nag_tsa_cp_binary_user (g13nec) 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 binary segmentation for 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 cost function, , nag_tsa_cp_binary_user (g13nec) gives an approximate solution to
where is a penalty term used to control the number of change points. The solution is obtained in an iterative manner as follows:
Set , and
Set . If , where is a user-supplied control parameter, then terminate the process for this segment.
Find that minimizes
If inequality (1) is false then the process is terminated for this segment.
If inequality (1) is true, then is added to the set of change points, and the segment is split into two subsegments, and . The whole process is repeated from step 2 independently on each subsegment, with the relevant changes to the definition of and (i.e., is set to when processing the left hand subsegment and is set to when processing the right hand subsegment.
The change points are ordered to give .
Chen J and Gupta A K (2010) Parameteric Statisical Change Point Analysis With Applications to GeneticsMedicine and FinanceSecond Edition Birkhäuser
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 maximum depth for the iterative process, which in turn puts an upper limit on the number of change points with .
If then no limit is put on the depth of the iterative process and no upper limit is put on the number of change points, other than that inherent in the length of the series and the value of minss.
– function, supplied by the userExternal Function
chgpfn must calculate a proposed change point, and the associated costs, within a specified segment.
if then , the proposed change point. That is, the value which minimizes
for to .
On exit: costs associated with the proposed change point, .
If then and the remaining two elements of cost need not be set.
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to chgpfn.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling nag_tsa_cp_binary_user (g13nec) you may allocate memory and initialize these pointers with various quantities for use by chgpfn when called from nag_tsa_cp_binary_user (g13nec) (see Section 220.127.116.11 in the Essential Introduction).
– Integer *Input/Output
On entry: .
On exit: in most circumstances info should remain unchanged.
If info is set to a strictly positive value then nag_tsa_cp_binary_user (g13nec) terminates with NE_USER_STOP.
If info is set to a strictly negative value the current segment is skipped (i.e., no change points are considered in this segment) and nag_tsa_cp_binary_user (g13nec) continues as normal. If info was set to a strictly negative value at any point and no other errors occur then nag_tsa_cp_binary_user (g13nec) will terminate with NW_POTENTIAL_PROBLEM.
– Integer *Output
On exit: , the number of change points detected.
Note: the dimension, dim, of the array tau
must be at least
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 18.104.22.168 in the Essential Introduction).
– NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 22.214.171.124 in the Essential Introduction 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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
User requested termination by setting .
User requested a segment to be skipped by setting .
8 Parallelism and Performance
nag_tsa_cp_binary_user (g13nec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9 Further Comments
nag_tsa_cp_binary (g13ndc) 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_binary (g13ndc) 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 .