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.
The function may be called by the names: g13nec or nag_tsa_cp_binary_user.
3Description
Let ${y}_{1:n}=\{{y}_{j}:j=1,2,\dots ,n\}$ denote a series of data and $\tau =\{{\tau}_{i}:i=1,2,\dots ,m\}$ denote a set of $m$ ordered (strictly monotonic increasing) indices known as change points with $1\le {\tau}_{i}\le n$ and ${\tau}_{m}=n$. For ease of notation we also define ${\tau}_{0}=0$. The $m$ change points, $\tau $, split the data into $m$ segments, with the $i$th segment being of length ${n}_{i}$ and containing ${y}_{{\tau}_{i-1}+1:{\tau}_{i}}$.
Given a cost function, $C\left({y}_{{\tau}_{i-1}+1:{\tau}_{i}}\right)$, g13nec gives an approximate solution to
5.If inequality (1) is false then the process is terminated for this segment.
6.If inequality (1) is true, then $v$ is added to the set of change points, and the segment is split into two subsegments, ${y}_{u:v}$ and ${y}_{v+1:w}$. The whole process is repeated from step 2 independently on each subsegment, with the relevant changes to the definition of $u$ and $w$ (i.e., $w$ is set to $v$ when processing the left-hand subsegment and $u$ is set to $v+1$ when processing the right-hand subsegment.
The change points are ordered to give $\tau $.
4References
Chen J and Gupta A K (2010) Parametric Statistical Change Point Analysis With Applications to GeneticsMedicine and FinanceSecond Edition Birkhäuser
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the length of the time series.
Constraint:
${\mathbf{n}}\ge 2$.
2: $\mathbf{beta}$ – doubleInput
On entry: $\beta $, the penalty term.
There are a number of standard ways of setting $\beta $, including:
where $p$ is the number of parameters being treated as estimated in each segment. The value of $p$ will depend on the cost function being used.
If no penalty is required then set $\beta =0$. Generally, the smaller the value of $\beta $ the larger the number of suggested change points.
3: $\mathbf{minss}$ – IntegerInput
On entry: the minimum distance between two change points, that is ${\tau}_{i}-{\tau}_{i-1}\ge {\mathbf{minss}}$.
Constraint:
${\mathbf{minss}}\ge 2$.
4: $\mathbf{mdepth}$ – IntegerInput
On entry: $K$, the maximum depth for the iterative process, which in turn puts an upper limit on the number of change points with $m\le {2}^{K}$.
If $K\le 0$ 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.
5: $\mathbf{chgpfn}$ – function, supplied by the userExternal Function
chgpfn must calculate a proposed change point, and the associated costs, within a specified segment.
On entry: flag indicating what chgpfn must calculate and at which point of the Binary Segmentation it has been called.
${\mathbf{side}}=\mathrm{Nag\_FirstSegCall}$
only $C\left({y}_{u:w}\right)$ need be calculated and returned in ${\mathbf{cost}}\left[0\right]$, neither v nor the other elements of cost need be set. In this case, $u=1$ and $w=\mathrm{n}$.
${\mathbf{side}}=\mathrm{Nag\_SecondSegCall}$
all elements of cost and v must be set. In this case, $u=1$ and $w=\mathrm{n}$.
${\mathbf{side}}=\mathrm{Nag\_LeftSubSeg}$
the segment, ${y}_{u:w}$, is a left-hand side subsegment from a previous iteration of the Binary Segmentation algorithm. All elements of cost and v must be set.
${\mathbf{side}}=\mathrm{Nag\_RightSubSeg}$
the segment, ${y}_{u:w}$, is a right-hand side subsegment from a previous iteration of the Binary Segmentation algorithm. All elements of cost and v must be set.
The distinction between ${\mathbf{side}}=\mathrm{Nag\_LeftSubSeg}$ and $\mathrm{Nag\_RightSubSeg}$ may allow for chgpfn to be implemented in a more efficient manner.
The first call to chgpfn will always have ${\mathbf{side}}=\mathrm{Nag\_FirstSegCall}$ and the second call will always have ${\mathbf{side}}=\mathrm{Nag\_SecondSegCall}$. All subsequent calls will be made with ${\mathbf{side}}=\mathrm{Nag\_LeftSubSeg}$ or $\mathrm{Nag\_RightSubSeg}$.
2: $\mathbf{u}$ – IntegerInput
On entry: $u$, the start of the segment of interest.
3: $\mathbf{w}$ – IntegerInput
On entry: $w$, the end of the segment of interest.
4: $\mathbf{minss}$ – IntegerInput
On entry: the minimum distance between two change points, as passed to g13nec.
5: $\mathbf{v}$ – Integer *Output
On exit: if ${\mathbf{side}}=\mathrm{Nag\_FirstSegCall}$ then v need not be set.
if ${\mathbf{side}}\ne \mathrm{Nag\_FirstSegCall}$ then $v$, the proposed change point. That is, the value which minimizes
for $v=u+{\mathbf{minss}}-1$ to $w-{\mathbf{minss}}$.
6: $\mathbf{cost}\left[3\right]$ – doubleOutput
On exit: costs associated with the proposed change point, $v$.
If ${\mathbf{side}}=\mathrm{Nag\_FirstSegCall}$ then ${\mathbf{cost}}\left[0\right]=C\left({y}_{u:w}\right)$ and the remaining two elements of cost need not be set.
If ${\mathbf{side}}\ne \mathrm{Nag\_FirstSegCall}$ then
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 g13nec you may allocate memory and initialize these pointers with various quantities for use by chgpfn when called from g13nec (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
8: $\mathbf{info}$ – Integer *Input/Output
On entry: ${\mathbf{info}}=0$.
On exit: in most circumstances info should remain unchanged.
If info is set to a strictly positive value then g13nec terminates with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$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 g13nec continues as normal. If info was set to a strictly negative value at any point and no other errors occur then g13nec will terminate with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NW_POTENTIAL_PROBLEM.
Note:chgpfn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g13nec. If your code inadvertently does return any NaNs or infinities, g13nec is likely to produce unexpected results.
6: $\mathbf{ntau}$ – Integer *Output
On exit: $m$, the number of change points detected.
On exit: the first $m$ elements of tau hold the location of the change points. The $i$th segment is defined by ${y}_{({\tau}_{i-1}+1)}$ to ${y}_{{\tau}_{i}}$, where ${\tau}_{0}=0$ and ${\tau}_{i}={\mathbf{tau}}\left[i-1\right],1\le i\le m$.
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
9: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{minss}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{minss}}\ge 2$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n}}\ge 2$.
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_USER_STOP
User requested termination by setting ${\mathbf{info}}=\u27e8\mathit{\text{value}}\u27e9$.
NW_POTENTIAL_PROBLEM
User requested a segment to be skipped by setting ${\mathbf{info}}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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.
9Further Comments
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 g13ndc can be used without the need to provide a cost function routine.
10Example
This example identifies changes in the scale parameter, under the assumption that the data has a gamma distribution, for a simulated dataset with $100$ observations. A penalty, $\beta $ of $3.6$ is used and the minimum segment size is set to $3$. The shape parameter is fixed at $2.1$ across the whole input series.