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tonalli: an asexual genetic code to characterise APOGEE-2 stellar spectra. I. Validation with synthetic and solar spectra

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read An asexual genetic algorithm extracts four stellar parameters from APOGEE-2 spectra with no appreciable bias for stars from 3200 to 6250 K.

desk verdict A careful APOGEE-2 fitting pipeline whose headline four-parameter operating range is not backed by the synthetic tests, because only Teff and log(g) are varied at solar abundances. read the letter →

arxiv 2411.15342 v1 pith:6NLH5MYV submitted 2024-11-22 astro-ph.IM astro-ph.GAastro-ph.SR

Lucía Adame , Carlos Román-Zúñiga , Jesús Hernández , Ricardo López-Valdivia , Edilberto Sánchez This is my paper · ORCID
classification astro-ph.IMastro-ph.GAastro-ph.SR
keywords asexualgeneticalgorithmAPOGEE-2stellarspectraatmosphericparametersMARCSsyntheticpre-main-sequencestarsspectrumfittingMonteCarlouncertaintiescontinuumnormalization
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 presents tonalli, a Python code that pulls effective temperature ($T_{\rm eff}$), surface gravity ($\log(g)$), overall metallicity ($[{\rm M/H}]$), and $\alpha$-element abundance ($[\alpha/{\rm M}]$) out of APOGEE-2 near-infrared spectra by direct fitting against an interpolated MARCS synthetic library. The central claim under validation is that the code recovers all four stellar parameters with no appreciable bias for stars with $T_{\rm eff}$ between roughly 3200 and 6250 K, demonstrated on noisy synthetic spectra and on the APOGEE-2 solar spectrum reflected by Vesta. If that claim holds, the code offers a model-based parameter route for cool dwarfs and subgiants, including pre-main-sequence stars, where the standard APOGEE-2 pipeline was not optimized. The paper also details a Monte Carlo repetition scheme that converts the stochastic optimizer output into per-star credible intervals, and it releases the continuum-normalized MARCS library it constructs.

What carries the argument

The load-bearing mechanism is the asexual genetic algorithm of Cantó et al. (2009), which differs from classical genetic algorithms by generating each new generation within a shrinking hyper-cube around each of the $N_p$ fittest parents, with side lengths decreasing as $(\Delta x_i)_n = (\Delta x_i)_0 \, p^n$ for a convergence factor $p\in(0,1)$. Each offspring's synthetic spectrum is formed by 4D linear interpolation over the nearest $N_{\rm interpol}$ MARCS grid points, convolved to APOGEE-2 resolution, rotationally broadened, and Doppler-shifted, and its fitness is the $\chi^2$ between that spectrum and the observed one. Before fitting, both observed and synthetic spectra are mapped onto a common pseudo-continuum by an iterative BIC-selected polynomial fit with asymmetric $\sigma$-clipping, and the wavelength window for the figure of merit excludes the chip edges where the Brackett lines distort the normalization. A $k$-nearest-neighbours classifier on Mg i, Al i, and CO equivalent widths gates high-temperature and emission-line stars, and the whole fine search is repeated in a Monte Carlo loop so that the reported parameters are the median and interquartile range of the resulting distributions.

What would settle it

Apply tonalli to APOGEE-2 spectra of benchmark cool stars with independent effective temperatures from interferometry and surface gravities from asteroseismology, spanning 3200 to 6250 K, and check whether the recovered medians agree with the independent values within the claimed bias of less than half a MARCS grid step; a systematic offset that grows as $T_{\rm eff}$ approaches 6250 K, or an offset on real stars that is absent in the synthetic recovery tests, would refute the working-range claim.

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Extended reading notes

Core claim

On its own terms, the paper's contribution is the demonstration that an asexual genetic algorithm can reliably locate the best-fitting MARCS model for an APOGEE-2 spectrum without appreciable bias in a defined working range. In the noisy synthetic experiments, the reported bias stays below half the MARCS grid step for the four parameters over roughly 3200 to 6250 K, with the temperature bias growing at higher temperatures because the continuum normalization degrades near the Brackett lines at the chip edges. For the Vesta solar spectrum, the adopted univariate medians are $T_{\rm eff}=5779$ K, $\log(g)=4.51$ dex, $[{\rm M/H}]=-0.03$ dex, and $[\alpha/{\rm M}]=0.00$ dex, with credible intervals on the order of a few hundred kelvin and a few tenths of a dex, and the paper finds these consistent with published spectroscopic determinations of the Sun.

