nag_tsa_cp_binary (g13ndc) : NAG Library, Mark 26

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_binary (g13ndc) 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

nag_tsa_cp_binary (g13ndc) 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 $y_{i}$ .

5 Arguments

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

n

– IntegerInput

On entry:

n

, the length of the time series.

Constraint:

n \geq 2

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.

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.

minss

– IntegerInput

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

τ_{i} - τ_{i - 1} \geq minss

Constraint:

minss \geq 2

param [1]

– const doubleInput

On entry:

ϕ

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

ctype \neq Nag_GammaScale

, param may be set to NULL.

ctype = Nag_NormalMean

if param is NULL, $σ$ , the standard deviation of the Normal distribution, is estimated from the full input data. Otherwise $σ = param [0]$ .

ctype = Nag_NormalStd

If param is NULL, $μ$ , the mean of the Normal distribution, is estimated from the full input data. Otherwise $μ = param [0]$ .

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

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.

ntau

– Integer *Output

On exit:

m

, the number of change points detected.

tau [\dim]

– IntegerOutput

Note: the dimension, dim, of the array tau must be at least

$\min (⌈ \frac{n}{minss} ⌉, 2^{mdepth})$ when $mdepth > 0$ ;
$⌈ \frac{n}{minss} ⌉$ 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], 1 \leq i \leq m

The remainder of tau is used as workspace.

10:

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.

11:

fail

– NagError *Input/Output

The NAG error argument (see Section 2.7 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.

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

Nag_GammaScale

and

param is not NULL

, then

param [0] > 0.0

NE_REAL_ARRAY

On entry,

ctype = 〈value〉

and

y [〈value〉] = 〈value〉

.
Constraint: if

ctype = Nag_GammaScale

Nag_ExponentialLambda

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.

8 Parallelism and Performance

nag_tsa_cp_binary (g13ndc) 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.

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.

10.1 Program Text

10.2 Program Data

10.3 Program Results

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

NAG Library Function Document

nag_tsa_cp_binary (g13ndc)

▸▿ 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

NAG Library Function Documentnag_tsa_cp_binary (g13ndc)

▸▿ 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

NAG Library Function Document

nag_tsa_cp_binary (g13ndc)