Symbolic regression 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.) New equations are then formed by recombining previous equations, using genetic programming.
By not requiring a specific model to be specified, 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, 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 a priori assumptions, and instead infers the model from the data. In other words, attempts to discover both model structures and model parameters.
This approach has, of course, the disadvantage of having a much larger space to search — in fact, 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 much 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 an approximation that better fits their needs in terms of accuracy and simplicity.
- Eureqa, software that implements symbolic regression
- DataMelt, software that implements symbolic regression in Java and Python
-  software for comparing various optimization techniques, including symbolic regression
- RGP package for symbolic regression in R
- GPTIPS software that implements symbolic regression
- Closed-form expression#Conversion from numerical forms
- Genetic programming
- Kolmogorov complexity
- Mathematical optimization
- Regression analysis
- 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 (IEEE) 13 (2): 333–349. doi:10.1109/tevc.2008.926486.
- 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 (446). IEE. pp. 314–319.
- Michael Schmidt; Hod Lipson (2009). "Distilling free-form natural laws from experimental data" (PDF). Science (AAAS) 324 (5923): 81–85. doi:10.1126/science.1165893.
- 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.