Load-bearing premise

The load-bearing premise is that the iterative continuum normalization places real and synthetic spectra on a common pseudo-continuum across all three APOGEE-2 chips; if it distorts line-to-continuum ratios, especially near the Brackett lines at the chip edges, the chi-squared fit is biased, and the synthetic validation against the same MARCS library measures internal consistency rather than the fidelity of the models to real stars.

Editorial extensions

If this is right

  • APOGEE-2 spectra of cool dwarfs and subgiants can be parameterized entirely by direct synthetic fitting, with no training labels and with per-star Monte Carlo uncertainties.
  • The working range from 3200 to 6250 K includes pre-main-sequence stars, giving a model-based route for studying young stellar populations where the standard APOGEE-2 pipeline is known to struggle.
  • The released continuum-normalized MARCS library and the user-selectable library interface allow the same pipeline to be re-run against other model grids, such as BT-NextGen, BOSZ, PHOENIX, or SpecModels, to expose model dependence.
  • The stopping criterion based on the temperature hyper-volume shrinking to 1 K, together with the recommended input parameters, provides a reproducible recipe for parameter recovery.
  • The median of the one-dimensional Monte Carlo distribution is argued to be a robust statistic when parameter degeneracies make the distribution multimodal, giving a concrete reporting convention.

Reading between the lines

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

  • Beyond the paper, the synthetic validation uses spectra drawn from the same MARCS library that tonalli fits, so the reported accuracy is primarily internal consistency; a cross-check against asteroseismic or interferometric benchmarks for cool dwarfs would test the model-fidelity part of the claim.
  • The paper's own Vesta experiment shows parameter degeneracies, including a shift in $\log(g)$ when the radial velocity is optimized, so adding priors as the authors say they plan could shrink credible intervals substantially without changing the method.
  • The continuum-normalization failure above roughly 6250 K is tied to Brackett lines near chip edges, so restricting the figure-of-merit windows or adopting a different normalization for early-type stars might extend the same machinery to hotter stars.
  • The classifier gating is trained on young stars in the Pleiades and W3/4/5, so applying tonalli to field cool stars assumes those equivalent-width classes transfer; re-training on field stars would harden the pipeline for general surveys.
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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

4 major / 4 minor

Summary. The manuscript presents tonalli, a Python code implementing an asexual genetic algorithm (Cantó et al. 2009) to fit APOGEE-2 spectra against a continuum-normalized, resolution-matched MARCS synthetic library. The fitted parameters are Teff, log(g), [M/H], [alpha/M], v sin(i), and RV, with limb darkening fixed at a chosen value. The paper describes the algorithm, the iterative BIC-based sigma-clipping continuum normalization, a k-nearest-neighbour spectral classifier, and a Monte Carlo repetition scheme for uncertainties. Validation is performed with 50 Monte Carlo realizations on 66 synthetic MARCS spectra at solar abundances, in three configurations (M0: already-normalized library spectra; M1: re-normalized spectra; M2: M1 plus noise at several S/N), and on the APOGEE-2 DR17 Vesta solar spectrum with RV fixed and optimized. The central claim is that tonalli can recover all four stellar parameters without appreciable bias for stars with effective temperatures between roughly 3200 and 6250 K.

Significance. If the stated operating range were fully established, tonalli would be a useful open tool for cool-dwarf and pre-main-sequence parameter estimation from APOGEE-2 spectra, a regime in which ASPCAP is known to be less reliable. The paper has real strengths: the algorithm is described in unusual detail with pseudo-code, the parameter-control exploration in Appendix B is systematic, the Monte Carlo uncertainty treatment is honest, and the continuum-normalized MARCS library is publicly released with a DOI. The Vesta analysis openly reports broad credible intervals, multimodal distributions, and model degeneracy. However, the headline accuracy claim is broader than the validation actually supports: the Section 3 synthetic experiment varies only Teff and log(g) while fixing [M/H]=[alpha/M]=0, and all targets are drawn from the same MARCS library used for fitting. These two issues make the four-parameter, 3200-6250 K claim load-bearing and currently under-supported.

