Package {IDetect}

IDetect: Multiple generalised change-point detection using the Isolate-Detect methodology

Description

The IDetect package implements the Isolate-Detect methodology for multiple generalised change-point detection in one-dimensional data following the “deterministic signal + noise” model. The different structures that are implemented are: piecewise-constant mean signal, piecewise-constant mean signal with heavy tailed noise, piecewise-linear mean and continuous signal, and piecewise-linear mean and continuous signal with heavy-tailed noise. The main routine of the package is ID.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

References

“Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint.

See Also

ID, ID_pcm, ID_plm, ht_ID_pcm, and ht_ID_plm.

Examples

#See Examples for ID.

Multiple change-point detection in the mean or the slope of a vector using the Isolate-Detect methodology

Description

This is the main, general function of the package. It employs more specialised functions in order to estimate the number and locations of multiple change-points in either piecewise-constant or piecewise-linear mean of a noisy input vector xd. The noise can either follow the Gaussian distribution or not. Further to the estimated change-points, ID, returns the estimated signal, as well as the solution path. For more information and the relevant literature reference, see Details.

Usage

ID(
  xd,
  th.cons = 1,
  th.cons_lin = 1.4,
  th.ic = 0.9,
  th.ic.lin = 1.25,
  lam = 3,
  lam.ic = 10,
  contrast = c("mean", "slope"),
  ht = FALSE,
  scale = 3
)

Arguments

Details

The data points provided in xd are assumed to follow

X_t = f_t + \sigma\epsilon_t; t = 1,2,...,T,

where T is the total length of the data sequence, X_t are the observed data, f_t is an one-dimensional, deterministic signal with abrupt structural changes at certain points, and \epsilon_t are independent and identically distributed random variables with mean zero and variance equal to one. In this function, the following scenarios for f_t are implemented.

• Piecewise-constant signal with Gaussian noise.

  Use contrast = "mean" and ht = FALSE here.

• Piecewise-constant signal with heavy-tailed noise.

  Use contrast = "mean" and ht = TRUE here.

• Piecewise-linear and continuous signal with Gaussian noise.

  Use contrast = "slope" and ht = FALSE here.

• Piecewise-linear and continuous signal with heavy-tailed noise.

  Use contrast = "slope" and ht = TRUE here.

Value

A list with the following components:

cpt A vector with the detected change-points.

no_cpt The number of change-points detected.

fit A numeric vector with the estimated piecewise-linear mean signal.

solution_path A vector containing the solution path.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

ID_pcm, ID_plm, ht_ID_pcm, and ht_ID_plm, which are the functions that are employed in in ID, depending on which scenario is imposed by the input arguments.

Examples

single.cpt.mean <- c(rep(4,3000),rep(0,3000))
single.cpt.mean.normal <- single.cpt.mean + rnorm(6000)
single.cpt.mean.student <- single.cpt.mean + rt(6000, df = 5)
cpt.single.mean.normal <- ID(single.cpt.mean.normal)
cpt.single.mean.student <- ID(single.cpt.mean.student, ht = TRUE)

single.cpt.slope <- c(seq(0, 1999, 1), seq(1998, -1, -1))
single.cpt.slope.normal <- single.cpt.slope + rnorm(4000)
single.cpt.slope.student <- single.cpt.slope + rt(4000, df = 5)
cpt.single.slope.normal <- ID(single.cpt.slope.normal, contrast = "slope")
cpt.single.slope.student <- ID(single.cpt.slope.student, contrast = "slope", ht = TRUE)

Multiple change-point detection in the mean of a vector using the Isolate-Detect method

Description

This function estimates the number and locations of multiple change-points in the piecewise-constant mean of the noisy input vector x, using the Isolate-Detect methodology. It also gives the estimated signal, as well as the solution path (see Details for the relevant literature reference).

