signal.Therefore the maximum predictable weighting of the signal is equal to the square sum of the power spectrum of the signal.Its residual value as the minimum weighting of the unpredictable portion of the signal can be considered as the predicting entropy of the random signal to measure its uncertainty.Its entropy function of the frequency component is set to the product of the predictable proportion at this frequency and the cumulative sum of the other components’power spectrum except this one’s.The expectation of model predicting and power spectrum estimating are both to achieve minimum predicting entropy of the signal as well.