# NAG Library Function Document

## 1Purpose

nag_tsa_cp_binary (g13ndc) 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.

## 2Specification

 #include #include
 void nag_tsa_cp_binary (Nag_TS_ChangeType ctype, Integer n, const double y[], double beta, Integer minss, const double param[], Integer mdepth, Integer *ntau, Integer tau[], double sparam[], NagError *fail)

## 3Description

Let ${y}_{1:n}=\left\{{y}_{j}:j=1,2,\dots ,n\right\}$ denote a series of data and $\tau =\left\{{\tau }_{i}:i=1,2,\dots ,m\right\}$ denote a set of $m$ ordered (strictly monotonic increasing) indices known as change points, with $1\le {\tau }_{i}\le n$ and ${\tau }_{m}=n$. For ease of notation we also define ${\tau }_{0}=0$. The $m$ change points, $\tau$, split the data into $m$ segments, with the $i$th segment being of length ${n}_{i}$ and containing ${y}_{{\tau }_{i-1}+1:{\tau }_{i}}$.
Given a cost function, $C\left({y}_{{\tau }_{i-1}+1:{\tau }_{i}}\right)$, nag_tsa_cp_binary (g13ndc) gives an approximate solution to
 $minimize m,τ ∑ i=1 m Cyτi-1+1:τi + β$
where $\beta$ is a penalty term used to control the number of change points. The solution is obtained in an iterative manner as follows:
1. Set $u=1$, $w=n$ and $k=0$
2. Set $k=k+1$. If $k>K$, where $K$ is a user-supplied control parameter, then terminate the process for this segment.
3. Find $v$ that minimizes
 $Cyu:v + Cyv+1:w$
4. Test
 $Cyu:v + Cyv+1:w + β < Cyu:w$ (1)
5. If inequality (1) is false then the process is terminated for this segment.
6. 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 $\tau$.
nag_tsa_cp_binary (g13ndc) supplies four families of cost function. Each cost function assumes that the series, $y$, comes from some distribution, $D\left(\Theta \right)$. The parameter space, $\Theta =\left\{\theta ,\varphi \right\}$ is subdivided into $\theta$ containing those parameters allowed to differ in each segment and $\varphi$ 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:
 $C y τ i-1 + 1 : τi = -2 ⁢ log⁡ L θ^i , ϕ | y τ i-1 + 1 : τi$
where the ${\stackrel{^}{\theta }}_{i}$ is the maximum likelihood estimate of $\theta$ within the $i$th segment. Four distributions are available; Normal, Gamma, Exponential and Poisson distributions. Letting
 $Si= ∑ j=τi-1 τi yj$
the log-likelihoods and cost functions for the four distributions, and the available subdivisions of the parameter space are:
• Normal distribution: $\Theta =\left\{\mu ,{\sigma }^{2}\right\}$
 $-2⁢log⁡L = ∑ i=1 m ∑ j=τi-1 τi log2⁢π + logσi2 + yj-μi2 σi2$
• Mean changes: $\theta =\left\{\mu \right\}$
 $Cyτi-1+1:τi = ∑ j=τi-1 τi yj - ni-1 ⁢ Si 2 σ2$
• Variance changes: $\theta =\left\{{\sigma }^{2}\right\}$
 $Cyτi-1+1:τi = ni ⁢ log ∑ j=τi-1 τi yj-μ 2 - log⁡ni$
• Both mean and variance change: $\theta =\left\{\mu ,{\sigma }^{2}\right\}$
 $Cyτi-1+1:τi = ni ⁢ log ∑ j=τi-1 τi yj- ni-1 ⁢ Si 2 - log⁡ni$
• Gamma distribution: $\Theta =\left\{a,b\right\}$
 $-2⁢log⁡L = 2× ∑ i=1 m ∑ j=τi-1 τi log⁡Γai+ ai⁢log⁡bi+ 1-ai⁢log⁡yj+ yj bi$
• Scale changes: $\theta =\left\{b\right\}$
 $Cyτi-1+1:τi = 2⁢ a⁢ ni log⁡Si - log a⁢ ni$
• Exponential Distribution: $\Theta =\left\{\lambda \right\}$
 $- 2⁢log⁡L = 2× ∑ i=1 m ∑ j=τi-1 τi log⁡λi+ yj λi$
• Mean changes: $\theta =\left\{\lambda \right\}$
 $Cyτi-1+1:τi = 2⁢ ni log⁡Si - log⁡ni$
• Poisson distribution: $\Theta =\left\{\lambda \right\}$
 $-2⁢log⁡L = 2× ∑ i=1 m ∑ j=τi-1 τi λi- ⌊yj+0.5⌋⁢log⁡λi+ log⁡Γ⌊yj+0.5⌋+1$
• Mean changes: $\theta =\left\{\lambda \right\}$
 $Cyτi-1+1:τi = 2⁢ Si ⁢ log⁡ni - log⁡Si$
when calculating ${S}_{i}$ for the Poisson distribution, the sum is calculated for $⌊{y}_{i}+0.5⌋$ rather than ${y}_{i}$.

