maxsmooth: Derivative Constrained Function Fitting

Introduction

maxsmooth:	Derivative Constrained Function Fitting
Author:	Harry Thomas Jones Bevins
Version:	1.2.1
Homepage:	https://github.com/htjb/maxsmooth
Documentation:	https://maxsmooth.readthedocs.io/

Installation

In the following two sections we highlight the purpose of maxsmooth and show an example. To install the software follow these instructions:

The software can be pip installed from the PYPI repository like so,

pip install maxsmooth

or alternatively it can be installed from the git repository via,

git clone https://github.com/htjb/maxsmooth.git
cd maxsmooth
python setup.py install --user

Derivative Constrained Functions and `maxsmooth`

maxsmooth is an open source software, written in Python (supporting version 3 upwards), for fitting derivative constrained functions (DCFs) such as Maximally Smooth Functions (MSFs) to data sets. MSFs are functions for which there are no zero crossings in derivatives of order m >= 2 within the domain of interest. More generally for DCFs the minimum constrained derivative order, m can take on any value or a set of specific high order derivatives can be constrained. They are designed to prevent the loss of signals when fitting out dominant smooth foregrounds or large magnitude signals that mask signals of interest. Here "smooth" means that the foregrounds follow power law structures in the band of interest. In some cases DCFs can be used to highlight systematics in the data.

maxsmooth uses quadratic programming implemented with CVXOPT to fit data subject to a fixed linear constraint, Ga <= 0, where the product Ga is a matrix of derivatives. The constraint on an MSF are not explicitly linear and each constrained derivative can be positive or negative. maxsmooth is, however, designed to test the <= 0 constraint multiplied by a positive or negative sign. Where a positive sign in front of the m^th order derivative forces the derivative to be negative for all x. For an N^th order polynomial maxsmooth can test every available sign combination but by default it implements a sign navigating algorithm. This is detailed in the maxsmooth paper (see citation), is summarized below and in the software documentation.

The available sign combinations act as discrete parameter spaces all with global minima and maxsmooth is capable of finding the minimum of these global minima by implementing a cascading algorithm which is followed by a directional exploration. The cascading routine typically finds an approximate to the global minimum and then the directional exploration is a complete search of the sign combinations in the neighbourhood of that minimum. The searched region is limited by factors that encapsulate enough of the neighbourhood to confidently return the global minimum.

The sign navigating method is reliant on the problem being "well defined" but this is not always the case and it is in these instances it is possible to run the code testing every available sign combination on the constrained derivatives. For a definition of a "well defined" problem and it's counter part see the maxsmooth paper and the documentation.

maxsmooth features a built in library of DCFs or allows the user to define their own. The addition of possible inflection points and zero crossings in higher order derivatives is also available to the user. The software has been designed with these two applications in mind and is a simple interface.

Example Fit

Shown below is an example MSF fit performed with maxsmooth to data that follows a y = x^-2.5 power law with a randomly generated Gaussian noise with a standard deviation 0.02. The top panel shows the data and the bottom panel shows the residual after subtraction of the MSF fit alongside the actual noise in the data. The software using the default built-in DCF model is shown to be capable of recovering the random noise.

Further examples can be found in the Documentation (https://maxsmooth.readthedocs.io/) and in the github repository in the files 'example_codes/' and 'example_notebooks/' (notebooks can also be accessed online here).

Licence and Citation

The software is free to use on the MIT open source license. However if you use the software for academic purposes we request that you cite the maxsmooth papers. They are detailed below.

MNRAS paper (referred to throughout the documentation as the maxsmooth paper),

H. T. J. Bevins et al., maxsmooth: Rapid maximally smooth function fitting with applications in Global 21-cm cosmology, Monthly Notices of the Royal Astronomical Society, 2021;, stab152, https://doi.org/10.1093/mnras/stab152

Below is the BibTex citation,

@article{10.1093/mnras/stab152,
  author = {Bevins, H T J and Handley, W J and Fialkov, A and Acedo, E de Lera and Greenhill, L J and Price, D C},
  title = "{maxsmooth: rapid maximally smooth function fitting with applications in Global 21-cm cosmology}",
  journal = {Monthly Notices of the Royal Astronomical Society},
  year = {2021},
  month = {01},
  issn = {0035-8711},
  doi = {10.1093/mnras/stab152},
  url = {https://doi.org/10.1093/mnras/stab152},
  note = {stab152},
  eprint = {https://academic.oup.com/mnras/advance-article-pdf/doi/10.1093/mnras/stab152/35931358/stab152.pdf},
}

JOSS paper,

Bevins, H. T., (2020). maxsmooth: Derivative Constrained Function Fitting. Journal of Open Source Software, 5(54), 2596, https://doi.org/10.21105/joss.02596

and the BibTex,

@article{Bevins2020,
    doi = {10.21105/joss.02596},
    url = {https://doi.org/10.21105/joss.02596},
    year = {2020},
    publisher = {The Open Journal},
    volume = {5},
    number = {54},
    pages = {2596},
    author = {Harry T. j. Bevins},
    title = {maxsmooth: Derivative Constrained Function Fitting},
    journal = {Journal of Open Source Software}
}

Contributing

Contributions to maxsmooth are welcome and can be made via:

Opening an issue to purpose new features/report bugs.
Making a pull request. Please consider opening an issue to discuss any proposals beforehand and ensure that your PR will be accepted.

An example contribution may be the addition of a basis function into the standard library.

Documentation

The documentation is available at: https://maxsmooth.readthedocs.io/

Alternatively, it can be compiled locally from the git repository and requires sphinx to be installed. You can do this via:

cd docs/
make SOURCEDIR=source html

cd docs/
make SOURCEDIR=source latexpdf

The resultant docs can be found in the docs/_build/html/ and docs/_build/latex/ respectively.

Requirements

To run the code you will need the following additional packages:

When installing via pip or from source using the setup.py file the above packages will also be installed if absent.

To compile the documentation locally you will need:

To run the test suit you will need:

pytest

Basin-hopping/Nelder-Mead Code

In the maxsmooth MNRAS paper and JOSS paper we provide a comparison of maxsmooth to a Basin-hopping/Nelder-Mead approach for fitting DCFs. For completeness we provide in this repo the code used to make this comparison in the file 'Basin-hopping_Nelder_Mead/'.

The code times_chis.py is used to call maxsmooth and the Basin-hopping methods (in the file 'BHNM/'). It will plot the recorded times and objective function evaluations.

The Basin-hopping/Nelder-Mead code is designed to fit MSFs and is not generalised to all types of DCF. It is also not documented, however there are minor comments in the script and it should be self explanatory. Questions on this are welcome and can be posted as an issue or by contacting the author.

htjb/maxsmooth