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Impacts and Statistical Mitigation of Missing Data on the 21cm Power Spectrum: A Case Study with the Hydrogen Epoch of Reionization Array

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Pith's one-line read This paper establishes that partially flagged data combined with night-to-night systematic variations can ring bright foregrounds into the 21cm EoR window, and that DPSS inpainting with a carefully built covariance matrix restores honest…

desk verdict Real effect, solid demonstration, but the inpainting covariance in Eq. (44) is an unvalidated mix of frequentist and Bayesian pieces; needs a Monte Carlo check before the error bars are trusted. read the letter →

arxiv 2411.10529 v2 pith:RNLYLRWG submitted 2024-11-15 astro-ph.CO

Kai-Feng Chen , Michael J. Wilensky , Adrian Liu , Joshua S. Dillon , Jacqueline N. Hewitt , Tyrone Adams , James E. Aguirre , Rushelle Baartman
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Adam P. Beardsley Lindsay M. Berkhout Gianni Bernardi Tashalee S. Billings Judd D. Bowman Philip Bull Jacob Burba Ruby Byrne Steven Carey Samir Choudhuri Tyler Cox David R. DeBoer Matt Dexter Nico Eksteen John Ely Aaron Ewall-Wice Steven R. Furlanetto Kingsley Gale-Sides Hugh Garsden Bharat Kumar Gehlot Adélie Gorce Deepthi Gorthi Ziyaad Halday Bryna J. Hazelton Jack Hickish Daniel C. Jacobs Alec Josaitis Nicholas S. Kern Joshua Kerrigan Piyanat Kittiwisit Matthew Kolopanis Paul La Plante Adam Lanman Yin-Zhe Ma David H. E. MacMahon Lourence Malan Cresshim Malgas Keith Malgas Bradley Marero Zachary E. Martinot Lisa McBride Andrei Mesinger Nicel Mohamed-Hinds Mathakane Molewa Miguel F. Morales Steven G. Murray Hans Nuwegeld Aaron R. Parsons Robert Pascua Yuxiang Qin Eleanor Rath Nima Razavi-Ghods James Robnett Mario G. Santos Peter Sims Saurabh Singh Dara Storer Hilton Swarts Jianrong Tan Pieter van Wyngaarden Haoxuan Zheng
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classification astro-ph.CO
keywords 21cmcosmologyepochofreionizationdelaypowerspectrumradiofrequencyinterferencedatainpaintingdiscreteprolatespheroidalsequencesquadraticestimatorHERA
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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that radio-frequency-interference (RFI) flagging does more than remove data: when the underlying visibility also has night-to-night systematic variations, averaging over partially flagged data makes bright foregrounds ring into the EoR window of the 21cm delay power spectrum, even when every channel retains most of its observations. In a realistic HERA-like simulation, the paper shows that inpainting the flagged channels with a discrete prolate spheroidal sequence (DPSS) basis removes this foreground ringing. The central contribution is a covariance model for the inpainted visibility, $N_{\rm inp} = O_{\rm inp} N_u O_{\rm inp}^{\dagger} + N_f$, which adds the intrinsic noise of the flagged channels to the propagated uncertainty of the inpainting operator, and then plugs that covariance into a quadratic estimator so that error bars and window functions are statistically honest. Applied to HERA Phase II data, the framework shows that nightly inpainting is necessary and delineates when simple approximate error bars can be trusted.

What carries the argument

The central object is the discrete prolate spheroidal sequence (DPSS), a set of band-limited basis functions whose Fourier transforms concentrate within a chosen delay interval, here $\pm 500\,{\rm ns}$. The inpainting operator is $O_{\rm inp} = W + (I-W)A(A^{\dagger} N_u^{-1} A)^{+} A^{\dagger} N_u^{-1}$, where $W$ selects unflagged channels and $A$ evaluates the DPSS basis; this linear operator maps the observed visibility to a smooth foreground fit that fills the gaps. The load-bearing identity is the covariance model $N_{\rm inp} = O_{\rm inp} N_u O_{\rm inp}^{\dagger} + N_f$, which combines the propagated noise from the observed channels with the intrinsic thermal noise of the flagged channels themselves. Substituting $C_{\rm inp} = O_{\rm inp} C_{\rm sig} O_{\rm inp}^{\dagger} + N_{\rm inp}$ into the quadratic estimator's expectation and covariance formulas produces modified window functions and power spectrum errors; this is what turns inpainting from an empirical fix into a statistically quantifiable operation.

