g13nbf detects change points in a univariate time series, that is, the time points at which some feature of the data, for example the mean, changes. Change points are detected using the PELT (Pruned Exact Linear Time) algorithm and a user-supplied cost function.
Let denote a series of data and denote a set of ordered (strictly monotonic increasing) indices known as change points with and . For ease of notation we also define . The change points, , split the data into segments, with the th segment being of length and containing .
Given a user-supplied cost function, g13nbf solves
where is a penalty term used to control the number of change points. This minimization is performed using the PELT algorithm of Killick et al. (2012). The PELT algorithm is guaranteed to return the optimal solution to (1) if there exists a constant such that
Chen J and Gupta A K (2010) Parametric Statistical Change Point Analysis With Applications to GeneticsMedicine and FinanceSecond Edition Birkhäuser
Killick R, Fearnhead P and Eckely I A (2012) Optimal detection of changepoints with a linear computational cost Journal of the American Statistical Association107:500 1590–1598
1: – IntegerInput
On entry: , the length of the time series.
2: – Real (Kind=nag_wp)Input
On entry: , the penalty term.
There are a number of standard ways of setting , including:
SIC or BIC
where is the number of parameters being treated as estimated in each segment. The value of will depend on the cost function being used.
If no penalty is required then set . Generally, the smaller the value of the larger the number of suggested change points.
3: – IntegerInput
On entry: the minimum distance between two change points, that is .
4: – Real (Kind=nag_wp)Input
On entry: , the constant value that satisfies equation (2). If exists, it is unlikely to be unique in such cases, it is recommened that the largest value of , that satisfies equation (2), is chosen. No check is made that is the correct value for the supplied cost function.
5: – Subroutine, supplied by the user.External Procedure
The cost function, . costfn must calculate a vector of costs for a number of segments.
On exit: set info to a nonzero value if you wish g13nbf to terminate with .
costfn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which g13nbf is called. Arguments denoted as Input must not be changed by this procedure.
Note:costfn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g13nbf. If your code inadvertently does return any NaNs or infinities, g13nbf is likely to produce unexpected results.
6: – IntegerOutput
On exit: , the number of change points detected.
7: – Integer arrayOutput
On exit: the first elements of tau hold the location of the change points. The th segment is defined by to , where and .
y is not used by g13nbf, but is passed directly to costfn and may be used to pass information to this routine. y will usually be used to pass (functions of) the time series, of interest.
9: – Integer arrayUser Workspace
10: – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by g13nbf, but are passed directly to costfn and may be used to pass information to this routine.
11: – IntegerInput/Output
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, . Constraint: .
On entry, . Constraint: .
User requested termination.
An unexpected error has been triggered by this routine. Please
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
Parallelism and Performance
g13nbf is not threaded in any implementation.
g13naf performs the same calculations for a cost function selected from a provided set of cost functions. If the required cost function belongs to this provided set then g13naf can be used without the need to provide a cost function routine.
This example identifies changes in the scale parameter, under the assumption that the data has a gamma distribution, for a simulated dataset with observations. A penalty, of is used and the minimum segment size is set to . The shape parameter is fixed at across the whole input series.
The cost function used is
where is a shape parameter that is fixed for all segments and .