3 Symbolic Regression

In many scientific disciplines, finding a model explaining the relationship of large and complex sets of data is required. However, the task is often difficult and many methods have been employed to uncover models of data. The domain of symbolic regression is one method where input and output pairs are used to infer a functional model, typically consisting of a function and its coefficients. Keijzer (2001) researched the application of genetic programming on the Regression domain in the context of ``scientific discovery'', demonstrating several ways to improve its application.

The Regression problem attempts to find a program that approximates a target function. A target function $f(x)=y$ is applied to domain values, $x$, in a pre-determined range. The resulting $y$ values are then compared with the candidate program's value upon the same $x$ values. This thesis uses the Quartic polynomial,

$$f(x) = x^4 +x^3 + x^2 + x,$$

the Rastrigin function,

$$f(x) = 3.0n + \sum_{i=1}^{n}{x^2_i - 3.0 \textrm{cos} (2\pi x_i)},$$

the Binomial-3 polynomial,

$$f(x) = (1+x)^3,$$

and several random polynomials (described in Chapter 5). For the Rastrigin instance, $x$ is in the range $[-5.12, 5.12]^n$, for the Quartic, Binomial-3 and random polynomial instances, $x$ is in the range $[-1.00, 1.00]$. The number of ($x,y$) pairs is $20$ for both the Quartic and Rastrigin instances and $50$ for the Binomial-3 and random polynomial instances. Figure 2.3 shows the Quartic polynomial and the Rastrigin function ($n=1$).