PoissonRatioUQ: An R package for band ratio uncertainty quantification

Matthew LeDuc; Tomoko Matsuo

REVIEW 3 major objections 6 minor 35 references

PoissonRatioUQ: An R package for band ratio uncertainty quantification

T0 review · 3 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Closed-form posterior for the ratio of Poisson means, delivered as an R package

desk verdict A useful R package for ratio UQ that overstates its closed-form posteriors; the math is mostly right, but the UQ needs a calibration check. read the letter →

arxiv 2602.07165 v3 pith:IRASRYUZ submitted 2026-02-06 stat.CO physics.data-anstat.ME

Matthew LeDuc , Tomoko Matsuo This is my paper · ORCID

classification stat.COphysics.data-anstat.ME MSC 62F1562M30

keywords PoissonratiopermanentalprocessBayesianuncertaintyquantificationgeneralizedBetaPrimedistributionGaussiancountdataRpackageremotesensing

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 introduces PoissonRatioUQ, an R package for Bayesian uncertainty quantification of count ratios. The core idea is that the physically meaningful quantity is the ratio of Poisson means, not the ratio of noisy counts. By modeling the latent intensities with a permanental process—a squared Gaussian process driving Poisson counts—the package derives closed-form posterior distributions for intensity ratios and for quantities of interest linked to those ratios through a power-law forward model. The result is fast, analytic uncertainty quantification that avoids Markov chain Monte Carlo, making full posterior retrievals feasible for problems with thousands of spatial bins.

What carries the argument

The central object is the permanental process, where the Poisson intensity is λ(s) = (c/2) f(s)^2 with f drawn from a Gaussian process. The posterior of f is approximated as Gaussian via the Laplace method; squaring and scaling turns each bin intensity Λ_i into a Gamma distribution by moment matching. The ratio of two independent Gammas is then a generalized Beta Prime distribution, and the known algebraic relationship between Beta Prime and Beta distributions enables exact CDF, quantile, and random-number generation. This chain of exact analytic transformations is what carries the argument from count data to closed-form posteriors.

What would settle it

Simulate binned Poisson count data with mean counts well below one per bin (generating many zero counts), then compare the package's closed-form 95% HPD intervals for the intensity ratio Z against a long-run MCMC posterior. If empirical coverage of the HPD intervals falls far below 95% (e.g., below 80%) across repeated simulations, the central approximation is falsified.

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Extended reading notes

Core claim

The paper shows that under a permanental process model with a Laplace approximation to the latent Gaussian field and a moment-matched Gamma approximation for each bin intensity, the posterior distribution of the intensity ratio Z = Λ_a/Λ_b is a generalized Beta Prime distribution. When the forward model has the form Z = (mT + z0)^p, the quantity of interest T inherits a shifted generalized Beta Prime distribution. This yields analytic posteriors, highest posterior density sets, and scoring metrics without sampling, and the package implements the full pipeline: kernel-based spatial estimation, pointwise conjugate-Gamma estimation, ratio and T retrieval, and CRPS/HPD utilities.

Load-bearing premise

The load-bearing premise is that the log-posterior of the latent Gaussian field is close to quadratic, so the Laplace approximation and the moment-matched Gamma distribution for bin intensities adequately describe the posterior; with very low counts, many zero bins, or a mis-specified kernel, the closed-form Beta Prime posterior is only approximate and the reported uncertainty intervals may be miscalibrated.

Editorial extensions

If this is right

Users can obtain full posterior distributions, including HPD intervals, for intensity ratios and derived quantities in seconds—about 15 seconds for 1000 bins on a desktop—without MCMC.
Estimating the ratio of latent Poisson means rather than the ratio of observed counts reduces bias and properly propagates Poisson shot noise into uncertainty bounds.
The package supports both spatially correlated retrievals via the permanental process and pointwise retrievals via conjugate Gamma priors, and it can handle missing realizations using NaN placeholders.
For forward models of the form Z = (mT + z0)^p, the quantity of interest T has a closed-form shifted generalized Beta Prime posterior, enabling direct uncertainty quantification for nonlinear retrievals such as temperature from FUV radiance ratios.
Built-in CRPS and HPD functions provide model scoring and credible intervals for arbitrary parametric predictive distributions, including multimodal cases.

