NAG Library Routine Document

G13NEF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, the length of the time series.

Constraint: $\mathbf{N}\ge 2$.

2:     $\mathrm{BETA}$ – REAL (KIND=nag_wp)Input

On entry: $\beta$, the penalty term.

There are a number of standard ways of setting $\beta$, including:

SIC or BIC

$\beta =p\times \log \left(n\right)$.

AIC

$\beta =2p$.

Hannan-Quinn

$\beta =2p\times \log \left(\log \left(n\right)\right)$.

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:     $\mathrm{MINSS}$ – 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:     $\mathrm{MDEPTH}$ – 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:     $\mathrm{CHGPFN}$ – SUBROUTINE, supplied by the user.External Procedure

CHGPFN must calculate a proposed change point, and the associated costs, within a specified segment.

The specification of CHGPFN is:

SUBROUTINE CHGPFN ( SIDE, U, W, MINSS, V, COST, Y, IUSER, RUSER, INFO) INTEGER  SIDE, U, W, MINSS, V, IUSER(*), INFO REAL (KIND=nag_wp)  COST(3), Y(*), RUSER(*)

1:     $\mathrm{SIDE}$ – INTEGERInput

On entry: flag indicating what CHGPFN must calculate and at which point of the Binary Segmentation it has been called.

$\mathbf{SIDE}=-1$

only $C\left(y_{u:w}\right)$ need be calculated and returned in $\mathbf{COST}\left(1\right)$, neither V nor the other elements of COST need be set. In this case, $u=1$ and $w=\mathrm{n}$.

$\mathbf{SIDE}=0$

all elements of COST and V must be set. In this case, $u=1$ and $w=\mathrm{n}$.

$\mathbf{SIDE}=1$

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.

$\mathbf{SIDE}=2$

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 $\mathbf{SIDE}=1$ and $2$ may allow for CHGPFN to be implemented in a more efficient manner. See section Section 10 for one such example.

The first call to CHGPFN will always have $\mathbf{SIDE}=-1$ and the second call will always have $\mathbf{SIDE}=0$. All subsequent calls will be made with $\mathbf{SIDE}=1$ or $2$.

2:     $\mathrm{U}$ – INTEGERInput

On entry: $u$, the start of the segment of interest.

3:     $\mathrm{W}$ – INTEGERInput

On entry: $w$, the end of the segment of interest.

4:     $\mathrm{MINSS}$ – INTEGERInput

On entry: the minimum distance between two change points, as passed to G13NEF.

5:     $\mathrm{V}$ – INTEGEROutput

On exit: if $\mathbf{SIDE}=-1$ then V need not be set.

if $\mathbf{SIDE}\ne -1$ then $v$, the proposed change point. That is, the value which minimizes

$$\underset{v}{\mathrm{minimize}}C\left(y_{u:v}\right)+C\left(y_{v+1:w}\right)$$

for $v=u+\mathbf{MINSS}-1$ to $w-\mathbf{MINSS}$.

6:     $\mathrm{COST}\left(3\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: costs associated with the proposed change point, $v$.

If $\mathbf{SIDE}=-1$ then $\mathbf{COST}\left(1\right)=C\left(y_{u:w}\right)$ and the remaining two elements of COST need not be set.

If $\mathbf{SIDE}\ne -1$ then

• $\mathbf{COST}\left(1\right)=C\left(y_{u:v}\right)+C\left(y_{v+1:w}\right)$.
• $\mathbf{COST}\left(2\right)=C\left(y_{u:v}\right)$.
• $\mathbf{COST}\left(3\right)=C\left(y_{v+1:w}\right)$.

7:     $\mathrm{Y}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Data

CHGPFN is called with Y as supplied to G13NEF. You are free to use the array Y to supply information to CHGPFN.

Y is supplied in addition to IUSER and RUSER for ease of coding as in most cases CHGPFN will require (functions of) the time series, $y$.

8:     $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace

9:     $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

CHGPFN is called with the parameters IUSER and RUSER as supplied to G13NEF. You are free to use the arrays IUSER and RUSER to supply information to CHGPFN as an alternative to using COMMON global variables.

10:   $\mathrm{INFO}$ – INTEGERInput/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 G13NEF terminates with $\mathbf{IFAIL}=\mathbf{51}$.

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 G13NEF continues as normal. If INFO was set to a strictly negative value at any point and no other errors occur then G13NEF will terminate with $\mathbf{IFAIL}=\mathbf{52}$.

CHGPFN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G13NEF is called. Parameters denoted as Input must not be changed by this procedure.

6:     $\mathrm{NTAU}$ – INTEGEROutput

On exit: $m$, the number of change points detected.

7:     $\mathrm{TAU}\left(*\right)$ – INTEGER arrayOutput

Note: the dimension of the array TAU must be at least $\min \left(\lceil \frac{\mathbf{N}}{\mathbf{MINSS}}\rceil ,2^{\mathbf{MDEPTH}}\right)$ if $\mathbf{MDEPTH}>0$, and at least $\lceil \frac{\mathbf{N}}{\mathbf{MINSS}}\rceil$ otherwise.

On exit: the first $m$ elements of TAU hold the location of the change points. The $i$th segment is defined by $y_{\left({\tau }_{i-1}+1\right)}$ to $y_{{\tau }_i}$, where ${\tau }_0=0$ and ${\tau }_i=\mathbf{TAU}\left(i\right),1\le i\le m$.

The remainder of TAU is used as workspace.

8:     $\mathrm{Y}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Data

Y is not used by G13NEF, but is passed directly to CHGPFN and may be used to pass information to this routine. Y will usually be used to pass (functions of) the time series, $y$ of interest.

9:     $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace

10:   $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by G13NEF, but are passed directly to CHGPFN and may be used to pass information to this routine as an alternative to using COMMON global variables.

11:   $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=11$

On entry, $\mathbf{N}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{N}\ge 2$.

$\mathbf{IFAIL}=31$

On entry, $\mathbf{MINSS}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{MINSS}\ge 2$.

$\mathbf{IFAIL}=51$

User requested termination by setting $\mathbf{INFO}=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=52$

User requested a segment to be skipped by setting $\mathbf{INFO}=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-999$

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (g13nefe.f90)

10.2  Program Data

Program Data (g13nefe.d)

10.3  Program Results

Program Results (g13nefe.r)