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Interpretability in symbolic regression: a benchmark of explanatory methods using the Feynman data set

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Abstract

In some situations, the interpretability of the machine learning models plays a role as important as the model accuracy. Interpretability comes from the need to trust the prediction model, verify some of its properties, or even enforce them to improve fairness. Many model-agnostic explanatory methods exists to provide explanations for black-box models. In the regression task, the practitioner can use white-boxes or gray-boxes models to achieve more interpretable results, which is the case of symbolic regression. When using an explanatory method, and since interpretability lacks a rigorous definition, there is a need to evaluate and compare the quality and different explainers. This paper proposes a benchmark scheme to evaluate explanatory methods to explain regression models, mainly symbolic regression models. Experiments were performed using 100 physics equations with different interpretable and non-interpretable regression methods and popular explanation methods, evaluating the performance of the explainers performance with several explanation measures. In addition, we further analyzed four benchmarks from the GP community. The results have shown that Symbolic Regression models can be an interesting alternative to white-box and black-box models that is capable of returning accurate models with appropriate explanations. Regarding the explainers, we observed that Partial Effects and SHAP were the most robust explanation models, with Integrated Gradients being unstable only with tree-based models. This benchmark is publicly available for further experiments.

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Notes

  1. Open source module available at https://github.com/gAldeia/iirsBenchmark.

  2. Also called Marginal Effects in the literature, but this term can be misleading, as mentioned in [80].

  3. https://github.com/gAldeia/itea-python.

  4. https://github.com/heal-research/operon.

  5. https://github.com/scikit-learn/scikit-learn.

References

  1. M. Medvedeva, M. Vols, M. Wieling, Using machine learning to predict decisions of the European Court of Human Rights. Artif. Intell. Law 28(2), 237–266 (2020). https://doi.org/10.1007/s10506-019-09255-y

    Article  Google Scholar 

  2. G. Winter, Machine learning in healthcare A review. Br. J. Health Care Manag. 25(2), 100–101 (2019). https://doi.org/10.12968/bjhc.2019.25.2.100

    Article  Google Scholar 

  3. R. Roscher, B. Bohn, M.F. Duarte, J. Garcke, Explainable machine learning for scientific insights and discoveries. IEEE Access 8, 42200–42216 (2020). https://doi.org/10.1109/ACCESS.2020.2976199arXiv:1905.08883

    Article  Google Scholar 

  4. C. Modarres, M. Ibrahim, M. Louie, J. Paisley, Towards explainable deep learning for credit lending: a case study, 1–8 (2018) arXiv:1811.06471

  5. S. Yoo, X. Xie, F.-C. Kuo, T.Y. Chen, M. Harman, Human competitiveness of genetic programming in spectrum-based fault localisation: theoretical and empirical analysis. ACM Trans. Softw. Eng. Methodol. (2017). https://doi.org/10.1145/3078840

    Article  Google Scholar 

  6. M.A. Lones, J.E. Alty, J. Cosgrove, P. Duggan-Carter, S. Jamieson, R.F. Naylor, A.J. Turner, S.L. Smith, A new evolutionary algorithm-based home monitoring device for parkinson’s dyskinesia. J. Med. Syst. (2017). https://doi.org/10.1007/s10916-017-0811-7

    Article  Google Scholar 

  7. D. Lynch, M. Fenton, D. Fagan, S. Kucera, H. Claussen, M. O’Neill, Automated self-optimization in heterogeneous wireless communications networks. IEEE/ACM Trans. Netw. 27(1), 419–432 (2019). https://doi.org/10.1109/TNET.2018.2890547

    Article  Google Scholar 

  8. D. Izzo, L.F. Simões, M. Märtens, G.C.H.E. de Croon, A. Heritier, C.H. Yam, Search for a grand tour of the jupiter galilean moons. In: proceedings of the 15th annual conference on genetic and evolutionary computation. GECCO ’13, pp. 1301–1308. Association for Computing Machinery, New York, NY, USA (2013). https://doi.org/10.1145/2463372.2463524

  9. Y. Semet, B. Berthelot, T. Glais, C. Isbérie, A. Varest, Expert competitive traffic light optimization with evolutionary algorithms. In: VEHITS, pp. 199–210 (2019)

