Advertisement

Bivariate Function Extensions

  • Jaroslaw Harezlak
  • David Ruppert
  • Matt P. Wand
Chapter
Part of the Use R! book series (USE R)

Abstract

We now focus on models for the joint effect of two continuous predictor variables. Additive models are convenient, but there is no reason to assume that they are always adequate. In the general bivariate models studied in this chapter, the joint effect of the two variables is a smooth, but otherwise unrestricted, function of these variables. Therefore, these models allow interactions so that the effect of one predictor depends upon the value of the other predictor.

References

  1. Adler, D., Murdoch, D. and others (2017). rgl: 3D visualization using OpenGL. R package version 0.98.22.https://r-forge.r-project.org/projects/rgl/.
  2. Bivand, R.S., Pebesma, E.J. and Gómez-Rubio, V. (2008). Applied Spatial Data Analysis with R. New York: Springer.zbMATHGoogle Scholar
  3. Brinkman, N.D. (1981). Ethanol fuel-A single-cylinder engine study of efficiency and exhaust emissions. SAE Transactions, 90, 1410–1424.Google Scholar
  4. Carroll, R.J. and Ruppert, D. (1988). Transformation and Weighting in Regression. New York: Chapman & Hall.CrossRefGoogle Scholar
  5. Cressie, N.A.C. (2015). Statistics for Spatial Data, Revised Edition. New York: Wiley.zbMATHGoogle Scholar
  6. Cressie, N. and Wikle, C. (2011). Statistics for Spatio-Temporal Data. Hoboken, New Jersey: John Wiley & Sons.zbMATHGoogle Scholar
  7. Diggle, P.J. and P.J. Ribeiro Jr. (2007). Model-based Geostatistics. New York: Springer.zbMATHGoogle Scholar
  8. Duan, N. (1983). Smearing estimate: a nonparametric retransformation method. Journal of the American Statistical Association, 78, 605–610.MathSciNetCrossRefGoogle Scholar
  9. Fahrmeir, L. and Kneib, T. (2011). Bayesian Smoothing and Regression for Longitudinal, Spatial, and Event History Data. Oxford, U.K.: Oxford University Press.CrossRefGoogle Scholar
  10. Goldsmith, J., Scheipl, F., Huang, L., Wrobel, J., Gellar, J., Harezlak, J., McLean, M.W., Swihart, B., Xiao, L., Crainiceanu, C., Reiss, P., Chen, (2016). refund: Regression with functional data. R package version 0.1. http://www.r-project.org.
  11. Green, P.J. and Silverman, B.W. (1994). Nonparametric Regression and Generalized Linear Models. London: Chapman and Hall.CrossRefGoogle Scholar
  12. Gu, C. (2017). gss: General smoothing splines. R package version 2.1. http://www.r-project.org.
  13. Hastie, T. (2017a). gam: Generalized additive models. R package version 1.14. http://www.r-project.org.
  14. Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models. Journal of the Royal Statistical Society, Series B, 55, 757–796.MathSciNetzbMATHGoogle Scholar
  15. Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. New York: Springer.CrossRefGoogle Scholar
  16. Kammann, E.E. and Wand, M.P. (2003). Geoadditive models. Journal of the Royal Statistical Society, Series C, 52, 1–18.MathSciNetCrossRefGoogle Scholar
  17. Landman, B.A., Huang, A.J., Gifford, A., Vikram, D.S., Lim, I.A.L, Farrell, J.A.D., Bogovic, J.A., Hua, J., Chen, M., Jarso, S., Smith, S.A., Joel, S., Mori, S., Pekar, J.J., Barker, P.B., Prince, J.L. and van Zijl, P.C.M. (2010). Multi-parametric neuroimaging reproducibility: A 3T resource study. NeuroImage, 54, 2854–2866.CrossRefGoogle Scholar
  18. Nychka, D., Furrer, R., Paige, J. and Sain, S. (2017). fields: Tools for spatial data. R package version 9.0. http://www.r-project.org.
  19. O’Connell, M.A. and Wolfinger, R.D. (1997). Spatial regression models, response surfaces, and process optimization. Journal of Computational and Graphical Statistics, 6, 224–241.zbMATHGoogle Scholar
  20. Ramsay, J.O., Hooker, G. and Graves, S. (2009). Functional Data Analysis with R and MATLAB. New York: Springer.CrossRefGoogle Scholar
  21. Ramsay, J.O. and Silverman, B.W. (2006). Functional Data Analysis, Second Edition. New York: Springer.Google Scholar
  22. Ramsay, J.O., Wickham, H., Graves, S. and Hooker, G. (2017). fda: Functional data analysis. R package version 2.4.7. http://www.functionaldata.org.
  23. Rasmussen, C.E. and Williams, C.K.I. (2006). Gaussian Processes for Machine Learning. Cambridge, Massachusetts: MIT Press.zbMATHGoogle Scholar
  24. Ribeiro, P.J. and Diggle, P.J. (2016). geoR: Analysis of geostatistical data. R package version 1.7. http://www.leg.ufpr.br/geoR.
  25. Ruppert, D. and Matteson, D.S. (2015). Statistics and Data Analysis for Financial Engineering, Second Edition. New York: Springer.zbMATHGoogle Scholar
  26. Ruppert, D., Wand, M.P. and Carroll, R.J. (2003). Semiparametric Regression. Cambridge, U.K.: Cambridge University Press.CrossRefGoogle Scholar
  27. Sarkar, D. (2008). Lattice. New York: Springer.CrossRefGoogle Scholar
  28. Wood, S.N. (2003). Thin plate regression splines. Journal of the Royal Statistical Society, Series B, 65, 95–114.MathSciNetCrossRefGoogle Scholar
  29. Wood, S.N. (2006a). Generalized Additive Models. Boca Raton, Florida: Chapman & Hall/CRC.CrossRefGoogle Scholar
  30. Wood, S.N. (2006b). Low rank scale invariant tensor product smooths for generalized additive mixed models. Biometrics, 62, 1025–1036.MathSciNetCrossRefGoogle Scholar
  31. Wood, S.N. (2017). mgcv: Mixed GAM computation vehicle with GCV/AIC/REML smoothness estimation. R package version 1.8.http://cran.r-project.org.
  32. Wood, S.N., Li, Z., Shaddick, G. and Augustin, N.H. (2017). Generalized additive models for gigadata: modelling the U.K. black smoke network daily data. Journal of the American Statistical Association, 112, 1199–1210.MathSciNetCrossRefGoogle Scholar
  33. Xiao, L., Li, Y. and Ruppert, D. (2013). Fast bivariate P-splines: the sandwich smoother. Journal of the Royal Statistical Society, Series B, 75, 577–599.MathSciNetCrossRefGoogle Scholar
  34. Xiao, L., Ruppert, D., Zipunnikov, V. and Crainiceanu, C. (2016). Fast covariance estimation for high-dimensional function data. Statistics and Computing, 26, 409–421.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Jaroslaw Harezlak
    • 1
  • David Ruppert
    • 2
  • Matt P. Wand
    • 3
  1. 1.School of Public HealthIndiana University BloomingtonBloomingtonUSA
  2. 2.Department of Statistical ScienceCornell UniversityIthacaUSA
  3. 3.School of Mathematical and Physical SciencesUniversity of Technology SydneyUltimoAustralia

Personalised recommendations