We investigate the addition of constraints on the function image and its derivatives for the incorporation of prior knowledge in symbolic regression. The approach is called shape-constrained symbolic regression and allows us to enforce, for example, monotonicity of the function over selected inputs. The aim is to find models which conform to expected behavior and which have improved extrapolation capabilities. We demonstrate the feasibility of the idea and propose and compare two evolutionary algorithms for shape-constrained symbolic regression: (i) an extension of tree-based genetic programming which discards infeasible solutions in the selection step, and (ii) a two-population evolutionary algorithm that separates the feasible from the infeasible solutions. In both algorithms we use interval arithmetic to approximate bounds for models and their partial derivatives. The algorithms are tested on a set of 19 synthetic and four real-world regression problems. Both algorithms are able to identify models which conform to shape constraints which is not the case for the unmodified symbolic regression algorithms. However, the predictive accuracy of models with constraints is worse on the training set and the test set. Shape-constrained polynomial regression produces the best results for the test set but also significantly larger models.

中文翻译：

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形状约束的符号回归——利用先验知识改进外推

我们研究了对函数图像及其导数的约束，以便将先验知识纳入符号回归中。这种方法被称为形状约束符号回归，它允许我们在选定的输入上强制执行例如函数的单调性。目的是找到符合预期行为并具有改进的外推能力的模型。我们证明了这个想法的可行性，并提出并比较了两种用于形状约束符号回归的进化算法：（i）基于树的遗传编程的扩展，它在选择步骤中丢弃了不可行的解决方案，以及（ii）两个种群进化将可行解与不可行解分开的算法。在这两种算法中，我们都使用区间算术来近似模型及其偏导数的界限。这些算法在一组 19 个合成和四个真实世界的回归问题上进行了测试。两种算法都能够识别符合形状约束的模型，而未修改的符号回归算法则不然。然而，有约束的模型在训练集和测试集上的预测准确性较差。形状约束多项式回归为测试集产生了最好的结果，但也产生了显着更大的模型。有约束的模型在训练集和测试集上的预测精度更差。形状约束多项式回归为测试集产生了最好的结果，但也产生了显着更大的模型。有约束的模型在训练集和测试集上的预测精度更差。形状约束多项式回归为测试集产生了最好的结果，但也产生了显着更大的模型。