Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty

Torsten A. Enßlin and Mona Frommert

Phys. Rev. D 83, 105014 – Published 19 May 2011

Abstract

The optimal reconstruction of cosmic metric perturbations and other signals requires knowledge of their power spectra and other parameters. If these are not known a priori, they have to be measured simultaneously from the same data used for the signal reconstruction. We formulate the general problem of signal inference in the presence of unknown parameters within the framework of information field theory. To solve this, we develop a generic parameter-uncertainty renormalized estimation (PURE) technique. As a concrete application, we address the problem of reconstructing Gaussian signals with unknown power-spectrum with five different approaches: (i) separate maximum-a-posteriori power-spectrum measurement and subsequent reconstruction, (ii) maximum-a-posteriori reconstruction with marginalized power-spectrum, (iii) maximizing the joint posterior of signal and spectrum, (iv) guessing the spectrum from the variance in the Wiener-filter map, and (v) renormalization flow analysis of the field-theoretical problem providing the PURE filter. In all cases, the reconstruction can be described or approximated as Wiener-filter operations with assumed signal spectra derived from the data according to the same recipe, but with differing coefficients. All of these filters, except the renormalized one, exhibit a perception threshold in case of a Jeffreys prior for the unknown spectrum. Data modes with variance below this threshold do not affect the signal reconstruction at all. Filter (iv) seems to be similar to the so-called Karhune-Loève and Feldman-Kaiser-Peacock estimators for galaxy power spectra used in cosmology, which therefore should also exhibit a marginal perception threshold if correctly implemented. We present statistical performance tests and show that the PURE filter is superior to the others, especially if the post-Wiener-filter corrections are included or in case an additional scale-independent spectral smoothness prior can be adopted.

Received 15 February 2010

DOI:https://doi.org/10.1103/PhysRevD.83.105014

Authors & Affiliations

Torsten A. Enßlin and Mona Frommert

Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany

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Vol. 83, Iss. 10 — 15 May 2011

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Images

Figure 1
Parameter $δ_{i}$ and $ε_{i}$ of the five different filters for a Jeffreys prior in the representation of the generic filter formula Eq. (37). The parameters of the displayed filter are derived in the following sections: the critical filter in Sec. 3d, the classical filter in Sec. 3g, the joint MAP filter in Sec. 3e, the MAP-spectrum filter in Sec. 3f, and the PURE filter in Sec. 5e. The critical line between filters with and without a perception threshold as given by Eq. (64) is also shown.Reuse & Permissions
Figure 2
The effective signal Hamiltonian, Eq. (40), without the normalization constant $H_{0}$ in the case of a Jeffreys prior and for a single, independent signal $s = s_{i}$ and data point $d = d_{i}$ . The parameters are $R_{i j} = N_{i j} = δ_{i j}$ and $S_{i j} = p_{i} δ_{i j}$ . The different curves show the Hamiltonian for representative data values. The triangle symbols mark the results of the inverse response estimator $m_{ir} = R^{- 1} d$ on the corresponding curves. The large open and small filled circles mark the renormalized and classical map estimator results, respectively. The existence of a classical perception threshold can be seen: for $- 2 < d < 2$ , the classical map is exactly zero since no nontrivial stationary point of the Hamiltonian exists. The thin dotted line shows the renormalized Hamiltonian for the case $d = 3$ , as provided by $\frac{1}{2} (s - m_{p})^{†} D_{p}^{- 1} (s - m_{p})$ .Reuse & Permissions
Figure 3
Left: Used spectral power $y$ of the filters as a function of $x$ , the data power in noise power units. A spectral bandwidth of $ϱ_{i} = 8$ was assumed. Right: The same for the signal pass through $R_{i}^{'}$ .Reuse & Permissions
Figure 4
First panel: Test data points according to $d = R s + n$ and signal-response $R s$ in a setting with periodic boundary conditions. Second panel: Signal realization $s$ and the reconstructions as labeled in the fourth panel. The four MAP reconstructions (joined MAP, MAP spectrum, classical, and critical filter) are shown with the same line since they are very similar. Also the reconstruction using the exact spectrum is displayed. Third panel: The same as above, just enlarged and with the signal subtracted to highlight the difference in the reconstruction errors. Fourth panel: Response $R$ and line key for the panels above. Fifth panel: Error variance $⟨ (s_{x} - m_{x})^{2} ⟩_{(d, s)}$ of the filters from 700 signal and data realizations in logarithmic units to show the average fidelity of the individual filters. The order of the line keys reflects roughly the order of the average error of the different methods. The color/grey-scale areas (in online/printed version) should only help to guide the eye.Reuse & Permissions
Figure 5
Signal and noise spectra in comparison to the assumed spectra of the five filters for the data sets displayed in Fig. 4 in double logarithmic units. $k$ -vectors in units of the Nyquist wave vector of $k_{Ny} \approx 256 π$ . The filter spectra for the individual data sets of Fig. 4 are shown in the top panel, the average filter spectra for the 700 signal and data realizations also shown in Fig. 4 are displayed in the bottom panel. The presence of perception thresholds in many of the presented filters is clearly visible by the many missing frequencies in the top panel and also as the general down trend of the average spectra close to the crossing of signal and noise spectra. The order of the line keys reflects the order of the average spectral amplitudes of the different filters at $k = 0.3$ .Reuse & Permissions

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