Reading between the lines

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

Because the posterior is a standard parametric family, downstream tasks like hierarchical modeling, Bayesian updating, or importance sampling become straightforward—one can analytically marginalize or reweight without drawing samples from the raw count process.
The same Beta-Prime change-of-variable trick should extend to other monotone forward models beyond the power law (e.g., exponential or log-linear), giving closed-form T posteriors whenever the inverse transformation is tractable.
A practical diagnostic for the package would be a built-in coverage check: simulate data under known parameters, compare the claimed 95% HPD intervals to empirical coverage, and warn when the Laplace/Gamma approximation breaks down in low-count regimes.
The method is naturally portable to any photon- or particle-counting instrument with band ratios—X-ray hardness ratios, lidar ratios, isotope-ratio mass spectrometry—beyond the atmospheric remote sensing applications cited.

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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

This is a package paper for a useful R tool, and the math is mostly right, but the central UQ claim is under-validated. The package does what it says, but the paper oversells the word 'closed-form' and never checks whether the uncertainty intervals are actually calibrated.

What's new: PoissonRatioUQ bundles the permanental process estimator from prior work (Flaxman, Walder & Bishop, and the authors' own ref [7]) into an R workflow, adds a pointwise Gamma-Beta-prime route that handles unequal numbers of realizations, and provides transformation to a quantity of interest of the form Z=(mT+z0)^p. The derivations for the shifted generalized Beta Prime distribution are correct, and the CDF/quantile/random-number functions are sensible, using the Beta relationship. The timing is good - 1000 bins in about 15 seconds - and the code examples are reproducible.

Where it's soft, in proportion. First, Eq. (2.7) has a typo: with f = K̃ψ the quadratic penalty should be -1/2⟨ψ, K̃ψ⟩, not -1/2⟨ψ, K̃^{-1}ψ⟩; the gradient in (2.8) matches the corrected form, so it's a presentation error, not a substantive one. Second, the 'closed-form' language. Eq. (2.9) is a Laplace approximation and (2.10) is a moment-matched Gamma; the abstract and intro call them closed-form, while the body (Section 2) says 'well-approximated'. That is an overstatement. Third, and most important, there is no calibration or coverage check anywhere. The demonstration reports only CRPS and MAE of the MAP. In the low-count regime that motivates this work - Poisson means near or below one, many zero bins - the log-posterior is far from quadratic near f_i=0, and the Gamma approximation could be badly skewed. The authors should add a small simulation comparing their approximate intervals to exact sampling (MCMC or a dense grid) and report empirical coverage. This is not a dealbreaker; the package is likely fine for moderate counts, but UQ is the selling point, so it needs the check.

Who it's for: applied statisticians and remote-sensing/x-ray astronomers who need quick Bayesian ratio intervals. It deserves a serious referee, but the revision should fix the typo and add the calibration evidence. I'd send it out.

Referee Report

3 major / 6 minor

Summary. The manuscript introduces an R package, PoissonRatioUQ, for Bayesian estimation and uncertainty quantification of ratios of Poisson intensities from binned count data. The modeling uses a permanental process with a latent Gaussian field and the quadratic link Lambda_i = c/2 f_i^2; after a Laplace approximation for the latent field and a moment-matched Gamma approximation for each Lambda_i, the intensity ratio is modeled as a generalized Beta Prime distribution, and quantities of interest T defined by Z=(mT+z0)^p are obtained by a location-scale transformation. The paper describes the mathematical derivation, package functions for optimization, generalized Beta Prime calculations, CRPS, and HPD intervals, and gives two toy demonstrations. A timing experiment reports that a full posterior for 1000 bins takes roughly 15 seconds.

Significance. If the approximations are well calibrated, the package would fill a practical gap in spatial count-ratio analysis: it would provide fast, analytic UQ without MCMC. The paper's strengths are the explicit formulas for the generalized Beta Prime distribution, the analytic CDF/quantile/random-generation functions, the HPD and CRPS utilities, and the public package implementation. These are genuinely useful and clearly described. However, the central claim that the package yields closed-form posterior distributions is stated strongly, while the actual posterior is only approximate, and no calibration evidence is provided. The authors should be credited for making the code and examples available, but the UQ claim is not yet supported.

major comments (3)

