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Slope Estimation in Noisy Piecewise Linear Functions.

Atul Ingle1, James Bucklew2, William Sethares2

  • 1Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison WI 53706, USA. ; Department of Medical Physics, University of Wisconsin-Madison, Madison WI 53705, USA.

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Summary
This summary is machine-generated.

This study introduces MAPSlope, a novel algorithm for estimating slopes in noisy, piecewise linear data. It accurately identifies unknown breakpoints using a Bayesian approach and hidden Markov models.

Keywords:
EM algorithmMAP estimationalternating maximizationdynamic programming optimizationpiecewise linear function

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Area of Science:

  • Signal Processing
  • Statistical Modeling
  • Machine Learning

Background:

  • Estimating slopes in data with unknown change points is challenging, especially with Gaussian noise.
  • Piecewise linear data is common in various scientific and financial applications.
  • Existing methods may struggle with unknown breakpoint locations and number.

Purpose of the Study:

  • To develop a robust slope estimation algorithm (MAPSlope) for noisy piecewise linear data.
  • To handle scenarios where slope change points (breakpoints) are unknown.
  • To provide a Bayesian maximum a posteriori (MAP) approach for accurate slope estimation.

Main Methods:

  • Utilized a stochastic hidden Markov model (HMM) to represent transitions between slope values.
  • Employed a Bayesian maximum a posteriori (MAP) estimation framework.
  • Developed a dynamic programming algorithm for posterior density maximization after discretizing slope values.

Main Results:

  • Numerical simulations validated the algorithm's performance and informed the choice of quantization levels.
  • An alternating maximization algorithm was proposed for estimating unknown model parameters, with a convergence guarantee.
  • Demonstrated practical utility across diverse applications including political science, finance, and medical imaging.

Conclusions:

  • MAPSlope offers an effective solution for slope estimation in challenging piecewise linear datasets.
  • The Bayesian MAP approach combined with HMM provides a powerful framework for breakpoint estimation.
  • The algorithm's applicability is confirmed by successful real-world data analysis.