Error formulae for the energy-dependent cross-spectrum

Adam Ingram

REVIEW 3 major objections 4 minor 50 references

Error formulae for the energy-dependent cross-spectrum

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The standard error bars used for energy-dependent X-ray cross-spectra are too large, and this paper derives the correct analytic replacements.

desk verdict Useful and mostly correct fix for an over-fitting bug in X-ray spectral-timing, but the modulus/phase formulas overclaim 'any coherence' — they are high-coherence approximations. read the letter →

arxiv 1909.01385 v1 pith:Y7LFAJOF submitted 2019-09-03 astro-ph.HE astro-ph.IM

Adam Ingram This is my paper · ORCID

classification astro-ph.HEastro-ph.IM

keywords cross-spectrumerroranalysisX-raytimingcoherencermsspectrumphaselagMonteCarlosimulationaccretingcompactobjects

verification ladder T0 review T1 audit T2 compute T3 formal

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 derives analytic error formulae for the energy-dependent cross-spectrum and rms spectrum, the Fourier statistics used to study rapid X-ray variability from accreting compact objects. It shows that the textbook errors of Bendat & Piersol (2010), which were designed for a single cross-spectrum, overestimate uncertainties when many subject bands are referenced to one common band, causing systematic over-fitting. The new formulae cover the real part, imaginary part, modulus, phase, and rms spectrum, and remain valid for any intrinsic coherence between energy bands. They are no harder to implement than the old ones, and a worked Cygnus X-1 example shows they bring a reverberation-model fit from reduced chi-squared 0.76 to an acceptable value.

What carries the argument

The central object is the energy-dependent cross-spectrum with a common reference band, modeled as complex Gaussian Fourier transforms with additive, uncorrelated noise. The argument splits the variance into a 'k variance' over frequency realizations and an 'n variance' over energy channels; in the energy-dependent limit, the shared reference band contributes zero n-variance to the real and imaginary parts, while subject-band and noise terms contribute $1/(2N)$ variances. This reduction makes the new errors on the real, imaginary, and modulus parts identical, and it sets the phase-error and rms-error formulae. The broad-reference-band correction subtracts the subject band's own Poisson noise contribution from the cross-spectrum.

What would settle it

Simulate light curves with log-normal, non-Gaussian flux variability, apply equations (18), (19), and (22) with the same N=500 averaging, and check whether 1-sigma error bars bracket the true model in about 68 percent of energy channels and whether the reduced chi-squared distribution matches the theoretical chi-squared with Ne-1 degrees of freedom; systematic deviations would show that the Gaussian assumption is load-bearing.

Watch

Extended reading notes

Core claim

The central claim is that equations (18), (19), and (22) give the correct 1-sigma errors for the energy-dependent cross-spectrum and rms spectrum, in the limit where averaged Fourier estimates are Gaussian. The key is that the same reference band appears in every subject-band cross-spectrum, and that subject bands are correlated with one another, so reference-band and shared-signal fluctuations act as a common systematic rather than independent scatter. Consequently, fitting the correct model with these errors yields reduced chi-squared with expectation value unity, whereas Bendat & Piersol (2010) errors yield over-fitting. The formulae also handle a broad reference band built from the sum of subject bands by subtracting a small noise term, eliminating the need to excise the current subject band from the reference.

Load-bearing premise

The formulae assume that the measured Fourier transforms behave like complex Gaussian random variables with signal and noise adding independently, and that averages over at least about 40 realizations are Gaussian; real variability from accreting sources is known to be log-normal and non-linear.

Editorial extensions

If this is right

Energy-dependent spectral-timing fits that use the new uncertainties will no longer be systematically over-fit: the expected reduced chi-squared for a correct model is unity, not below it.
A broad reference band formed by summing all subject bands can be used directly by subtracting the subject band's own Poisson noise contribution; the common practice of excising the subject band from the reference is unnecessary.
For rms spectra, the formula interpolates between the old single-band error at $γ=0$ and a corrected $γ=1$ limit that differs from a commonly quoted version by a factor $√2$.
The example fit to Cygnus X-1 with the reltrans reverberation model moves from reduced chi-squared 0.76 to 42.9/36, with best-fit parameters consistent within 1 sigma.
The new formulae also improve the expected sensitivity of X-ray polarimetry-timing searches for quasi-periodic oscillations in polarization degree and angle.

