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# math::changepoint(n) 0.2 tcllib "Tcl Math Library"

## Name

math::changepoint - Change point detection methods

## Synopsis

- package require Tcl 8.6 9
- package require TclOO
- package require math::statistics
- package require math::changepoint ?0.2?

- ::math::changepoint::cusum-detect data ?args?
- ::math::changepoint::cusum-online ?args?
- $cusumObj examine value
- $cusumObj reset
- ::math::changepoint::binary-segmentation data ?args?

## Description

The math::changepoint package implements a number of well-known methods to determine if a series of data contains a shift in the mean or not. Note that these methods only indicate if a shift in the mean is probably. Due to the stochastic nature of the data that will be analysed, false positives are possible. The CUSUM method is implemented in both an "offline" and an "online" version, so that it can be used either for a complete data series or for detecting changes in data that come in one by one. The implementation has been based on these websites mostly:

- https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc323.htm

- https://en.wikipedia.org/wiki/CUSUM

Basically, the deviation of the data from a given target value is accumulated and when the total deviation becomes too large, a change point is reported. A second method, binary segmentation, is implemented only as an "offline" method, as it needs to examine the data series as a whole. In the variant contained here the following ideas have been used:

- The segments in which the data series may be separated shold not be too short, otherwise the ultimate result could be segments of only one data point long. So a minimum length is used.

- To make the segmentation worthwhile there should be a minimum gain in reducing the cost function (the sum of the squared deviations from the mean for each segment).

This may not be in agreement with the descriptions of the method found in various publications, but it is simple to understand and intuitive. One publication that provides more information on the method in general is "Selective review of offline change point detection methods" by Truong et al. https://arxiv.org/abs/1801.00718.

## PROCEDURES

The package defines the following public procedures:

::math::changepoint::cusum-detect data ?args?

  Examine a given data series and return the location of the first change (if any)

  double data

    Series of data to be examined

  list args

    Optional list of key-value pairs:

    -target value

      The target (or mean) for the time series

    -tolerance value

      The tolerated standard deviation

    -kfactor value

      The factor by which to multiply the standard deviation (defaults to 0.5, typically between 0.5 and 1.0)

    -hfactor value

      The factor determining the limits betweem which the "cusum" statistic is accepted (typicaly 3.0-5.0, default 4.0)

::math::changepoint::cusum-online ?args?

  Class to examine data passed in against expected properties. At least the keywords -target and -tolerance must be given.

  list args

    List of key-value pairs:

    -target value

      The target (or mean) for the time series

    -tolerance value

      The tolerated standard deviation

    -kfactor value

      The factor by which to multiply the standard deviation (defaults to 0.5, typically between 0.5 and 1.0)

    -hfactor value

      The factor determining the limits betweem which the "cusum" statistic is accepted (typicaly 3.0-5.0, default 4.0)

$cusumObj examine value

  Pass a value to the cusum-online object and examine it. If, with this new value, the cumulative sum remains within the bounds, zero (0) is returned, otherwise one (1) is returned.

  double value

    The new value

$cusumObj reset

  Reset the cumulative sum, so that the examination can start afresh.

::math::changepoint::binary-segmentation data ?args?

  Apply the binary segmentation method recursively to find change points. Returns a list of indices of potential change points

  list data

    Data to be examined

  list args

    Optional key-value pairs:

    -minlength number

      Minimum number of points in each segment (default: 5)

    -threshold value

      Factor applied to the standard deviation functioning as a threshold for accepting the change in cost function as an improvement (default: 1.0)

## Keywords

control, statistics

## Category

Mathematics

## Copyright

Copyright © 2020 by Arjen Markus