Symbolically Regressing Fish Biomass Spectral Data: A Linear Genetic Programming Method with Tunable Primitives

Bing Xue; Daniel P. Killeen; Jeremy S. Ronney; Keith C. Gordon; Mengjie Zhang; Zhixing Huang

REVIEW 5 major objections 3 minor 30 references

Symbolically Regressing Fish Biomass Spectral Data: A Linear Genetic Programming Method with Tunable Primitives

T0 review · 5 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Adding tunable coefficients to genetic programming yields compact models that beat six regressors on fish biomass spectra.

desk verdict A genuine, useful extension of LGP with tunable primitives, but the significance testing on the fish data is anticonservative and needs a cluster-level correction before the empirical claims can be trusted. read the letter →

arxiv 2505.21901 v1 pith:VD2FQVPU submitted 2025-05-28 cs.NE

Zhixing Huang , Bing Xue , Mengjie Zhang , Jeremy S. Ronney , Keith C. Gordon , Daniel P. Killeen This is my paper · ORCID

classification cs.NE

keywords lineargeneticprogrammingtunableprimitivessymbolicregressionRamanspectroscopyfishbiomasspredictionspectraldataanalysisinterpretableSRBench

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 tries to establish that a linear genetic programming method with tunable primitives (LGP-TP) is a practical, interpretable alternative for predicting fish biomass composition from small, noisy Raman spectral datasets. It frames the task as symbolic regression: evolution searches over short register-based programs while newly added tunable primitives fit their internal coefficients to the data. Across ten biomass targets measured from 39 fish, LGP-TP reports the best overall mean rank of test $R^2$ among seven methods, including a fish-specific neural network, and the ablation study claims the tunable primitives drive the gain, with LGP-TP significantly outperforming basic LGP ($p=0.019$). If the paper is right, compact human-readable formulas could replace black-box models for spectral quality control while highlighting which spectral peaks actually carry the signal.

What carries the argument

The load-bearing machinery is the tunable primitive: a function $T(X,W,\cdot)$ that takes immediate input values $X$, adjustable coefficients $W$, and, for terminals, two feature indices $\alpha,\beta$, and returns a vector meant to approximate the target $Y$. Three kinds are added to a linear genetic programming register machine: tunable terminals that fit linear regression or derived spectral statistics (average, standard deviation, fluctuation, slope, peak, valley, peak location) over a contiguous feature range $x_{\alpha:\beta}$; tunable functions that fit residual error of intermediate register values with linear, sine, exponential, or power forms; and a multivariate linear regression function appended at the end that maps several register values to the single output. Coefficients are tuned by least squares for the linear primitives and by normalized gradient descent for the trigonometric, exponential, and power primitives, so genetic search handles symbolic structure while local optimizers set the numbers.

What would settle it

Re-run the comparison with six-fold cross-validation in which each fish's three repeated spectra are explicitly assigned to the same fold, then compare mean test $R^2$ and the Friedman/Wilcoxon ranks. If LGP-TP no longer holds the best overall rank or the significant win counts reported in Fig. 2, the empirical claim fails. A second check is to run LGP-TP without the final multivariate linear regression function and see whether the mean-rank gain from LGP-TTF to LGP-TP disappears.

Watch

Extended reading notes

Core claim

The paper's central claim is that giving linear genetic programming three kinds of tunable primitives—tunable terminals that fit linear or shape-based features over a wavenumber range, tunable functions that fit residual error at intermediate program points, and a final multivariate linear regression over registers—lets evolution synthesize compact symbolic regressors that predict ten fish biomass components from Raman spectra better than PLSR, KNN, MLP, XGBoost, random forests, and FishCNN. On the paper's own terms, LGP-TP has the best overall mean rank (5.45) in ten runs of six-fold cross-validation, is statistically better than each compared method on at least four of the ten targets, and grows consistently in performance as the three primitive types are added one by one. The same setup transfers to three other spectral preprocessing treatments and, after removing spectral-specific terminals, ranks second in test $R^2$ among the 24 benchmark methods on the SRBench symbolic-regression benchmark while maintaining among the smallest model sizes and shortest training times.

