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Simple and multiple P‐splines regression with shape constraints
Author(s) -
Bollaerts Kaatje,
Eilers Paul H. C.,
Mechelen Iven
Publication year - 2006
Publication title -
british journal of mathematical and statistical psychology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.157
H-Index - 51
eISSN - 2044-8317
pISSN - 0007-1102
DOI - 10.1348/000711005x84293
Subject(s) - monotone polygon , mathematics , monotonic function , simple (philosophy) , regression , sign (mathematics) , parametric statistics , regression analysis , mathematical optimization , nonparametric regression , additive model , regular polygon , econometrics , statistics , mathematical analysis , philosophy , geometry , epistemology
In many research areas, especially within social and behavioural sciences, the relationship between predictor and criterion variables is often assumed to have a particular shape, such as monotone, single‐peaked or U‐shaped. Such assumptions can be transformed into (local or global) constraints on the sign of the n th‐order derivative of the functional form. To check for such assumptions, we present a non‐parametric regression method, P‐splines regression, with additional asymmetric discrete penalties enforcing the constraints. We show that the corresponding loss function is convex and present a Newton–Raphson algorithm to optimize. Constrained P‐splines are illustrated with an application on monotonicity‐constrained regression with both one and two predictor variables, using data from research on the cognitive development of children.