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Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
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Parameter Estimation with Data-Driven Nonparametric Likelihood Functions.

Shixiao W Jiang1, John Harlim1,2,3

  • 1Department of Mathematics, the Pennsylvania State University, 109 McAllister Building, University Park, PA 16802-6400, USA.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces a novel data-driven surrogate modeling approach for nonparametric likelihood functions, utilizing spectral expansion on manifolds. The method demonstrates robust parameter estimation, outperforming standard models, especially for complex data geometries.

Keywords:
Bayesian inferenceMCMCdiffusion mapskernel embedding of the conditional distributionnonparametric likelihood functionreproducing kernel Hilbert spacesurrogate modeling

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Area of Science:

  • Computational Mathematics
  • Statistical Modeling
  • Machine Learning

Background:

  • Surrogate modeling is crucial for complex systems where direct simulation is computationally expensive.
  • Traditional parametric likelihood functions often fail to capture the underlying geometry of data.
  • Data-driven approaches are needed to construct accurate likelihood functions without prior assumptions.

Purpose of the Study:

  • To develop a data-driven nonparametric likelihood function using manifold learning and spectral expansion.
  • To demonstrate the robustness and accuracy of this approach for parameter estimation.
  • To compare the proposed method against existing parametric and non-parametric surrogate models.

Main Methods:

  • Constructed a nonparametric likelihood function on a data manifold using kernel embedding of the conditional distribution.
  • Employed diffusion maps to obtain data-driven basis functions that respect data geometry.
  • Utilized spectral expansion for representing the likelihood function.

Main Results:

  • The data-driven likelihood function's error bound is independent of basis function variance, enabling controlled data requirements.
  • The proposed method shows superior performance compared to standard parametric models when data manifold dimension is lower than ambient space.
  • Achieved comparable estimation accuracy to direct Markov Chain Monte Carlo (MCMC) with significantly fewer function evaluations (8 vs. 4000).

Conclusions:

  • The proposed manifold-based, data-driven likelihood function offers a robust and accurate surrogate modeling approach.
  • The method is particularly effective for complex, non-smooth, and unknown data manifolds.
  • This approach significantly reduces computational cost for parameter estimation in complex systems.