## 4References

Chen J and Gupta A K (2010) Parametric Statistical 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

## 5Arguments

1:    $\mathbf{ctype}$Nag_TS_ChangeTypeInput
On entry: a flag indicating the assumed distribution of the data and the type of change point being looked for.
${\mathbf{ctype}}=\mathrm{Nag_NormalMean}$
Data from a Normal distribution, looking for changes in the mean, $\mu$.
${\mathbf{ctype}}=\mathrm{Nag_NormalStd}$
Data from a Normal distribution, looking for changes in the standard deviation $\sigma$.
${\mathbf{ctype}}=\mathrm{Nag_NormalMeanStd}$
Data from a Normal distribution, looking for changes in the mean, $\mu$ and standard deviation $\sigma$.
${\mathbf{ctype}}=\mathrm{Nag_GammaScale}$
Data from a Gamma distribution, looking for changes in the scale parameter $b$.
${\mathbf{ctype}}=\mathrm{Nag_ExponentialLambda}$
Data from an exponential distribution, looking for changes in $\lambda$.
${\mathbf{ctype}}=\mathrm{Nag_PoissonLambda}$
Data from a Poisson distribution, looking for changes in $\lambda$.
Constraint: ${\mathbf{ctype}}=\mathrm{Nag_NormalMean}$, $\mathrm{Nag_NormalStd}$, $\mathrm{Nag_NormalMeanStd}$, $\mathrm{Nag_GammaScale}$, $\mathrm{Nag_ExponentialLambda}$ or $\mathrm{Nag_PoissonLambda}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the length of the time series.
Constraint: ${\mathbf{n}}\ge 2$.
3:    $\mathbf{y}\left[{\mathbf{n}}\right]$const doubleInput
On entry: $y$, the time series.
if ${\mathbf{ctype}}=\mathrm{Nag_PoissonLambda}$, that is the data is assumed to come from a Poisson distribution, $⌊y+0.5⌋$ is used in all calculations.
Constraints:
• if ${\mathbf{ctype}}=\mathrm{Nag_GammaScale}$, $\mathrm{Nag_ExponentialLambda}$ or $\mathrm{Nag_PoissonLambda}$, ${\mathbf{y}}\left[\mathit{i}-1\right]\ge 0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$;
• if ${\mathbf{ctype}}=\mathrm{Nag_PoissonLambda}$, each value of y must be representable as an integer;
• if ${\mathbf{ctype}}\ne \mathrm{Nag_PoissonLambda}$, each value of y must be small enough such that${{\mathbf{y}}\left[\mathit{i}-1\right]}^{2}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$, can be calculated without incurring overflow.
4:    $\mathbf{beta}$doubleInput
On entry: $\beta$, the penalty term.
There are a number of standard ways of setting $\beta$, including:
SIC or BIC
$\beta =p×\mathrm{log}\left(n\right)$
AIC
$\beta =2p$
Hannan-Quinn
$\beta =2p×\mathrm{log}\left(\mathrm{log}\left(n\right)\right)$
where $p$ is the number of parameters being treated as estimated in each segment. This is usually set to $2$ when ${\mathbf{ctype}}=\mathrm{Nag_NormalMeanStd}$ and $1$ otherwise.
If no penalty is required then set $\beta =0$. Generally, the smaller the value of $\beta$ the larger the number of suggested change points.
5:    $\mathbf{minss}$IntegerInput
On entry: the minimum distance between two change points, that is ${\tau }_{i}-{\tau }_{i-1}\ge {\mathbf{minss}}$.
Constraint: ${\mathbf{minss}}\ge 2$.
6:    $\mathbf{param}\left[1\right]$const doubleInput
On entry: $\varphi$, values for the parameters that will be treated as fixed. If ${\mathbf{ctype}}\ne \mathrm{Nag_GammaScale}$, param may be set to NULL.
If ${\mathbf{ctype}}=\mathrm{Nag_NormalMean}$
• if param is NULL, $\sigma$, the standard deviation of the Normal distribution, is estimated from the full input data. Otherwise $\sigma ={\mathbf{param}}\left[0\right]$.
If ${\mathbf{ctype}}=\mathrm{Nag_NormalStd}$
• If param is NULL, $\mu$, the mean of the Normal distribution, is estimated from the full input data. Otherwise $\mu ={\mathbf{param}}\left[0\right]$.
If ${\mathbf{ctype}}=\mathrm{Nag_GammaScale}$, ${\mathbf{param}}\left[0\right]$ must hold the shape, $a$, for the Gamma distribution, otherwise param is not referenced.
