Abstract
A nonparametric Bayesian method for regression under combinations of local shape constraints is proposed. The shape constraints considered include monotonicity, concavity (or convexity), unimodality, and in particular, combinations of several types of range-restricted constraints. By using a B-spline basis, the support of the prior distribution is included in the set of piecewise polynomial functions. The first novelty is that, thanks to the local support property of B-splines, many combinations of constraints can easily be considered by identifying B-splines whose support intersects with each constrained region. Shape constraints are included in the coefficients prior using a truncated Gaussian distribution. However, the prior density is only known up to the normalizing constant, which does change with the dimension of coefficients. The second novelty is that we propose to simulate from the posterior distribution by using a reversible jump MCMC slice sampler, for selecting the number and the position of the knots, coupled to a simulated annealing step to project the coefficients on the constrained space. This method is valid for any combination of local shape constraints and particular attention is paid to the construction of a trans-dimensional MCMC scheme.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abraham, C.: Bayesian regression under combinations of constraints. J. Stat. Plan. Inference 142, 2672–2687 (2012)
Bartoli, N., Moral, P.D.: Simulation et algoritmes stochastiques. Cepadues-Editions (2001)
de Boor, C.: A Practical Guide to Splines. Springer-Verlag, New-York (2001)
Delecroix, M., Thomas-Agnan, C.: Spline and Kernel Regression under Shape Restrictions. Wiley, New-York (2000)
Denison, D., Mallick, B., Smith, A.: Automatic Bayesian curve fitting. J. Royal Stat. Soc. (B) 60, 333–350 (1998)
DiMatteo, I., Genovese, C., Kass, R.: Bayesian curve-fitting with free-knot splines. Biometrika 88, 1055–1071 (2001)
Gelfand, A., Kuo, L.: Nonparametric Bayesian bioassay including ordered polytomous response. Biometrika 78, 355–366 (1991)
Gelfand, A., Smith, A., Lee, T.: Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling. J. Am. Stat. Assoc. 87, 523–532 (1992)
Green, P.: Reversible jump Markov chain monte carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995).
Gunn, L., Dunson, D.: A transformation approach for incorporating monotone or unimodal constraints. Biostatistics 6, 434–449 (2005)
Holmes, C., Heard, N.: Generalized monotonic regression using random change points. Stat. Med. 22, 623–638 (2003)
Ibrahim, J., Chen, M.: Power prior distributions for regression models. Stat. Sci. 15, 46–60 (2000)
Jeanson, S., Hilgert, N., Coquillard, M., Seukpanya, C., Faiveley, M., Neuveu, P., Abraham, C., Georgescu, V., Fourcassie, P., Beuvier, E.: Milk acidification by Lactococcus lactis is improved by decreasing the level of dissolved oxygen rather than decreasing redox potential in the potential in the milk prior to inoculation. Int. J. Food Microbiol. 131, 75–81 (2009)
Lavine, M., Mockus, A.: A nonparametric Bayes method for isotonic regression. J. Stat. Plan. Inference 46, 235–248 (1995)
Mammen, E., Thomas-Agnan, C.: Smoothing splines and shape restrictions. Scand. J. Stat. 26, 239–252 (1999)
Mammen, E., Marron, J., Turlach, B., Wand, M.: A general projection framework for constrained smoothing. Stat. Sci. 16, 232–248 (2001)
Meyer, M.: Inference using shape-restricted regression splines. Ann. Appl. Stat. 2, 1013–1033 (2008)
Mukerjee, H.: Monotone nonparametric regression. Ann. Stat. 16, 741–750 (1988)
Neelon, B., Dunson, D.: Bayesian isotonic regression and trend analysis. Biometrics 60, 398–406 (2004)
Ramgopal, P., Laud, P., Smith, A.: Nonparametric Bayesian bioassay with prior constraints on the shape of the potency curve. Biometrika 80, 489–498 (1993)
Ramsay, J.: Estimating smooth monotone functions. J. Royal Stat. Soc. (B) 60, 365–375 (1998)
Shively, T., Sager, T.: A Bayesian approach to non-parametric monotone function estimation. J. Royal Stat. Soc. (B) 71, 159–175 (2009)
Shively, T., Walker, S., Damien, P.: Nonparametric function estimation subject to monotonicity, convexity and other shape constraints. J. Econom. 161, 166–181 (2011)
Tierney, L.: Markov chains for exploring posterior distributions. Ann. Stat. 22, 1701–1762 (1994)
Villalobos, M., Wahba, G.: Inequality constrained multivariate smoothing splines with application to the estimation of posterior probability. J. Am. Stat. Assoc. 82, 239–248 (1987)
Wang, X.: Bayesian free-knot monotone cubic spline regression. J. Comput. Graphical Stat. 17, 373–387 (2008)
Wright, I., Wegman, E.: Nonparametric regression under qualitative smoothness assumptions. Ann. Stat. 8, 1023–1035 (1980)
Acknowledgments
This article was mainly prepared while the author was working at UMR Mistea of SupAgro Montpellier. I would like to thank Christophe Abraham for his suggestion to study this topic and for his always accurate insights. The author is very grateful to the referees for many useful comments that improved the clarity of the chapter.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Khadraoui, K. (2015). Nonparametric Bayesian Regression Under Combinations of Local Shape Constraints. In: Polpo, A., Louzada, F., Rifo, L., Stern, J., Lauretto, M. (eds) Interdisciplinary Bayesian Statistics. Springer Proceedings in Mathematics & Statistics, vol 118. Springer, Cham. https://doi.org/10.1007/978-3-319-12454-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-12454-4_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12453-7
Online ISBN: 978-3-319-12454-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)