NAG Library Function Document
nag_tsa_cp_pelt (g13nac)
1
Purpose
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.
2
Specification
#include <nag.h> |
#include <nagg13.h> |
|
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,
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:
- 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_ChangeTypeInput
-
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:
– IntegerInput
-
On entry: , the length of the time series.
Constraint:
.
- 3:
– 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.
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:
– doubleInput
-
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:
– IntegerInput
-
On entry: the minimum distance between two change points, that is .
Constraint:
.
- 6:
– const doubleInput
-
On entry:
, values for the parameters that will be treated as fixed. If
,
param may be set to
NULL.
If
- if param is NULL, , the standard deviation of the Normal distribution, is estimated from the full input data. Otherwise .
If
- 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.
Constraint:
if or , .
- 7:
– Integer *Output
-
On exit: , the number of change points detected.
- 8:
– IntegerOutput
-
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:
– doubleOutput
-
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 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
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.
- NE_REAL
-
On entry, and .
Constraint: if or and , 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
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
nag_tsa_cp_pelt (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
Program Text (g13nace.c)
10.2
Program Data
Program Data (g13nace.d)
10.3
Program Results
Program Results (g13nace.r)
This example plot shows the original data series, the estimated change points and the estimated mean in each of the identified segments.