[Section 2, Eq. (2.7)] There is an internal inconsistency in the penalty term. With f = Ktilde psi, the quadratic form in Eq. (2.5) becomes -(1/2)<psi, Ktilde psi>, not -(1/2)<psi, Ktilde^{-1} psi>. The gradient in Eq. (2.8) matches the corrected form, so this is likely a typographical error, but it appears in the central derivation and should be fixed.
[Section 2, Eqs. (2.9)-(2.10); Section 3.1] The abstract and Section 2 describe the method as yielding 'closed-form posterior distributions,' but Eq. (2.9) is a Laplace approximation and Eq. (2.10) is a moment-matched Gamma approximation. These are not exact. The log-posterior (2.4) contains terms a_i log(f_i^2), which are strongly non-quadratic near zero, so in low-count or zero-bin regimes the HPD intervals may be miscalibrated. Section 3.1 reports only CRPS and relative MAE of the MAP, not interval coverage or comparison to an exact posterior sampler. I recommend a simulation study reporting empirical coverage of the 95% HPD intervals over repeated datasets with Poisson means near or below 1 and with zero bins, and, if feasible, a comparison to MCMC.
[Section 2, Eq. (2.9)] The covariance formula defines D = diag(psi_i^2 / (2 a_i)). For zero-count bins, a_i=0, this quantity is undefined. Since zero counts are common in the motivating low-count applications, the manuscript should state how zero counts are treated in the optimization and in constructing the Laplace covariance, e.g., whether D^{-1} is set to zero. Without this, the package's behavior for the key low-count regime is ambiguous.

minor comments (6)

[Abstract and Section 2] Consider replacing 'closed-form posterior distributions' with 'analytic approximations to the posterior' or 'closed-form approximate posterior distributions' to avoid overstating the Laplace/Gamma approximations.
[Eq. (2.10)] The notation in the denominator of alpha_i and beta_i appears as '2 mu_i2 + sigma_i^2'; this should be '2 mu_i^2 + sigma_i^2'.
[Section 3.1] The text contains a typo: 'Conitnuous Rank Probability Score' should be 'Continuous Rank Probability Score.' Also report the count levels used in the toy example (e.g., average counts per bin) so readers can judge whether the low-count regime is exercised.
[Figure 3.1(a)] The legend repeats 'True ratio function / Observed count ratios / Estimated ratio function' three times. The legend should be cleaned up.
[Reference [34]] The second author's name is misspelled: it should be 'Raftery,' not 'Rafferty.'
[Section 5] There is a typo: 'involve' should be 'involve' in 'Two immediate applications for future work invole the ability...'

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the closed-form posterior chain is derived in-line and tested against independent toy truth; the self-citation to [7] is provenance, not load-bearing.

full rationale

I find no circular step. The central derivation in Section 2 is presented in the paper itself: log-posterior (2.4)-(2.5), equivalent kernel (2.6), MAP/gradient (2.7)-(2.8), Laplace covariance (2.9), Gamma moment matching (2.10), Beta Prime ratio (2.11)-(2.12), and the transformation to T (2.13). These are mathematical consequences of the stated model, not definitions of the target quantity. The toy demonstrations generate Poisson data from an independent true ratio curve, and the hyperparameters (kernel width 0.75, gamma=c=1) are chosen by hand, not tuned to the truth; CRPS, MAE, and timing are evaluated against that independent synthetic truth. The package functions implement published algorithms [10], [11], [24], [26], and the algorithm is attributed to [7] by the same first author, but the self-citation is provenance rather than the load-bearing proof because the equations are restated inline. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is hidden behind a citation. There is a non-circular correctness caveat: Eq. (2.7) writes the penalty as -1/2 <psi, Ktilde^{-1} psi> while the gradient in Eq. (2.8) and the substitution f = Ktilde psi imply -1/2 <psi, Ktilde psi>; this appears to be a typographical inconsistency, and the absence of interval-coverage validation is an accuracy concern, not a circularity. The score of 2 reflects only a minor self-citation that is not load-bearing.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The model depends on user-supplied hyperparameters (γ, c, kernel width) that are not inferred; the likelihood and prior are standard assumptions from the permanental process literature. No new physical or statistical entities (particles, forces, latent dimensions) are introduced.

free parameters (3)

γ (marginal precision) = default 1
Prior precision in f ~ N(0, γ^{-1}K), user-supplied with no data-driven selection or sensitivity analysis.
c (intensity scaling) = default 1
Scales intensity as Λ_i = (c/2) f_i^2; user-supplied, default 1, not inferred.
Wendland kernel support width = 0.75 in §3.1 demo
Kernel hyperparameter chosen by hand for the toy problem; no sensitivity study, though accuracy and uncertainty intervals likely depend on it.