What would settle it

Construct a simulation whose flagged-channel noise is not spectrally smooth (for example, residual RFI with a narrow spectral feature), apply the proposed inpainting covariance, and compare the resulting error bars against the scatter of many noise realizations; if the coverage is off, the interpolation assumption for $N_f$ is the breaking point.

Watch

Extended reading notes

Core claim

The paper claims that the most damaging effect of RFI flags arises from the convolution of a nightly varying flag mask with nightly varying systematic effects such as gain errors, feed perturbations, and mutual coupling. In the sidereal-day-averaged visibility this produces terms proportional to $\varepsilon_i K_i \circledast (s+e)$, so spectrally smooth foregrounds leak into high-delay Fourier modes even when no channel is fully lost. Because the DPSS inpainting fits band-limited foreground structure from unflagged channels and fills the gaps linearly, it removes the discontinuity and suppresses the ringing. To keep the statistical inference correct, the paper treats inpainting as a linear operator $O_{\rm inp}$ and assigns the inpainted data the covariance $N_{\rm inp} = O_{\rm inp} N_u O_{\rm inp}^{\dagger} + N_f$, where $N_f$ is the noise variance in the flagged channels obtained by interpolating the smooth autocorrelations; this added term prevents inpainting from artificially increasing sensitivity. Inserting this covariance into the quadratic estimator yields power spectrum error bars and window functions that account for the filling-in process. On HERA Phase II data, the inpainted delay spectrum reaches the expected radiometer noise floor, whereas the un-inpainted flagged data exceed that floor by over an order of magnitude.

Load-bearing premise

The load-bearing premise is that the thermal-noise variance in the flagged channels, $N_f$, can be reliably interpolated from the smooth autocorrelation functions; if residual RFI leaves non-smooth noise in flagged channels, the quoted error bars and window functions become biased.

Editorial extensions

If this is right

  • Foreground leakage from the flags-systematics interplay can appear even when fewer than ten percent of data are flagged, so aggressive RFI flagging alone is not sufficient.
  • Inpainting with the proposed covariance can yield error bars larger than either a naive propagation or a conservative diagonal approximation, because off-diagonal frequency covariances amplify Fourier-space variance.
  • For HERA Phase II data, nightly DPSS inpainting brings the delay spectrum down to the expected radiometer noise floor, whereas not inpainting leaves it more than an order of magnitude above that floor.
  • When wide flags affect all nights, the power spectrum window function develops appreciable off-diagonal structure, so the inpainted covariance must be included in any interpretation of the measured bands.
  • A simple conservative covariance approximation is adequate when flags are narrow and affect only a single night, but it overestimates the noise when channels are completely flagged across all nights.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same covariance accounting should apply to other linear gap-filling methods, such as linear least-squares spectral analysis or Gaussian process regression with a fixed kernel, because the argument relies only on linearity of the inpainted visibility in the observed data.
  • If the interpolation of $N_f$ from smooth autocorrelations ever fails, the error bars would be biased, but the covariance structure itself could be repaired by substituting a more detailed noise model, so the main claim is more robust than that particular interpolation step.
  • A testable extension for future surveys is to compute the full inpainted covariance once per field and use the simple approximation only when the resulting window-function distortion is smaller than the expected EoR signal level.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper studies the impact of missing frequency channels on 21 cm delay power spectra, with application to HERA Phase II data. The authors first argue analytically (Sec. 2) that when visibilities are LST-averaged with nightly varying systematics, the flag-dependent sampling kernel convolves bright foreground modes into the EoR window. They verify this with a realistic simulation of a seven-element HERA-like array (Sec. 3), showing that partially flagged data combined with nightly gain, beam, and coupling systematics raise the noise floor by roughly an order of magnitude. They then develop DPSS inpainting (Sec. 4.1), derive a Bayesian posterior predictive variance for flagged channels (Appendix A, Eqs. 29-33), and propose a full frequency-frequency covariance for the inpainted visibility (Eqs. 34 and 44). This covariance is inserted into a quadratic estimator to compute power spectrum error bars and window functions (Secs. 4.3 and 5). The framework is applied to 14 nights of HERA Phase II data, and artificial flag injection (Figs. 10-11) is used to test a cheaper approximate treatment.

