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A prediction perspective on the Wiener–Hopf equations for time series
Journal of Time Series Analysis ( IF 0.9 ) Pub Date : 2022-04-11 , DOI: 10.1111/jtsa.12648
Suhasini Subba Rao, Junho Yang

The Wiener–Hopf equations are a Toeplitz system of linear equations that naturally arise in several applications in time series. These include the update and prediction step of the stationary Kalman filter equations and the prediction of bivariate time series. The celebrated Wiener–Hopf technique is usually used for solving these equations and is based on a comparison of coefficients in a Fourier series expansion. However, a statistical interpretation of both the method and solution is opaque. The purpose of this note is to revisit the (discrete) Wiener–Hopf equations and obtain an alternative solution that is more aligned with classical techniques in time series analysis. Specifically, we propose a solution to the Wiener–Hopf equations that combines linear prediction with deconvolution. The Wiener–Hopf solution requires the spectral factorization of the underlying spectral density function. For ease of evaluation it is often assumed that the spectral density is rational. This allows one to obtain a computationally tractable solution. However, this leads to an approximation error when the underlying spectral density is not a rational function. We use the proposed solution with Baxter's inequality to derive an error bound for the rational spectral density approximation.

中文翻译:

时间序列 Wiener-Hopf 方程的预测视角

Wiener–Hopf 方程是 Toeplitz 线性方程组,在时间序列的多个应用中自然出现。这些包括固定卡尔曼滤波器方程的更新和预测步骤以及双变量时间序列的预测。著名的 Wiener–Hopf 技术通常用于求解这些方程,它基于傅里叶级数展开式中系数的比较。然而,方法和解决方案的统计解释是不透明的。本说明的目的是重新审视(离散的)Wiener-Hopf 方程并获得与时间序列分析中的经典技术更一致的替代解决方案。具体来说,我们提出了一种将线性预测与反卷积相结合的 Wiener-Hopf 方程的解。Wiener–Hopf 解需要基础谱密度函数的谱分解。为了便于评估,通常假设谱密度是合理的。这允许人们获得计算上易于处理的解决方案。然而,当基础谱密度不是有理函数时,这会导致近似误差。我们将建议的解决方案与 Baxter 不等式一起使用,以得出有理谱密度近似的误差界限。
更新日期:2022-04-11
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