assumptions (7)

domain assumption Counts a_i are independent Poisson(Λ_i) random variables.
The likelihood Eq. (2.3) assumes this; no overdispersion check is reported.
domain assumption The latent field f has prior f ~ N(0, γ^{-1}K) and intensity is Λ_i = (c/2) f_i^2.
Equation (2.4); a modeling choice inherited from the permanental process literature rather than derived from data or physics.
domain assumption The log-posterior of f is well approximated by a Gaussian (Laplace approximation).
Used in §2.2 around Eq. (2.9); can fail at low counts or with many zero-count bins.
domain assumption The forward model is exactly Z = (mT + z0)^p with known m, z0, p.
Equation (2.13) assumes a known deterministic transformation; misspecification is not modeled.
domain assumption The ratio of Poisson means is the physical quantity of interest, not the ratio of counts.
Framing assumption in the abstract and §1; standard in the cited literature but unverifiable from counts alone.
standard math The Representer Theorem applies, giving f = K̃ψ.
Equation (2.7) relies on standard nonparametric regression theory.
standard math Known relationships between generalized Beta Prime and Beta distributions.
Used in §4.1.1 for CDF, quantile, and random number generation; standard results.

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Cite this review

Pith. "Pith review of PoissonRatioUQ: An R package for band ratio uncertainty quantification." pith.science (2026). https://pith.science/paper/IRASRYUZ

@misc{pith2026260207165,
  author       = {Pith},
  title        = {Pith review of: PoissonRatioUQ: An R package for band ratio uncertainty quantification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRASRYUZ}},
  note         = {Machine review of arXiv:2602.07165}
}

abstract

We introduce an R package for Bayesian modeling and uncertainty quantification for problems involving count ratios. The modeling relies on the assumption that the quantity of interest is the ratio of Poisson means rather than the ratio of counts. We provide multiple different options for retrieval of this quantity for problems with and without spatial information included. Some added capability for uncertainty quantification for problems of the form $Z=(mT+z_0)^{p}$, where $Z$ is the intensity ratio and $T$ the quantity of interest, is included.

Figures

Figures reproduced from arXiv: 2602.07165 by the authors.

**Figure 3.1.** Example retrieval and timing study using the permanental process model to retrieve [PITH_FULL_IMAGE:figures/full_fig_p006_3_1.png] view at source ↗

**Figure 3.2.** Estimation results from two realizations of the Poisson data for the nonlinear trans [PITH_FULL_IMAGE:figures/full_fig_p007_3_2.png] view at source ↗

**Figure 4.1.** Comparison: Empirical distribution, CDF, and quantiles from [PITH_FULL_IMAGE:figures/full_fig_p009_4_1.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.2.** Figure 4.2: Demonstration of the algorithm for calculating the highest-posterior density sets on a [PITH_FULL_IMAGE:figures/full_fig_p014_4_2.png]

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Reference graph

Works this paper leans on

35 extracted references

[7]

2025.url:https://doi

Matthew LeDuc, Tomoko Matsuo, and William Kleiber.A New Approach to Inversion of Multi-spectral Data and Applications to FUV Remote Sensing. 2025.url:https://doi. org/10.5194/EGUSPHERE-2025-5570

doi:10.5194/egusphere-2025-5570 2025
[1]

Thermospheric Composition and Solar EUV Flux From the Global-Scale Observations of the Limb and Disk (GOLD) Mission

J. Correira et al. “Thermospheric Composition and Solar EUV Flux From the Global-Scale Observations of the Limb and Disk (GOLD) Mission”. In:Journal of Geophysical Research: Space Physics(2021).doi:https : / / doi . org / 10 . 1029 / 2021JA029517.url:https : //agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021JA029517

doi:10.1029/2021ja029517 2021
[2]

The Thermospheric Column O/N2 Ratio

R. R. Meier. “The Thermospheric Column O/N2 Ratio”. In:Journal of Geophysical Re- search: Space Physics(2021).url:https://doi.org/10.1029/2020JA029059

doi:10.1029/2020ja029059 2021
[3]

Satellite remote sensing of thermospheric O/N2 and solar EUV: 1. Theory

D. J. Strickland, J. S. Evans, and L. J. Paxton. “Satellite remote sensing of thermospheric O/N2 and solar EUV: 1. Theory”. In:Journal of Geophysical Research: Space Physics(1995). url:https://doi.org/10.1029/95JA00574

doi:10.1029/95ja00574 1995
[4]