major comments (4)
  1. [Section 3.2, Figure 6, Table 3] The synthetic validation set is described as having "zero metal and alpha-elements abundances" with only Teff and log(g) varied (Section 3.2, second paragraph: 66 models with [M/H]=[alpha/M]=0). Consequently the heat maps in Figure 6 and the temperature ranges in Table 3 exercise recovery of Teff and log(g) at solar composition; they do not test recovery of [M/H] or [alpha/M] at non-solar values. The sentence in Section 3.2.3 stating that tonalli "can recover with success the four stellar parameters in this experiment for stars with effective temperatures between ~3200 and ~6250 K" is therefore not established for the two abundance parameters. Because abundance changes alter line depths and molecular opacities and can trade off against Teff and log(g), the absence of non-solar-abundance tests is load-bearing. I request adding M1/M2-style experiments at non-zero abundances, for example [M/H] = -0.5, -0.25, +0.25 and [alpha/M] = +0.2 or +0.4, or, failing that, explicitly restricting the claimed operating range to solar-composition recovery.
  2. [Section 3.2 and Section 2.3.8] All three validation experiments (M0, M1, M2) generate the target spectra from the same MARCS library that tonalli fits against. These experiments therefore measure internal consistency of the interpolator, the optimizer, the continuum-normalization procedure, and the noise handling; they cannot detect systematic errors of the MARCS models relative to real stars. The paper acknowledges model dependence in Section 2.3.8, but the abstract and Section 3.2.3 present the M2 results as accuracy. The solar Vesta comparison is an external anchor, but its default credible intervals are broad (Table 5: Teff = 5779+372/-304 K, log(g) = 4.51+0.40/-0.33, [M/H] = -0.03+/-0.15, [alpha/M] = 0.00+0.16/-0.14). Please either reword the accuracy claims as "internal recovery within the MARCS grid" or add independent validation such as a second synthetic grid (PHOENIX, BT-Settl, or BOSZ) or a set of benchmark stars with interferometric or asteroseismic parameters.
  3. [Section 4.3.2, Table 5, Equation (10)] The second row of Table 5, with much tighter quoted intervals (Teff = 5780+55/-51 K, log(g) = 4.44+0.06/-0.03, [M/H] = -0.028+0.029/-0.033, [alpha/M] = -0.024+0.020/-0.025), is not the default tonalli result. It is the model selected by Equation (10) as minimizing a chi-square against the IAU solar values among the Appendix B2 models. This is a post hoc model-selection step that uses the same target values as the objective, so the resulting tight intervals quantify proximity to the chosen solar priors rather than independent precision. I advise presenting Equation (10) explicitly as a calibration or selection diagnostic and avoiding the selected row as evidence of nominal accuracy in the abstract or conclusions.
  4. [Section 3, Abstract] The abstract states that tonalli efficiently predicts rotational and radial velocities in addition to the four stellar parameters, but Section 3 contains no synthetic recovery test for v sin(i) or RV. The Vesta run reports v sin(i) values near the APOGEE-2 resolution limit and an RV close to the header value, but these are not accuracy tests. Either add synthetic experiments that vary v sin(i) and RV and report their biases, or remove them from the abstract-level claim of predicted parameters.
minor comments (4)
  1. [Figure 2 caption and Section 2.3.3] The text defines label 2 as emission-line stars, while the Figure 2 caption refers to "label 3 (emission line) stars"; this inconsistency should be corrected.
  2. [Section 1 and Section 3.2] The introduction states the method is applicable for 3.0 <= log(g) <= 6.0, but the Section 3.2 synthetic grid uses log(g) = 3, 4, and 5 dex only. Either extend the validation to log(g) = 6 or adjust the stated gravity range.
  3. [Section 2.3.7 and Appendix B] The adopted zero-generation population is given as N0 = 240 in Section 2.3.7 and Table 3, while Appendix B1 states N0 = 250 and refers to "our adopted input parameters N0 = 240"; the two values should be reconciled.
  4. [Section 2.3.3] The text mentions the option "weigthts" in KNeighborsClassifier; this is a typo for "weights".

