Python module for performing linear regression for data with measurement errors and intrinsic scatter

Overview

Linear regression for data with measurement errors and intrinsic scatter (BCES)

Python module for performing robust linear regression on (X,Y) data points where both X and Y have measurement errors.

The fitting method is the bivariate correlated errors and intrinsic scatter (BCES) and follows the description given in Akritas & Bershady. 1996, ApJ. Some of the advantages of BCES regression compared to ordinary least squares fitting (quoted from Akritas & Bershady 1996):

  • it allows for measurement errors on both variables
  • it permits the measurement errors for the two variables to be dependent
  • it permits the magnitudes of the measurement errors to depend on the measurements
  • other "symmetric" lines such as the bisector and the orthogonal regression can be constructed.

In order to understand how to perform and interpret the regression results, please read the paper.

Installation

Using pip:

pip install bces

If that does not work, you can install it using the setup.py script:

python setup.py install

You may need to run the last command with sudo.

Alternatively, if you plan to modify the source then install the package with a symlink, so that changes to the source files will be immediately available:

python setup.py develop

Usage

import bces.bces as BCES
a,b,aerr,berr,covab=BCES.bcesp(x,xerr,y,yerr,cov)

Arguments:

  • x,y : 1D data arrays
  • xerr,yerr: measurement errors affecting x and y, 1D arrays
  • cov : covariance between the measurement errors, 1D array

If you have no reason to believe that your measurement errors are correlated (which is usually the case), you can provide an array of zeroes as input for cov:

cov = numpy.zeros_like(x)

Output:

  • a,b : best-fit parameters a,b of the linear regression such that y = Ax + B.
  • aerr,berr : the standard deviations in a,b
  • covab : the covariance between a and b (e.g. for plotting confidence bands)

Each element of the arrays a, b, aerr, berr and covab correspond to the result of one of the different BCES lines: y|x, x|y, bissector and orthogonal, as detailed in the table below. Please read the original BCES paper to understand what these different lines mean.

Element Method Description
0 y|x Assumes x as the independent variable
1 x|y Assumes y as the independent variable
2 bissector Line that bisects the y|x and x|y. This approach is self-inconsistent, do not use this method, cf. Hogg, D. et al. 2010, arXiv:1008.4686.
3 orthogonal Orthogonal least squares: line that minimizes orthogonal distances. Should be used when it is not clear which variable should be treated as the independent one

By default, bcesp run in parallel with bootstrapping.

Examples

bces-example.ipynb is a jupyter notebook including a practical, step-by-step example of how to use BCES to perform regression on data with uncertainties on x and y. It also illustrates how to plot the confidence band for a fit.

If you have suggestions of more examples, feel free to add them.

Running Tests

To test your installation, run the following command inside the BCES directory:

pytest -v

Requirements

See requirements.txt.

Citation

If you end up using this code in your paper, you are morally obliged to cite the following works

I spent considerable time writing this code, making sure it is correct and user-friendly, so I would appreciate your citation of the second paper in the above list as a token of gratitude.

If you are really happy with the code, you can buy me a beer.

Misc.

This python module is inspired on the (much faster) fortran routine originally written Akritas et al. I wrote it because I wanted something more portable and easier to use, trading off speed.

For a general tutorial on how to (and how not to) perform linear regression, please read this paper: Hogg, D. et al. 2010, arXiv:1008.4686. In particular, please refrain from using the bisector method.

If you want to plot confidence bands for your fits, have a look at nmmn package (in particular, modules nmmn.plots.fitconf and stats).

Bayesian linear regression

There are a couple of Bayesian approaches to perform linear regression which can be more powerful than BCES, some of which are described below.

A Gibbs Sampler for Multivariate Linear Regression: R code, arXiv:1509.00908. Linear regression in the fairly general case with errors in X and Y, errors may be correlated, intrinsic scatter. The prior distribution of covariates is modeled by a flexible mixture of Gaussians. This is an extension of the very nice work by Brandon Kelly (Kelly, B. 2007, ApJ).

LIRA: A Bayesian approach to linear regression in astronomy: R code, arXiv:1509.05778 Bayesian hierarchical modelling of data with heteroscedastic and possibly correlated measurement errors and intrinsic scatter. The method fully accounts for time evolution. The slope, the normalization, and the intrinsic scatter of the relation can evolve with the redshift. The intrinsic distribution of the independent variable is approximated using a mixture of Gaussian distributions whose means and standard deviations depend on time. The method can address scatter in the measured independent variable (a kind of Eddington bias), selection effects in the response variable (Malmquist bias), and departure from linearity in form of a knee.

AstroML: Machine Learning and Data Mining for Astronomy. Python example of a linear fit to data with correlated errors in x and y using AstroML. In the literature, this is often referred to as total least squares or errors-in-variables fitting.

Todo

If you have improvements to the code, suggestions of examples,speeding up the code etc, feel free to submit a pull request.

  • implement weighted least squares (WLS)
  • implement unit testing: bces
  • unit testing: bootstrap

Visit the author's web page and/or follow him on twitter (@nemmen).