Significance. The paper addresses a timely and practical problem for 21 cm cosmology. The core empirical findings—that modest flagging fractions combined with realistic nightly systematics can produce significant foreground ringing, and that DPSS inpainting suppresses it—are convincingly demonstrated with a controlled simulation in which noise and systematics realizations are held fixed across flagging patterns. The Gaussian integration leading to the posterior predictive distribution for the inpainted channels is a useful contribution, and the application to HERA Phase II data gives the paper immediate relevance. However, the central statistical claim—that the proposed covariance in Eq. (44) correctly propagates inpainting uncertainty into QE error bars and window functions—is not yet validated against the sampling distribution of the actual estimator, and the model bias acknowledged in Fig. 4 is not represented in the covariance. With additional validation, the framework would be a valuable methodological contribution.

major comments (3)
  1. [Sec. 4.3, Eq. (44)] The covariance Cinp = OinpCsigOinp† + OinpNuOinp† + Nf is not derived from a single generative model. In the actual pipeline vinp = Oinpvobs, the repeated-experiment sampling covariance of the estimator is exactly OinpCobsOinp†; the flagged-channel noise Nf is not a property of the estimator. The flagged-block term N'_f + O'_inpNuO'_inp† is a Bayesian posterior predictive variance for an unobserved quantity, not the sampling variance of the statistic. The off-diagonal blocks of Eq. (44) are inherited from the frequentist operator while the flagged block is Bayesian, and no joint distribution produces all blocks simultaneously. Consequently PSN in Eq. (47) is not the standard deviation of the QE under repetitions of the experiment. The validation in Figs. 5-6 and 9-11 compares the full covariance only with the paper's own approximations, not with the empirical scatter of recovered band powers for known input spectra. In addition, the model bias visible in Fig. 4 is not included in the covariance. Please either add a Monte Carlo validation (many noise and flag realizations; compare empirical band-power scatter and empirical window functions with the predicted covariance) or explicitly reframe the error bars as a conservative Bayesian predictive statement rather than a sampling uncertainty.
  2. [Sec. 4.2, Eqs. (29)-(33)] The derivation of N'_inp by Gaussian integration is correct, but the step 'we can safely assume that we can interpolate the auto-correlations over the flagged channels and infer N'_f with very low uncertainty' is load-bearing: N'_f enters every error bar and window function through Eqs. (34), (45), and (48). If RFI-flagged channels have noise statistics that differ from the smooth interpolation of the surrounding auto-correlations (e.g., due to system-state changes or flagging-induced decorrelation), the resulting uncertainties are biased. The paper should quantify the sensitivity of PSN and the window functions to mismodeled N'_f in the simulation, or justify the smoothness assumption with data from the HERA auto-correlations in the flagged channels.
  3. [Sec. 5.2, Figs. 10-11] The artificial flag injection tests are performed on real HERA data, where the true sky signal and noise are unknown. The quantity plotted as 'Full Covariance PSN' is a model prediction, not an empirical error, and the comparison with the 'Approximation' only establishes internal consistency between two model-based estimators. To support the claim that the full covariance correctly captures the impact of wide flags, the same injection protocol should be run on the Sec. 3 simulation (or on a realistic mock data cube with known input power spectrum), and the predicted PSN and window functions should be compared with the scatter of recovered band powers across noise realizations.
minor comments (5)
  1. [Eq. (29)] The printed definition N'_f = Pf N Pu† appears to be a typo; it should be Pf N Pf† for the noise covariance of the hypothetical RFI-free data in the flagged channels, since the subsequent derivation treats N'_f as a covariance on the flagged subspace.
  2. [Appendix A] The footnote assumption that A†Nu^-1A is invertible is stated without proof or explicit conditions. Since Figs. 10-11 explore heavy flagging, the paper should state when this holds or use a regularized inverse consistently.
  3. [Fig. 5] The middle panel of Fig. 5 labels the visibility variance in mK, while the simulation description and Fig. 9 use Jy; please harmonize the units and state the conversion.
  4. [Sec. 4.3 and Conclusion] The heuristic nature of Eq. (34) ('we propose the following modification') should be more prominently reflected in the abstract and conclusion, where the framework is described as 'rigorous'; the current wording overstates the status of the covariance model.
  5. [General] The paper does not state whether the analysis code will be released; for a methodology paper, a code release or a statement of availability would aid reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the Bayesian predictive covariance derivation is self-contained, and the simulation is a controlled falsifiable test; only a minor self-referential T=500 ns choice from same-collaboration HERA data.

