piecewise-regression 1.0.3

piecewise (segmented) regression in python

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 “Estimating regression models with unknown break-points” (2003). For example:

Installation

You can install piecewise-regression using python’s pip package index

pip install piecewise-regression

The package was developed and tested on Python 3.7.

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 starting breakpoints until it finds a solution that converges (up to n_boot times). This is a good option if the algorithm is otherwise not converging.

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 not as thorough as the main Fit tool:

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

Testing

The package includes comprehensive tests.

To run all tests, from the main directory run:

python3 -m "nose"

Note: This requires nosetests, can be downloaded from apt with:

sudo apt install python3-nose

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”.

Documentation

Full docs, including an API reference.