  10. M. Fernández-Delgado, E. Cernadas, S. Barro, D. Amorim, Do we need hundreds of classifiers to solve real world classification problems? J. Mach. Learn. Res. 15(90), 3133–3181 (2014)

    MathSciNet  MATH  Google Scholar 

  11. R. Guidotti, A. Monreale, F. Giannotti, D. Pedreschi, S. Ruggieri, F. Turini, Factual and counterfactual explanations for black box decision making. IEEE Intell. Syst. 34(6), 14–23 (2019). https://doi.org/10.1109/MIS.2019.2957223

    Article  Google Scholar 

  12. S.M. Lundberg, S.-I. Lee, A unified approach to interpreting model predictions. In: proceedings of the 31st international conference on neural information processing systems. NIPS’17, pp. 4768–4777. Curran Associates Inc., Red Hook, NY, USA (2017)

  13. A. Adadi, M. Berrada, Peeking inside the black-box: a survey on explainable artificial intelligence (XAI). IEEE Access 6, 52138–52160 (2018). https://doi.org/10.1109/ACCESS.2018.2870052

    Article  Google Scholar 

  14. G.S.I. Aldeia, F.O. de França, Measuring feature importance of symbolic regression models using partial effects. In: proceedings of the genetic and evolutionary computation conference. GECCO ’21. ACM, New York, NY, USA (2021). https://doi.org/10.1145/3449639.3459302

  15. P. Orzechowski, W.L. Cava, J.H. Moore, Where are we now?: A large benchmark study of recent symbolic regression methods. In: proceedings of the genetic and evolutionary computation conference. GECCO ’18, pp. 1183–1190. ACM, New York, NY, USA (2018). https://doi.org/10.1145/3205455.3205539

  16. W.L. Cava, P. Orzechowski, B. Burlacu, F.O. de França, M. Virgolin, Y. JIN, M. Kommenda, J.H. Moore, Contemporary symbolic regression methods and their relative performance. In: thirty-fifth conference on neural information processing systems datasets and benchmarks track (Round 1) (2021). https://openreview.net/forum?id=xVQMrDLyGst

  17. G. Kronberger, F.O. de França, B. Burlacu, C. Haider, M. Kommenda, Shape-constrained symbolic regression–improving extrapolation with prior knowledge. Evolutionary Computation, 1–24

  18. M. Affenzeller, S.M. Winkler, G. Kronberger, M. Kommenda, B. Burlacu, S. Wagner, Gaining deeper insights in symbolic regression, in Genetic Programming Theory and Practice XI. ed. by R. Riolo, J.H. Moore, M. Kotanchek (Springer, New York, NY, 2014), pp. 175–190. https://doi.org/10.1007/978-1-4939-0375-7_10

    Chapter  Google Scholar 

  19. F.O. de França, A greedy search tree heuristic for symbolic regression. Inf. Sci. 442–443, 18–32 (2018). https://doi.org/10.1016/j.ins.2018.02.040

    Article  MathSciNet  MATH  Google Scholar 

  20. L.A. Ferreira, F.G. Guimaraes, R. Silva, Applying genetic programming to improve interpretability in machine learning models. In: 2020 IEEE Congress on Evolutionary Computation (CEC). IEEE, New York (2020). https://doi.org/10.1109/cec48606.2020.9185620

  21. F.O. de França, G.S.I. Aldeia, Interaction-transformation evolutionary algorithm for symbolic regression. Evolut. Comput. 29(3), 367–390 (2021). https://doi.org/10.1162/evco_a_00285

    Article  Google Scholar 

  22. F.O. de França, M.Z. de Lima, Interaction-transformation symbolic regression with extreme learning machine. Neurocomputing 423, 609–619 (2021)

    Article  Google Scholar 

  23. D. Kantor, F.J. Von Zuben, F.O. de França, Simulated annealing for symbolic regression. In: proceedings of the genetic and evolutionary computation conference, pp. 592–599 (2021)

  24. R.M. Filho, A. Lacerda, G.L. Pappa, Explaining symbolic regression predictions. In: 2020 IEEE Congress on Evolutionary Computation (CEC). IEEE, New York (2020). https://doi.org/10.1109/cec48606.2020.9185683

