Shape-constrained Symbolic Regression integrates prior knowledge about the function shape into the symbolic regression model. This can be used to enforce that the model has desired properties such as monotonicity, or convexity, among others. Shape-constrained Symbolic Regression can also help to create models with better extrapolation behavior and reduced sensitivity to noise. The constraint evaluation can be challenging because exact evaluation of constraints may require a search for the extrema of non-convex functions. Approximations via interval arithmetic allow to efficiently find bounds for the extrema of functions. However, interval arithmetic can lead to overly wide bounds and therefore produces a pessimistic estimation. Another possibility is to use sampling which underestimates the true range. Sampling therefore produces an optimistic estimation. In this paper we evaluate both methods and compare them on different problem instances. In particular we evaluate the sensitivity to noise and the extrapolation capabilities in combination with noise data. The results indicate that the optimistic approach works better for predicting out-of-domain points (extrapolation) and the pessimistic approach works better for high noise levels.

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.

UNLABELLED: We present TreeDom, a web tool for graphically analysing the evolutionary history of domains in multi-domain proteins. Individual domains on the same protein chain may have distinct evolutionary histories, which is important to grasp in order to understand protein function. For instance, it may be important to know whether a domain was duplicated recently or long ago, to know the origin of inserted domains, or to know the pattern of domain loss within a protein family. TreeDom uses the Pfam database as the source of domain annotations, and displays these on a sequence tree. An advantage of TreeDom is that the user can limit the analysis to N sequences that are most similar to a query, or provide a list of sequence IDs to include. Using the Pfam alignment of the selected sequences, a tree is built and displayed together with the domain architecture of each sequence.Availablility and implementation: http://TreeDom.sbc.su.se CONTACT: [email protected].