piecewise-regression 0.2.2

piecewise (segmented) regression in python

Piecewise (aka segmented) regression in Python. Simultaneously find breakpoints and straightline segments between those breakpoints. Based on Muggeo “Estimating regression models with unknown break-points” (2003)

Installation

You can install piecewise-regression from PyPI

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:

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 genreating some data with a breakpoint, for demonstration purposes:

import piecewise_regression
import numpy as np

alpha_1 = -4
alpha_2 = -2
intercept = 100
breakpoint_1 = 7
n_points = 200
np.random.seed(0)
xx = np.linspace(0, 20, n_points)
yy = intercept + 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
bp_fit = Fit(xx, yy, start_values=[5], n_breakpoints=1)

# Print a summary of the fit
bp_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)

There are also tools for plotting data:

import matplotlib.pyplot as plt

# Plot the data, fit, breakpoints and confidence intervals
bp_fit.plot_data(color="grey", s=20)
# Pass in standard matplotlib keywords to control any of the plots
bp_fit.plot_fit(color="red", linewidth=4)
bp_fit.plot_breakpoints()
bp_fit.plot_breakpoint_confidence_intervals()
plt.xlabel("x")
plt.ylabel("y")
plt.show()
plt.close()

How It Works

The package implements Muggeo’s iterative algorithm (Muggeo “Estimating regression models with unknown break-points” (2003)), to quickly find breakpoints. The Fit method also implements a non-parametric bootstrap restarting to escape local minima, this can be controlled with n_boot. To run the Fit without bootstrap restarting, set n_boot=0. Muggeo’s algorthm does not always converge. In this case, the Fit method will keep trying to find a fit using bootstrap restarting n_boot times.

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 slution 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 = 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 simluation 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.