Copyright (c) 2021, Rodrigo Nemmen. All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Owner
Rodrigo Nemmen
Professor of Astronomy & Astrophysics
Rodrigo Nemmen
Dieses Projekt ermöglicht es den Smartmeter der EVN (Netz Niederösterreich) über die Kundenschnittstelle auszulesen.

SmartMeterEVN Dieses Projekt ermöglicht es den Smartmeter der EVN (Netz Niederösterreich) über die Kundenschnittstelle auszulesen. Smart Meter werden

greenMike 43 Dec 04, 2022
Multiple Linear Regression using the LinearRegression class from sklearn.linear_model library

Multiple-Linear-Regression-master - A python program to implement Multiple Linear Regression using the LinearRegression class from sklearn.linear model library

Kushal Shingote 1 Feb 06, 2022
Machine-learning-dell - Repositório com as atividades desenvolvidas no curso de Machine Learning

📚 Descrição Neste curso da Dell aprofundamos nossos conhecimentos em Machine Learning. 🖥️ Aulas (Em curso) 1.1 - Python aplicado a Data Science 1.2

Claudia dos Anjos 1 Jan 05, 2022
Dragonfly is an open source python library for scalable Bayesian optimisation.

Dragonfly is an open source python library for scalable Bayesian optimisation. Bayesian optimisation is used for optimising black-box functions whose

744 Jan 02, 2023
This repository demonstrates the usage of hover to understand and supervise a machine learning task.

Hover Example Apps (works out-of-the-box on Binder) This repository demonstrates the usage of hover to understand and supervise a machine learning tas

Pavel 43 Dec 03, 2021
Nevergrad - A gradient-free optimization platform

Nevergrad - A gradient-free optimization platform nevergrad is a Python 3.6+ library. It can be installed with: pip install nevergrad More installati

Meta Research 3.4k Jan 08, 2023
Bayesian Additive Regression Trees For Python

BartPy Introduction BartPy is a pure python implementation of the Bayesian additive regressions trees model of Chipman et al [1]. Reasons to use BART

187 Dec 16, 2022
Upgini : data search library for your machine learning pipelines

Automated data search library for your machine learning pipelines → find & deliver relevant external data & features to boost ML accuracy :chart_with_upwards_trend:

Upgini 175 Jan 08, 2023
Given the names and grades for each student in a class N of students, store them in a nested list and print the name(s) of any student(s) having the second lowest grade.

Hackerank-Nested-List Given the names and grades for each student in a class N of students, store them in a nested list and print the name(s) of any s

Sangeeth Mathew John 2 Dec 14, 2021
Falken provides developers with a service that allows them to train AI that can play their games

Falken provides developers with a service that allows them to train AI that can play their games. Unlike traditional RL frameworks that learn through rewards or batches of offline training, Falken is

Google Research 223 Jan 03, 2023
pandas, scikit-learn, xgboost and seaborn integration

pandas, scikit-learn and xgboost integration.

299 Dec 30, 2022
Empyrial is a Python-based open-source quantitative investment library dedicated to financial institutions and retail investors

By Investors, For Investors. Want to read this in Chinese? Click here Empyrial is a Python-based open-source quantitative investment library dedicated

Santosh 640 Dec 31, 2022
Transpile trained scikit-learn estimators to C, Java, JavaScript and others.

sklearn-porter Transpile trained scikit-learn estimators to C, Java, JavaScript and others. It's recommended for limited embedded systems and critical

Darius Morawiec 1.2k Jan 05, 2023
WAGMA-SGD is a decentralized asynchronous SGD for distributed deep learning training based on model averaging.

WAGMA-SGD is a decentralized asynchronous SGD based on wait-avoiding group model averaging. The synchronization is relaxed by making the collectives externally-triggerable, namely, a collective can b

Shigang Li 6 Jun 18, 2022
Machine learning template for projects based on sklearn library.

Machine learning template for projects based on sklearn library.

Janez Lapajne 17 Oct 28, 2022
The Simpsons and Machine Learning: What makes an Episode Great?

The Simpsons and Machine Learning: What makes an Episode Great? Check out my Medium article on this! PROBLEM: The Simpsons has had a decline in qualit

1 Nov 02, 2021
DeepSpeed is a deep learning optimization library that makes distributed training easy, efficient, and effective.

DeepSpeed is a deep learning optimization library that makes distributed training easy, efficient, and effective. 10x Larger Models 10x Faster Trainin

Microsoft 8.4k Dec 30, 2022
A Python package to preprocess time series

Disclaimer: This package is WIP. Do not take any APIs for granted. tspreprocess Time series can contain noise, may be sampled under a non fitting rate

Maximilian Christ 57 Dec 17, 2022
Intel(R) Extension for Scikit-learn is a seamless way to speed up your Scikit-learn application

Intel(R) Extension for Scikit-learn* Installation | Documentation | Examples | Support | FAQ With Intel(R) Extension for Scikit-learn you can accelera

Intel Corporation 858 Dec 25, 2022
BigDL: Distributed Deep Learning Framework for Apache Spark

BigDL: Distributed Deep Learning on Apache Spark What is BigDL? BigDL is a distributed deep learning library for Apache Spark; with BigDL, users can w

4.1k Jan 09, 2023