Differentiation Of Noisy Data Research Articles

Multiquadric trigonometric spline quasi-interpolation for numerical differentiation of noisy data: a stochastic perspective

Based on multiquadric trigonometric spline quasi-interpolation, the paper proposes a scheme for numerical differentiation of noisy data, which is a well-known ill-posed problem in practical applications. In addition, in the perspective of kernel regression, the paper studies its large sample properties including optimal bandwidth selection, convergence rate, almost sure convergence, and uniformly asymptotic normality. Simulations are provided at the end of the paper to demonstrate features of the scheme. Both theoretical results and simulations show that the scheme is simple, easy to compute, and efficient for numerical differentiation of noisy data.

A Fictitious Time Integration Method for the Numerical Solution of the Fredholm Integral Equation and for Numerical Differentiation of Noisy Data, and Its Relation to the Filter Theory

The Fictitious Time Integration Method (FTIM) previously developed by Liu and Atluri (2008a) is employed here to solve a system of ill-posed linear algebraic equations, which may result from the discretization of a first-kind lin- ear Fredholm integral equation. We rationalize the mathematical foundation of the FTIM by relating it to the well-known filter theory. For the linear ordinary differ- ential equations which are obtained through the FTIM (and which are equivalently used in FTIM to solve the ill-posed linear algebraic equations), we find that the fictitous time plays the role of a regularization parameter, and its filtering effect is better than that of the Tikhonov and the exponential filters. Then, we apply this new method to solve the problem of numerical differentiation of noisy data (such as finding da=dN in fatigue, where a is the measured crack-length and N is the number of load cycles), and the inversion of the Abel integral equation under noise. It is established that the numerical method of FTIM is robust against the noise.

Estimation of air‐sea heat flux from ocean measurements: An ill‐posed problem

In this paper we addressed that the estimation of ocean surface heat flux from ocean temperature data is an ill‐posed inverse problem, just like a well‐known ill‐posed problem: differentiation of noisy data. We reviewed engineering literature on such problems. Using a Mellor‐Yamada level 2.5 closure model and simulated temperature data, we conducted numerical experiments of retrieving heat fluxes using variational data assimilation. This study shows that estimated nonsolar heat fluxes with high time resolution have large errors even if the observation errors are small. We also discussed the ill‐posedness based on sensitivity coefficients. However, by applying some regularization methods and a longer assimilation window, it is feasible to estimate heat fluxes with reasonable accuracies, at least at some lower temporal resolution. On the basis of our model and experiment configurations, the high nonlinearity of the ocean mixed layer model also can cause difficulty in optimization when the assimilation window is long (e.g., 7 days). We proposed a modified an adjoint method approach and yielded good results.

Smoothing and differentiating load–displacement data using a low‐pass filter for improved crack opening load estimates

The presence of even small amounts of noise in experimental data significantly influences numerically determined derivatives. Dynamic programming is used to construct a low‐pass filter enabling the smoothing and differentiation of noisy data. The filter is used to smooth and differentiate load–displacement data for a propagating fatigue crack to obtain smoothed compliance as a function of applied load. The smoothed compliance is shown to allow for improved subsequent estimates of the crack opening load.

Digital differentiation of noisy data measured through a dynamic system

The design of discrete time differentiating filters, in the presence of colored noise and nonneglectable transducer dynamics, is investigated. The signal and noise are described by autoregressive moving average (ARMA) models, possibly with poles on mod z mod =1. The mean square error (MSE) optimal filter, based on a discrete time approximation of the derivative operator, is given by a spectral factorization and a linear polynomial equation. >

Numerical differentiation of noisy data

Numerical differentiation of noisy data

Smoothing noisy data using dynamic programming and generalized cross-validation.

Smoothing and differentiation of noisy data using spline functions requires the selection of an unknown smoothing parameter. The method of generalized cross-validation provides an excellent estimate of the smoothing parameter from the data itself even when the amount of noise associated with the data is unknown. In the present model only a single smoothing parameter must be obtained, but in a more general context the number may be larger. In an earlier work, smoothing of the data was accomplished by solving a minimization problem using the technique of dynamic programming. This paper shows how the computations required by generalized cross-validation can be performed as a simple extension of the dynamic programming formulas. The results of numerical experiments are also included.