Load-bearing premise

The reported test scores assume that the six cross-validation folds are split by fish rather than by individual measurement, so that spectra from the same fish never appear in both training and test sets; the paper never explicitly states this.

Editorial extensions

If this is right

Fish production lines could estimate water, protein, fat, and seven other biomass components from Raman spectra with compact formulas instead of black-box models, across several spectral preprocessing protocols.
The frequency heat maps identify the first three and the last peaks of the 1891.58–580.109 cm$^{-1}$ range as the informative regions, giving chemists specific wavenumbers to investigate.
The ablation result implies that each of the three tunable-primitive types contributes to the gain, with the final multivariate linear regression function producing the largest jump in mean rank.
The SRBench result suggests the tunable-primitive strategy generalizes beyond spectroscopy to tabular symbolic regression with competitive accuracy, smaller models, and faster training.
Because the synthesized programs are compact and their coefficients are explicit, a domain scientist can read the model and reason about whether the highlighted features make chemical sense.

Reading between the lines

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

Editorial inference: the performance advantage on small noisy datasets may come less from the evolutionary search itself than from the built-in least-squares coefficient fitting, which gives every candidate program a strong local head start; a testable extension would compare LGP-TP against a version with only random coefficients.
Editorial inference: the paper never explicitly states that the six-fold split groups the three repeated measurements of each fish into the same fold, and the reported test $R^2$ would be optimistically biased if spectra from the same fish leaked across training and test folds; reporting the exact split procedure would settle this.
Editorial inference: the same tunable-primitive recipe could be dropped into other high-dimensional, small-sample spectral domains such as soil, food, or pharmaceutical analysis, where interpretability is as valuable as accuracy.
Editorial inference: the shared heat-map pattern across water, protein, and lipids yield suggests one multi-output model might exploit the same spectral peaks for all three targets; an explicit check is whether such a model retains the per-target accuracy of LGP-TP.

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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

Short version: this is a legitimate extension of linear genetic programming with a genuinely new set of tunable primitives, and the empirical story on the fish data is mostly believable. But the statistics are sloppier than the paper lets on, and a couple of references are wrong. It deserves a serious referee, not a desk reject.

What's new: the three kinds of tunable primitives—window-based linear terminals, residual-approximating functions, and a final multivariate linear regression—are a real combination not present in the cited LGP baseline. The ablation study shows monotone improvement from basic LGP to LGP-TT to LGP-TTF to LGP-TP, which is good evidence that the proposed pieces are doing work. The SRBench result, where LGP-TP ranks second in test R2 while staying small and fast, is impressive and strengthens the generality claim. The interpretability analysis, including the feature-frequency heat maps and the compact program sizes, is a genuine asset for the chemometrics audience.

Soft spots, in proportion. First, the ten repeated six-fold CV runs are treated as independent samples in the Friedman and Wilcoxon tests. They are not independent: the same 39 fish recur in the test folds across runs, so the reported p-values, including the ablation p=0.019, are anticonservative. This is not a fatal flaw, but it means the "significantly better" claims are less secure than presented. A cluster-level or fish-level analysis is needed. Second, the multiple-testing correction is ambiguous: the paper says "with the Bonferroni correction" in one sentence, but it is not clear whether the pairwise Wilcoxon comparisons across ten targets and seven methods are adjusted at all, and the reported threshold p<0.0162 appears without derivation. Third, no code or pseudocode is provided, and the fish data are commercially restricted, so the main experimental result is not directly reproducible. Fourth, the reference list has at least one clear error: reference [4] is about nuclear shapes in 62Cr, unrelated to fish biomass, and reference [11] is about krill oil adulteration rather than SVR on fish spectra. These are minor but should be fixed.