O/N2 changes during 1–4 October 2002 storms: IMAGE SI-13 and TIMED/GUVI observations

Y. Zhang et al. “O/N2 changes during 1–4 October 2002 storms: IMAGE SI-13 and TIMED/GUVI observations”. In:Journal of Geophysical Research: Space Physics(2004).doi:https : / / doi . org / 10 . 1029 / 2004JA010441.url:https : / / agupubs . onlinelibrary . wiley . com/doi/abs/10.1029/2004JA010441

doi:10.1029/2004ja010441 2002
[5]

New approaches for quantifying and understanding thermosphere temper- ature variability from far ultraviolet dayglow

Clayton Cantrall. “New approaches for quantifying and understanding thermosphere temper- ature variability from far ultraviolet dayglow”. PhD thesis. University of Colorado-Boulder, 2022

2022
[6]

Deriving column-integrated thermospheric tem- perature with the N2 Lyman–Birge–Hopfield (2,0) band

Clayton Cantrall and Tomoko Matsuo. “Deriving column-integrated thermospheric tem- perature with the N2 Lyman–Birge–Hopfield (2,0) band”. In:Atmospheric Measurement Techniques(2021).doi:10.5194/amt-14-6917-2021

doi:10.5194/amt-14-6917-2021 2021
[8]

Deriving Thermospheric Tem- perature From Observations by the Global Ultraviolet Imager on the Thermosphere Iono- sphere Mesosphere Energetics and Dynamics Satellite

Yongliang Zhang, Larry J. Paxton, and Robert K. Schaefer. “Deriving Thermospheric Tem- perature From Observations by the Global Ultraviolet Imager on the Thermosphere Iono- sphere Mesosphere Energetics and Dynamics Satellite”. In:Journal of Geophysical Research: Space Physics(2019)

2019

Show all 35 references

[9]

Spatial analysis of the polycyclic aromatic hydrocarbon features southeast of the Orion Bar

C Boersma, RH Rubin, and LJ Allamandola. “Spatial analysis of the polycyclic aromatic hydrocarbon features southeast of the Orion Bar”. In:The Astrophysical Journal(2012)

2012
[10]

Hardness Ratio estimation in Low Counting X-Ray Photometry

Y.K. Jin, S.N. Zhang, and J.F. Wu. “Hardness Ratio estimation in Low Counting X-Ray Photometry”. In:The Astrophysical Journal(2006).doi:10.1086/508677

2006 doi
[11]

Bayesian Estimation of Hardness Ratios: Modeling and Computa- tions

Taeyoung Park et al. “Bayesian Estimation of Hardness Ratios: Modeling and Computa- tions”. In:The Astrophysical Journal(2006).doi:10.1086/507406.url:https://dx. doi.org/10.1086/507406

2006 doi
[12]

Analysis of bright source hardness ratios in the 4 yr Insight-HXMT galactic plane scanning survey catalog

Chen Wang et al. “Analysis of bright source hardness ratios in the 4 yr Insight-HXMT galactic plane scanning survey catalog”. In:Research in Astronomy and Astrophysics(2024). 16

2024
[13]

Upper Atmosphere Radi- ance Data Assimilation: A Feasibility Study for GOLD Far Ultraviolet Observations

Clayton E. Cantrall, Tomoko Matsuo, and Stanley C. Solomon. “Upper Atmosphere Radi- ance Data Assimilation: A Feasibility Study for GOLD Far Ultraviolet Observations”. In: Journal of Geophysical Research: Space Physics(2019).url:https://agupubs.onlinelibrary. wiley.com/doi/abs...

2019 doi
[14]

Statistical Bias in Isotope Ratios

Christopher D. Coath, Robert C. J. Steele, and W. Fred Lunnon. “Statistical Bias in Isotope Ratios”. In:Journal of Analytical Atomic Spectrometry(2013)

2013
[15]

A Generalized Beta Prime Distribution as the Ratio Prob- ability Density Function for Change Detection Between Two SAR Intensity Images With Different Number of Looks

Gerard Gallardo i Peres et al. “A Generalized Beta Prime Distribution as the Ratio Prob- ability Density Function for Change Detection Between Two SAR Intensity Images With Different Number of Looks”. In:IEEE Transactions on Geoscience and Remote Sensing (2024).doi:10.1109/TGR...

