TY - JOUR

T1 - On Estimating the Hurst Parameter from Least-Squares Residuals. Case Study

T2 - Correlated Terrestrial Laser Scanner Range Noise

AU - Kermarrec, Gaël

N1 - Funding information: The publication of this article was funded by the Open Access fund of Leibniz Universität Hannover. The author gratefully acknowledge the funding by the Deutsche Forschungsgemeinschaft under the label KE 2453/2-1. The publication of this article was funded by the Open Access fund of Leibniz Universit?t Hannover. The author gratefully acknowledge the funding by the Deutsche Forschungsgemeinschaft under the label KE 2453/2-1.

PY - 2020/4/29

Y1 - 2020/4/29

N2 - Many signals appear fractal and have self-similarity over a large range of their power spectral densities. They can be described by so-called Hermite processes, among which the first order one is called fractional Brownian motion (fBm), and has a wide range of applications. The fractional Gaussian noise (fGn) series is the successive differences between elements of a fBm series; they are stationary and completely characterized by two parameters: the variance, and the Hurst coefficient (H). From physical considerations, the fGn could be used to model the noise of observations coming from sensors working with, e.g., phase differences: due to the high recording rate, temporal correlations are expected to have long range dependency (LRD), decaying hyperbolically rather than exponentially. For the rigorous testing of deformations detected with terrestrial laser scanners (TLS), the correct determination of the correlation structure of the observations is mandatory. In this study, we show that the residuals from surface approximations with regression B-splines from simulated TLS data allow the estimation of the Hurst parameter of a known correlated input noise. We derive a simple procedure to filter the residuals in the presence of additional white noise or low frequencies. Our methodology can be applied to any kind of residuals, where the presence of additional noise and/or biases due to short samples or inaccurate functional modeling make the estimation of the Hurst coefficient with usual methods, such as maximum likelihood estimators, imprecise. We demonstrate the feasibility of our proposal with real observations from a white plate scanned by a TLS.

AB - Many signals appear fractal and have self-similarity over a large range of their power spectral densities. They can be described by so-called Hermite processes, among which the first order one is called fractional Brownian motion (fBm), and has a wide range of applications. The fractional Gaussian noise (fGn) series is the successive differences between elements of a fBm series; they are stationary and completely characterized by two parameters: the variance, and the Hurst coefficient (H). From physical considerations, the fGn could be used to model the noise of observations coming from sensors working with, e.g., phase differences: due to the high recording rate, temporal correlations are expected to have long range dependency (LRD), decaying hyperbolically rather than exponentially. For the rigorous testing of deformations detected with terrestrial laser scanners (TLS), the correct determination of the correlation structure of the observations is mandatory. In this study, we show that the residuals from surface approximations with regression B-splines from simulated TLS data allow the estimation of the Hurst parameter of a known correlated input noise. We derive a simple procedure to filter the residuals in the presence of additional white noise or low frequencies. Our methodology can be applied to any kind of residuals, where the presence of additional noise and/or biases due to short samples or inaccurate functional modeling make the estimation of the Hurst coefficient with usual methods, such as maximum likelihood estimators, imprecise. We demonstrate the feasibility of our proposal with real observations from a white plate scanned by a TLS.

KW - B-spline approximation

KW - Fractional gaussian noise

KW - Generalized hurst estimator

KW - Hurst exponent

KW - Stochastic model

KW - Terrestrial laser scanner

UR - http://www.scopus.com/inward/record.url?scp=85085579100&partnerID=8YFLogxK

U2 - 10.3390/MATH8050674

DO - 10.3390/MATH8050674

M3 - Article

VL - 8

JO - Mathematics

JF - Mathematics

IS - 5

M1 - 674

ER -