Constraint: if ${\mathbf{ctype}}=\mathrm{Nag_NormalMean}$ or $\mathrm{Nag_GammaScale}$, ${\mathbf{param}}\left[0\right]>0.0$.
7:    $\mathbf{mdepth}$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.
8:    $\mathbf{ntau}$Integer *Output
On exit: $m$, the number of change points detected.
9:    $\mathbf{tau}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array tau must be at least
• $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(⌈\frac{{\mathbf{n}}}{{\mathbf{minss}}}⌉,{2}^{{\mathbf{mdepth}}}\right)$ when ${\mathbf{mdepth}}>0$;
• $⌈\frac{{\mathbf{n}}}{{\mathbf{minss}}}⌉$ otherwise.
On exit: the first $m$ elements of tau hold the location of the change points. The $i$th segment is defined by ${y}_{\left({\tau }_{i-1}+1\right)}$ to ${y}_{{\tau }_{i}}$, where ${\tau }_{0}=0$ and ${\tau }_{i}={\mathbf{tau}}\left[i-1\right],1\le i\le m$.
The remainder of tau is used as workspace.
10:  $\mathbf{sparam}\left[\mathit{dim}\right]$doubleOutput
On exit: the estimated values of the distribution parameters in each segment
${\mathbf{ctype}}=\mathrm{Nag_NormalMean}$, $\mathrm{Nag_NormalStd}$ or $\mathrm{Nag_NormalMeanStd}$
${\mathbf{sparam}}\left[2i-2\right]={\mu }_{i}$ and ${\mathbf{sparam}}\left[2i-1\right]={\sigma }_{i}$ for $i=1,2,\dots ,m$, where ${\mu }_{i}$ and ${\sigma }_{i}$ is the mean and standard deviation, respectively, of the values of $y$ in the $i$th segment.
It should be noted that ${\sigma }_{i}={\sigma }_{j}$ when ${\mathbf{ctype}}=\mathrm{Nag_NormalMean}$ and ${\mu }_{i}={\mu }_{j}$ when ${\mathbf{ctype}}=\mathrm{Nag_NormalStd}$, for all $i$ and $j$.
${\mathbf{ctype}}=\mathrm{Nag_GammaScale}$
${\mathbf{sparam}}\left[2i-2\right]={a}_{i}$ and ${\mathbf{sparam}}\left[2i-1\right]={b}_{i}$ 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 ${a}_{i}={\mathbf{param}}\left[0\right]$ for all $i$.
${\mathbf{ctype}}=\mathrm{Nag_ExponentialLambda}$ or $\mathrm{Nag_PoissonLambda}$
${\mathbf{sparam}}\left[i-1\right]={\lambda }_{i}$ for $i=1,2,\dots ,m$, where ${\lambda }_{i}$ is the mean of the values of $y$ in the $i$th segment.
The remainder of sparam is used as workspace.
11:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{minss}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{minss}}\ge 2$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2$.
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, ${\mathbf{ctype}}=〈\mathit{\text{value}}〉$ and ${\mathbf{param}}\left[0\right]=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{ctype}}=\mathrm{Nag_NormalMean}$ or $\mathrm{Nag_GammaScale}$ and ${\mathbf{param}}\phantom{\rule{0.25em}{0ex}}\text{is not}\phantom{\rule{0.25em}{0ex}}\mathbf{NULL}$, then ${\mathbf{param}}\left[0\right]>0.0$.
NE_REAL_ARRAY
On entry, ${\mathbf{ctype}}=〈\mathit{\text{value}}〉$ and ${\mathbf{y}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{ctype}}=\mathrm{Nag_GammaScale}$, $\mathrm{Nag_ExponentialLambda}$ or $\mathrm{Nag_PoissonLambda}$ then ${\mathbf{y}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
On entry, ${\mathbf{y}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$, 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.

## 7Accuracy

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.

## 8Parallelism 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.

None.

## 10Example

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 $\beta =\mathrm{log}n\approx 4.6$, the minimum segment size is set to $2$ and the variance is fixed at $1$ across the whole input series.

### 10.1Program Text

Program Text (g13ndce.c)

### 10.2Program Data

Program Data (g13ndce.d)

### 10.3Program Results

Program Results (g13ndce.r)

This example plot shows the original data series, the estimated change points and the estimated mean in each of the identified segments.
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017