Reading between the lines

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

If the Gaussian-plus-uncorrelated-noise model is adequate, the same variance-splitting argument should apply to other correlated Fourier statistics, such as energy-dependent covariance spectra, by simple rescaling; the paper already gives the rescaling for complex covariance.
The formulae expose a clean separation between statistical scatter among energy channels and a normalization systematic coming from the shared reference band, which matters when model normalizations are tied across frequency ranges.
A natural stress test is to apply the formulae to simulated log-normal, non-linear variability; if coverage departs from nominal, the Gaussian assumption would need a multiplicative correction or a different noise model.
The new understanding of which fluctuations are systematic rather than statistical could sharpen frequentist model comparison in multi-band timing fits, not just single-model chi-squared.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The headline: this is a genuinely useful methods paper that corrects a real over-fitting problem in X-ray spectral-timing, but the modulus and phase error formulas are not valid for arbitrarily low intrinsic coherence as claimed. The real and imaginary part formulas and the rms formula are solid; the modulus/phase formulas are fine only in the high-coherence, high-signal-to-noise regime.

What's new: Ingram derives analytic errors for the energy-dependent cross-spectrum that correctly account for a common reference band and correlations between subject bands, which the standard Bendat & Piersol formulas don't. He also points out a factor-of-two error in the widely quoted rms error formula in Uttley et al. 2014. The broad-reference-band recipe with noise subtraction is elegant and removes the need to subtract each subject band from the reference. The Monte Carlo validation is extensive (50,000 realizations), and the chi-square distributions match theory for the cases tested. The Cygnus X-1 example is a nice demonstration. Code is provided. That's all real and worth having.

Where it's soft: the claim "valid for any value of intrinsic coherence" is not supported for modulus and phase. Section 4.2 derives d|G| = dRe = dIm by arguing that circular Gaussian error contours imply the same width in any direction. That is only true when the true cross-spectrum is several sigma from the origin. When gamma is low and |G| is comparable to or smaller than the scatter, the modulus follows a Rician/Rayleigh distribution whose standard deviation is noticeably smaller than the real/imaginary part error — for gamma=0 it's about 0.655 sigma versus sigma. The phase formula diverges as gamma goes to 0 while the actual phase distribution becomes uniform, so quoting a linear 1-sigma phase error is meaningless there. The Monte Carlo tests use gamma=0.6 with |G|/sigma ~ 27, which never probes this regime. The observational example is also high-coherence. So the abstract's "any value of intrinsic coherence" overclaims; the modulus and phase formulas are high-coherence approximations.

That said, the real and imaginary part formulas (Eqs. 18) and the rms formula (Eq. 22) appear correct for any coherence — the derivation for Re/Im is a straightforward variance sum and doesn't rely on the problematic contour argument. And in most practical X-ray timing applications, coherence is high enough that the modulus/phase formulas are fine. But anyone fitting low-coherence bands (e.g., high-frequency or faint channels) should be cautious.

Verdict: worthy of serious peer review as a methods paper. The core statistical correction is important and the derivations are mostly sound. I'd ask the author to either prove the modulus/phase formulas for low coherence (which I doubt they can) or explicitly restrict the claim to |G| of order a few sigma and above, and to add a simulation at gamma=0 or gamma=0.1 to show the limits.

Bring it to reading group; it'll spark a good discussion about when Gaussian error propagation breaks down.

Referee Report

3 major / 4 minor

Summary. The paper presents analytic error formulae for the energy-dependent cross-spectrum (real part, imaginary part, modulus, phase) and for the rms spectrum, explicitly accounting for the use of a common reference band and for correlations between subject bands. The central claim is that equations (18), (19) and (22) give correct 1σ uncertainties for any intrinsic coherence, in contrast to the Bendat & Piersol (2010) formulae, which overestimate errors in the energy-dependent limit and cause overfitting. The paper validates the formulae with 50,000 Monte Carlo realizations under a Gaussian signal-plus-noise model, demonstrates the method on RXTE Cygnus X-1 data, discusses the optimal reference band and the bias correction for coherence, and provides a Fortran implementation.

