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Numerical differentiation in python.

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Documentation Status PyPI MIT License

Numerical differentiation of noisy time series data in python

derivative is a Python package for differentiating noisy data. The package showcases a variety of improvements that can be made over finite differences when data is not clean.

Want to see an example of how derivative can help? This package is part of PySINDy (, a sparse-regression framework for discovering nonlinear dynamical systems from data.

This package binds common differentiation methods to a single easily implemented differentiation interface to encourage user adaptation. Numerical differentiation methods for noisy time series data in python includes:

  1. Symmetric finite difference schemes using arbitrary window size.
  2. Savitzky-Galoy derivatives (aka polynomial-filtered derivatives) of any polynomial order with independent left and right window parameters.
  3. Spectral derivatives with optional filter.
  4. Spline derivatives of any order.
  5. Polynomial-trend-filtered derivatives generalizing methods like total variational derivatives.
  6. Kalman derivatives find the maximum likelihood estimator for a derivative described by a Brownian motion.
from derivative import dxdt
import numpy as np

t = np.linspace(0,2*np.pi,50)
x = np.sin(x)

# 1. Finite differences with central differencing using 3 points.
result1 = dxdt(x, t, kind="finite_difference", k=1)

# 2. Savitzky-Golay using cubic polynomials to fit in a centered window of length 1
result2 = dxdt(x, t, kind="savitzky_golay", left=.5, right=.5, order=3)

# 3. Spectral derivative
result3 = dxdt(x, t, kind="spectral")

# 4. Spline derivative with smoothing set to 0.01
result4 = dxdt(x, t, kind="spline", s=1e-2)

# 5. Total variational derivative with regularization set to 0.01
result5 = dxdt(x, t, kind="trend_filtered", order=0, alpha=1e-2)

# 6. Kalman derivative with smoothing set to 1
result6 = dxdt(x, t, kind="kalman", alpha=1)


Thanks to the members of the community who have contributed!

Jacob Stevens-Haas Kalman derivatives #12


[1] Numerical differentiation of experimental data: local versus global methods- K. Ahnert and M. Abel

[2] Numerical Differentiation of Noisy, Nonsmooth Data- Rick Chartrand

[3] The Solution Path of the Generalized LASSO- R.J. Tibshirani and J. Taylor

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