Signal Processing Algorithms

A suite of algorithms implementing Energy Statistics, E-Divisive with Means and Generalized ESD Test for Outliers in python.

Getting Started - Users

pip install signal-processing-algorithms

Getting Started - Developers

Getting the code:

$ git clone git@github.com:mongodb/signal-processing-algorithms.git
$ cd signal-processing-algorithms

Installation

$ pip install poetry
$ poetry install

Testing/linting:

$ poetry run pytest

Running the slow tests:

$ poetry run pytest --runslow

Some of the larger tests can take a significant amount of time (more than 2 hours).

Energy statistics

Energy Statistics is the statistical concept of Energy Distance and can be used to measure how similar/different two distributions are.

For statistical samples from two random variables X and Y: x1, x2, ..., xn and y1, y2, ..., yn

E-Statistic is defined as:

$E_{n,m}(X,Y):=2A-B-C$

where,

$A:={\frac {1}{nm}}\sum _{i=1}^{n}\sum _{j=1}^{m}\|x_{i}-y_{j}\|,B:={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\|x_{i}-x_{j}\|,C:={\frac {1}{m^{2}}}\sum _{i=1}^{m}\sum _{j=1}^{m}\|y_{i}-y_{j}\|$

T-statistic is defined as:

$T={\frac {nm}{n+m}}E_{{n,m}}(X,Y)$

E-coefficient of inhomogeneity is defined as:

$H=\frac{2E||X-Y|| - E||X-X'|| - E||Y-Y'||}{2E||X-Y||}$

from signal_processing_algorithms.energy_statistics import energy_statistics
from some_module import series1, series2

# To get Energy Statistics of the distributions.
es = energy_statistics.get_energy_statistics(series1, series2)

# To get Energy Statistics and permutation test results of the distributions.
es_with_probabilities = energy_statistics.get_energy_statistics_and_probabilities(series1, series2, permutations=100)

Intro to E-Divisive

Detecting distributional changes in a series of numerical values can be surprisingly difficult. Simple systems based on thresholds or mean values can be yield false positives due to outliers in the data, and will fail to detect changes in the noise profile of the series you are analyzing.

One robust way of detecting many of the changes missed by other approaches is to use E-Divisive with Means, an energy statistic based approach that compares the expected distance (Euclidean norm) between samples of two portions of the series with the expected distance between samples within those portions.

That is to say, assuming that the two portions can each be modeled as i.i.d. samples drawn from distinct random variables (X for the first portion, Y for the second portion), you would expect the E-statistic to be non-zero if there is a difference between the two portions:

$E_{n,m}(X,Y):=2A-B-C$ where A, B and C are as defined in the Energy Statistics above.

One can prove that ${E_{n,m}(X,Y)\geq 0}$ and that the corresponding population value is zero if and only if X and Y have the same distribution. Under this null hypothesis the test statistic

$T={\frac {nm}{n+m}}E_{{n,m}}(X,Y)$

converges in distribution to a quadratic form of independent standard normal random variables. Under the alternative hypothesis T tends to infinity. This makes it possible to construct a consistent statistical test, the energy test for equal distributions

Thus for a series Z of length L,

$Z = \{Z_{1}, ..., Z_{\tau} , ... , Z_{L}\}, X =\{Z_{1},...,Z_{\tau}\}, Y=\{Z_{\tau+1}\,...,Z_{L}\}$

we find the most likely change point by solving the following for τ such that it has the maximum T statistic value.

Multiple Change Points

The algorithm for finding multiple change points is also simple.

Assuming you have some k known change points:

Partition the series into segments between/around these change points.
Find the maximum value of our divergence metric within each partition.
Take the maximum of the maxima we have just found --> this is our k+1th change point.
Return to step 1 and continue until reaching your stopping criterion.

Stopping Criterion

In this package we have implemented a permutation based test as a stopping criterion:

After step 3 of the multiple change point procedure above, randomly permute all of the data within each cluster, and find the most likely change point for this permuted data using the procedure laid out above.

After performing this operation z times, count the number of permuted change points z' that have higher divergence metrics than the change point you calculated with un-permuted data. The significance level of your change point is thus z'/(z+1).

We allow users to configure a permutation tester with pvalue and permutations representing the significance cutoff for algorithm termination and permutations to perform for each test, respectively.

Example

from signal_processing_algorithms.energy_statistics import energy_statistics
from some_module import series

change_points = energy_statistics.e_divisive(series, pvalue=0.01, permutations=100)

angus-c / signal-processing-algorithms