Symbolic regression

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[]

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 end 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.

Software[]

End-user software[]

• HeuristicLab, a software environment for heuristic and evolutionary algorithms, including symbolic regression (free, open source)
• Discipulus Linear Genetic Programming Software (commercial)
• Gene expression programming#Software, GeneXProTools - an implementation of Gene expression programming technique for various problems including symbolic regression (commercial)
• Multi expression programming#MEPX, an implementation of Multi expression programming for symbolic regression and classification (free, open source)
• TuringBot, a symbolic regression software based on simulated annealing (commercial).

Libraries[]

• Eureqa, a symbolic regression engine
• Microgp, An evolutionary tool able to generate assembly programs (free, open source)
• gramEvol, Grammatical Evolution in R (free, open source)
• RGP, a Genetic Programming (GP) framework in R that supports symbolic regression
• GPTIPS, a Genetic Programming and Symbolic Data Mining Platform for MATLAB
• dcgp, an open source symbolic regression toolbox.
• Glyph, a python 3 library based on deap providing abstraction layers for symbolic regression problems
• AI-Feynman python3 + pytorch code for A physics-inspired method for symbolic regression.

See also[]

• Closed-form expression § Conversion from numerical forms
• Genetic programming^[3]
• Gene expression programming
• Kolmogorov complexity
• Linear genetic programming
• Mathematical optimization
• Multi expression programming
• Regression analysis
• Reverse mathematics

References[]

Further reading[]

• Mark J. Willis; Hugo G. Hiden; Ben McKay; Gary A. Montague; Peter Marenbach (1997). "Genetic programming: An introduction and survey of applications" (PDF). IEE Conference Publications. IEE. pp. 314–319.
• Wouter Minnebo; Sean Stijven (2011). "Chapter 4: Symbolic Regression" (PDF). Empowering Knowledge Computing with Variable Selection (M.Sc. thesis). University of Antwerp.
• John R. Koza; Martin A. Keane; James P. Rice (1993). "Performance improvement of machine learning via automatic discovery of facilitating functions as applied to a problem of symbolic system identification" (PDF). IEEE International Conference on Neural Networks. San Francisco: IEEE. pp. 191–198.

External links[]

• Ivan Zelinka (2004). "Symbolic regression — an overview".
• Hansueli Gerber (1998). "Simple Symbolic Regression Using Genetic Programming". (Java applet) — approximates a function by evolving combinations of simple arithmetic operators, using algorithms developed by John Koza.
• Katya Vladislavleva. "Symbolic Regression: Function Discovery & More". Archived from the original on 2014-12-18.