shapeconstrained

Shape-constrained refers to statistical and machine learning methods that impose qualitative restrictions on the shape of a function or probability distribution during estimation. The goal is to encode prior knowledge about how a quantity should behave, which can improve interpretability and stability when data are limited or noisy.

Common shape constraints include monotonicity (nondecreasing or nonincreasing), convexity or concavity, unimodality (a single peak), symmetry, and log-concavity of a density. Monotonicity ensures a function moves in one direction with the input; convexity imposes curvature restrictions that affect how estimates respond to changes in input; log-concavity constrains the log of a density to be concave, which implies unimodality and tail behavior.

Shape-constrained methods appear in regression, density estimation, and survival analysis, among other areas. Examples include isotonic regression for monotone relationships, convex regression, and shape-constrained spline approaches that enforce monotone or convex fits. In density estimation, log-concave maximum likelihood methods provide nonparametric estimates with automatic regularization dictated by the constraint.

Computationally, shape-constrained problems are often formulated as convex optimization tasks and solved via projection techniques, interior-point methods, or specialized algorithms such as the pool-adjacent-violators algorithm for isotonic regression. Bayesian variants also exist, incorporating shape constraints through priors or constrained posterior sampling.

Applications span economics, biostatistics, reliability engineering, and environmental science, where incorporating shape information yields more plausible models and can reduce variance. Challenges include choosing appropriate constraints, balancing bias and variance, and ensuring that imposed shapes reflect true underlying relationships.