用于平稳过程的自适应多窗谱估计
工程技术TOPEI检索SCI升级版 工程技术1区IF 7.9高亮
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自适应多窗方法在频谱估计中采用每个频率点变化的窗数量。 - •
所提出的方法在两个具有不同光谱特性的例子中被证明是有效的。 - •
所提出的方法可以平衡光谱的局部分辨率和方差。
摘要
多窗谱估计方法在减少平稳随机过程谱的偏差和方差方面是有效的。该方法中的窗数量(NoT)控制着谱的偏差和方差之间的权衡。通常,NoT 是经验确定的,并且在所有频率下保持恒定,限制了局部偏差和方差的调整。本文提出了一种自适应多窗方法,该方法在每个频率点使用不同的 NoT 来估计多元平稳过程的功率谱密度(PSD)和相干函数。提出了一种带有停止准则的迭代过程,以在每个频率下优化 NoT,而无需手动调整。采用具有简单解析表达式和满意的泄漏保护的正弦窗。所提出的自适应多窗方法应用于三个数值示例:建筑物模型的结构响应、风速过程和自回归过程,这些示例表现出不同的谱特征。 所有示例的结果表明,所提出的方法在估计功率谱密度(PSD)和相干性时优于 Welch 方法和传统多谱图方法,具有更小的偏差和方差。
关键词
1. 引言
各种具有随机特征的物理和工程事件,如风速和在随机激励下的结构响应[1],[2],[3],[4],[5],可以视为平稳随机过程的一种实现。与时间域方法相比,通过频谱分析获得的频域表示更具信息量,允许分析能量分布[6],[7],[8]。
由于频域表示通常是从有限数据估计得出的,所获得的频谱通常会受到偏差和误差方差的影响[9]。几种非参数频谱估计器,包括 Welch 方法[10]和多谱估计方法[11],已被广泛用于减少估计频谱的偏差和方差[12],[13],[14],[15]。不幸的是,这些方法涉及偏差-方差权衡的困境,即低偏差估计器往往具有高方差,而低方差估计器通常具有高偏差[16]。多谱估计方法采用多个正交序列(加窗函数)进行频域平均以估计频谱[17]。 它相对于 Welch 方法表现出更好的性能,因为每个加窗的周期图都是从所有数据中获得的[18],[19],[20]。在实践中,Welch 方法可以被视为多窗谱估计技术的一种特殊情况,其中窗函数是相同的但时间上是移位的[20]。尽管这些方法可以缓解偏差和方差的问题,但它们通常包含难以调整的特定参数,例如 Welch 方法中的合适分段数和多窗方法中的窗函数数。
由自谱和互谱导出的相干性是量化两个时间序列在频域中线性耦合强度的重要工具[21]。频谱中的小低方差可能会导致较大的相干性方差。在工程应用中,准确估计相干性(如风相干性)具有挑战性,因为 Welch 方法和多谱估计方法是频率无关的,从而无法适应不同的频率分辨率[22]。
在最近几十年里,人们提出了几种方法来解决多谱估计法的偏差-方差权衡问题[19],[23],[24],[25],[26],[27],[28]。Thomson 提出了离散球面椭圆序列(也称为 Slepian 序列)作为一组具有较小泄漏的正交谱估计器,用于估计平稳过程的频谱[11],[29]。在设定时频积之后,可以计算出实现最佳方差的谱估计器数量。然而,控制频率分辨率和方差减少的带宽通常是由用户定义的,而没有系统的选取方法。 为解决这一问题,Haley 和 Anitescu [26] 提出了一种通过最小化对数谱的均方误差(MSE)来确定最优带宽的优化方法。尽管 Thomson 的多谱估计方法应用广泛,但 Slepian 窗的计算需要进行数值特征值分解,对于长时间序列来说计算成本较高 [30]。这种窗还需要在不同的带宽下重新计算。Riedel 和 Sidorenko [23] 提出了多谱估计中的正弦窗,它具有简单的解析表达式,无需设置和求解特征值方程。特别地,正弦窗的带宽仅取决于所用窗的数量。这意味着谱的分辨率和方差减少之间的内在权衡可以通过仅改变正弦窗的数量来控制。具体来说,较大的窗数量可以减少方差,但会伴随较高的偏差。 Riedel 和 Sidorenko [23] 还给出了通过最小化均方误差(MSE)来确定每个频率下使用的谱线数的公式。利用这个公式,Barbour 和 Parker [30] 开发了一种迭代程序来优化每个频率下采用的谱线数。尽管他们的方法生成了令人满意的频谱,但 迭代次数 是由用户根据经验确定的。最近,一种多谱线 S 变换(MTST)方法被提出用于 平稳和非平稳过程的谱估计,该方法结合了正交时频 Hermite 窗口和 S 变换方法 [27], [28]。该方法通过改变 Hermite 窗口的形状来降低不同频率下的谱和相干性的偏差和方差。 然而,MTST 方法在各种频谱形状上的适用性是有限的,因为其性能主要取决于定义的窗口形状函数,并且 NoT 在所有频率上仍然是恒定的。
在这项研究中,我们提出了一种自适应多谱估计方法,用于多变量平稳过程的谱估计和相干性估计,无需人工干预。与传统多谱估计方法固定 NoT 不同,我们提出了一种带有停止准则的优化策略来选择频率依赖的 NoT。因此,它使谱估计器能够在局部分辨率和谱方差之间取得平衡。通过三个数值例子验证了自适应多谱方法在处理不同谱特征方面的有效性,即建筑物模型的结构响应、风速过程和自回归(AR)过程。本方法减少了 NoT 选择的主观性,并平衡了局部谱的偏差-方差权衡。它还为具有各种谱形状的工程应用中的谱估计和相干性估计提供了一种有前景的方法。
2. 静态过程的多窗谱估计
2.1. 多变量平稳过程
设 为一个零均值多变量平稳过程,具有 , 。存在一个定义在区间 上的正交过程 ,使得[16] (1) 其中 T 表示转置算子, 为采样间隔, 为采样频率, 是奈奎斯特频率。 是一个复值正交增量过程,具有以下性质[16] (2) (3) (4) 其中*表示复共轭操作, 是克罗内克δ函数(当 时其值为 1,否则为 0),并且 被假定为纯粹连续的。
2.2. Multi-taper spectral estimation
3. Adaptive multi-taper approach
3.1. Formula of the adaptive multi-taper method
3.2. Bias and variance of the adaptive multi-taper method
3.3. The optimal NoT
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr1.jpg)
Fig. 1. Optimization procedure for the NoT.
