NAME

       math::changepoint - Change point detection methods

SYNOPSIS

       package require Tcl  8.6

       package require TclOO

       package require math::statistics

       package require math::changepoint  ?0.1?

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

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

       $cusumObj examine value

       $cusumObj reset

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

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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 (c) 2020 by Arjen Markus