NAG C Library Function Document

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>

void	nag_tsa_cp_pelt (Nag_TS_ChangeType ctype, Integer n, const double y[], double beta, Integer minss, const double param[], Integer ntau, Integer tau[], double sparam[], NagError fail)

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}})

nag_tsa_cp_pelt (g13nac) 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

nag_tsa_cp_pelt (g13nac) 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) 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: $ctype$ – Nag_TS_ChangeTypeInput

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

$ctype = Nag_NormalMean$: Data from a Normal distribution, looking for changes in the mean, $μ$ .
$ctype = Nag_NormalStd$: Data from a Normal distribution, looking for changes in the standard deviation $σ$ .
$ctype = Nag_NormalMeanStd$: Data from a Normal distribution, looking for changes in the mean, $μ$ and standard deviation $σ$ .
$ctype = Nag_GammaScale$: Data from a Gamma distribution, looking for changes in the scale parameter $b$ .
$ctype = Nag_ExponentialLambda$: Data from an exponential distribution, looking for changes in $λ$ .
$ctype = Nag_PoissonLambda$: Data from a Poisson distribution, looking for changes in $λ$ .

Constraint:

ctype = Nag_NormalMean

Nag_NormalStd

Nag_NormalMeanStd

Nag_GammaScale

Nag_ExponentialLambda

Nag_PoissonLambda

2: $n$ – IntegerInput

On entry:

n

, the length of the time series.

Constraint:

n \geq 2

3: $y [n]$ – const doubleInput

On entry:

y

, the time series.

ctype = Nag_PoissonLambda

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

⌊ y + 0.5 ⌋

is used in all calculations.

Constraints:

if $ctype = Nag_GammaScale$ , $Nag_ExponentialLambda$ or $Nag_PoissonLambda$ , $y [i - 1] \geq 0$ , for $i = 1, 2, \dots, n$ ;
if $ctype = Nag_PoissonLambda$ , each value of y must be representable as an integer;
if $ctype \neq Nag_PoissonLambda$ , each value of y must be small enough such that ${y [i - 1]}^{2}$ , for $i = 1, 2, \dots, n$ , can be calculated without incurring overflow.

4: $beta$ – doubleInput

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 = Nag_NormalMeanStd

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: $param [1]$ – const doubleInput

On entry:

ϕ

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

ctype = Nag_GammaScale

then param must be supplied, otherwise param may be NULL.

If supplied, then when

$ctype = Nag_NormalMean$: $param [0] = σ$ , the standard deviation of the normal distribution. If not supplied then $σ$ is estimated from the full input data,
$ctype = Nag_NormalStd$: $param [0] = μ$ , the mean of the normal distribution. If not supplied then $μ$ is estimated from the full input data,
$ctype = Nag_GammaScale$: $param [0]$ must hold the shape, $a$ , for the gamma distribution,
otherwise: param is not referenced.

Constraint: if

ctype = Nag_NormalMean

Nag_GammaScale

param [0] > 0.0

7: $ntau$ – Integer *Output

On exit:

m

, the number of change points detected.

8: $tau [\dim]$ – IntegerOutput

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], 1 \leq i \leq m

The remainder of tau is used as workspace.

9: $sparam [\dim]$ – doubleOutput

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

$ctype = Nag_NormalMean$ , $Nag_NormalStd$ or $Nag_NormalMeanStd$: $sparam [2 i - 2] = μ_{i}$ and $sparam [2 i - 1] = σ_{i}$ 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 $ctype = Nag_NormalMean$ and $μ_{i} = μ_{j}$ when $ctype = Nag_NormalStd$ , for all $i$ and $j$ .
$ctype = Nag_GammaScale$: $sparam [2 i - 2] = a_{i}$ and $sparam [2 i - 1] = b_{i}$ 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 $a_{i} = param [0]$ for all $i$ .
$ctype = Nag_ExponentialLambda$ or $Nag_PoissonLambda$: $sparam [i - 1] = λ_{i}$ 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.

10: $fail$ – 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 $〈value〉$ had an illegal value.
NE_INT: On entry, $minss = 〈value〉$ .
Constraint: $minss \geq 2$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .
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 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, $ctype = 〈value〉$ and $param [0] = 〈value〉$ .
Constraint: if $ctype = Nag_NormalMean$ or $Nag_GammaScale$ and param has been supplied, then $param [0] > 0.0$ .
NE_REAL_ARRAY: On entry, $ctype = 〈value〉$ and $y [〈value〉] = 〈value〉$ .
Constraint: if $ctype = Nag_GammaScale$ , $Nag_ExponentialLambda$ or $Nag_PoissonLambda$ then $y [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n$ .

On entry, $y [〈value〉] = 〈value〉$ , is too large.
NW_TRUNCATED: To avoid overflow some truncation occurred when calculating the cost function, $C$ . 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,

C

and the parameter estimates returned in sparam, this function 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 nag_tsa_cp_pelt (g13nac) should be sufficient for the majority of datasets. If a more stable method of calculating

C

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.

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.

NAG C Library Manual, Mark 26

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

g13 Chapter Contents

g13 Chapter Introduction

NAG C Library Function Document

nag_tsa_cp_pelt (g13nac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

2

Specification

3

Description

4

References

5

Arguments

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