I do not share the reader's worry that spectra from the same fish leak across train and test folds. The text says each fold has "6 or 7 fish samples," which strongly suggests fish-level grouping. The pseudoreplication issue is the real concern, and it is distinct.

Who is this for? Applied GP researchers and spectral chemometrics practitioners. It is a solid applied paper with a novel method and rich empirical comparison. Send it to peer review, but the reviewers should require a cluster-aware significance test and clearer reporting of the multiple-testing procedure before acceptance.

Referee Report

5 major / 3 minor

Summary. The paper proposes LGP-TP, a linear genetic programming method with three kinds of tunable primitives (tunable terminals, tunable functions, and a multivariate linear regression function) for symbolic regression on fish biomass Raman spectra. The method is evaluated on ten biomass targets from 117 spectra of 39 fish using ten repeated six-fold cross-validation runs, and compared against six baselines, reporting test R2 values. The paper claims LGP-TP achieves the best overall prediction performance, with compact and interpretable models, and that the tunable primitives are responsible for significant improvements over basic LGP. Generality is further tested on three additional spectral data treatments and on the SRBench symbolic regression benchmark.

Significance. If the central claim holds, LGP-TP offers a practical, interpretable alternative for spectral regression with small, noisy datasets, and the tunable-primitive idea could transfer to other symbolic regression tasks. Strengths of the paper include the use of held-out test folds, a real-world spectral dataset, an ablation study, and an extension to a public benchmark with 122 problems. The authors provide a clear description of the algorithm and make the fish data available on reasonable request. However, the statistical inference supporting the superiority claims is weakened by pseudoreplication in the repeated cross-validation protocol, and several details of the method and comparison require clarification.

major comments (5)

[Section 2.3] The Wilcoxon and Friedman tests in Section 2.3 treat the ten repeated six-fold cross-validation runs as ten independent observations. Because each run re-splits the same 39 fish (117 spectra), and the three measurements of each fish are kept together as implied by Section 2.1, the same fish appears in test folds multiple times across runs, so the R2 values entering the tests are correlated. This pseudoreplication makes the reported p-values (e.g., the ablation p=0.019 and the per-target p<0.0162 thresholds) anticonservative. The authors should re-analyze with a clustering-aware procedure (e.g., cluster bootstrap or permutation at the fish level), or justify why treating the ten runs as independent is valid.
[Section 2.1] The cross-validation splitting is not fully specified: the text says each fold has approximately 20 instances (6 or 7 fish samples) but never states explicitly that the three repeated measurements of each fish are always placed in the same fold. If instances are split at the measurement level, spectra from the same fish would leak across training and test folds and inflate test R2. Please state explicitly that no fish appears in both training and test sets in any fold and describe how the folds are constructed.
[Section 2.3] The description of the statistical adjustments is ambiguous: the sentence 'Friedman's test with a significance level of 0.05 and with the Bonferroni correction and the Wilcoxon rank-sum test with a significance level of 0.05' does not say which family of comparisons the Bonferroni correction applies to, and the threshold p<0.0162 used for per-target comparisons is not derived. Please specify the number of comparisons, the method of correction, and whether the pairwise Wilcoxon tests are adjusted across the ten targets, across the seven methods, or both.
[Section 2.2, Table 1] The baseline methods KNN, MLP, XGB, and RF use default hyperparameters, whereas PLSR and FishCNN use recommended settings. Because the central claim is that LGP-TP achieves the best overall prediction performance, a comparison using untuned defaults for four of the six baselines may not reflect their true capabilities. The authors should either tune all baselines on the training folds (with nested cross-validation) or clearly state that the comparison is against default configurations and discuss the risk that tuned baselines would close the gap.
[Section 4.2.2] The derivatives for the tunable functions contain an error: for ExpoRF, the derivative with respect to omega_1 includes the term (omega_1^2 + 1) x_j^{omega_1 - 1}, which is not the derivative of omega_0 + (omega_1^2 + 1) x_j + x_j with respect to omega_1; the correct derivative is 2 omega_1 x_j. Since the gradient descent update is used to tune these primitives, the method as described is not reproducible. Please correct the equation and, if the implementation differs, state the exact update used.

