信噪比在AR模型定阶方法选择中的研究

摘要/Abstract

摘要：

AR模型有多种定阶方法，针对特定的时间序列，不同方法得到的结果会有差异，如何适应性地选择合适的定阶方法是一个重要的问题。该文针对低阶自回归模型，在考虑噪声标准差、序列长度和特征根的影响下，引入一种估计模型信噪比的方法，并将其作为评价AIC、BIC和FPE准则定阶准确度的标准.实验表明：当模型的特征根满足|λ₁|=|λ₂|=…=|λ_p|=|λ_max|时，准确率达到该最大特征根条件下的最大值；定阶准确率与序列长度、特征根相对于单位圆心的距离呈正相关，与噪声标准差无关.在此基础上，提出一种利用参考模型信噪比选择定阶方法的方案，为不同定阶方法优劣的比较提供了新的视角.

关键词: 自回归模型, 信噪比, 噪声标准差, 序列长度, 特征根, 参考模型

Abstract:

There are many methods can be used to determine the order of AR models. For specific time series, different method may provide different results. How to select method adaptively for particular series is an important problem, especially in big data era. In this paper, we introduce a method to estimate the signal-to-noise ratio (SNR) of the AR model in low-order noisy environments. It takes the influence of noise standard deviation, series length and eigenvalue of the model into consideration, which can be used as a criterion to evaluate the accuracy of AIC, BIC and FPE. The experimental results show that when the eigenvalue satisfies|λ₁|=|λ₂|=…=|λ_p|=|λ_max|, the order determination accuracy reaches the maximum under the condition of maximum eigenvalue. The accuracy is positively correlated with the series length and the distance of eigenvalue from origin, independent of noise standard deviation. Finally, based on the experimental results, we can select the order determination method of AR model according to the SNR of converted reference model, which provides a new perspective on the comparison of the advantages and disadvantages in different order determination methods.

Key words: Autoregressive model, Signal-to-noise ratio, Noise standard deviation, Series length, Eigenvalue, Reference model

中图分类号:

O212.1

王志刚,丁义明. 信噪比在AR模型定阶方法选择中的研究[J]. 数学物理学报, 2020, 40(3): 811-823.

Zhigang Wang,Yiming Ding. Research on Signal-to-Noise Ratio in Order Selection of AR Model[J]. Acta mathematica scientia,Series A, 2020, 40(3): 811-823.

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参考文献 9

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序列长度	$\lambda_{\max }=0.4$		$\lambda_{\max }=0.5$		$\lambda_{\max }=0.6$
	准确率(%)	信噪比	准确率(%)	信噪比	准确率(%)	信噪比
300	62.0	36.831	88.4	38.611	90.5	41.012
400	73.8	36.838	90.6	38.627	91.2	41.038
500	81.5	36.842	90.8	38.642	91.1	41.064
600	85.2	36.862	91.3	38.648	90.8	41.072
700	87.6	36.864	91.3	38.656	91.0	41.078

序列长度	$\lambda_{\max }=0.4$		$\lambda_{\max }=0.5$		$\lambda_{\max }=0.6$
	准确率(%)	信噪比	准确率(%)	信噪比	准确率(%)	信噪比
300	18.0	36.831	73.5	38.611	99.4	41.012
400	27.6	36.838	88.9	38.627	99.9	41.038
500	38.6	36.842	95.7	38.642	100.0	41.064
600	50.2	36.862	98.4	38.648	100.0	41.072
700	60.9	36.864	99.5	38.656	100.0	41.078

准确度	AR(2)		AR(3)		AR(4)		AR(5)
	上界	下界	上界	下界	上界	下界	上界	下界
$F>A>B$	$33.9_{0.0}$	$37.4_{29.1}$	$33.9_{0.0}$	$47.8_{32.4}$	$33.9_{0.0}$	$57.9_{25.4}$	$33.9_{0.0}$	$78.6_{31.9}$
$A>F>B$	$37.4_{29.1}$	$40.7_{73.0}$	$47.8_{32.4}$	$54.2_{71.0}$	$57.9_{25.4}$	$71.0_{71.5}$	$78.6_{31.9}$	$92.6_{69.3}$
$A>B>F$	$40.7_{73.0}$	$44.1_{88.5}$	$54.2_{71.0}$	$62.1_{88.2}$	$71.0_{71.5}$	$82.1_{88.6}$	$92.6_{69.3}$	$108.1_{86.0}$
$B>A>F$	$44.1_{88.5}$	$63.4_{99.1}$	$62.1_{88.2}$	$85.5_{99.3}$	$82.1_{88.6}$	$107.8_{99.2}$	$108.1_{86.0}$	$129.6_{99.9}$

准确率	AR(2)		AR(3)		AR(4)		AR(5)
	上界	下界	上界	下界	上界	下界	上界	下界
$F>A>B$	$33.9_{0.0}$	$36.1_{27.9}$	$33.9_{0.0}$	$43.7_{30.1}$	$34.0_{0.0}$	$55.3_{33.4}$	$34.1_{0.0}$	$70.8_{36.7}$
$A>F>B$	$36.1_{27.9}$	$38.6_{83.7}$	$43.7_{30.1}$	$49.2_{78.6}$	$55.3_{33.4}$	$64.0_{79.2}$	$70.8_{36.7}$	$83.3_{82.2}$
$A>B>F$	$38.6_{83.7}$	$40.8_{90.0}$	$49.2_{78.6}$	$54.1_{89.9}$	$64.0_{79.2}$	$71.6_{90.2}$	$83.3_{82.2}$	$94.0_{90.9}$
$B>A>F$	$40.8_{90.0}$	$52.0_{99.5}$	$54.1_{89.9}$	$89.5_{100.0}$	$71.6_{90.2}$	$113.7_{99.7}$	$94.0_{90.9}$	$137.8_{99.9}$

准确率	AR(2)		AR(3)		AR(4)		AR(5)
	上界	下界	上界	下界	上界	下界	上界	下界
$F>A>B$	$34.0_{0.0}$	$35.5_{34.5}$	$34.0_{0.0}$	$41.4_{33.6}$	$34.1_{0.0}$	$51.0_{34.7}$	$34.1_{0.0}$	$64.8_{38.4}$
$A>F>B$	$35.5_{34.5}$	$37.2_{84.1}$	$41.4_{33.6}$	$46.1_{86.2}$	$51.0_{34.7}$	$58.8_{86.3}$	$64.8_{38.4}$	$74.7_{85.1}$
$A>B>F$	$37.2_{84.1}$	$38.5_{92.0}$	$46.1_{86.2}$	$49.0_{90.7}$	$58.8_{86.3}$	$63.1_{91.4}$	$74.7_{85.1}$	$80.8_{92.8}$
$B>A>F$	$38.5_{92.0}$	$66.4_{99.5}$	$49.0_{90.7}$	$90.8_{100.0}$	$63.1_{91.4}$	$115.6_{99.9}$	$80.8_{92.8}$	$140.8_{100.0}$