Significance. If the real/imag and rms formulae are correct, they fill a practical gap: fitting energy-dependent timing spectra is increasingly common, and the paper correctly identifies why single-spectrum error bars cause overfitting. The Monte Carlo validation is extensive, the observational fit is consistent, the correction to the Uttley et al. (2014) rms error formula is valuable, and the treatment of broad reference bands is elegant. The main weakness is that the modulus and phase formulae are derived from a small-scatter linearization that fails at low intrinsic coherence, while the paper claims validity for any coherence; this needs to be fixed before publication.

major comments (3)

[§4.2, Eq. (18)] The equality d|G| = dRe = dIm is derived from concentric circular error contours, which is only an approximation for a Gaussian cloud that does not contain the origin. Under the paper's own model with γ = 0, G̃ is a zero-mean complex Gaussian and |G̃| has a Rayleigh distribution with standard deviation ≈ 0.655 σ, where σ = dRe from Eq. (18); equation (18) therefore overestimates the modulus error by roughly a factor 1.5, and a constant-modulus fit to 50 channels would yield reduced χ² ≈ 0.43 rather than 1. The Section 3 simulation uses γ = 0.6 with |G|/σ ≈ 27, so it does not probe this regime, and the claim that the formulae are valid 'for any value of intrinsic coherence' (abstract and conclusions) is thus not supported.
[§4.2, Eq. (19)] The phase error diverges as γ → 0 because the denominator |G̃|² − b̃² tends to zero, whereas the true phase distribution under the Gaussian model tends to a uniform distribution on [0, 2π), for which a linear 1σ error is not meaningful. Equation (44), taken from Bendat & Piersol, is a small-angle linearization and is not valid when the Gaussian cloud contains the origin; the paper should either restrict Eq. (19) to |G|/dRe ≫ 1 or provide a proper wrapped/Rician treatment.
[§4.2, around Eq. (46)] The assertion that the 'other 35 terms' have zero k covariance does not follow from the fact that the total k covariance equals the first term's contribution; the remaining terms could cancel. Since the conclusion Cov_n{Re[G̃], Im[G̃]} = 0 rests on this shortcut, the derivation needs a direct evaluation or an explicit symmetry argument for the cross terms.

minor comments (4)

[Abstract and Section 7] The phrase 'valid for any value of intrinsic coherence' should be qualified, since the modulus and phase formulae are only demonstrated for high coherence; the real/imag and rms formulae appear more general.
[Section 3] Please report the ratio |G|/σ for the simulated and observed data; without this, the reader cannot easily see that the validation is confined to the high-coherence, high-signal-to-noise regime.
[Section 5] There are two typos: 'This is not indented as a means' should be 'intended', and 'logarithmicaly' should be 'logarithmically'.
[Section 2.4] It would help to state explicitly that d|G| in Eq. (18) is a large-|G|/σ approximation rather than an exact equality for all coherence values.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the new error formulae are derived from an explicit Gaussian signal-plus-noise model and validated against the external chi-square distribution, with no fitted parameters presented as predictions.

full rationale

The derivation chain in Section 4 is self-contained. Starting from the explicit generative model in equations (23) and (24), the paper expands the estimated cross-spectrum real and imaginary parts, computes the n-variance of each term from standard Gaussian sampling statistics (equations 37-42), sums them to obtain equation (43), and rearranges to get equation (18). The phase and rms formulae (19) and (22) follow from the same variance bookkeeping. No parameter is fit to a data subset and no fitted quantity is renamed as a prediction; the formulae are expressed directly in terms of the estimated power spectra, coherence, and noise levels. The Monte Carlo verification uses the same Gaussian model, so it is an in-sample check, but it is not circular because the error formulae are not used to generate the simulated data; the chi-square histograms are compared with the theoretical chi-square distribution with 49 degrees of freedom, which is an external benchmark. The paper's self-citations (e.g., reltrans in Section 5 and the discussion in Section 6.2) are applications or context, not load-bearing premises of the derivation. The passage in Section 6.2 noting that reference-band uncertainty is treated as a systematic rather than statistical error is a stated limitation, not a circular step; similarly, the skeptical concern that the modulus/phase formula may fail at low intrinsic coherence is a possible correctness issue in a nonlinear regime, not a reduction of the claimed result to its inputs. Overall, the central claim is independently derived and does not assume the target error formulas.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The formulas require no fitted parameters; all inputs are measured or known noise levels. The main assumptions are Gaussianity, stationarity, and additive uncorrelated noise, which are standard in this field though not exact. No new physical entities are introduced; the 'k variance' and 'n variance' are statistical conventions, not entities.

