New Approaches to Regression by Generalized Additive Models and Continuous Optimization for Modern Applications in Finance, Science and Techology

Generalized additive models belong to modern techniques from statistical learning, and are applicable in many areas of prediction, e.g., in financial mathematics, computational biology, medicine, chemistry and environmental protection. These models have the form ( ) ( ) 0 1 m j j j G( ( X ) ) ψ X β f X μ = = = +∑ , where ψ is a function of the predictors. These models are fitted through local scoring algorithm using a scatterplot smoother as building blocks proposed by Hastie and Tibshirani (1987). In this paper, we first give a short introduction and review. Then, we present a mathematical modeling by splines based on a new clustering approach for the input data x, their density, and the variation of the output data y. We contribute to regression with generalized additive models by bounding (penalizing) second order terms (curvature) of the splines, leading to a more robust approximation. Previously, we proposed a refining modification and investigation of the backfitting algorithm, applied to additive models. Then, because of drawbacks of the modified backfitting algorithm, we solve this problem using continuous optimization techniques which will become an important complementary technology and alternative to the concept of modified backfitting algorithm [24]. In particular, we model and treat the constrained residual sum of squares by the elegant framework of conic quadratic programming.