Circularity Check

3 steps flagged · score 6.0 of 10

Partial circularity: the refined solar parameters are selected to match IAU values, and the synthetic validation is a same-library closed-box test that never varies [M/H] or [alpha/M].

  1. fitted input called prediction [Section 4.3.2, Eq. (10), Table 5 note d]
    "The second row of Table 5 presents the model with the minimum chi-squared of all the models computed in Appendix B2 with IQR <= Delta-X/2 ... where we adopt the solar values of the IAU definition (Prsa et al. 2016), and the subscript m refers to the tonalli model."

    The 'refined' tonalli solar parameters (Teff = 5780 K, log(g) = 4.44, [M/H] = -0.028, [alpha/M] = -0.024) are the single model among 108 hyperparameter combinations whose median lies closest to the IAU solar values, as measured by Eq. (10). Agreement with the IAU scale is therefore a selection criterion, not an independent prediction. Presenting this row as a tonalli spectroscopic determination in Table 5 means the agreement with the reference values is forced by construction rather than demonstrated.

  2. self definitional [Section 3.2, Section 3.2.3]
    "For this, we select synthetic spectra with zero metal and alpha-elements abundances, effective temperatures of 3000-4000 K (in steps of 100 K) and 4250-7000 K (in steps of 250 K), and log(g) of 3, 4, 5 dex, resulting in a set of 66 synthetic stars."

    The accuracy test feeds tonalli synthetic spectra drawn from the same continuum-normalised MARCS library against which tonalli fits. The 'true' parameters are grid labels of the fitting library itself, so the reported biases measure interpolation and optimizer convergence, not the fidelity of MARCS to real stellar spectra. The claim that 'tonalli can recover with success the four stellar parameters in this experiment for stars with effective temperatures between ~3200 and ~6250 K' is a self-consistency check within one model family; moreover, [M/H] and [alpha/M] are never varied from zero, so the recovery of those two parameters is only tested at a single point.

1 more flagged steps
  1. fitted input called prediction [Appendix B1, Section 2.3.7, Section 4]
    "The following experiments were conducted with the APOGEE-2 DR17 solar spectrum reflected by Vesta as our sample spectrum. ... From the experiments carried out and detailed in Appendix B, we suggest the following values for the input parameters ... N0 = 240, Np = 10, p = 0.4."

    The adopted AGA hyperparameters are selected by scanning 108 configurations on the Vesta spectrum and comparing the resulting parameters with the IAU solar values (X_sun - X_tonalli). Section 4 then validates tonalli on the same Vesta spectrum using those hyperparameters. The solar 'validation' is therefore performed on the tuning set; part of the agreement with solar reference values is inherited from the hyperparameter search rather than being an out-of-sample test.

full rationale

The paper is transparent about the model-dependent nature of its results and provides a genuinely implemented algorithm, Monte Carlo uncertainty estimation, and a public continuum-normalised library; those components are not circular. However, several load-bearing claims reduce partially to their own inputs. First, the refined solar parameters in Table 5 row 2 are selected as the model minimizing Eq. (10), a chi-squared distance to the IAU solar values, so their agreement with those values is a selection artifact. Second, the central accuracy and operating-range claims for 3200-6250 K rest on synthetic spectra drawn from the same MARCS library used for fitting; this validates internal consistency of the optimizer and interpolator, not model truth, and the test never varies [M/H] or [alpha/M] away from zero despite claiming recovery of all four stellar parameters. Third, the AGA hyperparameters are tuned on the same Vesta spectrum later used as the external solar validation, further reducing the independence of that anchor. These issues do not make the code useless, but the headline 'accuracy and precision' statements are partly circular, supporting a score of 6 rather than a fully clean result.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the MARCS library's fidelity, on the common continuum-normalisation scheme, and on the AGA plus interpolation pipeline. The free parameters are algorithm controls and a fixed limb-darkening value, tuned in Appendix B. No new physical entities are introduced.

