Stochastic Functional Data Analysis: A Diffusion Model-Based Approach

Authors

  • Bin Zhu,

    Corresponding author
    1. Department of Statistical Science, Duke University, Durham, North Carolina 27708, U.S.A.
    2. Center for Human Genetics, Duke University Medical Center, Durham, North Carolina 27710, U.S.A.
      email: bin.zhu@duke.edu
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  • Peter X.-K. Song,

    Corresponding author
    1. Department of Biostatistics, University of Michigan, Ann Arbor, Michigan 48109, U.S.A.
      email: pxsong@umich.edu
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  • Jeremy M.G. Taylor

    Corresponding author
    1. Department of Biostatistics, University of Michigan, Ann Arbor, Michigan 48109, U.S.A.
      email: jmgt@umich.edu
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email:bin.zhu@duke.edu

email:pxsong@umich.edu

email:jmgt@umich.edu

Abstract

Summary This article presents a new modeling strategy in functional data analysis. We consider the problem of estimating an unknown smooth function given functional data with noise. The unknown function is treated as the realization of a stochastic process, which is incorporated into a diffusion model. The method of smoothing spline estimation is connected to a special case of this approach. The resulting models offer great flexibility to capture the dynamic features of functional data, and allow straightforward and meaningful interpretation. The likelihood of the models is derived with Euler approximation and data augmentation. A unified Bayesian inference method is carried out via a Markov chain Monte Carlo algorithm including a simulation smoother. The proposed models and methods are illustrated on some prostate-specific antigen data, where we also show how the models can be used for forecasting.

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