nag_tsa_cp_pelt_user (g13nbc) 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, nag_tsa_cp_pelt_user (g13nbc) 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
On entry: , the length of the time series.
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.
On entry: the minimum distance between two change points, that is .
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.
– function, supplied by the userExternal Function
The cost function, . costfn must calculate a vector of costs for a number of segments.
On entry: time points which, along with ts, define the segments being considered, for .
On exit: the cost function, , with
where and .
It should be noted that if for any value of then it will be true for all values of . Therefore the inequality need only be tested once per call to costfn.
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to costfn.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling nag_tsa_cp_pelt_user (g13nbc) you may allocate memory and initialize these pointers with various quantities for use by costfn when called from nag_tsa_cp_pelt_user (g13nbc) (see Section 18.104.22.168 in How to Use the NAG Library and its Documentation).
– Integer *Input/Output
On entry: .
On exit: set info to a nonzero value if you wish nag_tsa_cp_pelt_user (g13nbc) to terminate with NE_USER_STOP.
Note:costfn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_tsa_cp_pelt_user (g13nbc). If your code inadvertently does return any NaNs or infinities, nag_tsa_cp_pelt_user (g13nbc) is likely to produce unexpected results.
– Integer *Output
On exit: , the number of change points detected.
On exit: the first elements of tau hold the location of the change points. The th segment is defined by to , where and .
The NAG communication argument (see Section 22.214.171.124 in How to Use the NAG Library and its Documentation).
– NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 126.96.36.199 in How to Use the NAG Library and its Documentation for further information.
On entry, argument had an illegal value.
On entry, . Constraint: .
On entry, . Constraint: .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
User requested termination.
Parallelism and Performance
nag_tsa_cp_pelt_user (g13nbc) is not threaded in any implementation.
nag_tsa_cp_pelt (g13nac) 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 nag_tsa_cp_pelt (g13nac) 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 .