2024
[16]

Interstellar Dust Experiment (IDEX) onboard NASA’s Interstellar Mapping and Acceleration Probe (IMAP)

Mih´ aly Hor´ anyi et al. “Interstellar Dust Experiment (IDEX) onboard NASA’s Interstellar Mapping and Acceleration Probe (IMAP)”. In:Space Science Reviews(2025)

2025
[17]

Atmospheric temperature measurements at altitudes of 5-30km with a double-grating-based pure rotational Raman lidar

Jingyu Jia and Fan Yi. “Atmospheric temperature measurements at altitudes of 5-30km with a double-grating-based pure rotational Raman lidar”. In:Appl. Opt.(2014).doi:10.1364/ AO.53.005330.url:https://opg.optica.org/ao/abstract.cfm?URI=ao-53-24-5330

2014
[18]

Improving Solar Flare Now- casting with the Hot Onset Precursor Event Technique

Anant Telikicherla, Thomas N Woods, and Bennet D Schwab. “Improving Solar Flare Now- casting with the Hot Onset Precursor Event Technique”. In:The Astrophysical Journal (2025)

2025
[19]

A new strategy for ionospheric remote sensing using the 130.4/135.6 nm airglow intensity ratios

XiaoHan Yin, JianQi Qin, and Larry J. Paxton. “A new strategy for ionospheric remote sensing using the 130.4/135.6 nm airglow intensity ratios”. In:Earth and Planetary Physics (2023).doi:10.26464/epp2023042

2023 doi
[20]

The Permanental Process

Peter McCullagh and Jesper Møller. “The Permanental Process”. In:Advances in Applied Probability(2006).url:https://doi.org/10.1017/S0001867800001361

2006 doi
[21]

Properties of spatial Cox process models

Jesper Møller. “Properties of spatial Cox process models”. In:Journal of Statistical Research of Iran(2005)

2005
[22]

R Foundation for Statistical Computing

R Core Team.R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria, 2025.url:https://www.R-project.org/

2025
[23]

Streit.Poisson Point Processes:Imaging, Tracking, and Sensing

Roy L. Streit.Poisson Point Processes:Imaging, Tracking, and Sensing. Springer, 2010

2010
[24]

Poisson intensity estimation with reproducing kernels

Seth Flaxman, Yee Whye Teh, and Dino Sejdinovic. “Poisson intensity estimation with reproducing kernels”. In:Electronic Journal of Statistics(2017)

2017
[25]

Regularization and Approximation of Linear Operator Equations in Reproducing Kernel Spaces

M.Z. Nashed and Grace Wahba. “Regularization and Approximation of Linear Operator Equations in Reproducing Kernel Spaces”. In:Bulletin of the American Mathematical Society (1974)

1974
[26]

Fast Bayesian Intensity Estimation for the Per- manental Process

Christian J. Walder and Adrian N. Bishop. “Fast Bayesian Intensity Estimation for the Per- manental Process”. In:Proceedings of the 34th International Conference on Machine Learn- ing. Ed. by Doina Precup and Yee Whye Teh. Proceedings of Machine Learning Research. PMLR, 2017.u...

2017
[27]

Carl Edward Rasmussen and Christopher K. I. Williams.Gaussian Processes for Machine Learning. The MIT Press, 2005.doi:10.7551/mitpress/3206.001.0001.url:https: //doi.org/10.7551/mitpress/3206.001.0001. 17

2005 doi
[28]

Satellite drag coefficient modeling for thermosphere science and mission operations

Piyush M Mehta et al. “Satellite drag coefficient modeling for thermosphere science and mission operations”. In:Advances in Space Research(2023)

2023
[29]

Modelling of space weather effects on satellite drag

E Doornbos and H Klinkrad. “Modelling of space weather effects on satellite drag”. In: Advances in Space Research(2006)

2006
[30]

Impact of tidal density variability on orbital and reentry predictions

J. M. Leonard, J. M. Forbes, and G. H. Born. “Impact of tidal density variability on orbital and reentry predictions”. In:Space Weather(2012).doi:https : / / doi . org / 10 . 1029 / 2012SW000842

2012
[31]

Satellite orbital drag

Eftyhia Zesta and Cheryl Y Huang. “Satellite orbital drag”. In:Space weather fundamentals. CRC Press, 2016

2016
[32]

Ant´ onia Amaral Turkman, Carlos Daniel Paulino, and Peter M¨ uller.Computational Bayesian Statistics: An Introduction