Usage

ID_pcm(x, thr_id = 1, th_ic_id = 0.9, pointsth = 3, pointsic = 10)

Arguments

Details

Firstly, this function detects the change-points using wind_pcm_th. If the estimated number of change-points is larger than 100, then the result is returned and we stop. Otherwise, ID_pcm proceeds to detect the change-points using cpt_ic_pcm and this is what is returned. To sum up, ID_pcm returns a result based on cpt_ic_pcm if the estimated number of change-points is less than 100. Otherwise, the result comes from thresholding. More details can be found in “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint.

Value

A list with the following components:

cpt A vector with the detected change-points.

no_cpt The number of change-points detected.

fit A numeric vector with the estimated piecewise-constant mean signal.

solution_path A vector containing the solution path.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

wind_pcm_th and cpt_ic_pcm which are the functions that ID_pcm is based on. In addition, see ID_plm for the case of detecting changes in the slope of a piecewise-linear and continuous signal. The main function ID of the package employs ID_pcm.

Examples

single.cpt <- c(rep(4,1000),rep(0,1000))
single.cpt.noise <- single.cpt + rnorm(2000)
cpts_detect <- ID_pcm(single.cpt.noise)

three.cpt <- c(rep(4,500),rep(0,500),rep(-4,500),rep(1,500))
three.cpt.noise <- three.cpt + rnorm(2000)
cpts_detect_three <- ID_pcm(three.cpt.noise)

multi.cpt <- rep(c(rep(0,50),rep(3,50)),20)
multi.cpt.noise <- multi.cpt + rnorm(2000)
cpts_detect_multi <- ID_pcm(multi.cpt.noise)

Multiple change-point detection in the slope of a vector using the Isolate-Detect method

Description

This function estimates the number and locations of multiple change-points in the slope of a continuous piecewise-linear mean of the noisy input vector x, using the Isolate-Detect methodology. It also gives the estimated signal, as well as the solution path (see Details for the relevant literature reference).

Usage

ID_plm(x, thr_id = 1.4, th_ic_id = 1.25, pointsth = 3, pointsic = 10)

Arguments

Details

Firstly, this function detects the change-points using wind_plm_th. If the estimated number of change-points is larger than 100, then the result is returned and we stop. Otherwise, ID_plm proceeds to detect the change-points using cpt_ic_plm and this is what is returned. To sum up, ID_plm returns a result based on cpt_ic_plm if the estimated number of change-points is less than 100. Otherwise, the result comes from thresholding. More details can be found in “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint.

Value

A list with the following components:

cpt A vector with the detected change-points.

no_cpt The number of change-points detected.

fit A numeric vector with the estimated continuous piecewise-linear mean signal.

solution_path A vector containing the solution path.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

wind_plm_th and cpt_ic_plm which are the functions that ID_plm is based on. In addition, see ID_pcm for the case of detecting changes in the mean of a piecewise-constant signal. The main function ID of the package employs ID_plm.

Examples

single.cpt <- c(seq(0, 999, 1), seq(998.5, 499, -0.5))
single.cpt.noise <- single.cpt + rnorm(2000)
cpt.single <- ID_plm(single.cpt.noise)

three.cpt <- c(seq(0, 499, 1), seq(498.5, 249, -0.5), seq(250,1249,2), seq(1248,749,-1))
three.cpt.noise <- three.cpt + rnorm(2000)
cpt.three <- ID_plm(three.cpt.noise)

multi.cpt <- rep(c(seq(0,49,1), seq(48,0,-1)),20)
multi.cpt.noise <- multi.cpt + rnorm(1980)
cpt.multi <- ID_plm(multi.cpt.noise)

Multiple change-point detection in the mean via minimising an information criterion

Description

This function performs the Isolate-Detect methodology based on an information criterion approach, in order to detect multiple change-points in the mean of a given data sequence. The relevant literature reference is given in details.

Usage

cpt_ic_pcm(
  x,
  th_const = 0.9,
  Kmax = 200,
  penalty = c("ssic_pen", "sic_pen"),
  points = 10
)

Arguments

Details

The approach followed in cpt_ic_pcm in order to detect the change-points is based on identifying the set of change-point that minimise an information criterion. The obtained set of change-points is a subset of the solution path, which is given by sol_path_pcm. More details can be found in “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint.

