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Functional regression is a dynamic research field within functional data analysis, with a wide range of applications across different areas. This paper aims to expand the existing framework of shape-constrained generalized additive models to functional generalized additive models with shape constraints. We introduce an extension of the shape-constrained P-spline (SCOP-spline) approach to a broad class of functional regression models with various shape constraints. Our framework includes parametric and a mixture of constrained and unconstrained smooth effects of functional and scalar covariates. Estimation and inference in this framework build upon the shape-constrained generalized additive models, enabling the use of wellestablished, robust, and flexible procedures. The described methods are implemented in the user-friendly R package scam. Simulation shows performance improvements when modelling with the proposed approach compared to the unconstrained method.

Regression models that incorporate smooth functions of predictor variables to explain the relationships with a response variable have gained widespread usage and proved successful in various applications. By incorporating smooth functions of predictor variables, these models can capture complex relationships between the response and predictors while still allowing for interpretation of the results. In situations where the relationships between a response variable and predictors are explored, it is not uncommon to assume that these relationships adhere to certain shape constraints. Examples of such constraints include monotonicity and convexity. The scam package for R has become a popular package to carry out the full fitting of exponential family generalized additive modelling with shape restrictions on smooths. The paper aims to extend the existing framework of shape-constrained generalized additive models (SCAM) to accommodate smooth interactions of covariates, linear functionals of shape-constrained smooths and incorporation of residual autocorrelation. The methods described in this paper are implemented in the recent version of the package scam, available on the Comprehensive R Archive Network (CRAN).

scam provides functions for generalized additive modelling under shape constraints on the component functions of the linear predictor of the GAM. Models can contain multiple shape constrained and unconstrained terms as well as bivariate smooths with double or single monotonicity.

We introduce an algorithm which can be directly used to feasible and optimum search in linear programming. Starting from an initial point the algorithm iteratively moves a point in a direction to resolve the violated constraints. At the same time, it ensures that previously fulfilled constraints are not breached during this process. The method is based on geometrical properties of n-dimensional space and can be used on any type of linear constraints (>, =, ≥), moreover it can be used when the feasible region is non-full-dimensional.

One of the primary concerns of product quality control in the automotive industry is an automated detection of defects of small sizes on specular car body surfaces. A new statistical learning approach is presented for surface finish defect detection based on spline smoothing method for feature extraction and k-nearest neighbour probabilistic classifier. Since the surfaces are specular, structured lightning reflection technique is applied for image acquisition. Reduced rank cubic regression splines are used to smooth the pixel values while the effective degrees of freedom of the obtained smooths serve as components of the feature vector. A key advantage of the approach is that it allows reaching near zero misclassification error rate when applying standard learning classifiers. In addition, probability based performance evaluation metrics have been proposed as alternatives to the conventional metrics. The usage of those provides the means for uncertainty estimation of the predictive performance of a classifier. Experimental classification results on the images obtained from the pilot system located at Volvo GTO Cab plant in Umeå, Sweden, show that the proposed approach is much more efficient than the compared methods.