piecewise-regression (aka segmented regression) in python
piecewise-regression: | fitting straight line models with breakpoints |
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Author: | Charlie Pilgrim |
Version: | 1.1.2 |
Github: | https://github.com/chasmani/piecewise-regression |
Documentation: | https://piecewise-regression.readthedocs.io/en/master/index.html |
Easy-to-use piecewise regression (aka segmented regression) in Python. For fitting straight lines to data where there is one or more changes in gradient (known as breakpoints). Based on Muggeo's paper "Estimating regression models with unknown break-points" (2003).
For example:
There are some code examples below, and more in this Google Colab Jupyter Notebook.
Installation
You can install piecewise-regression using python's pip package index
pip install piecewise-regression
The package is tested on Python 3.7, 3.8 and 3.9.
Getting started
The package requires some x and y data to fit. You also need to specify either a) some initial breakpoint guesses as start_values or b) how many breakpoints you want to fit as n_breakpoints (or both). Here is a very simple example, assuming we already have some data x and y:
import piecewise_regression pw_fit = piecewise_regression.Fit(x, y, n_breakpoints=2) pw_fit.summary()
Example
Here is a more detailed example. We start off generating some data with a breakpoint. This is for demonstration purposes, normally you will have your own data to fit:
import piecewise_regression import numpy as np alpha_1 = -4 alpha_2 = -2 constant = 100 breakpoint_1 = 7 n_points = 200 np.random.seed(0) xx = np.linspace(0, 20, n_points) yy = constant + alpha_1*xx + (alpha_2-alpha_1) * np.maximum(xx - breakpoint_1, 0) + np.random.normal(size=n_points)
Now we fit the model:
# Given some data, fit the model pw_fit = piecewise_regression.Fit(xx, yy, start_values=[5], n_breakpoints=1) # Print a summary of the fit pw_fit.summary()
Example output:
Breakpoint Regression Results ==================================================================================================== No. Observations 200 No. Model Parameters 4 Degrees of Freedom 196 Res. Sum of Squares 193.264 Total Sum of Squares 46201.8 R Squared 0.995817 Adjusted R Squared 0.995731 Converged: True ==================================================================================================== ==================================================================================================== Estimate Std Err t P>|t| [0.025 0.975] ---------------------------------------------------------------------------------------------------- const 100.726 0.244 413.63 3.1e-290 100.25 101.21 alpha1 -4.21998 0.0653 -64.605 4.37e-134 -4.3488 -4.0912 beta1 2.18914 0.0689 31.788 - 2.0533 2.325 breakpoint1 6.48706 0.137 - - 6.2168 6.7573 ---------------------------------------------------------------------------------------------------- These alphas(gradients of segments) are estimated from betas(change in gradient) ---------------------------------------------------------------------------------------------------- alpha2 -2.03084 0.0218 -93.068 3.66e-164 -2.0739 -1.9878 ==================================================================================================== Davies test for existence of at least 1 breakpoint: p=5.13032e-295 (e.g. p<0.05 means reject null hypothesis of no breakpoints at 5% significance)
This includes estimates for all the model variables, along with confidence intervals. The Davies test is a hypothesis test for the existence of at least one breakpoint, against the null hypothesis of no breakpoints.
There are also tools for plotting data:
import matplotlib.pyplot as plt # Plot the data, fit, breakpoints and confidence intervals pw_fit.plot_data(color="grey", s=20) # Pass in standard matplotlib keywords to control any of the plots pw_fit.plot_fit(color="red", linewidth=4) pw_fit.plot_breakpoints() pw_fit.plot_breakpoint_confidence_intervals() plt.xlabel("x") plt.ylabel("y") plt.show() plt.close()
You can extract data as well:
# Get the key results of the fit pw_results = pw_fit.get_results() pw_estimates = pw_results["estimates"]
How It Works
The package implements Muggeo's iterative algorithm (Muggeo "Estimating regression models with unknown break-points" (2003)), to quickly find breakpoints. That method simultaneously fits breakpoint positions and the linear models for the different segments of the fit. This method is quick and it gives confidence intervals for all the model estimates. See the accompanying paper for more details.
Muggeo's method doesn't always converge on the best solution - sometimes it finds a locally optimal solution or doesn't converge at all. For this reason the Fit method also implements a process called bootstrap restarting. This involves taking a bootstrap resample of the data, then using this bootstrapped data to try and find a better solution. The number of times this runs can be controlled with n_boot. To run the Fit without bootstrap restarting, set n_boot=0.
If you don't have good guesses for inital breakpoints, you can just set the number of e.g. n_breakpoints=3. In this case the algorithm will randomly generate start_values. The bootstrap restarting algorithm will now, with a 50% chance, either use the best currently converged breakpoints or randomly generate new start_values. In this way local optima are escaped in two ways and the algorithm will usually converge to a global optima. Even with these measures the algorithm is not completely guranteed to converge to global optima, as is often the case with fitting non-linear models. Increasing 'n_boot' increases the probability of global convergence, at the cost of computation time.
Model Selection
In addition to the main Fit tool, the package also offers a ModelSelection option based on the Bayesian Information Criterion. This is experimental and is not as thorough as the main Fit function. In particular, the models are generated with random start_values which can influence the model fit and give different values for the BIC. The tool can be useful for exploring posisble models, but should not at this point be used to choose the best model.
ms = piecewise_regression.ModelSelection(x, y, max_breakpoints=6)
This gives the following example output:
Breakpoint Model Comparision Results ==================================================================================================== n_breakpoints BIC converged RSS ---------------------------------------------------------------------------------------------------- 0 421.09 True 1557.4 1 14.342 True 193.26 2 22.825 True 191.23 3 24.169 True 182.59 4 29.374 True 177.73 5 False 6 False Minimum BIC (Bayesian Information Criterion) suggests the best model
The data of the model fits can be accessed in
ms.models
For a robust comparision, one could run the ModelSelection tools many times and take the lowest BIC for each model.
Testing
The package includes comprehensive tests.
To run all tests, from the main directory run (requires the pytest library):
pytest
To get code coverage, run (requires pytest and pytest-cov libraries):
pytest --cov=./
There are also a series of simulation tests that check the estimates have realistic confidence intervals, and the Davies test gives realistic p-values. These can be found in the folder "tests-manual".
Requirements
See requirements.txt for specific version numbers. Required packages, and their uses are:
- matplotlib for plotting.
- numpy for simple data handling and data transformations.
- scipy for statistical tests including using t-distributions and Gaussians.
- statsmodels for performing ordinary least squares.
Community Guidelines and Contributing
I welcome community participation:
- Open an issue on github if you want to suggest a new feature or report a bug
- If you want to make changes yourself, that is welcome via a pull request.
- Ideally, open an issue first before making a pull request with major changes.
Installing From Source
To install from source:
git clone https://github.com/chasmani/piecewise-regression cd piecewise_regression python3 setup.py install --user