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In scar: Shape-Constrained Additive Regression: a Maximum Likelihood Approach

Shape-Constrained Additive (index) Regression: a maximum likelihood approach

Description

scar computes the maximum likelihood estimator of the generalised additive and index regression with shape constraints. Each component of the additive function of the predictors is assumed to belong to one of the nine possible shape restrictions: linear, increasing, decreasing, convex, convex and increasing, convex and decreasing, concave, concave and increasing, or concave and decreasing. For the generalised additive regression, the problem is transformed into a convex optimisation problem and the active set algorithm is used to find the optimum. We emphasise that unlike most of the other nonparametric methods, this approach is free of tuning parameters.

Furthermore, we can extend our findings to the generalised additive index regression, where a stochastic search algorithm is proposed to solve the problem.

Details

Package:	scar
Type:	Package
Version:	0.2-2
Date:	2022-05-25
License:	GPL(>=2)

This package contains a selection of functions for maximum likelihood estimation of the generalised additive (and additive index) regression under shape constraints:

scar computes the maximum likelihood estimator (specified via its value at the observed covariates). Output is a list of class scar which is used as input to various auxiliary functions.

plot.scar produces plots of the maximum likelihood estimator produced by scar on the scale of the additive predictors.

predict.scar obtains predictions either on the scale of the additive predictors or on the scale of the response variable from a fitted scar object.

scair tries to find the maximum likelihood estimator (specified via its value at the observed indices). Output is a list of class scair which is used as input to various auxiliary functions.

plot.scair produces plots of the maximum likelihood estimator produced by scair on the scale of the additive index predictors.

predict.scair obtains predictions either on the scale of the additive index predictors or on the scale of the response variable from a fitted scair object.

The methods proposed here were applied to the following datasets: PhDPublications, decathlon.

Note

The authors gratefully acknowledge the assistance of Ming Yuan for his insights into this problem.

Thanks also go to Mary Meyer for kindly providing the authors with her manuscript prior to its publication.

Author(s)

Yining Chen (maintainer) [e-mail]

Richard Samworth [e-mail]

References

Chen, Y. and Samworth, R. J. (2016). Generalized additive and index models with shape constraints. Journal of the Royal Statistical Society: Series B, 78, 729-754.

Groeneboom, P., Jongbloed, G. and Wellner, J.A. (2008). The support reduction algorithm for computing non-parametric function estimates in mixture models. Scandinavian Journal of Statistics, 35, 385-399.

Hastie, T. and Tibshirani, R. (1990) Generalized Additive Models. Chapman and Hall, London.

Meyer, M. C. (2013) Semi-parametric additive constrained regression. Journal of nonparametric statistics, 25, 715-743.

Nocedal, J., and Wright, S. J. (2006) Numerical Optimization, 2nd edition. Springer, New York.

Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Springer, New York.

Wood, S. N. (2004) Stable and efficient multiple smoothing parameter estimation for generalized additive models. Journal of American Statistical Association, 99, 673-686.

See Also

scar, scair, scam, glm

Examples

## See examples provided in functions scar and scair

scar documentation built on May 28, 2022, 1:22 a.m.