Numerical Differentiation of Noisy, Nonsmooth Data

By Rick Chartrand. 2011

495 citations recorded by the source corpus.

We consider the problem of differentiating a function specified by noisy data. Regularizing the differentiation process avoids the noise amplification of finite-difference methods. We use total-variation regularization, which allows for discontinuous solutions. The resulting simple algorithm accurately differentiates noisy functions, including those which have a discontinuous derivative.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Mathematical Physics

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Citations in the corpus

Discovering governing equations from data by sparse identification of nonlinear dynamical systems (2015, methodology): “To counteract this, we use the total variation regularized derivative [10] to de-noise the derivative.”
Data Driven Governing Equations Approximation Using Deep Neural Networks (2018, background): “Details of experimental measurement can be found in [10].”
Inferring Biological Networks by Sparse Identification of Nonlinear Dynamics (2016, methodology): “variation regularized derivative [62] provides a numerically robust method to compute derivatives from noisy data.”
Constrained sparse Galerkin regression (2016, methodology): “In practice, this may be computed directly from the data in X; for noisy data, the total-variation regularized derivative tends to provide numerically robust derivatives (Chartrand, 2011).”
METHODS FOR NUMERICAL DIFFERENTIATION OF NOISY DATA (2014, methodology): “A simple smoothing method consists of fitting the data points in a least squares sense with a sequence of polynomials f of degree d for d = 0, 1, . . . , k, where k is the smallest value for which the residual norm is bounded by a tolerance.”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “To approximate derivatives from noisy state measurements, the SINDY algorithm uses the total variation regularized derivative (Rudin et al., 1992; Chartrand, 2011).”
Data-driven discovery of Koopman eigenfunctions for control (2017, background): “The total variation derivative [28] is recommended for noise-corrupted measurements.”
Data-driven discovery of coordinates and governing equations (2019, background): “While this represents a challenge, approaches for estimating derivatives from noisy data such as the total variation regularized derivative can prove useful in providing derivative estimates [82].”
SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics (2020, methodology): “We investigate several approaches, including ﬁnite-difference and total-variation regularized difference (TVRegDiff) [54] derivatives.”
Learning partial differential equations via data discovery and sparse optimization (2017, methodology): “Smoothing approaches using other regularizers, such as the total variation semi-norm, could be used to better approximate the derivative of noisy data (see [38]).”