  25. R. Guidotti, A. Monreale, S. Ruggieri, F. Turini, F. Giannotti, D. Pedreschi, A survey of methods for explaining black box models. ACM Comput. Surv. 51(5), 1–45 (2018). https://doi.org/10.1145/3236009arXiv:1802.01933

    Article  Google Scholar 

  26. L. Ljung, Perspectives on system identification. Ann. Rev. Control 34(1), 1–12 (2010). https://doi.org/10.1016/j.arcontrol.2009.12.001

    Article  Google Scholar 

  27. R. Marcinkevičs, J.E. Vogt, Interpretability and Explainability: A Machine Learning Zoo Mini-tour, 1–24 (2020) arXiv:2012.01805

  28. C. Rudin, Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Machine Intelligence 1(5), 206–215 (2019) arXiv:1811.10154. https://doi.org/10.1038/s42256-019-0048-x

  29. Z.F. Wu, J. Li, M.Y. Cai, Y. Lin, W.J. Zhang, On membership of black-box or white-box of artificial neural network models. In: 2016 IEEE 11th conference on industrial electronics and applications (ICIEA), pp. 1400–1404 (2016). https://doi.org/10.1109/ICIEA.2016.7603804

  30. O. Loyola-González, Black-box vs. white-box: understanding their advantages and weaknesses from a practical point of view. IEEE Access 7, 154096–154113 (2019). https://doi.org/10.1109/ACCESS.2019.2949286

    Article  Google Scholar 

  31. S.M. Julia Angwin, J. Larson, P. Lauren Kirchner, Machine bias: there’s software used across the country to predict future criminals. and it’s biased against blacks (2016)

  32. A. Datta, M.C. Tschantz, A. Datta, Automated experiments on ad privacy settings: A tale of opacity, choice, and discrimination. CoRR abs/1408.6491 (2014) arXiv:1408.6491

  33. Z.C. Lipton, The mythos of model interpretability: in machine learning, the concept of interpretability is both important and slippery. Queue 16(3), 31–57 (2018). https://doi.org/10.1145/3236386.3241340

    Article  Google Scholar 

  34. D.V. Carvalho, E.M. Pereira, J.S. Cardoso, Machine learning interpretability: a survey on methods and metrics. Electronics 8(8), 832 (2019). https://doi.org/10.3390/electronics8080832

    Article  Google Scholar 

  35. A.B. Arrieta, N. D íaz-Rodríguez, J.D. Ser, A. Bennetot, S. Tabik, A. Barbado, S. Garcia, S. Gil-Lopez, D. Molina, R. Benjamins, R. Chatila, F. Herrera, Explainable artificial intelligence (XAI): concepts, taxonomies, opportunities and challenges toward responsible AI. Information Fusion 58, 82–115 (2020). https://doi.org/10.1016/j.inffus.2019.12.012

  36. F. Doshi-Velez, B. Kim, Towards a rigorous science of interpretable machine learning. arXiv preprint arXiv:1702.08608 (2017)

  37. L.H. Gilpin, D. Bau, B.Z. Yuan, A. Bajwa, M. Specter, L. Kagal, Explaining explanations: an overview of interpretability of machine learning. Proceedings—2018 IEEE 5th international conference on data science and advanced analytics, DSAA 2018, 80–89 (2019) arXiv:1806.00069. https://doi.org/10.1109/DSAA.2018.00018

  38. M. Sendak, M.C. Elish, M. Gao, J. Futoma, W. Ratliff, M. Nichols, A. Bedoya, S. Balu, C. O’Brien, The human body is a black box. In: proceedings of the 2020 conference on fairness, accountability, and transparency. ACM, New York, NY, USA (2020). https://doi.org/10.1145/3351095.3372827

  39. M. Ghassemi, L. Oakden-Rayner, A.L. Beam, The false hope of current approaches to explainable artificial intelligence in health care. Lancet Digit. Health 3(11), 745–750 (2021). https://doi.org/10.1016/s2589-7500(21)00208-9