full rationale

No circular step rises to the level of a prediction reducing to an input. The Bayesian posterior predictive derivation (Eq. 29; Appendix A) is a self-contained calculation from stated Gaussian and flat-prior assumptions, yielding N'_f + O'_inp N_u O'†_inp. Eq. (44) is explicitly introduced as a proposed model ("we propose the following modification"), and the reported PSN and window functions are obtained by substituting that model into the standard QE covariance Eq. (42); this is model application, not fitting a parameter and then "predicting" the same quantity. The Sec. 3 simulation is a controlled falsifiable test: the same noise and systematic realizations are used with and without flags, and the inpainted spectra are compared to an external radiometer noise floor, so the ringing-mitigation claim does not reduce to the simulation inputs. The one self-referential element is the DPSS width: "we choose T = 500 ns as it has been shown in data obtained by the Phase II HERA observations that foreground can leak to such a delay due to mutual coupling of antennas (Rath et al. 2024)." This is a parameter-setting measurement by the same collaboration applied to the same HERA Phase II data, but it is an externally falsifiable characterization rather than a fitted parameter renamed as a prediction, and the central derivation does not depend on it. The skeptic's concern that Eq. (44) mixes frequentist sampling covariance with Bayesian predictive variance and that PSN is not validated against empirical scatter is a calibration/correctness limitation, not a circularity.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central claim rests on several modeling choices. The DPSS hyperparameters (T=500 ns, eigenvalue cutoff 10^-12) are hand-picked based on prior HERA characterization, not derived. The covariance model in Eq. (44) is a proposal, not a theorem. The simulation systematics amplitudes are inputs, not measurements. The axioms enumerate the statistical assumptions (diagonal noise, interpolable flagged noise, invertibility) that are load-bearing for the uncertainty quantification.

free parameters (6)
  • DPSS inpainting width T = 500 ns
    Hand-chosen to cover foreground leakage identified in HERA Phase II due to mutual coupling (Rath et al. 2024), Sec. 4.1. Larger than the horizon scale of 50 ns to absorb systematics leakage.
  • DPSS eigenvalue cutoff = 1e-12
    Hand-chosen threshold to retain enough basis functions while limiting high-delay structure, Sec. 4.1.
  • Simulation gain uncertainty amplitude = 5 percent (a,b in [-0.05,0.05])
    Chosen to mimic residual calibration errors, Sec. 3 item 1. Not fitted, but an input to the demonstration.
  • Simulation beam feed perturbation sigma = 2 cm
    Chosen from HERA feed motion memo (Rath et al. 2021), Sec. 3 item 2.
  • Simulation mutual coupling coefficient = 1 percent (a,b in [-0.01,0.01])
    Chosen to simulate weak antenna coupling, Sec. 3 item 3.
  • Nsample regularization for fully flagged channels = minimum non-zero Nsample across band
    Ad hoc fix to avoid division by zero in Eq. (46), Sec. 5.2.
assumptions (7)
  • domain assumption The noise covariance Nu is diagonal with variances from the radiometer equation using autocorrelations (Eq. 25).
    Used throughout Sec. 4; noise correlations between channels are neglected.
  • domain assumption The noise covariance in flagged channels N'_f can be inferred by interpolating the smooth autocorrelation functions.
    Sec. 4.2, 'we can safely assume'. This is load-bearing for the error bars.
  • ad hoc to paper The matrix A†N_u^-1A is invertible (only a small fraction of channels is flagged).
    Appendix A footnote 7. If many channels are flagged, the pseudo-inverse used in Eq. (24) does not have this simple Gaussian posterior.
  • standard math DPSS eigenvectors form a complete basis and their eigenvalues concentrate near 0 or 1.
    Slepian 1978; Karnik et al. 2020, Sec. 4.1.
  • standard math A flat prior on the DPSS coefficients b is appropriate for the Bayesian derivation.
    Eq. (31). Not a physical axiom but a statistical modeling choice.
  • domain assumption The signal covariance model in Eq. (39) approximates the primary beam as compact and uses narrow band-power bins.
    Sec. 4.3, standard in the QE formalism.
  • domain assumption Noise is independent across different times and nights (Eq. 45).
    Used to express the time-averaged covariance as a sum; ignores potential correlations from the inpainting operator across time (since flags vary with time).

