P-spline Varying Coefficient Models for Complex Data



Although the literature on varying coefficient models (VCMs) is vast, we believe that there remains room to make these models more widely accessible and provide a unified and practical implementation for a variety of complex data settings. The adaptive nature and strength of P-spline VCMs allow a full range of models: from simple to additive structures, from standard to generalized linear models, from one-dimensional coefficient curves to two-dimensional (or higher) coefficient surfaces, among others, including bilinear models and signal regression. As P-spline VCMs are grounded in classical or generalized (penalized) regression, fitting is swift and desirable diagnostics are available. We will see that in higher dimensions, tractability is only ensured if efficient array regression approaches are implemented. We also motivate our approaches through several examples, most notably the German deep drill data, to highlight the breadth and utility of our approach.


Tensor Product Effective Dimension Smooth Term Difference Penalty Cataclastic Rock 
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I would like to thank Paul H.C. Eilers for his generous time and his numerous thought provoking conversations with me that led to a significantly improved presentation.


  1. Currie, I. D., Durbán, M. & Eilers, P. H. C. (2006). Generalized linear array models with applications to multidimensional smoothing. Journal of the Royal Statistical Society: Series B, 68(2): 259–280.MATHCrossRefMathSciNetGoogle Scholar
  2. Dierckx, P. (1995). Curve and Surface Fitting with Splines. Clarendon Press, Oxford.MATHGoogle Scholar
  3. Eilers, P. H. C., Gampe, J., Marx, B. D. & Rau, R. (2008). Modulation models for seasonal life tables. Statistics in Medicine, 27(17): 3430–3441.CrossRefMathSciNetGoogle Scholar
  4. Eilers, P. H. C. & Marx, B. D. (2003). Multivariate calibration with temperature interaction using two-dimensional penalized signal regression. Chemometrics and Intelligent Laboratory Systems, 66: 159–174.Google Scholar
  5. Eilers, P. H. C. & Marx, B. D. (2002). Generalized linear additive smooth structures. Journal of Computational and Graphical Statistics, 11(4): 758–783.CrossRefMathSciNetGoogle Scholar
  6. Eilers, P. H. C. & Marx, B. D. (1996). Flexible smoothing with B-splines and penalties (with comments and rejoinder). Statistical Science, 11: 89–121.MATHCrossRefMathSciNetGoogle Scholar
  7. Fahrmeir, L., Kneib, T. & Lang, S. (2004). Penalized structured additive regression for space-time data: a Bayesian perspective. Statistica Sinica, 14: 731–761.MATHMathSciNetGoogle Scholar
  8. Farhmeir, L. & Tutz, G. (2001). Multivariate Statistical Modelling Based on Generalized Linear Models (2nd Edition). Springer, New York.Google Scholar
  9. Hastie, T. & Tibshirani, R. (1990). Generalized Additive Models. Chapman and Hall, London.MATHGoogle Scholar
  10. Hastie, T. & Tibshirani, R. (1993). Varying-coefficient models. Journal of the Royal Statistical Society, B, 55: 757–796.MATHMathSciNetGoogle Scholar
  11. Heim, S., Fahrmeir, L., Eilers, P. H. C., & Marx, B. D. (2007). Space-varying coefficient models for brain imaging. Computational Statistics and Data Analysis, 51: 6212–6228.MATHCrossRefMathSciNetGoogle Scholar
  12. Kauermann, G. & Küchenhoff, K. (2003). Modelling data from inside the Earth: local smoothing of mean and dispersion structure in deep drill data. Statistical Modelling, 3: 43–64.MATHCrossRefMathSciNetGoogle Scholar
  13. Lang, S. & Brezger, A. (2004). Bayesian P-splines. Journal of Computational and Graphical Statistics, 13(1): 183–212.CrossRefMathSciNetGoogle Scholar
  14. Marx, B. D. & Eilers, P. H. C. (1999). Generalized linear regression on sampled signals and curves: a P-spline approach. Technometrics, 41: 1–13.CrossRefGoogle Scholar
  15. Marx, B. D., Eilers, P. H. C., Gampe, J. & Rau, R. (2010). Bilinear modulation models for seasonal tables of counts. Statistics and Computing. In press.Google Scholar
  16. Ruppert, D., Wand, M. P. & Carroll, R. J. (2003). Semiparametric Regression, Cambridge University Press, New York.MATHGoogle Scholar
  17. Schall, R. (1991). Estimation in generalized linear models with random effects. Biometrika, 78: 719–727.MATHCrossRefGoogle Scholar
  18. Winter, H., Adelhardt, S., Jerak, A. & Küchenhoff, H. (2002). Characteristics of cataclastic shear zones of the ktb deep drill hole by regression analysis of drill cuttings data. Geophysics Journal International, 150: 1–9.CrossRefGoogle Scholar
  19. Zeger, S. L. (1988). A regression model for time series of counts. Biometrika, 75: 621–629.MATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of Experimental StatisticsLouisiana State UniversityBaton RougeUSA

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