4. Case study 1: A 4-DOF model
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr2.jpg)
Fig. 2. The four-story shear building model.
4.1. The spectrum estimate
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr3.jpg)
Fig. 3. Optimal NoT.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr4.jpg)
Fig. 4. PSD of the structural responses estimated using different techniques.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr5.jpg)
Table 1. DNRE and RPE for different methods.
Empty Cell | Methods | DNRE (%) | (%) | (%) |
---|---|---|---|---|
First floor | Welch | 4.95 | 3.02 | 13.79 |
Traditional multi-taper | 6.29 | 2.37 | 12.68 | |
Adaptive multi-taper | 3.81 | 0.98 | 12.09 | |
Second floor | Welch | 4.64 | 4.77 | 4.88 |
Traditional multi-taper | 6.06 | 3.13 | 3.97 | |
Adaptive multi-taper | 3.70 | 2.47 | 3.38 |
4.2. The coherence estimate
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr6.jpg)
Fig. 6. Estimated coherence of the structural responses of the first two floors.
4.3. Multiple realizations
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr7.jpg)
Fig. 7. Comparison of the theoretical and averaged optimal NoT.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr8.jpg)
Fig. 8. The derivatives of the theoretical PSD.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr9.jpg)
Fig. 9. Comparison of the constrained theoretical and averaged optimal NoT.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr10.jpg)
Fig. 10. Averaged PSD of the structural response using the adaptive multi-taper method.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr11.jpg)
Fig. 11. Averaged coherence of the structural response using the adaptive multi-taper method.
Table 2. AAB, ACoV, and RPE of the estimated results.
Estimate | Adaptive multi-taper | ||
---|---|---|---|
Theoretical NoT | Optimal NoT | ||
PSD for first floor | AAB | 0.358 | 0.359 |
ACoV | −0.078 | −0.088 | |
(%) | 4.425 | 4.816 | |
(%) | 0.915 | 1.954 | |
PSD for second floor | AAB | 0.353 | 0.359 |
ACoV | 0.027 | −0.004 | |
(%) | 5.776 | 5.828 | |
(%) | 2.939 | 3.736 | |
Coherence | AAB | 0.075 | 0.106 |
ACoV | −0.088 | −0.095 |
5. Case study 2: A wind speed process
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr12.jpg)
Fig. 12. Simulated bivariate wind speed time series.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr13.jpg)
Fig. 13. Optimal NoT.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr14.jpg)
Fig. 14. Estimated PSD of the wind speed process.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr15.jpg)
Fig. 15. DSE of the estimated PSD of the wind speed process.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr16.jpg)
Fig. 16. Estimated coherence of the wind speed process.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr17.jpg)
Fig. 17. Averaged NoT over 100 wind speed realizations.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr18.jpg)
Fig. 18. Averaged PSD of the wind speed using the adaptive multi-taper method.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr19.jpg)
Fig. 19. Averaged coherence of the wind speed using the adaptive multi-taper method.
Table 3. AAB, ACoV, and RPE of the estimated results over 100 realizations.
Estimate | Adaptive MT | ||
---|---|---|---|
Theoretical NoT | Optimal NoT | ||
PSD for Sample 1 | AAB | 0.129 | 0.169 |
ACoV | 8.610 × 10-5 | 1.464 × 10-4 | |
(%) | 1.816 | 0.002 | |
(%) | 3.841 | 2.721 | |
PSD for Sample 2 | AAB | 0.121 | 0.161 |
ACoV | 0.002 | 0.015 | |
(%) | 1.870 | 0.094 | |
(%) | 3.656 | 2.553 | |
Coherence | AAB | 0.193 | 0.255 |
ACoV | −0.131 | −0.151 |
6. Case study 3: AR (4) process
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr20.jpg)
Fig. 20. Optimal NoT.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr21.jpg)
Fig. 21. Estimated PSD of the AR(4) process.
![](https://ars.els-cdn.com/content/image/1-s2.0-S0888327022007178-gr22.jpg)
Fig. 22. Averaged PSD of the AR(4) process using the multi-taper method.
7. Conclusions
CRediT authorship contribution statement
Declaration of Competing Interest
Acknowledgements
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