Nonparametric Curve Estimation by Kernel Smoothers under Correlated Errors
Résumé
http://demonstrations.wolfram.com/NonparametricCurveEstimationByKernelSmoothersUnderCorrelated/
Wolfram Demonstrations Project.
This mathematica demonstration provides interactive assessments of the statistical efficiency of a modified Mallows' $C_L$ smoothing-parameter selector, where the modification consists of an adaptive whitening which is required as soon as the observation errors are correlated. More precisely this selector is either the bandwidth which minimizes a classic estimate of the Kullback–Leibler loss function as described in [1], or an alternative where the previous minimization w.r.t. the underlying correlation (whitening parameter) is replaced by the solution of the Gibbs energy-variance (GE-V, see [2–5]) matching equation.
Such assessments can be done with reasonably fast interactivity on a current personal computer (with the free Mathematica-Player) for rather large data-sizes, and for various underlying curves, correlation values and noise-levels.
References:
[1] C. Gu, Smoothing Spline Anova Models, 2nd ed., New York: Springer, 2013.
[2] D. A. Girard, "Estimating a Centered Ornstein–Uhlenbeck Process under Measurement Errors" from the Wolfram Demonstrations Project.
[3] D. A. Girard, "Estimating a Centered Matérn (1) Process: Three Alternatives to Maximum Likelihood via Conjugate Gradient Linear Solvers" from the Wolfram Demonstrations Project.
[4] D. A. Girard, "Asymptotic Near-Efficiency of the 'Gibbs-Energy and Empirical-Variance' Estimating Functions for Fitting Matérn Models — I: Densely Sampled Processes," Statistics and Probability Letters, 110, 2016, pp. 191–197.
[5] D. A. Girard, "Efficiently Estimating Some Common Geostatistical Models by 'Energy–Variance Matching' or Its Randomized 'Conditional–Mean' Versions," Spatial Statistics, 21, 2017 pp. 1–26.
Wolfram Demonstrations Project.
This mathematica demonstration provides interactive assessments of the statistical efficiency of a modified Mallows' $C_L$ smoothing-parameter selector, where the modification consists of an adaptive whitening which is required as soon as the observation errors are correlated. More precisely this selector is either the bandwidth which minimizes a classic estimate of the Kullback–Leibler loss function as described in [1], or an alternative where the previous minimization w.r.t. the underlying correlation (whitening parameter) is replaced by the solution of the Gibbs energy-variance (GE-V, see [2–5]) matching equation.
Such assessments can be done with reasonably fast interactivity on a current personal computer (with the free Mathematica-Player) for rather large data-sizes, and for various underlying curves, correlation values and noise-levels.
References:
[1] C. Gu, Smoothing Spline Anova Models, 2nd ed., New York: Springer, 2013.
[2] D. A. Girard, "Estimating a Centered Ornstein–Uhlenbeck Process under Measurement Errors" from the Wolfram Demonstrations Project.
[3] D. A. Girard, "Estimating a Centered Matérn (1) Process: Three Alternatives to Maximum Likelihood via Conjugate Gradient Linear Solvers" from the Wolfram Demonstrations Project.
[4] D. A. Girard, "Asymptotic Near-Efficiency of the 'Gibbs-Energy and Empirical-Variance' Estimating Functions for Fitting Matérn Models — I: Densely Sampled Processes," Statistics and Probability Letters, 110, 2016, pp. 191–197.
[5] D. A. Girard, "Efficiently Estimating Some Common Geostatistical Models by 'Energy–Variance Matching' or Its Randomized 'Conditional–Mean' Versions," Spatial Statistics, 21, 2017 pp. 1–26.