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Cite this review

Pith. "Pith review of Impacts and Statistical Mitigation of Missing Data on the 21cm Power Spectrum: A Case Study with the Hydrogen Epoch of Reionization Array." pith.science (2026). https://pith.science/paper/RNLYLRWG

@misc{pith2026241110529,
  author       = {Pith},
  title        = {Pith review of: Impacts and Statistical Mitigation of Missing Data on the 21cm Power Spectrum: A Case Study with the Hydrogen Epoch of Reionization Array},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RNLYLRWG}},
  note         = {Machine review of arXiv:2411.10529}
}
read the original abstract

The precise characterization and mitigation of systematic effects is one of the biggest roadblocks impeding the detection of the fluctuations of cosmological 21cm signals. Missing data in radio cosmological experiments, often due to radio frequency interference (RFI), poses a particular challenge to power spectrum analysis as it could lead to the ringing of bright foreground modes in Fourier space, heavily contaminating the cosmological signals. Here we show that the problem of missing data becomes even more arduous in the presence of systematic effects. Using a realistic numerical simulation, we demonstrate that partially flagged data combined with systematic effects can introduce significant foreground ringing. We show that such an effect can be mitigated through inpainting the missing data. We present a rigorous statistical framework that incorporates the process of inpainting missing data into a quadratic estimator of the 21cm power spectrum. Under this framework, the uncertainties associated with our inpainting method and its impact on power spectrum statistics can be understood. These results are applied to the latest Phase II observations taken by the Hydrogen Epoch of Reionization Array, forming a crucial component in power spectrum analyses as we move toward detecting 21cm signals in the ever more noisy RFI environment.

Figures

Figures reproduced from arXiv: 2411.10529 by the authors.

Figure 1
Figure 1. Left: An example interferometric array layout we simulate in this work. antennas are placed on the vertices and the center of a hexagon with a side of 14.6 meters. Center: An example of the simulated flagging patterns we draw in this work. The top panel shows the flagging patterns for the baseline formed by antenna 0 and 1 summed over all nights of observation. Darker color indicates a higher amount of flags. The lo… view at source ↗
Figure 2
Figure 2. shows the delay spectra for a single 14.6 me￾ter baseline following Eq. (2). These delay spectra are formed by first coherently averaging 300 seconds of vis￾ibilities after phasing them to a common pointing cen￾ter. The delay spectra within the 1.5 hour window of observations are then averaged incoherently to further 0 1000 2000 3000 Delays [ns] 10−6 10−4 10−2 100 Peak Normalized Delay Spectrum No flagged channels N… view at source ↗
Figure 3
Figure 3. Power spectra with or without data inpainting under different simulated flagging patterns. Similar to [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Uncertainties in the inpainted visibilities for var￾ious types of flagged channels. The solid black line is the simulated visibility at a single time instance at a single night from a single 14.6 meter baseline. We flag the simulated visibility with two 5-channel wide …
Figure 5
Figure 5. Figure 5: shows the noise (co)variance of the in￾painted visibility calculated through the last two terms in Eq. (45). Here, the visibility is from a single 14.6 meter baseline in our simulation after coherently av￾eraging across sidereal days and a 300-second window. The percen…
Figure 6
Figure 6. Figure 6: Delay power spectrum and power spectrum win￾dow function from an inpainted visibility in our simulation. The delay spectrum is obtained with the inpainted visibil￾ity of a single 14.6 meter baseline after coherently averaging across sidereal days and a 300-second windo…
Figure 7
Figure 7. Figure 7: Percentage of good data for the shortest east-west baseline group across 14 nights of observations with the HERA Phase II instruments. The upgraded Vivaldi feed extends our observation range to roughly from 50 to 250 MHz. The grey region is excluded due to heavy contam…
Figure 8
Figure 8. Figure 8: Delay power spectra from the H6C Phase II data with (solid blue) or without (dashed orange) nightly inpaint￾ing. The power spectra are derived from 14 nights of data observed by all the 14.6 meter east-west baselines with a 300- second coherent time average and an one-…
Figure 9
Figure 9. Figure 9: Statistical properties of nightly inpainted visibility from the HERA Phase II data. Similar to [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: Changes in power spectrum statistics as a func￾tion of flagged channel width. Here, we artificially flag chan￾nels in 1 of the 14 nights in the observed HERA Phase II data and inpaint over them to investigate the impact of in￾painting over gaps with different width. T…

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    2017 doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.