minor comments (3)

[Section 2.1] The phrase 'wavenumber range of 1891.58-580.109 (cm−1)' lists the upper bound first, which is easy to misread; consider reordering to 580.109-1891.58 (cm−1). Also, Section 2.6 contains a typo '1064num' that should read '1064nm'.
[Section 4.2.1] The definition of Fluctuate has a typo: it reads sum_{i=alpha+1}^{beta} |(x_j)_i - (x_j)_i|, likely intended as |(x_j)_i - (x_j)_{i-1}|, since otherwise the term is identically zero. Please correct the formula.
[Section 2.7] The model size for LGP-TP in the SRBench comparison is computed by multiplying the number of effective instructions by 2.0, but it is not clear how this factor is chosen or how it aligns with the model-size metric used for other benchmark methods. Please clarify to make the Pareto plot in Figure 5b interpretable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central claim rests on out-of-sample cross-validation and an independent ablation, not on definitional equivalence or self-citation.

full rationale

The paper's central claim is empirical: LGP-TP predicts ten fish biomass targets and is compared with six baselines under six-fold cross-validation. All tunable-primitive parameters are fitted on training folds (Eqs. 1-3 and the least-squares solution for MVLR in Section 4.2) and the reported R2 values are computed on held-out test folds. No test target value is used to fit any coefficient, so the reported predictions do not reduce by construction to a fitted value. The ablation study compares basic LGP with crippled variants and LGP-TP under the same CV protocol; this is a genuine controlled comparison rather than a renamed fit. The paper does cite prior work by the same authors (Refs. 13, 16, 18), but those citations are used for contextual statements such as LGP having promising learning ability or for parameter settings; they are not load-bearing for the superiority claim, which is supported by the experiments. The reader-raised concern about fold grouping and the skeptic's pseudoreplication point are statistical validity or correctness risks, not circularity: even if the p-values are anticonservative, the test R2 values themselves are still out-of-sample estimates, and the claimed reduction of a prediction to its own input is absent. Therefore no circular step is identified.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The method is an empirical machine-learning system; its central claim is supported by experimental evaluation rather than derivation. The listed free parameters are algorithm hyperparameters and fitted primitive coefficients. The axioms are standard assumptions about data, preprocessing, and numerical stability. No new physical entities are introduced.

free parameters (6)

LGP hyperparameters (fish data) = popsize=250, n_gen=100, max_programsize=50, n_registers=30
Set in Table 1. These hyperparameters were chosen by the authors and affect the search and final model complexity.
LGP hyperparameters (SRBench) = popsize=500, n_gen=200, max_programsize=50, n_registers=8
Section 2.7. Chosen for the SRBench experiments; affects performance.
Gradient descent step size lambda = 0.1
Section 4.2.2, used to tune SinRF, ExpoRF, PowRF parameters.
Tunable parameter bounds = [-3, 3]
Section 4.2.2, limits the optimized coefficients in tunable functions.
Feature window cap = 50% of features for fish, 10% for SRBench
Section 4.2.1, restricts the [alpha..beta] range for tunable terminals.
Fitted primitive coefficients W = varies per individual and run
Tunable terminals/functions/MVLR coefficients are fitted to training data via least squares or gradient descent; they are central to the model's predictive performance.

assumptions (4)

domain assumption The design matrices in the closed-form least-squares solutions (Eqs. 2-4) are full rank and numerically invertible after data augmentation.
The paper uses the normal equations without regularization; with highly collinear spectral features this may be violated, though augmentation to ~5000 samples helps.
domain assumption The six-fold split keeps all three measurements of a fish in the same fold, so training and test sets do not share fish.
Section 2.1 says each fold has 6-7 fish samples, but the split is not explicitly described as fish-level; if it is measurement-level, results are optimistically biased.
domain assumption Data augmentation (spectral augmentation, mixup, Gaussian noise) preserves the true spectrum-to-biomass relationship.
Section 2.1: 50x augmented training data; if augmentation distorts chemistry, test performance could be misleading.
ad hoc to paper Standard LGP evolutionary search with the given primitive set and program-size limits can reach competitive models within 100 generations.
Section 4: the method's success depends on the chosen search space and settings, which are not derived from theory.