assumptions (5)

domain assumption Fourier transforms of the observed light curves are modeled as complex Gaussian signal plus uncorrelated Gaussian noise (equations 23 and 24).
All error derivations in Section 4 use the statistical structure of this model; real X-ray variability is known to be non-linear and log-normal (Uttley et al. 2005), so this is an idealization.
domain assumption For N at least 40 realizations, the estimated cross-spectral quantities are Gaussian distributed (Huppenkothen and Bachetti 2018).
The paper explicitly limits itself to this asymptotic regime in Section 2; the formulas may not hold for small N.
domain assumption The underlying power spectra, coherence, and phase lags are stationary over the observation time T.
Stationarity is assumed in Section 2.1 when defining underlying quantities and ensemble averaging.
domain assumption Signal and noise contributions are completely uncorrelated with one another.
Stated in Section 2 as 'almost always the case in X-ray astronomy' but not generally valid, for example atmospheric noise in optical light curves.
domain assumption When the reference band is the sum of subject bands, the noise contribution is accounted for by subtracting the term N(k,n) derived in Section 2.2 (equations 6 to 11).
This construction relies on Poisson counting noise properties and the assumption that the reference band is a sum of the subject bands.

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Pith. "Pith review of Error formulae for the energy-dependent cross-spectrum." pith.science (2026). https://pith.science/paper/Y7LFAJOF

@misc{pith2026190901385,
  author       = {Pith},
  title        = {Pith review of: Error formulae for the energy-dependent cross-spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y7LFAJOF}},
  note         = {Machine review of arXiv:1909.01385}
}

read the original abstract

I present analytic error formulae for the energy-dependent cross-spectrum and rms spectrum, which are Fourier statistics widely used to probe the rapid X-ray variability observed from accreting compact objects. The new formulae cover the modulus, phase, real and imaginary parts of the cross-spectrum, and are valid for any value of intrinsic coherence between variability in different energy bands. I show that existing error formulae (including that for the phase lag), which are valid for a single cross-spectrum or power spectrum, lead to over-fitting when applied to the energy-dependent cross-spectrum - which consists of cross-spectra between individual energy channels and a common reference band. I also introduce an optimal, unbiased way to define the reference band and an accurate way to calculate the intrinsic coherence between energy bands. I find that the traditional use of the old formulae has likely had a rather benign impact on the literature, but recommend the use of the new formulae in future wherever appropriate, since they are more accurate and are no harder to implement than existing error estimates. A code to implement the new error formulae on observational data is available online.

Figures

Figures reproduced from arXiv: 1909.01385 by the authors.

**Figure 1.** Left: Example of simulated data with error bars calculated using the analytic expressions presented in this paper (black), along with the best-fitting model (red lines). The top two panels show the real and imaginary parts of the energy-dependent cross-spectrum, whereas the bottom two panels show the modulus and phase angle (the latter is in radians). The input model does not depend on channel number and the input p… view at source ↗

**Figure 2.** Left: Simulations of an RMS spectrum with error bars calculated using equation (22) (black), along with the best-fitting model (red lines). Right: Probability distributions of the χ 2 value calculated for each of 50,000 such simulations. Black and grey stepped lines respectively correspond to errors calculated using equation (22) and the Bendat & Piersol equivalent. The input parameters are the same as those used fo… view at source ↗

**Figure 3.** The same as [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Example of the real part of the cross-spectrum for RXTE PCA data from Cygnus X-1 (black points). The reference band is 0.14 keV ≤ E ≤ 4.34 keV and the subject band is 4.34 keV ≤ E ≤ 4.64 keV. The error bars are calculated from the Bendat & Piersol formulae (equation 13…

**Figure 6.** Figure 6: Example fit of the X-ray reverberation model reltrans to the 0.041 − 0.092 Hz energy-dependent cross-spectrum. Real and imaginary parts are in the top and bottom panels respectively. The data are unfolded around the best fitting model (xspec command iplot eeuf). The …

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write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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