Forecasting in the presence of scale-free noise

Abstract

The extraction of signals from noise is a common problem in all areas of science and engineering. A particularly useful version is that of forecasting: determining a causal filter that estimates a future value of a hidden process from past observations. Current techniques for deriving the filter require that the noise be well described by rational power spectra. However, scale-free noises, whose spectra scale as a non-integer power of frequency, are ubiquitous in practice. We establish a method, together with performance guarantees, that solves the forecasting problem in the presence of scale-free noise. Via the duality between estimation and control, our technique can be used to design control for distributed systems. These results will have wide-ranging applications in neuroscience, finance, fluid dynamics, and quantum measurements.

Figures (3)

• Figure 1: Causal Wiener filtering of scale-free data. (a,b) Grey shows a simulated trajectory of the noisy measurement $y(t) = x(t) + n(t)$ for $1600$ s, sampled at $512$. These simulations assume an underlying signal with spectrum assumed to be $S_{xx}=A \gamma^2/ ((|\omega|-\omega_c)^2+\gamma ^2)$ with parameters $\gamma = 2 \pi$ rad/s, $A=0.9$, and $\omega_c = 10 \cdot 2\pi$ rad/s -- a peak centered at $2\pi \cdot 10$ rad/s with half-width $2 \pi$ rad/s. The noise was chosen such that $S_{nn}=5/\omega^{1.8}+0.01$. (a) Zoomed-in time series of the measured signal $y(t)$, the true signal $x(t)$, and estimate $\hat{x}(t)$ from the recorded and real-time causal filters. (c) Estimates of the power spectral density $S_{yy}$ obtained from the simulated time series in (a) using Welch's method, with a Slepian window, 80–90% segment overlap, and segment length chosen so that approximately eight Fourier bins spanned the target linewidth $S_{xy}$ is presumed known from a model. (d) Causal Wiener filter in the "recorded" and "real-time" scenarios (orange and green respectively), compared against the non-causal Wiener filter $S_{xy}/S_{yy}$. Solid lines show squared magnitude and dashed line the phase of the filter. The causal filters are computed following the algorithm laid out in the main text: data is transformed as prescribed by \ref{['TheoremBound']} with $f(\omega) = e^{\mathrm{i} a /2} \omega^\beta (\mathrm{i} \omega_0 + \omega)^{\alpha - \beta}$ with $\alpha=0$, $\beta=0.9$, and $\omega_0 = 5$ rad/s. The scale frequency in \ref{['eq:tkCutoff']} was also chosen to be $5$ rad/s. (e) Relative error PSDs $S_{\varepsilon\varepsilon}/S_{yy}$ for the two causal filter scenarios and the non-causal filter. $S_{\varepsilon \varepsilon}$ was estimated by Welch’s method to the residual $x(t)-\hat{x}(t)$, averaging across approximately 250 logarithmically spaced bins in $\omega$, with outliers more than 10 standard deviations removed.
• Figure 2: Figure reproduces Fig. (1.a) and Fig. (1.b). The yellow "recorded" and magenta "non-causal" curves are the same as in Fig. 1. The green "analytic" curve is the Wiener filter obtained directly from analytic forms of the PSD's: $S_{xx}=A \gamma^2/ ((|\omega|-\omega_c)^2+\gamma ^2)$ with parameters $\gamma = 2 \pi$, $A=0.9$ and $\omega_c = 10 \cdot 2\pi$, and $S_{nn}=5/\omega^{1.8}+0.01$. The plot shows how sampling noise impacts the performance of the filter construction procedure by benchmarking it against the solution obtained from smooth PSDs.
• Figure 3: Figure reproduces Fig. (2.b). The "analytic" curve is the same as the "analytic" curve in Fig. 2. The "analytic" curve is obtained by preconditioning the smooth data and applying the filter computation procedure. The "lattice preconditioned" curve was obtained by solving the \ref{['Gplus', 'Gminus']} on an equally spaced frequency grid after preconditioning the data. The "lattice" curve was obtained by solving the \ref{['Gplus', 'Gminus']} on the same grid without preconditioning the data. The plot illustrates that the solution obtained with the proposed filter-construction procedure converges to a naive attempt to solve \ref{['Gplus', 'Gminus']}. The small difference at small frequencies is attributed to the finite step size of the lattice grid. The small difference at large frequencies is attributed to the growing unregulated logarithmic divergence of the Hilbert transform at the high-frequency edge of the finite frequency grid. The PSDs used to construct the plot are the same as in Fig 2: $S_{xx}=A \gamma^2/ ((|\omega|-\omega_c)^2+\gamma ^2)$ with parameters $\gamma = 2 \pi$, $A=0.9$ and $\omega_c = 10 \cdot 2\pi$, and $S_{nn}=5/\omega^{1.8}+0.01$. The plot shows how sampling noise impacts the performance of the filter construction procedure by benchmarking it against the solution obtained from smooth PSDs.

Theorems & Definitions (49)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition A.1
• Lemma A.1
• Definition A.2
• Theorem A.1: Proposition 2.40, Barbu2012-pn
• Theorem A.2