    Article  Google Scholar 

  40. I. Banerjee, A.R. Bhimireddy, J.L. Burns, L.A. Celi, L.-C. Chen, R. Correa, N. Dullerud, M. Ghassemi, S.-C. Huang, P.-C. Kuo, M.P. Lungren, L. Palmer, B.J. Price, S. Purkayastha, A. Pyrros, L. Oakden-Rayner, C. Okechukwu, L. Seyyed-Kalantari, H. Trivedi, R. Wang, Z. Zaiman, H. Zhang, J.W. Gichoya, Reading Race: AI Recognises Patient’s Racial Identity In Medical Images (2021)

  41. M. Yang, B. Kim, Benchmarking Attribution Methods with Relative Feature Importance (2019)

  42. O.-M. Camburu, E. Giunchiglia, J. Foerster, T. Lukasiewicz, P. Blunsom, The Struggles of Feature-Based Explanations: Shapley Values vs. Minimal Sufficient Subsets (2020)

  43. T. Laugel, M.-J. Lesot, C. Marsala, X. Renard, M. Detyniecki, The dangers of post-hoc interpretability: unjustified counterfactual explanations. In: proceedings of the twenty-eighth international joint conference on artificial intelligence, IJCAI-19, pp. 2801–2807. International joint conferences on artificial intelligence organization, California, USA (2019). https://doi.org/10.24963/ijcai.2019/388

  44. O. Camburu, E. Giunchiglia, J. Foerster, T. Lukasiewicz, P. Blunsom, Can I trust the explainer? verifying post-hoc explanatory methods. CoRR abs/1910.02065 (2019) arXiv:1910.02065

  45. D. Alvarez-Melis, T.S. Jaakkola, On the robustness of interpretability methods (Whi) (2018) arXiv:1806.08049

  46. G. Hooker, L. Mentch, Please stop permuting features: an explanation and alternatives, 1–15 (2019) arXiv:1905.03151

  47. M. Orcun Yalcin, X. Fan, On Evaluating Correctness of Explainable AI Algorithms: an Empirical Study on Local Explanations for Classification (April), 0–7 (2021)

  48. R. Caruana, Y. Lou, J. Gehrke, P. Koch, M. Sturm, N. Elhadad, Intelligible models for healthcare: predicting pneumonia risk and hospital 30-day readmission. In: proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining. KDD ’15, pp. 1721–1730. Association for Computing Machinery, New York, NY, USA (2015). https://doi.org/10.1145/2783258.2788613

  49. C. Molnar, G. König, J. Herbinger, T. Freiesleben, S. Dandl, C.A. Scholbeck, G. Casalicchio, Grosse-Wentrup, M., Bischl, B.: General pitfalls of model-agnostic interpretation methods for machine learning models (01) (2020) arXiv:2007.04131

  50. M. Yang, B. Kim, BIM: towards quantitative evaluation of interpretability methods with ground truth. CoRR abs/1907.09701 (2019) arXiv:1907.09701

  51. R. Guidotti, Evaluating local explanation methods on ground truth. Artif. Intell. 291, 103428 (2021). https://doi.org/10.1016/j.artint.2020.103428

    Article  MathSciNet  MATH  Google Scholar 

  52. S. Hooker, D. Erhan, P.-J. Kindermans, B. Kim, A benchmark for interpretability methods in deep neural networks. In: Wallach, H., Larochelle, H., Beygelzimer, A., d’ Alché-Buc, F., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 32, pp. 9737–9748. Curran Associates, Inc., Red Hook, NY, USA (2019). https://proceedings.neurips.cc/paper/2019/file/fe4b8556000d0f0cae99daa5c5c5a410-Paper.pdf

  53. J.W. Vaughan, H. Wallach, A human-centered agenda for intelligible machine learning (Getting Along with Artificial Intelligence, Machines We Trust, 2020)

  54. D.R. White, J. McDermott, M. Castelli, L. Manzoni, B.W. Goldman, G. Kronberger, W. Jaśkowski, U.-M. O’Reilly, S. Luke, Better GP benchmarks: community survey results and proposals. Genet. Program. Evolv. Mach. 14(1), 3–29 (2012). https://doi.org/10.1007/s10710-012-9177-2

