## 2 Classifiers and Symbolic Regression via Genetic Programming
### Symbolic Regression via Genetic Programming  
As briefly discussed in Section 1, the subject of this paper is the application of Symbolic Regression (SR) via Genetic Programming (GP) to identifying a physically meaningful equation for the boundary between safe and disruptive regions of the operational space. The main advantages of the proposed approach consist of: (a) flexibility in implementing the most appropriate assumptions about the mathematical form of the boundary; (b) allowing one to express the equations of the boundary in mathematical forms, which reflect the physics of disruptions and permit comparison with theory. The other specific characteristic of the proposed methodology resides in its capability to identify the most suitable mathematical expressions, for the boundary between safe and disruptive regions of the operational space, directly from the experimental data. So, no prior information about the mathematical form of the boundary is required because the developed tools allow extracting the most suited expression directly from the database.  
The technique is based on generating a high number of formulas, built combining basis functions covering a sufficiently wide spectrum to describe the phenomenon to be studied, and on determining which ones better fit the data. Such a task is achieved efficiently by Genetic Programming.  
Genetic Programming is a collective name comprising a series of mathematical techniques, in which programmes are represented as some sort of genes and then adapted to solving specific optimisation problems by evolutionary computational methods [36,37,38,39]. Genetic Programs (GPs) are meant to emulate the natural selection of the genotype in living organisms. GPs manipulate a population of individuals, which are mathematical expressions in our case. Each individual constitutes a potential equation for the boundary between the safe and disruptive regions of the operational space in the application presented in this paper. The quality of each individual, with respect to the available data, is assessed by means of a utility function that, in this context, is normally referred to as Fitness Function (FF). The FF is basically a metric determining how good an individual is in solvingthe problem at hand, given the data. The individuals of the current generation, with a better value of this metric, are granted a higher probability to pass their genes to descendants. In other words, the probability that the genes of an individual are inherited by the following generation is linked to its performance in terms of the FF.  
Once the final structure of the candidate models has been agreed upon, typically with the method of the Pareto Frontier, the parameters of the model, obtained with SR via GP, are typically refined with appropriate nonlinear fitting routines [40]. In addition to improving their values, this step allows associating confidence intervals to the parameters of the models. All the details about the methodology described in this Section can be found in [40; 41; 42].  
Identification of the Boundary Equation between Safe and Disruptive Regions of the Operational Space  
This section introduces the innovative part of the work presented in this paper: the mathematical procedure, to obtain the equation of the boundary between disruptive and non-disruptive operational regions, by applying symbolic regression to the outputs of machine learning tools, namely SVM and ensembles of CART classifiers (Sections 3.1 and 3.2, respectively). Numerical examples, to elucidate the details, and to prove that the approach works for problems of the same level of complexity as the experimental databases investigated in the rest of the paper, are reported in Appendix A. Some considerations about the computational requirements of the techniques are also mentioned (Section 3.3).