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Cite this review

Pith. "Pith review of Symbolically Regressing Fish Biomass Spectral Data: A Linear Genetic Programming Method with Tunable Primitives." pith.science (2026). https://pith.science/paper/VD2FQVPU

@misc{pith2026250521901,
  author       = {Pith},
  title        = {Pith review of: Symbolically Regressing Fish Biomass Spectral Data: A Linear Genetic Programming Method with Tunable Primitives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VD2FQVPU}},
  note         = {Machine review of arXiv:2505.21901}
}

read the original abstract

Machine learning techniques play an important role in analyzing spectral data. The spectral data of fish biomass is useful in fish production, as it carries many important chemistry properties of fish meat. However, it is challenging for existing machine learning techniques to comprehensively discover hidden patterns from fish biomass spectral data since the spectral data often have a lot of noises while the training data are quite limited. To better analyze fish biomass spectral data, this paper models it as a symbolic regression problem and solves it by a linear genetic programming method with newly proposed tunable primitives. In the symbolic regression problem, linear genetic programming automatically synthesizes regression models based on the given primitives and training data. The tunable primitives further improve the approximation ability of the regression models by tuning their inherent coefficients. Our empirical results over ten fish biomass targets show that the proposed method improves the overall performance of fish biomass composition prediction. The synthesized regression models are compact and have good interpretability, which allow us to highlight useful features over the spectrum. Our further investigation also verifies the good generality of the proposed method across various spectral data treatments and other symbolic regression problems.

Figures

Figures reproduced from arXiv: 2505.21901 by the authors.

**Figure 1.** Schematic diagram of applying LGP-TP for fish biomass prediction. Particularly, LGP-TP synthesizes regression models based on the spectral data (inputs) and the chemical ground truth of on-hand fish samples (target outputs). The synthesized regression model from LGP-TP predicts the biomass of unseen fish samples in real-world production. concise models for describing knowledge in the given data [17]. This relieves t… view at source ↗

**Figure 2.** [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Interpretability analysis of LGP-TP. (a) An example synthesized regression model by LGP-TP for predicting water in fish meat, outputting results from the first register R0. (b) Highlighted input features by the result models. The upper part of sub-figure b is the Raman spectra data with highlighted features in yellow. The three heat maps in the lower part show the frequency of input terminals over feature ranges (X-… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Training and test performance of LGP-TP and the general effectiveness across spectral data treatments. (a) The training and test R2 of LGP-TP in four example biomass targets. (b) Test R2 (X-axis) of the four compared methods for different data treatments over 10 six-fo…

**Figure 5.** Figure 5: Results of SRBench. (a) Comprehensive comparison of benchmark methods in terms of test R2 , model size, and training time. (b) Non-dominate sets over model size rank and test R2 rank. (c) Training time of benchmark methods over different numbers of training instances. …

**Figure 6.** Figure 6: Overview of LGP-TP. (a) shows an example LGP program, representing a formula of (x1+1) 2 x0 x1 . (b) is the training process of LGP-TP. The left yellow part is the evolutionary framework of LGP-TP. The right blue part details the production of the next-generation popul…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Gade, A. et al. In-beam spectroscopy reveals competing nuclear shapes in the rare isotope 62Cr. Nat. Phys. (2024)

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Ahmmed, F., Gordon, K. C., Killeen, D. P. & Fraser-Miller, S. J. Detection and Quantification of Adulteration in Krill Oil with Raman and Infrared Spectroscopic Methods. Molecules 28, 3695 (2023)

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