Explicit formulas for the inverses of Toeplitz matrices, with applications

Inoue, Akihiko

doi:10.1007/s00440-022-01162-9

Cited by 2 publications

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“…Remark Consider the

n \times n

Toeplitz matrix

T_{n} (f) = (c (s - t); 0 \leq s, t \leq n - 1)

and

d_{k, j}^{(n)} = {(T_{n} {(f)}^{- 1})}_{k, j}

. There are several different expressions for

d_{k, j}^{(n)}

including the Cholesky decomposition given in Akaike (1969), Pourahmadi (2001), and Jentsch and Meyer (2021) or expressions based on a dual process representation; Subba Rao and Yang (2021) and Inoue (2021). The arguments in this article can also be used to obtain an alternative expression for the inverse of a finite dimensional Toeplitz matrix.…”

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confidence: 99%

A prediction perspective on the Wiener–Hopf equations for time series

Rao

Yang

2022

Journal Time Series Analysis

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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.

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“…Remark Consider the