M. Ant´ onia Amaral Turkman, Carlos Daniel Paulino, and Peter M¨ uller.Computational Bayesian Statistics: An Introduction. Institute of Mathematical Statistics Textbooks. Cam- bridge University Press, 2019

2019
[33]

Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree

Holger Wendland. “Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree.” In:Adv Comput Math(1995)

1995
[34]

Strictly Proper Scoring Rules, Prediction, and Estimation

T. Gneiting and A. Rafferty. “Strictly Proper Scoring Rules, Prediction, and Estimation”. In:Journal of the American Statistical Association(2007).url:https://doi.org/10. 1198/016214506000001437. [35]NIST Digital Library of Mathematical Functions.https://dlmf.nist.gov/, Releas...

2007
[36]

XVI. Functions of positive and negative type, and their connection the theory of integral equations

James Mercer and Andrew Russell Forsyth. “XVI. Functions of positive and negative type, and their connection the theory of integral equations”. In:Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Char- acter(1...

1909

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [7]

2025.url:https://doi

Matthew LeDuc, Tomoko Matsuo, and William Kleiber.A New Approach to Inversion of Multi-spectral Data and Applications to FUV Remote Sensing. 2025.url:https://doi. org/10.5194/EGUSPHERE-2025-5570

doi:10.5194/egusphere-2025-5570 2025

[2] [1]

Thermospheric Composition and Solar EUV Flux From the Global-Scale Observations of the Limb and Disk (GOLD) Mission

J. Correira et al. “Thermospheric Composition and Solar EUV Flux From the Global-Scale Observations of the Limb and Disk (GOLD) Mission”. In:Journal of Geophysical Research: Space Physics(2021).doi:https : / / doi . org / 10 . 1029 / 2021JA029517.url:https : //agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021JA029517

doi:10.1029/2021ja029517 2021

[3] [2]

The Thermospheric Column O/N2 Ratio

R. R. Meier. “The Thermospheric Column O/N2 Ratio”. In:Journal of Geophysical Re- search: Space Physics(2021).url:https://doi.org/10.1029/2020JA029059

doi:10.1029/2020ja029059 2021

[4] [3]

Satellite remote sensing of thermospheric O/N2 and solar EUV: 1. Theory

D. J. Strickland, J. S. Evans, and L. J. Paxton. “Satellite remote sensing of thermospheric O/N2 and solar EUV: 1. Theory”. In:Journal of Geophysical Research: Space Physics(1995). url:https://doi.org/10.1029/95JA00574

doi:10.1029/95ja00574 1995

[5] [4]

O/N2 changes during 1–4 October 2002 storms: IMAGE SI-13 and TIMED/GUVI observations

Y. Zhang et al. “O/N2 changes during 1–4 October 2002 storms: IMAGE SI-13 and TIMED/GUVI observations”. In:Journal of Geophysical Research: Space Physics(2004).doi:https : / / doi . org / 10 . 1029 / 2004JA010441.url:https : / / agupubs . onlinelibrary . wiley . com/doi/abs/10.1029/2004JA010441

doi:10.1029/2004ja010441 2002

[6] [5]

New approaches for quantifying and understanding thermosphere temper- ature variability from far ultraviolet dayglow

Clayton Cantrall. “New approaches for quantifying and understanding thermosphere temper- ature variability from far ultraviolet dayglow”. PhD thesis. University of Colorado-Boulder, 2022

2022

[7] [6]

Deriving column-integrated thermospheric tem- perature with the N2 Lyman–Birge–Hopfield (2,0) band

Clayton Cantrall and Tomoko Matsuo. “Deriving column-integrated thermospheric tem- perature with the N2 Lyman–Birge–Hopfield (2,0) band”. In:Atmospheric Measurement Techniques(2021).doi:10.5194/amt-14-6917-2021

doi:10.5194/amt-14-6917-2021 2021

[8] [8]

Deriving Thermospheric Tem- perature From Observations by the Global Ultraviolet Imager on the Thermosphere Iono- sphere Mesosphere Energetics and Dynamics Satellite

Yongliang Zhang, Larry J. Paxton, and Robert K. Schaefer. “Deriving Thermospheric Tem- perature From Observations by the Global Ultraviolet Imager on the Thermosphere Iono- sphere Mesosphere Energetics and Dynamics Satellite”. In:Journal of Geophysical Research: Space Physics(2019)

2019