Value

A list with the following components:

sol_path A vector containing the solution path.

ic_curve A list with values of the chosen information criteria.

cpt_ic A list with the change-points detected for each information criterion considered.

no_cpt_ic The number of change-points detected for each information criterion considered.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

ID_pcm and ID, which employ this function. In addition, see cpt_ic_plm for the case of detecting changes in the slope of a piecewise-linear and continuous signal using the information criterion based approach.

Examples

single.cpt <- c(rep(4,1000),rep(0,1000))
single.cpt.noise <- single.cpt + rnorm(2000)
cpt.single.ic <- cpt_ic_pcm(single.cpt.noise)

three.cpt <- c(rep(4,500),rep(0,500),rep(-4,500),rep(1,500))
three.cpt.noise <- three.cpt + rnorm(2000)
cpt.three.ic <- cpt_ic_pcm(three.cpt.noise)

Multiple change-point detection in the slope of a continuous piecewise-linear mean signal via minimising an information criterion

Description

This function performs the Isolate-Detect methodology based on an information criterion approach, in order to detect multiple change-points in the slope of a given data sequence. The relevant literature reference is given in details.

Usage

cpt_ic_plm(
  x,
  th_const = 1.25,
  Kmax = 200,
  penalty = c("ssic_pen", "sic_pen"),
  points = 10
)

Arguments

Details

The approach followed in cpt_ic_plm in order to detect the change-points is based on identifying the set of change-point that minimise an information criterion. The obtained set of change-points is a subset of the solution path, which is given by sol_path_plm. More details can be found in “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint.

Value

A list with the following components:

sol_path A vector containing the solution path.

ic_curve A list with values of the chosen information criteria.

cpt_ic A list with the change-points detected for each information criterion considered.

no_cpt_ic The number of change-points detected for each information criterion considered.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

ID_plm and ID, which employ this function. In addition, see cpt_ic_pcm for the case of detecting changes in the mean of a piecewise-constant signal using the information criterion based approach.

Examples

single.cpt <- c(seq(0, 999, 1), seq(998.5, 499, -0.5))
single.cpt.noise <- single.cpt + rnorm(2000)
cpt.single.ic <- cpt_ic_plm(single.cpt.noise)

three.cpt <- c(seq(0, 499, 1), seq(498.5, 249, -0.5), seq(250,1249,2), seq(1248,749,-1))
three.cpt.noise <- three.cpt + rnorm(2000)
cpt.three.ic <- cpt_ic_plm(three.cpt.noise)

Calculate the contrast function that is used in continuous piecewise-linear mean signals

Description

This function returns the values of the contrast function, which is used for for change-point detection in continuous piecewise-linear mean signals. See Details for more information.

Usage

cumsum_lin(x)

Arguments

Details

The mathematical expression of the result returned by cumsum_lin is rather large. Therefore, for the exact formula please see the relevant subsection for piecewise-linearity in the preprint “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017).

Value

A numeric vector with the contrast function values at b = 1,2,...,T-1, where T is the length of x. Note that due to the structure of the signal (piecewise-linear mean), the value of the contrast function statistic at b=1 is equal to zero.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

cusum_function for the calculation of the CUSUM statistic, which is the contrast function used in the case of piecewise-constant mean signals.

Examples

no.cpt.noise <- rnorm(2000)
cf.no.cpt <- IDetect:::cumsum_lin(no.cpt.noise)

single.cpt <- c(seq(0, 999, 1), seq(998.5, 499, -0.5))
single.cpt.noise <- single.cpt + rnorm(2000)
cf.single.cpt <- IDetect:::cumsum_lin(single.cpt.noise)
#*** Notice that the maximum in absolute value of \code{csm.single.cpt}
#*** occurs in a neighbourhood of the true change-point, which is 1000.
which.max(abs(cf.single.cpt))

Calculate the CUMSUM statistic

Description

This function returns the CUMSUM statistic for a given data sequence. See Details for more information.

