NAG Library Routine Document

G13NAF

▸▿ 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

G13NAF 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

SUBROUTINE G13NAF (

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

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

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}})

G13NAF solves

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

(1)

where

β

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

K

such that

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

(2)

for all

u < v < w

G13NAF 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

{\hat{θ}}_{i}

is the maximum likelihood estimate of

θ

within the

i

th segment. In all four cases setting

K = 0

satisfies equation (2). Four distributions are available: Normal, Gamma, Exponential and Poisson. 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

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 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: $NTAU$ – INTEGEROutput

On exit:

m

, the number of change points detected.

9: $TAU (N)$ – INTEGER arrayOutput

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.

10: $SPARAM (2 \times N + 2)$ – 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.

11: $IFAIL$ – 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

For efficiency reasons, when calculating the cost functions,

C

and the parameter estimates returned in SPARAM, this routine makes use of the mathematical identities:

\sum_{j = u}^{v} {y_{j}}^{2} = \sum_{j = 1}^{v} {y_{j}}^{2} - \sum_{j = 1}^{u - 1} {y_{j}}^{2}

and

\sum_{j = 1}^{n} {(y_{j} - \bar{y})}^{2} = (\sum_{j = 1}^{n} y_{j}^{2}) - n {\bar{y}}^{2}

where

\bar{y} = n^{- 1} \sum_{j = 1}^{n} y_{j}

The input data,

y

, is scaled in order to try and mitigate some of the known instabilities associated with using these formulations. The results returned by G13NAF should be sufficient for the majority of datasets. If a more stable method of calculating

C

is deemed necessary, G13NBF can be used and the method chosen implemented in the user-supplied cost function.

8 Parallelism and Performance

Not applicable.

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.

G13NAF (PDF version)

G13 Chapter Contents

G13 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentG13NAF

▸▿ 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

G13NAF