free parameters (6)
  • Convergence factor p = 0.4 (0.8 for refined solar model)
    Controls shrinking of search hyper-volume; chosen from Appendix B experiments to balance accuracy and runtime.
  • Population size N0 = 240 (500 for refined solar model)
    Initial random individuals; tuned to ensure convergence within reasonable runtime.
  • Number of parents Np = 10 (10 for refined solar model; 6 used in an intermediate test)
    Number of best individuals kept per generation; chosen via Appendix B.
  • Interpolation stencil Ninterpol = 16 (coarse), 36-64 (fine)
    Number of nearest library spectra used in scipy griddata interpolation; affects precision and runtime.
  • Limb-darkening parameter epsilon = 0.4 fixed; 0.25 for the Sun
    Set by hand from literature (Gilhool et al. 2017; Jonsson et al. 2020), not fitted.
  • Sigma-clipping thresholds = delta_l=1.2, delta_u=3, box width 15
    Hand-set parameters of the continuum normalisation routine.
assumptions (5)
  • domain assumption MARCS synthetic spectra are accurate representations of real stellar H-band spectra for the target parameter range.
    The entire fitting and all parameter values are relative to the MARCS grid (Section 2.3.1); the paper explicitly notes results are model-dependent.
  • domain assumption scipy griddata linear interpolation in the 4D parameter space yields spectra sufficiently close to true library spectra for chi-square minimisation to locate the right parameters.
    Interpolation is central to every fitness evaluation (Section 2.2); the validation in Section 3 only checks consistency with the same library.
  • domain assumption The asexual genetic algorithm, with the adopted stopping criterion (Delta T <= 1 K), converges to the global chi-square minimum or a representative neighbourhood when repeated.
    AGA convergence is asserted from Canto et al. (2009) and verified empirically through repetitions, not proved for this pipeline (Sections 2.1, 2.3.6, 3.1).
  • domain assumption The observed and synthetic spectra can be continuum-normalised to a common pseudo-continuum using the same iterative sigma-clipping polynomial procedure.
    This assumption is load-bearing for the FOM (eq. 1); the paper itself restricts wavelength ranges and reports normalisation breakdowns above about 6250-6750 K (Section 2.3.2, Table 3).
  • domain assumption The k-NN equivalent-width classifier reliably separates low-temperature from high-temperature or emission-line stars so that the Teff search range can be restricted.
    Trained on 327 stars from Pleiades and W3/4/5 with partly incomplete spectral types (Section 2.3.3); misclassification would send the search outside the valid grid.

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Cite this review

Pith. "Pith review of tonalli: an asexual genetic code to characterise APOGEE-2 stellar spectra. I. Validation with synthetic and solar spectra." pith.science (2026). https://pith.science/paper/6NLH5MYV

@misc{pith2026241115342,
  author       = {Pith},
  title        = {Pith review of: tonalli: an asexual genetic code to characterise APOGEE-2 stellar spectra. I. Validation with synthetic and solar spectra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6NLH5MYV}},
  note         = {Machine review of arXiv:2411.15342}
}
abstract

We present tonalli, a spectroscopic analysis python code that efficiently predicts effective temperature, stellar surface gravity, metallicity, $\alpha$-element abundance, and rotational and radial velocities for stars with effective temperatures between 3200 and 6250 K, observed with the Apache Point Observatory Galactic Evolution Experiment 2 (APOGEE-2). tonalli implements an asexual genetic algorithm to optimise the finding of the best comparison between a target spectrum and the continuum-normalised synthetic spectra library from the Model Atmospheres with a Radiative and Convective Scheme (MARCS), which is interpolated in each generation. Using simulated observed spectra and the APOGEE-2 solar spectrum of Vesta, we study the performance, limitations, accuracy and precision of our tool. Finally, a Monte Carlo realisation was implemented to estimate the uncertainties of each derived stellar parameter. The ad hoc continuum-normalised library is publicly available on Zenodo (DOI 10.5281/zenodo.12736546).

Figures

Figures reproduced from arXiv: 2411.15342 by the authors.