Usage

cusum_function(x)

Arguments

Details

The CUSUM statistic for x at a location b is defined as

\tilde{X}_{s,e}^b = \sqrt{\frac{e-b}{n(b-s+1)}}\sum_{t=s}^{b}X_t - \sqrt{\frac{b-s+1}{n(e-b)}}\sum_{t=b+1}^{e}X_t,

where 1\le s \le b < e\le T and n=e-s+1. In cusum_function, we have s=1, e=T.

Value

A numeric vector with the CUSUM statistic values at b = 1,2,...,T-1, where T is the length of x.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

cumsum_lin for the calculation of the contrast function that is used in the case of piecewise-linear mean signals.

Examples

no.cpt.noise <- rnorm(2000)
csm.no.cpt <- IDetect:::cusum_function(no.cpt.noise)

single.cpt <- c(rep(4,1000),rep(0,1000))
single.cpt.noise <- single.cpt + rnorm(2000)
csm.single.cpt <- IDetect:::cusum_function(single.cpt.noise)
#*** Notice that the maximum in absolute value of \code{csm.single.cpt}
#*** occurs in a neighbourhood of the true change-point, which is 1000.
which.max(abs(csm.single.cpt))

Calculate the CUMSUM statistic at specific values

Description

This function returns the CUMSUM statistic at predefined positions of a given data sequence. The routine is typically not called directly by the user; its result is used in the derivation of the solution path in the case of a piecewise-constant mean signal, which is carried out in sol_path_pcm.

Usage

cusum_one(x, s, e, b)

Arguments

Value

A numeric vector of length l_b, of which the j^{th} element is the CUSUM statistic value at b_j, when the start- and end-points are s_j and e_j, respectively.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

cusum_function for the calculation of the CUSUM statistic for all data points of x. Also, see linear_contr_one for a function that has the same purpose, but for the case of the contrast function for continuous and piecewise-linear mean signals.

Examples

no.cpt.noise <- rnorm(2000)
ex1 <- IDetect:::cusum_one(no.cpt.noise, s = c(1, 5, 9), e = c(30, 56, 71), b = c(20, 40, 45))

Estimate the signal

Description

This function estimates the signal in a given data sequence x with change-points at cpt. The type of the signal depends on whether the change-points represent changes in the mean of a piecewise-constant signal or a piecewise-linear signal. For more information see Details below.

Usage

est_signal(x, cpt, type = c("mean", "slope"))

Arguments

Details

The data points provided in x are assumed to follow

X_t = f_t + \sigma\epsilon_t; t = 1,2,...,T

,

where T is the total length of the data sequence, X_t are the observed data, f_t is an one-dimensional, deterministic signal with abrupt structural changes at certain points, and \epsilon_t is white noise. We denote by r_1, r_2, ..., r_N the elements in cpt and by r_0 = 0 and r_{N+1} = T. Depending on the value that has been passed to type, the returned value is calculated as follows.

• For type = "mean", in each segment (r_j + 1, r_{j+1}), f_t for t \in (r_j + 1, r_{j+1}) is approximated by the mean of X_t calculated over t \in (r_j + 1, r_{j+1}).

• For type = "slope", f_t is approximated by the linear spline fit with knots at r_1, r_2, ..., r_N minimising the l_2 distance between the fit and the data.

