PyNumDiff

Python methods for numerical differentiation of noisy data, including multi-objective optimization routines for automated parameter selection.

Table of contents

Introduction

PyNumDiff is a Python package that implements various methods for computing numerical derivatives of noisy data, which can be a critical step in developing dynamic models or designing control. There are four different families of methods implemented in this repository: smoothing followed by finite difference calculation, local approximation with linear models, Kalman filtering based methods and total variation regularization methods. Most of these methods have multiple parameters involved to tune. 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. For more details, refer to this paper.

Structure

PyNumDiff/
  |- README.md
  |- pynumdiff/
     |- __init__.py
     |- __version__.py
     |- finite_difference/
     |- kalman_smooth/
     |- linear_model/
     |- smooth_finite_difference/
     |- total_variation_regularization/
     |- utils/
     |- optimize/
        |- __init__.py
        |- __optimize__.py
        |- finite_difference/
        |- kalman_smooth/
        |- linear_model/
        |- smooth_finite_difference/
        |- total_variation_regularization/
     |- tests/
  |- examples
     |- 1_basic_tutorial.ipynb
     |- 2a_optimizing_parameters_with_dxdt_known.ipynb
     |- 2b_optimizing_parameters_with_dxdt_unknown.ipynb
  |- docs/
     |- Makefile
     |- make.bat
     |- build/
     |- source/
        |- _static
        |- _summaries
        |- conf.py
        |- index.rst
        |- ...
  |- setup.py
  |- .gitignore
  |- .travis.yml
  |- LICENSE.txt
  |- requirements.txt

Getting Started

Prerequisite

PyNumDiff requires common packages like numpy, scipy, matplotlib, pytest (for unittests), pylint (for PEP8 style check). For a full list, you can check the file requirements.txt

In addition, it also requires certain additional packages for select functions, though these are not required for a successful install of PyNumDiff:

  • Total Variation Regularization methods: cvxpy

  • Linear Model Chebychev: pychebfun

When using cvxpy, our default solver is set to be MOSEK (highly recommended), you would need to download their free academic license from their website. Otherwise, you can also use other solvers which are listed here.

Installing

The code is compatible with >=Python 3.5. It can be installed using pip or directly from the source code. Basic installation options include:

  • From PyPI using pip: pip install pynumdiff. May require pre-installing numpy, scipy, matplotlib.

  • From source using pip git+: pip install git+https://github.com/florisvb/PyNumDiff

  • From local source code using setup.py: requires pre-installing numpy, scipy, matplotlib. Then run python ./setup.py install from inside this directory. See below for example.

Installation of the optional packages such as cvxpy can be tricky because cvxpy requires pythonX-dev packages. Depending on your version of Ubuntu it can be challenging to meet all the right requirements and installation options (e.g. it is difficult to install python3.6-dev on Ubuntu 16.04). Here are several tested example installation workflows:

Complete install on Ubuntu 16.04 using python3.5 in blank virtual environment using pip git+:

sudo apt-get install python3.5-dev
python3.5 -m venv ~/PYNUMDIFF35
source ~/PYNUMDIFF35/bin/activate
pip install --upgrade pip
pip install --upgrade pip
pip install git+https://github.com/florisvb/PyNumDiff
pip install git+https://github.com/pychebfun/pychebfun
pip install cvxpy
pip install git+http://github.com/MOSEK/Mosek.pip

Complete install on Ubuntu 18.04 using python3.6 in blank virtual environment using pip git+:

sudo apt-get install python3.6-dev
python3.6 -m venv ~/PYNUMDIFF36
source ~/PYNUMDIFF36/bin/activate
pip install --upgrade pip
pip install git+https://github.com/florisvb/PyNumDiff
pip install git+https://github.com/pychebfun/pychebfun
pip install cvxpy
pip install Mosek

Complete install on Ubuntu 16.04 using python3.5 in blank virtual environment using setup.py:

sudo apt-get install python3.5-dev
python3.5 -m venv ~/PYNUMDIFF35
source ~/PYNUMDIFF35/bin/activate
pip install --upgrade pip
pip install --upgrade pip
pip install numpy scipy matplotlib
python ./setup.py install
pip install git+https://github.com/pychebfun/pychebfun
pip install cvxpy
pip install git+http://github.com/MOSEK/Mosek.pip

Note: If using the optional MOSEK solver for cvxpy you will also need a MOSEK license, free academic license.

Usage

Basic usages

  • Basic Usage: you provide the parameters

    x_hat, dxdt_hat = pynumdiff.sub_module.method(x, dt, params, options)

  • Advanced usage: automated parameter selection through multi-objective optimization

    params, val = pynumdiff.optimize.sub_module.method(x, dt, params=None,
                                                   tvgamma=tvgamma, # hyperparameter
                                                   dxdt_truth=None, # no ground truth data
                                                   options={})
print('Optimal parameters: ', params)
x_hat, dxdt_hat = pynumdiff.sub_module.method(x, dt, params, options={'smooth': True})`

Notebook examples

Important notes

  • Larger values of tvgamma produce smoother derivatives

  • The value of tvgamma is largely universal across methods, making it easy to compare method results

  • The optimization is not fast. Run it on subsets of your data if you have a lot of data. It will also be much faster with faster differentiation methods, like savgoldiff and butterdiff, and probably too slow for sliding methods like sliding DMD and sliding LTI fit.

  • The following heuristic works well for choosing tvgamma, where cutoff_frequency is the highest frequency content of the signal in your data, and dt is the timestep: tvgamma=np.exp(-1.6*np.log(cutoff_frequency)-0.71*np.log(dt)-5.1)

Running the tests

We are using Travis CI for continuous intergration testing. You can check out the current status here.

To run tests locally, type:

> pytest pynumdiff

Citation

@ARTICLE{9241009, author={F. {van Breugel} and J. {Nathan Kutz} and B. W. {Brunton}}, journal={IEEE Access}, title={Numerical differentiation of noisy data: A unifying multi-objective optimization framework}, year={2020}, volume={}, number={}, pages={1-1}, doi={10.1109/ACCESS.2020.3034077}}

Developer’s Guide