Evolving multidimensional transformations for symbolic regression with M3GP

Multidimensional Multiclass Genetic Programming with Multidimensional Populations (M3GP) was originally proposed as a wrapper approach for supervised classification. M3GP searches for transformations of the form $$k:{\mathbb {R}}^p \rightarrow {\mathbb {R}}^d$$k:Rp→Rd, where p is the number of dimensions of the problem data, and d is the dimensionality of the transformed data, as determined by the search. This work extends M3GP to symbolic regression, building models that are linear in the parameters using the transformed data. The proposal implements a sequential memetic structure with Lamarckian inheritance, combining two local search methods: a greedy pruning algorithm and least squares parameter estimation. Experimental results show that M3GP outperforms several standard and state-of-the-art regression techniques, as well as other GP approaches. Using several synthetic and real-world problems, M3GP outperforms most methods in terms of RMSE and generates more parsimonious models. The performance of M3GP can be explained by the fact that M3GP increases the maximal mutual information in the new feature space.