Accuracy in Symbolic Regression

Abstract

This chapter asserts that, in current state-of-the-art symbolic regression engines, accuracy is poor. That is to say that state-of-the-art symbolic regression engines return a champion with good fitness; however, obtaining a champion with the correct formula is not forthcoming even in cases of only one basis function with minimally complex grammar depth.Ideally, users expect that for test problems created with no noise, using only functions in the specified grammar, with only one basis function and some minimal grammar depth, that state-of-the-art symbolic regression systems should return the exact formula (or at least an isomorph) used to create the test data. Unfortunately, this expectation cannot currently be achieved using published state-of-the-art symbolic regression techniques.Several classes of test formulas, which prove intractable, are examined and an understanding of why they are intractable is developed. Techniques in Abstract Expression Grammars are employed to render these problems tractable, including manipulation of the epigenome during the evolutionary process, together with breeding of multiple targeted epigenomes in separate population islands.A selected set of currently intractable problems are shown to be solvable, using these techniques, and a proposal is put forward for a discipline-wide program of improving accuracy in state-of-the-art symbolic regression systems.

Genetic programming needs better benchmarks

“…The 53 symbolic regression target functions in Table 3 (with function sets in Table 2) are drawn from several wellknown sources in the literature [28,30,32,45,50,64]. There is significant variance among them.…”

Better GP benchmarks: community survey results and proposals

“…The Keijzer-6 and Vladislavleva-4 problems require extrapolation, not just interpolation. The Korns-12 problem is the only problem unsolved in [27] despite the application of several specialized techniques. Note that the dataset is specified to have 5 variables, even though only 2 affect the output: the aim is to test the ability of the system to discard unimportant variables and avoid using them to over-fit.…”

Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction

“…The symbolic regression approach adopted herein [24][25][26][27][28][29] is based upon genetic programming wherein a population of functions is allowed to breed and mutate with the genetic propagation into subsequent generations based upon a survival-of-the-fittest criteria [30]. The main goal of this study is to make accurate and computationally cheap explicit approximations of the Colebrook equation, where computationally cheap means to contain the least possible number of logarithmic functions and non-integer powers [31][32][33][34][35][36].…”

“…Unfortunately, in our case using the input parameters in their raw form the accuracy was not at a high level without acceleration, so having previous experience with the same problem where we used Artificial Neural Network [15,16] to simulate results, we normalized parameters a=log10(Re), b=-log10(ε/D), in order to avoid discrepancy in the scale which are in raw form 1000<Re<10 8 and ε/D<<1 and after normalization 3.5<a<8 and 1.3<b<6.5 (Eureqa, software used a as genetic programming tool also suggested to us a data normalization process) [34][35][36]. The normalization gives relatively good results, and genetic programming tool generated more accurate results without knowing that the logarithmic form of the Colebrook equation was originally used but only knowing the predicted input and output datasets; Figure 3: Using our previous experience [15] with the training of the Artificial Neural Network where very good results were achieved through the normalization of parameters; a=log10(Re), b=-log10(ε/D), the genetic programming tool generated a dozen equations with different levels of accuracy and complexity, but fortunately none of them contain logarithms or non-integer power terms.…”