NAG CL Interface
g13nac (cp_pelt)
1
Purpose
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.
2
Specification
The function may be called by the names: g13nac or nag_tsa_cp_pelt.
3
Description
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,
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
.
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:
- Normal distribution:
- Mean changes:
- Variance changes:
- Both mean and variance change:
- Gamma distribution:
- Exponential Distribution:
- Poisson distribution:
- Mean changes:
when calculating for the Poisson distribution, the sum is calculated for rather than .
4
References
Chen J and Gupta A K (2010) Parametric Statistical Change Point Analysis With Applications to Genetics Medicine and Finance Second 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 Association 107:500 1590–1598
5
Arguments
-
1:
– Nag_TS_ChangeType
Input
-
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 .
Constraint:
, , , , or .
-
2:
– Integer
Input
-
On entry: , the length of the time series.
Constraint:
.
-
3:
– const double
Input
-
On entry: , the time series.
If , that is the data is assumed to come from a Poisson distribution, is used in all calculations.
Constraints:
- 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.
-
4:
– double
Input
-
On entry:
, the penalty term.
There are a number of standard ways of setting
, including:
- SIC or BIC
- AIC
- Hannan-Quinn
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.
-
5:
– Integer
Input
-
On entry: the minimum distance between two change points, that is .
Constraint:
.
-
6:
– const double
Input
-
On entry:
, values for the parameters that will be treated as fixed. If
then
param must be supplied, otherwise
param may be
NULL.
If supplied, then when
- , the standard deviation of the normal distribution. If not supplied then is estimated from the full input data,
- , the mean of the normal distribution. If not supplied then is estimated from the full input data,
- must hold the shape, , for the gamma distribution,
- otherwise
- param is not referenced.
Constraint:
if or , .
-
7:
– Integer *
Output
-
On exit: , the number of change points detected.
-
8:
– Integer
Output
-
On exit: the first
elements of
tau hold the location of the change points. The
th segment is defined by
to
, where
and
.
The remainder of
tau is used as workspace.
-
9:
– double
Output
-
On exit: the estimated values of the distribution parameters in each segment
- , or
- and for , where and is the mean and standard deviation, respectively, of the values of in the th segment.
It should be noted that when and when , for all and .
- and for , where and are the shape and scale parameters, respectively, for the values of in the th segment. It should be noted that for all .
- or
- for , where is the mean of the values of in the th segment.
The remainder of
sparam is used as workspace.
-
10:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_REAL
-
On entry,
and
.
Constraint: if
or
and
param has been supplied, then
.
- NE_REAL_ARRAY
-
On entry, and .
Constraint: if , or then , for .
On entry, , is too large.
- NW_TRUNCATED
-
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.
7
Accuracy
For efficiency reasons, when calculating the cost functions,
and the parameter estimates returned in
sparam, this function makes use of the mathematical identities:
and
where
.
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
g13nac should be sufficient for the majority of datasets. If a more stable method of calculating
is deemed necessary,
g13nbc can be used and the method chosen implemented in the user-supplied cost function.
8
Parallelism and Performance
Background information to multithreading can be found in the
Multithreading documentation.
g13nac is not threaded in any implementation.
None.
10
Example
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.
10.1
Program Text
10.2
Program Data
10.3
Program Results
This example plot shows the original data series, the estimated change points and the estimated mean in each of the identified segments.