Numerical Differentiation Of Noisy Data Research Articles

Numerical differentiation of noisy data: A unifying multi-objective optimization framework.

Computing derivatives of noisy measurement data is ubiquitous in the physical, engineering, and biological sciences, and it is often a critical step in developing dynamic models or designing control. Unfortunately, the mathematical formulation of numerical differentiation is typically ill-posed, and researchers often resort to an ad hoc process for choosing one of many computational methods and its parameters. In this work, we take a principled approach and propose a multi-objective optimization framework for choosing parameters that minimize a loss function to balance the faithfulness and smoothness of the derivative estimate. Our framework has three significant advantages. First, the task of selecting multiple parameters is reduced to choosing a single hyper-parameter. Second, where ground-truth data is unknown, we provide a heuristic for selecting this hyper-parameter based on the power spectrum and temporal resolution of the data. Third, the optimal value of the hyper-parameter is consistent across different differentiation methods, thus our approach unifies vastly different numerical differentiation methods and facilitates unbiased comparison of their results. Finally, we provide an extensive open-source Python library pynumdiff to facilitate easy application to diverse datasets (https://github.com/florisvb/PyNumDiff).

METHOD FOR NUMERICAL DIFFERENTIATION OF NOISY DATA WITH OUTLIERS

Context. Using of conventional methods of numerical differentiation to the noisy data with outliers leads to significant errors. The objectof this study is the process of numerical differentiation of such data. Objective. The goal of this work is the development of a method of numerical differentiation of the noisy data with outliers to obtain a smooth approximation of the first derivative of original data as well as a smooth approximation of the original data themselves. Method. The proposed method of numerical differentiation is based on solving the problem of minimizing the smoothing functional, which is built on the criteria of a minimum of extent of the solution residual and of an energy constraint of the first derivative of solution. The minimum-extent criterion defines the main part of functional and ensures its stable behavior with respect to the additive noise and outliers. The energy constraint defines the stabilizing part of the functional and provides a smooth solution of the problem. The contribution of these parts is controlled by a regularization parameter. Since the main part of smoothing functional is not convex, then the minimization problem is the non-convex nonlinear programming problem. For the numerical solution of this problem the conjugate gradient method is used. In this method the step size along the descent direction is defined on the set of test steps. These steps minimize the individual components of the main and stabilizing parts of the smoothing functional that allows to move from the one local minimum of the functional to another deeper local minimum. Results. Simulation of the problem of numerical differentiation of noisy data with outliers and processing of the experimental data, which are photoluminescence spectra with narrow line components in their compositions, confirmed the performance of the proposed method. Conclusions. The proposed method can be used for numerical differentiation of noisy data with outliers. It provides a smooth approximation of the first derivative of the original data, as well as a smooth approximation of the original data themselves. This method can be generalized to the case of non-smooth solutions by constructing a stabilizing part of the functional based on the criterion of minimum total variation.

Smoothing via elliptic operators with application to edge detection

We propose a new regularization scheme for stable numerical differentiation of noisy data defined on a bounded domain with . The method generates a sequence of smoothed (regularized) data obtained by solving a perturbed elliptic boundary-value problem. Assuming the measured data are in and the true underlying data are sufficiently smooth, we prove convergence results in the -norm, provided the noisy data converge to true data in the sense. Using the finite element method, we derive error bounds and prove convergence theorems in the case of discrete data. Numerical examples indicate noteworthy results and shed light on some possible applications in image processing and computer vision.

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.

Reconstruction of the perfusion coefficient from temperature measurements using the conjugate gradient method

ABSTRACTWe consider the inverse bio-heat transfer problem to determine the space- and time-dependent perfusion coefficient from temperature measurements. In this formulation, the problem is fully determined and the coefficient is identifiable if and only if the temperature has dense support. However, the problem is still ill-posed since small errors in the measured temperature cause large errors in the output perfusion coefficient due to the numerical differentiation of noisy data involved which represents an unstable procedure. In order to overcome this difficulty and restore stability, we employ for the first time the conjugate gradient method (CGM) for solving the inverse problem under investigation. Regularization is achieved by stopping the iteration process at an appropriate threshold dictated by the discrepancy principle. Numerical results show that the CGM is accurate and reasonably stable in retrieving the perfusion coefficient. Moreover, comparison with other methods shows improved efficiency and stability in inverting noisy data.

Numerical and experimental assessment of the modal curvature method for damage detection in plate structures

This paper is concerned with the use of numerically obtained modal curvatures for damage detection in both isotropic and composite laminated plates. Numerical simulations are carried out by using COMSOL Multiphysics as FEM solver of the governing equations, in which a Mindlin-Reissner plate model is assumed and defects are introduced as localized smoothed variations of the baseline (healthy) configuration. Experiments are also performed on steel and aluminum plates using scanning laser vibrometry. This study confirms that the central difference method greatly amplifies the measurement errors and its application leads to ineffective predictions for damage detection, even after denoising. As a consequence, different numerical techniques should be explored to allow the use of numerically obtained modal curvatures for structural health monitoring. Herein, the Savitzky-Golay filter (or least-square smoothing filter) is considered for the numerical differentiation of noisy data. Our results show that this filter is effective in identifying the damaged regions of plate structures.

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.

Time-Domain Measurement of the Frequency-Dependent Delay of Broadband Antennas

We present a new time-domain technique for direct measurement of the frequency-dependent delay of broadband antenna structures. By utilizing a series of fixed-bandwidth, variable center frequency waveforms synthesized in a photonics-based electromagnetic pulse shaper, we perform high time resolution (~54 ps over ~10 ns) measurements of several broadband antenna structures. Our results show excellent agreement with the delay extracted via time-domain impulse response measurements, however, compared to conventional impulse response measurements, our method ensures a higher signal-to-noise ratio because it relies on several relatively narrowband measurements as opposed to a single broadband measurement. In addition, the delay uncertainty in our technique (~54 ps) is significantly below that of impulse response measurements (~300 ps) where numerical differentiation of noisy data leads to substantial errors. Our technique is well suited as a diagnostic tool for future variable-frequency fixed-bandwidth systems.

A scheme for stable numerical differentiation

A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the examples considered its solution is computed analytically. Some numerical results of its application are presented. These examples show that the proposed method for stable numerical differentiation is numerically more efficient than some other methods, in particular, than variational regularization.

Numerical differentiation of noisy data

Numerical differentiation of noisy data

Use of optimal estimation theory, in particular the Kalman filter, in data analysis and signal processing

The Kalman filter, a powerful and useful optimal estimation technique, does not seem to be widely known among physicists. Here we outline the derivation of the algorithm, and give three examples of its use: (a) in estimating the value of a constant, with both system and measurement noise, (b) in numerical differentiation of noisy data, and (c) in optimally estimating the amplitude of a signal with arbitrary but known time dependence superimposed on a noisy background.