Penalized spline models and applications

Doctoral thesis English OPEN
Costa, Maria J. (Maria João)
  • Subject: QA

Penalized spline regression models are a popular statistical tool for curve fitting\ud problems due to their flexibility and computational efficiency. In particular, penalized\ud cubic spline functions have received a great deal of attention. Cubic splines\ud have good numerical properties and have proven extremely useful in a variety of\ud applications. Typically, splines are represented as linear combinations of basis functions.\ud However, such representations can lack numerical stability or be difficult to\ud manipulate analytically.\ud The current thesis proposes a different parametrization for cubic spline functions\ud that is intuitive and simple to implement. Moreover, integral based penalty\ud functionals have simple interpretable expressions in terms of the components of the\ud parametrization. Also, the curvature of the function is not constrained to be continuous\ud everywhere on its domain, which adds flexibility to the fitting process.\ud We consider not only models where smoothness is imposed by means of a single\ud penalty functional, but also a generalization where a combination of different measures\ud of roughness is built in order to specify the adequate limit of shrinkage for the\ud problem at hand.\ud The proposed methodology is illustrated in two distinct regression settings.
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    21 references, page 1 of 3

    1 Introduction 1 1.1 Penalized Spline Regression: A Brief Review . . . . . . . . . . . . . . 1 1.2 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2 Spline Models 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Definition and Properties . . . . . . . . . . . . . . . . . . . . . 11

    4 The Value-First Derivative Parametrization 34 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2 Definition of the Parametrization . . . . . . . . . . . . . . . . . . . . 35 4.3 Penalty Implementation and Interpretation . . . . . . . . . . . . . . . 38 4.4 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    5 Simulation Study 46 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 Bayesian Inference via MCMC . . . . . . . . . . . . . . . . . . . . . . 48 5.2.1 Single Penalty Models . . . . . . . . . . . . . . . . . . . . . . 48 5.2.2 Double Penalty Models . . . . . . . . . . . . . . . . . . . . . . 50 5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8 Further Topics 108 8.1 Spatially Adaptive Smoothing . . . . . . . . . . . . . . . . . . . . . . 109 8.2 Bivariate Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.3 Full Bayesian Inference for Double Penalty Models . . . . . . . . . . . 114 B Markov Chain Monte Carlo Algorithms 125 B.1 The Gibbs Sampler . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 B.2 The Metropolis-Hastings Algorithm . . . . . . . . . . . . . . . . . . . 127 122 129 135 7.1 Degrees of freedom (d.f.) and Akaike's information criteria (AIC) for different model specifications. . . . . . . . . . . . . . . . . . . . . . . 103 7.2 bilirubin (left plot) and age (right plot) corresponding to observed failures vs time t (in days). The solid lines in both plots correspond to 'lowess' smooths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 AIC(i2,i1) = -2*partial_likelihoodMD(eta,indexes, XcovFailures,Xcov)+(2*TRtotal(i2,i1));

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