Symbolic regression

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Expression tree as it can be used in symbolic regression to represent a function.

Symbolic regression (SR) is a type of regression analysis that searches the space of mathematical expressions to find the model that best fits a given dataset, both in terms of accuracy and simplicity.

No particular model is provided as a starting point to the algorithm. Instead, initial expressions are formed by randomly combining mathematical building blocks such as mathematical operators, analytic functions, constants, and state variables. Usually, a subset of these primitives will be specified by the person operating it, but that's not a requirement of the technique. The symbolic regression problem for mathematical functions has been tackled with a variety of methods, including recombining equations most commonly using genetic programming,[1] as well as more recently methods utilizing Bayesian methods[2] and physics-inspired AI.[3] Another non-classical alternative method to SR is called Universal Functions Originator (UFO), which has a different mechanism, search-space, and building strategy.[4] Further methods such as Exact Learning attempt to transform the fitting problem into a moments problem in a natural function space, usually built around generalisations of the Meijer-G function.[5]

By not requiring a priori specification of a model, symbolic regression isn't affected by human bias, or unknown gaps in domain knowledge. It attempts to uncover the intrinsic relationships of the dataset, by letting the patterns in the data itself reveal the appropriate models, rather than imposing a model structure that is deemed mathematically tractable from a human perspective. The fitness function that drives the evolution of the models takes into account not only error metrics (to ensure the models accurately predict the data), but also special complexity measures,[6] thus ensuring that the resulting models reveal the data's underlying structure in a way that's understandable from a human perspective. This facilitates reasoning and favors the odds of getting insights about the data-generating system.

Difference from classical regression[edit]

While conventional regression techniques seek to optimize the parameters for a pre-specified model structure, symbolic regression avoids imposing prior assumptions, and instead infers the model from the data. In other words, it attempts to discover both model structures and model parameters.

This approach has the disadvantage of having a much larger space to search, because not only the search space in symbolic regression is infinite, but there are an infinite number of models which will perfectly fit a finite data set (provided that the model complexity isn't artificially limited). This means that it will possibly take a symbolic regression algorithm longer to find an appropriate model and parametrization, than traditional regression techniques. This can be attenuated by limiting the set of building blocks provided to the algorithm, based on existing knowledge of the system that produced the data; but in the end, using symbolic regression is a decision that has to be balanced with how much is known about the underlying system.

Nevertheless, this characteristic of symbolic regression also has advantages: because the evolutionary algorithm requires diversity in order to effectively explore the search space, the result is likely to be a selection of high-scoring models (and their corresponding set of parameters). Examining this collection could provide better insight into the underlying process, and allows the user to identify an approximation that better fits their needs in terms of accuracy and simplicity.


End-user software[edit]

See also[edit]


  1. ^ Michael Schmidt; Hod Lipson (2009). "Distilling free-form natural laws from experimental data". Science. American Association for the Advancement of Science. 324 (5923): 81–85. Bibcode:2009Sci...324...81S. CiteSeerX doi:10.1126/science.1165893. PMID 19342586. S2CID 7366016.
  2. ^ Ying Jin; Weilin Fu; Jian Kang; Jiadong Guo; Jian Guo (2019). "Bayesian Symbolic Regression". arXiv:1910.08892 [stat.ME].
  3. ^ a b Silviu-Marian Udrescu; Max Tegmark (2020). "AI Feynman: A physics-inspired method for symbolic regression". Science_Advances. American Association for the Advancement of Science. 6 (16): eaay2631. doi:10.1126/sciadv.aay2631. PMC 7159912. PMID 32426452.
  4. ^ Ali R. Al-Roomi; Mohamed E. El-Hawary (2020). "Universal Functions Originator". Applied Soft Computing. Elsevier B.V. 94: 106417. doi:10.1016/j.asoc.2020.106417. ISSN 1568-4946. S2CID 219743405.
  5. ^ Benedict W. J. Irwin (2021). "Exact Learning" (PDF). doi:10.21203/ S2CID 234014141. {{cite journal}}: Cite journal requires |journal= (help)
  6. ^ Ekaterina J. Vladislavleva; Guido F. Smits; Dick Den Hertog (2009). "Order of nonlinearity as a complexity measure for models generated by symbolic regression via pareto genetic programming" (PDF). IEEE Transactions on Evolutionary Computation. 13 (2): 333–349. doi:10.1109/tevc.2008.926486. S2CID 12072764.
  7. ^ "'Machine Scientists' Distill the Laws of Physics From Raw Data". Quanta Magazine. May 10, 2022.
  8. ^ Cranmer, Miles; Tamayo, Daniel; Rein, Hanno; Battaglia, Peter; Hadden, Samuel; Armitage, Philip J.; Ho, Shirley; Spergel, David N. (October 5, 2021). "A Bayesian neural network predicts the dissolution of compact planetary systems". Proceedings of the National Academy of Sciences. 118 (40): e2026053118. doi:10.1073/pnas.2026053118. PMC 8501828. PMID 34599094 – via (Crossref).

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