NAG Library Routine Document

G13NDF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

G13NDF 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 using one of a provided set of cost functions.

2 Specification

SUBROUTINE G13NDF (

CTYPE, N, Y, BETA, MINSS, IPARAM, PARAM, MDEPTH, NTAU, TAU, SPARAM, IFAIL)

INTEGER	CTYPE, N, MINSS, IPARAM, MDEPTH, NTAU, TAU(*), IFAIL
REAL (KIND=nag_wp)	Y(N), BETA, PARAM(1), SPARAM(2*N)

3 Description

Let

y_{1 : n} = \{y_{j} : j = 1, 2, \dots, n\}

denote a series of data and

τ = \{τ_{i} : i = 1, 2, \dots, m\}

denote a set of

m

ordered (strictly monotonic increasing) indices known as change points, with

1 \leq τ_{i} \leq n

and

τ_{m} = n

. For ease of notation we also define

τ_{0} = 0

. The

m

change points,

τ

, split the data into

m

segments, with the

i

th segment being of length

n_{i}

and containing

y_{τ_{i - 1} + 1 : τ_{i}}

Given a cost function,

C (y_{τ_{i - 1} + 1 : τ_{i}})

, G13NDF gives an approximate solution to

\underset{m, τ}{minimize} \sum_{i = 1}^{m} (C (y_{τ_{i - 1} + 1 : τ_{i}}) + β)

where

β

is a penalty term used to control the number of change points. The solution is obtained in an iterative manner as follows:

Set

u = 1

w = n

and

k = 0

Set

k = k + 1

. If

k > K

, where

K

is a user-supplied control parameter, then terminate the process for this segment.

Find

v

that minimizes

C (y_{u : v}) + C (y_{v + 1 : w})

Test

C (y_{u : v}) + C (y_{v + 1 : w}) + β < C (y_{u : w})

(1)

If inequality (1) is false then the process is terminated for this segment.

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

τ

G13NDF supplies four families of cost function. Each cost function assumes that the series,

y

, comes from some distribution,

D (Θ)

. The parameter space,

Θ = \{θ, ϕ\}

is subdivided into

θ

containing those parameters allowed to differ in each segment and

ϕ

those parameters treated as constant across all segments. All four cost functions can then be described in terms of the likelihood function,

L

and are given by:

C (y_{(τ_{i - 1} + 1) : τ_{i}}) = - 2 \log L ({\hat{θ}}_{i}, ϕ | y_{(τ_{i - 1} + 1) : τ_{i}})

where the

{\hat{θ}}_{i}

is the maximum likelihood estimate of

θ

within the

i

th segment. Four distributions are available; Normal, Gamma, Exponential and Poisson distributions. Letting

S_{i} = \sum_{j = τ_{i - 1}}^{τ_{i}} y_{j}

the log likelihoods and cost functions for the four distributions, and the available subdivisions of the parameter space are:

Normal distribution:

Θ = \{μ, σ^{2}\}

- 2 \log L = \sum_{i = 1}^{m} \sum_{j = τ_{i - 1}}^{τ_{i}} \log (2 π) + \log (σ_{i}^{2}) + \frac{{(y_{j} - μ_{i})}^{2}}{σ_{i}^{2}}

Mean changes: $θ = \{μ\}$
$C (y_{τ_{i - 1} + 1 : τ_{i}}) = \sum_{j = τ_{i - 1}}^{τ_{i}} \frac{{(y_{j} - n_{i}^{- 1} S_{i})}^{2}}{σ^{2}}$
Variance changes: $θ = \{σ^{2}\}$
$C (y_{τ_{i - 1} + 1 : τ_{i}}) = n_{i} (\log (\sum_{j = τ_{i - 1}}^{τ_{i}} {(y_{j} - μ)}^{2}) - \log n_{i})$

Both mean and variance change:

θ = \{μ, σ^{2}\}

C (y_{τ_{i - 1} + 1 : τ_{i}}) = n_{i} (\log (\sum_{j = τ_{i - 1}}^{τ_{i}} {(y_{j} - n_{i}^{- 1} S_{i})}^{2}) - \log n_{i})