    Article  Google Scholar 

  55. J. McDermott, K.D. Jong, U.-M. O’Reilly, D.R. White, S. Luke, L. Manzoni, M. Castelli, L. Vanneschi, W. Jaskowski, K. Krawiec, R. Harper, Genetic programming needs better benchmarks. In: proceedings of the fourteenth international conference on genetic and evolutionary computation conference—GECCO ’12. ACM Press, New York, NY, USA (2012). https://doi.org/10.1145/2330163.2330273

  56. S.M., Udrescu, M. Tegmark, AI Feynman: A physics-inspired method for symbolic regression. Science Advances 6(16) (2020) arXiv:1905.11481. https://doi.org/10.1126/sciadv.aay2631

  57. S.-M. Udrescu, A. Tan, J. Feng, O. Neto, T. Wu, M. Tegmark, Ai feynman 2.0: pareto-optimal symbolic regression exploiting graph modularity. Adv. Neural Inf. Process. Syst. 33, 4860–4871 (2020)

    Google Scholar 

  58. Y. Yasui, X. Wang, Statistical Learning from a Regression Perspective 65, 1309–1310 (2009). https://doi.org/10.1111/j.1541-0420.2009.01343_5.x

  59. D. Kuonen, Regression modeling strategies: with applications to linear models. Logist. Regres. Surv. Anal. 13, 415–416 (2004). https://doi.org/10.1177/096228020401300512

    Article  Google Scholar 

  60. M.Z. Asadzadeh, H.-P. Gänser, M. Mücke, Symbolic regression based hybrid semiparametric modelling of processes: an example case of a bending process. Appl. Eng. Sci. 6, 100049 (2021). https://doi.org/10.1016/j.apples.2021.100049

    Article  Google Scholar 

  61. J.R. Koza, Genetic programming: on the programming of computers by means of natural selection. A Bradford book. Bradford, Bradford, PA (1992). https://books.google.com.br/books?id=Bhtxo60BV0EC

  62. M. Kommenda, B. Burlacu, G. Kronberger, M. Affenzeller, Parameter identification for symbolic regression using nonlinear least squares. Genet. Program. Evolv. Mach. 21(3), 471–501 (2019). https://doi.org/10.1007/s10710-019-09371-3

    Article  Google Scholar 

  63. M. Kommenda, B. Burlacu, G. Kronberger, M. Affenzeller, Parameter identification for symbolic regression using nonlinear least squares. Genet. Program. Evolv. Mach. 21(3), 471–501 (2020)

    Article  Google Scholar 

  64. B. Burlacu, G. Kronberger, M. Kommenda, Operon c++: an efficient genetic programming framework for symbolic regression. In: proceedings of the genetic and evolutionary computation conference companion. GECCO ’20, pp. 1562–1570. Association for Computing Machinery, New York, NY, USA (2020). https://doi.org/10.1145/3377929.3398099

  65. S. Luke, Two fast tree-creation algorithms for genetic programming. Trans. Evol. Comp. 4(3), 274–283 (2000). https://doi.org/10.1109/4235.873237

    Article  Google Scholar 

  66. G.S.I. Aldeia, Avaliação da interpretabilidade em regressão simbólica. Master’s thesis, Universide Federal do ABC, Santo André, SP (December 2021)

  67. L. Breiman, Random forests 45(1), 5–32 (2001). https://doi.org/10.1023/a:1010933404324

  68. M.T. Ribeiro, S. Singh, C. Guestrin, why should i trust you?: Explaining the predictions of any classifier. In: proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining. KDD ’16, pp. 1135–1144. Association for Computing Machinery, New York, NY, USA (2016). https://doi.org/10.1145/2939672.2939778

  69. R. Miranda Filho, A. Lacerda, G.L. Pappa, Explaining symbolic regression predictions. In: 2020 IEEE congress on evolutionary computation (CEC), pp. 1–8 (2020). IEEE

  70. I. Covert, S. Lundberg, S.-I. Lee, Understanding global feature contributions with additive importance measures (2020)

