nag_tsa_cp_pelt (g13nac) 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 using one of a provided set of cost functions.
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_pelt (g13nac) 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
for all .
nag_tsa_cp_pelt (g13nac) supplies four families of cost function. Each cost function assumes that the series, , comes from some distribution, . The parameter space, is subdivided into containing those parameters allowed to differ in each segment and those parameters treated as constant across all segments. All four cost functions can then be described in terms of the likelihood function, and are given by:
where is the maximum likelihood estimate of within the th segment. In all four cases setting satisfies equation (2). Four distributions are available: Normal, Gamma, Exponential and Poisson. Letting
the log likelihoods and cost functions for the four distributions, and the available subdivisions of the parameter space are:
Both mean and variance change:
when calculating for the Poisson distribution, the sum is calculated for rather than .
Chen J and Gupta A K (2010) Parameteric Statisical 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: a flag indicating the assumed distribution of the data and the type of change point being looked for.
Data from a Normal distribution, looking for changes in the mean, .
Data from a Normal distribution, looking for changes in the standard deviation .
Data from a Normal distribution, looking for changes in the mean, and standard deviation .
Data from a Gamma distribution, looking for changes in the scale parameter .
Data from an exponential distribution, looking for changes in .
Data from a Poisson distribution, looking for changes in .
, , , , or .
On entry: , the length of the time series.
– const doubleInput
On entry: , the time series.
if , that is the data is assumed to come from a Poisson distribution, is used in all calculations.
if , or , , for ;
if , each value of y must be representable as an integer;
if , each value of y must be small enough such that , for , can be calculated without incurring overflow.
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. This is usually set to when and otherwise.
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 .
– const doubleInput
On entry: , values for the parameters that will be treated as fixed. If , param may be set to NULL.
if param is NULL, , the standard deviation of the Normal distribution, is estimated from the full input data. Otherwise .
If param is NULL, , the mean of the Normal distribution, is estimated from the full input data. Otherwise .
If , must hold the shape, , for the Gamma distribution, otherwise param is not referenced.
if or , .
– 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 error argument (see Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 188.8.131.52 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.
On entry, and . Constraint: if or and , then .
On entry, and . Constraint: if , or then , for .
On entry, , is too large.
To avoid overflow some truncation occurred when calculating the cost function, . All output is returned as normal.
To avoid overflow some truncation occurred when calculating the parameter estimates returned in sparam. All output is returned as normal.
For efficiency reasons, when calculating the cost functions, and the parameter estimates returned in sparam, this function makes use of the mathematical identities:
The input data, , is scaled in order to try and mitigate some of the known instabilities associated with using these formulations. The results returned by nag_tsa_cp_pelt (g13nac) should be sufficient for the majority of datasets. If a more stable method of calculating is deemed necessary, nag_tsa_cp_pelt_user (g13nbc) can be used and the method chosen implemented in the user-supplied cost function.
8 Parallelism and Performance
9 Further Comments
This example identifies changes in the mean, under the assumption that the data is normally distributed, for a simulated dataset with observations. A BIC penalty is used, that is , the minimum segment size is set to and the variance is fixed at across the whole input series.