Gamma distribution: Θ=a,b
$- 2 \log L = 2 \times \sum_{i = 1}^{m} \sum_{j = τ_{i - 1}}^{τ_{i}} \log Γ (a_{i}) + a_{i} \log b_{i} + (1 - a_{i}) \log y_{j} + \frac{y_{j}}{b_{i}}$
- Scale changes: $θ = \{b\}$
  $C (y_{τ_{i - 1} + 1 : τ_{i}}) = 2 a n_{i} (\log S_{i} - \log (a n_{i}))$
Exponential Distribution: Θ=λ
$- 2 \log L = 2 \times \sum_{i = 1}^{m} \sum_{j = τ_{i - 1}}^{τ_{i}} \log λ_{i} + \frac{y_{j}}{λ_{i}}$
- Mean changes: $θ = \{λ\}$
  $C (y_{τ_{i - 1} + 1 : τ_{i}}) = 2 n_{i} (\log S_{i} - \log n_{i})$
Poisson distribution: Θ=λ
$- 2 \log L = 2 \times \sum_{i = 1}^{m} \sum_{j = τ_{i - 1}}^{τ_{i}} λ_{i} - ⌊ y_{j} + 0.5 ⌋ \log λ_{i} + \log Γ (⌊ y_{j} + 0.5 ⌋ + 1)$
- Mean changes: $θ = \{λ\}$
  $C (y_{τ_{i - 1} + 1 : τ_{i}}) = 2 S_{i} (\log n_{i} - \log S_{i})$
  when calculating $S_{i}$ for the Poisson distribution, the sum is calculated for $⌊ y_{i} + 0.5 ⌋$ rather than $y_{i}$ .

4 References

Chen J and Gupta A K (2010) Parameteric Statisical Change Point Analysis With Applications to Genetics Medicine and Finance Second Edition Birkhäuser

West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM 22 532–555

5 Parameters

1: $CTYPE$ – INTEGERInput

On entry: a flag indicating the assumed distribution of the data and the type of change point being looked for.

$CTYPE = 1$: Data from a Normal distribution, looking for changes in the mean, $μ$ .
$CTYPE = 2$: Data from a Normal distribution, looking for changes in the standard deviation $σ$ .
$CTYPE = 3$: Data from a Normal distribution, looking for changes in the mean, $μ$ and standard deviation $σ$ .
$CTYPE = 4$: Data from a Gamma distribution, looking for changes in the scale parameter $b$ .
$CTYPE = 5$: Data from an exponential distribution, looking for changes in $λ$ .
$CTYPE = 6$: Data from a Poisson distribution, looking for changes in $λ$ .

Constraint:

CTYPE = 1

2

3

4

5

6

2: $N$ – INTEGERInput

On entry:

n

, the length of the time series.

Constraint:

N \geq 2

3: $Y (N)$ – REAL (KIND=nag_wp) arrayInput

On entry:

y

, the time series.

CTYPE = 6

, that is the data is assumed to come from a Poisson distribution,

⌊ y + 0.5 ⌋

is used in all calculations.

Constraints:

if $CTYPE = 4$ , $5$ or $6$ , $Y (i) \geq 0$ , for $i = 1, 2, \dots, N$ ;
if $CTYPE = 6$ , each value of Y must be representable as an integer;
if $CTYPE \neq 6$ , each value of Y must be small enough such that ${Y (i)}^{2}$ , for $i = 1, 2, \dots, N$ , can be calculated without incurring overflow.

4: $BETA$ – REAL (KIND=nag_wp)Input

On entry:

β

, the penalty term.

There are a number of standard ways of setting

β

, including:

SIC or BIC: $β = p \times \log (n)$
AIC: $β = 2 p$
Hannan-Quinn: $β = 2 p \times \log (\log (n))$

where

p

is the number of parameters being treated as estimated in each segment. This is usually set to

2

when

CTYPE = 3

and

1

otherwise.

If no penalty is required then set

β = 0

. Generally, the smaller the value of

β

the larger the number of suggested change points.

5: $MINSS$ – INTEGERInput

On entry: the minimum distance between two change points, that is

τ_{i} - τ_{i - 1} \geq MINSS

Constraint:

MINSS \geq 2

6: $IPARAM$ – INTEGERInput

On entry: if

IPARAM = 1

distributional parameters have been supplied in PARAM.

Constraints:

if $CTYPE = 4$ , $IPARAM = 1$ ;
otherwise $IPARAM = 0$ or $1$ .

7: $PARAM (1)$ – REAL (KIND=nag_wp) arrayInput

On entry:

ϕ

, values for the parameters that will be treated as fixed. If

IPARAM = 0

then PARAM is not referenced.

CTYPE = 1

if $IPARAM = 0$ , $σ$ , the standard deviation of the Normal distribution, is estimated from the full input data. Otherwise $σ = PARAM (1)$ .

CTYPE = 2

If $IPARAM = 0$ , $μ$ , the mean of the Normal distribution, is estimated from the full input data. Otherwise $μ = PARAM (1)$ .