  71. M.D. Morris, Factorial sampling plans for preliminary computational experiments. Technometrics 33(2), 161–174 (1991)

    Article  Google Scholar 

  72. H. Nori, S. Jenkins, P. Koch, R. Caruana, Interpretml: a unified framework for machine learning interpretability. CoRR arxiv: abs/1909.09223 (2019)

  73. M. Sundararajan, A. Taly, Q. Yan, Axiomatic attribution for deep networks (2017)

  74. R.J. Aumann, L.S. Shapley, Values of Non-atomic Games (Princeton University Press, Princeton, NJ, USA, 2015)

    Book  Google Scholar 

  75. D. Lüdecke, ggeffects: tidy data frames of marginal effects from regression models. J. Open Source Softw. 3(26), 772 (2018)

    Article  Google Scholar 

  76. E.C. Norton, B.E. Dowd, M.L. Maciejewski, Marginal effects-quantifying the effect of changes in risk factors in logistic regression models. JAMA 321(13), 1304–1305 (2019). https://doi.org/10.1001/jama.2019.1954

    Article  Google Scholar 

  77. J.S. Long, S.A. Mustillo, Using predictions and marginal effects to compare groups in regression models for binary outcomes 50(3), 1284–1320 (2018). https://doi.org/10.1177/0049124118799374

  78. T.D. Mize, L. Doan, J.S. Long, A general framework for comparing predictions and marginal effects across models. Sociol. Methodol. 49(1), 152–189 (2019). https://doi.org/10.1177/0081175019852763

    Article  Google Scholar 

  79. E. Onukwugha, J. Bergtold, R. Jain, A primer on marginal effects—part i: theory and formulae. PharmacoEconomics 33(1), 25–30 (2015). https://doi.org/10.1007/s40273-014-0210-6

    Article  Google Scholar 

  80. A. Agresti, C. Tarantola, Simple ways to interpret effects in modeling ordinal categorical data. Stat. Neerl. 72(3), 210–223 (2018). https://doi.org/10.1111/stan.12130

    Article  MathSciNet  Google Scholar 

  81. E.C. Norton, B.E. Dowd, M.L. Maciejewski, Marginal effects—quantifying the effect of changes in risk factors in logistic regression models. JAMA 321(13), 1304 (2019). https://doi.org/10.1001/jama.2019.1954

    Article  Google Scholar 

  82. G. Plumb, M. Al-Shedivat, E.P. Xing, A. Talwalkar, Regularizing black-box models for improved interpretability. CoRR arxiv: abs/1902.06787 (2019)

  83. D. Alvarez-Melis, T.S. Jaakkola, Towards Robust Interpretability with Self-Explaining Neural Networks (2018)

  84. C.K. Yeh, C.Y. Hsieh, A.S. Suggala, D.I. Inouye, P. Ravikumar, On the (In)fidelity and sensitivity of explanations. Advances in Neural Information Processing Systems 32(NeurIPS) (2019) arXiv:1901.09392

  85. Z. Zhou, G. Hooker, F. Wang, S-lime. Proceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining (2021). https://doi.org/10.1145/3447548.3467274

  86. W.-L. Loh et al., On latin hypercube sampling. Ann. Stat. 24(5), 2058–2080 (1996)

    Article  MathSciNet  Google Scholar 

  87. J. Demšar, Statistical comparisons of classifiers over multiple data sets. J. Mach. Learn. Res. 7(1), 1–30 (2006)

    MathSciNet  MATH  Google Scholar 

  88. S. Lee, D.K. Lee, What is the proper way to apply the multiple comparison test? Korean J. Anesthesiol. 71(5), 353–360 (2018). https://doi.org/10.4097/kja.d.18.00242

    Article  Google Scholar 

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Acknowledgements

This work was funded by Federal University of ABC (UFABC), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Grant Number 2018/14173-8.

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  1. Center for Mathematics, Computation and Cognition, Federal University of ABC, Av. dos Estados, Santo Andre, SP, 09210580, Brazil

    Guilherme Seidyo Imai Aldeia & Fabrício Olivetti de França

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Aldeia, G.S.I., de França, F.O. Interpretability in symbolic regression: a benchmark of explanatory methods using the Feynman data set. Genet Program Evolvable Mach 23, 309–349 (2022). https://doi.org/10.1007/s10710-022-09435-x

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