Figure 1
Figure 1. Continuum normalised MARCS spectra (top panel: black line: continuum normalised flux) and residuals (bottom panel, gray line: residuals) for the APOGEE-2 blue chip wavelength region. The physical parameters of the star are listed above the spectrum: metallicity/𝛼-elements abundance/log(g) (dex) /effective temperature (in K). The telluric absorption lines (grey vertical regions) and the wavelength regions at the end … view at source ↗
Figure 2
Figure 2. Observed equivalent widths 𝑊(Mg i,), 𝑊(Al i,), and 𝑊(CO) of the stars comprising the machine-learning classifier training set implemented in tonalli; blue squares: label 0 (late type) stars; green triangles: label 1 (early type) stars; pale yellow circles: label 3 (emission line) stars. Top-left panel: correlation between the Al-a and Mg i equivalent widths. Top-right panel: correlation between the Al-a and CO equiv… view at source ↗
Figure 3
Figure 3. Flowchart of the code tonalli. Left: the optimisation begins with the preparation of the observed spectrum, and then a coarse best-fitting model is found by the comparison of randomly generated synthetic spectra with known parameters with the observed spectrum. Bottom right: the hyper-volume centred at the coarse best-fitting model parameters serves as a new search region. The optimisation is repeated with new param… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Histograms of the 500 best-fitting model parameters for the syn￾thetic star with [M/H] = 0, [𝛼/M] = 0, log(g) = 4.0 dex and Teff = 5500 K, obtained by tonalli. Vertical lines depict the true value (dotted line), the mean of the distribution (solid line), and the mode (…
Figure 5
Figure 5. Figure 5: Panels (a)-(d): Population mean (black solid line) of the difference between the best-fitting parameter, ⟨𝑋⟩𝑡 , and the true value, 𝑋∗, as function of the number of repeats, and Panel (e): Mean RMSE (black solid line) as function of the number of repeats, for the synth…
Figure 6
Figure 6. Figure 6: Recovery of the synthetic stars parameters: heat maps of the bias 𝐵 of tonalli. The shades of blue represent the interval where the median of the difference |𝑋∗ − 𝑋ton |, 𝐵𝑥, lies. The intervals, in fractions of the MARCS grid steps Δ𝑥, are shown in the colour bars. Li…
Figure 7
Figure 7. Figure 7: Recovery of the synthetic stars parameters: heat maps of the precision of tonalli, for synthetic models with log(g)∗ = 4. The shades of red represent the IQR of the distribution of each Monte Carlo realisation lies. The intervals, in fractions of the MARCS grid steps Δ…
Figure 8
Figure 8. Figure 8: Ridge plots showing the PDF of the parameters for a set of selected tonalli Monte Carlo realisations (left label in each PDF of Teff), for the APOGEE-2 Solar spectrum reflected by Vesta, with the radial velocity of the spectrum restricted to be 0. The solid line repres…
Figure 9
Figure 9. Figure 9: Ridge plots showing the PDF of the parameters for a set of selected tonalli Monte Carlo realisations (left label in each PDF of Teff), for the APOGEE-2 Solar spectrum reflected by Vesta, with RV optimised. See [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Variation of the mean value of the mean (blue solid line), median (magenta solid line), and mode (yellow solid line) distributions of the parameters Teff (upper left panel), log(g) (upper right panel, [M/H] (bottom left panel), and [𝛼/M] (bottom right panel), as funct…
Figure 11
Figure 11. Figure 11: Variation of the mean value of the mean (blue solid line), median (magenta solid line), and mode (yellow solid line) distributions of the parameters Teff (upper left panel), log(g) (upper right panel, [M/H] (bottom left panel), and [𝛼/M] (bottom right panel), as funct…
Figure 12
Figure 12. Figure 12: Corner plot of the resultant total population of the Monte Carlo realisations in tonalli for the APOGEE-2 Solar spectrum reflected by Vesta, assuming the radial velocity of the spectrum is RV = 0, and for 𝑁 = 50 realisations. The contour lines encloses the population …
Figure 13
Figure 13. Figure 13: Corner plot of the resultant total population of the Monte Carlo realisations in tonalli for the APOGEE-2 solar spectrum reflected by Vesta, assuming the radial velocity of the spectrum is optimised, for 𝑁 = 50 realisations. See [PITH_FULL_IMAGE:figures/full_fig_p022…
Figure 14
Figure 14. Figure 14: Comparison of the diverse statistics figures (1D: mean, median and binned mode; 4D: mean of the multivariate normal distribution, MND, indicated in the titles of each column) obtained by tonalli with the ASPCAP results (Synspec, spectroscopic parameters). Top panels: …

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Pith tools

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