CTYPE = 4

PARAM (1)

must hold the shape,

a

, for the Gamma distribution, otherwise PARAM is not referenced.

Constraint: if

CTYPE = 1

4

PARAM (1) > 0.0

8: $MDEPTH$ – 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 \leq 2^{K}

K \leq 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.

9: $NTAU$ – INTEGEROutput

On exit:

m

, the number of change points detected.

10: $TAU (*)$ – INTEGER arrayOutput

Note: the dimension of the array TAU must be at least

\min (⌈ \frac{N}{MINSS} ⌉, 2^{MDEPTH})

MDEPTH > 0

, and at least

⌈ \frac{N}{MINSS} ⌉

otherwise.

On exit: the first

m

elements of TAU hold the location of the change points. The

i

th segment is defined by

y_{(τ_{i - 1} + 1)}

y_{τ_{i}}

, where

τ_{0} = 0

and

τ_{i} = TAU (i), 1 \leq i \leq m

The remainder of TAU is used as workspace.

11: $SPARAM (2 \times N)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the estimated values of the distribution parameters in each segment

$CTYPE = 1$ , $2$ or $3$: $SPARAM (2 i - 1) = μ_{i}$ and $SPARAM (2 i) = σ_{i}$ for $i = 1, 2, \dots, m$ , where $μ_{i}$ and $σ_{i}$ is the mean and standard deviation, respectively, of the values of $y$ in the $i$ th segment.
It should be noted that $σ_{i} = σ_{j}$ when $CTYPE = 1$ and $μ_{i} = μ_{j}$ when $CTYPE = 2$ , for all $i$ and $j$ .
$CTYPE = 4$: $SPARAM (2 i - 1) = a_{i}$ and $SPARAM (2 i) = b_{i}$ for $i = 1, 2, \dots, m$ , where $a_{i}$ and $b_{i}$ are the shape and scale parameters, respectively, for the values of $y$ in the $i$ th segment. It should be noted that $a_{i} = PARAM (1)$ for all $i$ .
$CTYPE = 5$ or $6$: $SPARAM (i) = λ_{i}$ for $i = 1, 2, \dots, m$ , where $λ_{i}$ is the mean of the values of $y$ in the $i$ th segment.

The remainder of SPARAM is used as workspace.

12: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ 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 ​ 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 $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 11$: On entry, $CTYPE = 〈value〉$ .
Constraint: $CTYPE = 1$ , $2$ , $3$ , $4$ , $5$ or $6$ .

$IFAIL = 21$: On entry, $N = 〈value〉$ .
Constraint: $N \geq 2$ .

$IFAIL = 31$: On entry, $CTYPE = 〈value〉$ and $Y (〈value〉) = 〈value〉$ .
Constraint: if $CTYPE = 4$ , $5$ or $6$ then $Y (i) \geq 0.0$ , for $i = 1, 2, \dots, N$ .

$IFAIL = 32$: On entry, $Y (〈value〉) = 〈value〉$ , is too large.

$IFAIL = 51$: On entry, $MINSS = 〈value〉$ .
Constraint: $MINSS \geq 2$ .

$IFAIL = 61$: On entry, $IPARAM = 〈value〉$ .
Constraint: if $CTYPE \neq 4$ then $IPARAM = 0$ or $1$ .

$IFAIL = 62$: On entry, $IPARAM = 〈value〉$ .
Constraint: if $CTYPE = 4$ then $IPARAM = 1$ .

$IFAIL = 71$: On entry, $CTYPE = 〈value〉$ and $PARAM (1) = 〈value〉$ .
Constraint: if $CTYPE = 1$ or $4$ and $IPARAM = 1$ , then $PARAM (1) > 0.0$ .

$IFAIL = 200$: To avoid overflow some truncation occurred when calculating the cost function, $C$ . All output is returned as normal.

$IFAIL = 201$: To avoid overflow some truncation occurred when calculating the parameter estimates returned in SPARAM. All output is returned as normal.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The calculation of means and sums of squares about the mean during the evaluation of the cost functions are based on the one pass algorithm of West (1979) and are believed to be stable.

8 Parallelism and Performance

G13NDF 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example identifies changes in the mean, under the assumption that the data is normally distributed, for a simulated dataset with

100

observations. A BIC penalty is used, that is

β = \log n \approx 4.6

, the minimum segment size is set to

2

and the variance is fixed at

1

across the whole input series.

This example plot shows the original data series, the estimated change points and the estimated mean in each of the identified segments.

G13NDF (PDF version)

G13 Chapter Contents

G13 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentG13NDF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

G13NDF