Slope estimation in noisy piecewise linear functions

This paper discusses the development of a slope estimation algorithm called MAPSlope for piecewise linear data that is corrupted by Gaussian noise. The number and locations of slope change points (also known as breakpoints) are assumed to be unknown a priori though it is assumed that the possible range of slope values lies within known bounds. A stochastic hidden Markov model that is general enough to encompass real world sources of piecewise linear data is used to model the transitions between slope values and the problem of slope estimation is addressed using a Bayesian maximum a posteriori approach. The set of possible slope values is discretized, enabling the design of a dynamic programming algorithm for posterior density maximization. Numerical simulations are used to justify choice of a reasonable number of quantization levels and also to analyze mean squared error performance of the proposed algorithm. An alternating maximization algorithm is proposed for estimation of unknown model parameters and a convergence result for the method is provided. Finally, results using data from political science, finance and medical imaging applications are presented to demonstrate the practical utility of this procedure.

[1]  G. Trahey,et al.  Shear-wave generation using acoustic radiation force: in vivo and ex vivo results. , 2003, Ultrasound in medicine & biology.

[2]  Christophe Andrieu,et al.  Bayesian curve fitting using MCMC with applications to signal segmentation , 2002, IEEE Trans. Signal Process..

[3]  Sandra E. Ryan,et al.  A Tutorial on the Piecewise Regression Approach Applied to Bedload Transport Data , 2015 .

[4]  G.C. Goodwin,et al.  Approximate EM Algorithms for Parameter and State Estimation in Nonlinear Stochastic Models , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[5]  John Fox,et al.  What Is Nonparametric Regression , 2000 .

[6]  Tomy Varghese,et al.  Radio-frequency ablation electrode displacement elastography: a phantom study. , 2008, Medical physics.

[7]  Tomy Varghese,et al.  Shear Wave Velocity Imaging Using Transient Electrode Perturbation: Phantom and ex vivo Validation , 2011, IEEE Transactions on Medical Imaging.

[8]  René Garcia,et al.  Série Scientifique Scientific Series an Analysis of the Real Interest Rate under Regime Shifts , 2022 .

[9]  Ali Gholami,et al.  A balanced combination of Tikhonov and total variation regularizations for reconstruction of piecewise-smooth signals , 2013, Signal Process..

[10]  Stephen P. Boyd,et al.  Convex piecewise-linear fitting , 2009 .

[11]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[12]  Maurício Soares Bugarin,et al.  Should Voting Be Mandatory? the Effect of Compulsory Voting Rules on Candidates' Political Platforms , 2015 .

[13]  A. Savitzky,et al.  Smoothing and Differentiation of Data by Simplified Least Squares Procedures. , 1964 .

[14]  J. C. Dennison,et al.  Sampled data controller synthesis , 1979 .

[15]  J. Freidman,et al.  Multivariate adaptive regression splines , 1991 .

[16]  James A. Bucklew,et al.  Two results on the asymptotic performance of quantizers , 1984, IEEE Trans. Inf. Theory.

[17]  Charles Audet,et al.  Construction of sparse signal representations with adaptive multiscale orthogonal bases , 2012, Signal Process..

[18]  A. Tishler,et al.  A New Maximum Likelihood Algorithm for Piecewise Regression , 1981 .

[19]  Robert Nowak,et al.  Multiscale generalised linear models for nonparametric function estimation , 2005 .

[20]  Stephen P. Boyd,et al.  Convex Optimization: Convex optimization problems , 2004 .

[21]  New York Dover,et al.  ON THE CONVERGENCE PROPERTIES OF THE EM ALGORITHM , 1983 .

[22]  Adrian F. M. Smith,et al.  Automatic Bayesian curve fitting , 1998 .

[23]  D. Titterington,et al.  Parameter estimation for hidden Markov chains , 2002 .

[24]  R. Bellman,et al.  Curve Fitting by Segmented Straight Lines , 1969 .

[25]  James D. Hamilton A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle , 1989 .

[26]  H. Akaike A new look at the statistical model identification , 1974 .

[27]  W. Cleveland Robust Locally Weighted Regression and Smoothing Scatterplots , 1979 .

[28]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[29]  K J Parker,et al.  Imaging of the elastic properties of tissue--a review. , 1996, Ultrasound in medicine & biology.

[30]  A. Gallant,et al.  Fitting Segmented Polynomial Regression Models Whose Join Points Have to Be Estimated , 1973 .

[31]  R. Yeung The Blahut-Arimoto Algorithms , 2008 .

[32]  P. Perron,et al.  Computation and Analysis of Multiple Structural-Change Models , 1998 .

[33]  J Ophir,et al.  An analysis of elastographic contrast-to-noise ratio. , 1998, Ultrasound in medicine & biology.

[34]  Marc T Ratkovic,et al.  Finding Jumps in Otherwise Smooth Curves: Identifying Critical Events in Political Processes , 2010, Political Analysis.

[35]  T. Yen A majorization–minimization approach to variable selection using spike and slab priors , 2010, 1005.0891.

[36]  P. Perron,et al.  Estimating and testing linear models with multiple structural changes , 1995 .

[37]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[38]  Michael F. Insana,et al.  Covariance analysis of time delay estimates for strained signals , 1998, IEEE Trans. Signal Process..

[39]  W. Qian,et al.  Estimation of parameters in hidden Markov models , 1991, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[40]  Paul Fearnhead,et al.  Exact Bayesian curve fitting and signal segmentation , 2005, IEEE Transactions on Signal Processing.

[41]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[42]  Sean R Eddy,et al.  What is dynamic programming? , 2004, Nature Biotechnology.

[43]  D. Hudson Fitting Segmented Curves Whose Join Points Have to Be Estimated , 1966 .

[44]  Tomy Varghese,et al.  Radiofrequency electrode vibration-induced shear wave imaging for tissue modulus estimation: a simulation study. , 2010, The Journal of the Acoustical Society of America.