Value

A numeric vector with the estimated signal.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

Examples

single.cpt.pcm <- c(rep(4,1000),rep(0,1000))
single.cpt.pcm.noise <- single.cpt.pcm + rnorm(2000)
cpt.single.pcm <- ID_pcm(single.cpt.pcm.noise)
fit.cpt.single.pcm <- est_signal(single.cpt.pcm.noise, cpt.single.pcm$cpt, type = "mean")

three.cpt.pcm <- c(rep(4,500),rep(0,500),rep(-4,500),rep(1,500))
three.cpt.pcm.noise <- three.cpt.pcm + rnorm(2000)
cpt.three.pcm <- ID_pcm(three.cpt.pcm.noise)
fit.cpt.three.pcm <- est_signal(three.cpt.pcm.noise, cpt.three.pcm$pcm, type = "mean")

single.cpt.plm <- c(seq(0,999,1),seq(998.5,499,-0.5))
single.cpt.plm.noise <- single.cpt.plm + rnorm(2000)
cpt.single.plm <- ID_plm(single.cpt.plm.noise)
fit.cpt.single.plm <- est_signal(single.cpt.plm.noise, cpt.single.plm$cpt, type = "slope")

Apply the Isolate-Detect methodology for multiple change-point detection in the mean of a vector with non Gaussian noise

Description

Using the Isolate-Detect methodology, this function estimates the number and locations of multiple change-points in the piecewise-constant mean of a noisy input vector x, with noise that is not normally distributed. It also gives the estimated signal, as well as the solution path (see Details for the relevant literature reference).

Usage

ht_ID_pcm(
  x,
  s_ht = 3,
  l_ht = 300,
  ht_thr_id = 1,
  ht_th_ic_id = 0.9,
  p_thr = 1,
  p_ic = 3
)

Arguments

Details

Firstly, in this function we call normalise, in order to create a new data sequence, \tilde{x}, by taking averages of observations in x. Then, we employ link{ID_pcm} on \tilde{x}_q to obtain the change-points, namely \tilde{r}_1, \tilde{r}_2, ..., \tilde{r}_{\hat{N}} in an increasing order. To obtain the original location of the change-points with, on average, the highest accuracy we define

\hat{r}_k = (\tilde{r}_{k}-1)*s_ht + \lfloor s_ht/2 + 0.5 \rfloor, k=1, 2,..., \hat{N}.

More details can be found in “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint.

Value

A list with the following components:

cpt A vector with the detected change-points.

no_cpt The number of change-points detected.

fit A numeric vector with the estimated piecewise-constant mean signal.

solution_path A vector containing the solution path.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

ID_pcm and normalise, which are functions that are used in ht_ID_pcm. In addition, see ht_ID_plm for the case of continuous and piecewise-linear mean signals.

Examples

single.cpt <- c(rep(4,3000),rep(0,3000))
single.cpt.student <- single.cpt + rt(6000, df = 5)
cpts_detect <- ht_ID_pcm(single.cpt.student)

three.cpt <- c(rep(4,2000),rep(0,2000),rep(-4,2000),rep(0,2000))
three.cpt.student <- three.cpt + rt(8000, df = 5)
cpts_detect_three <- ht_ID_pcm(three.cpt.student)

Apply the Isolate-Detect methodology for multiple change-point detection in the slope of a vector with non Gaussian noise

Description

Using the Isolate-Detect methodology, this function estimates the number and locations of multiple change-points in the piecewise-linear mean of a noisy input vector x, with noise that is not normally distributed. It also gives the estimated signal, as well as the solution path (see Details for the relevant literature reference).

Usage

ht_ID_plm(
  x,
  s_ht = 3,
  l_ht = 300,
  ht_thr_id = 1.4,
  ht_th_ic_id = 1.25,
  p_thr = 1,
  p_ic = 3
)

Arguments

Details

Firstly, in this function we call normalise, in order to create a new data sequence, \tilde{x}, by taking averages of observations in x. Then, we employ link{ID_plm} on \tilde{x}_q to obtain the change-points, namely \tilde{r}_1, \tilde{r}_2, ..., \tilde{r}_{\hat{N}} in an increasing order. To obtain the original location of the change-points with, on average, the highest accuracy we define

\hat{r}_k = (\tilde{r}_{k}-1)*s_ht + \lfloor s_ht/2 + 0.5 \rfloor, k=1, 2,..., \hat{N}.

More details can be found in “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint.

Value

A list with the following components:

cpt A vector with the detected change-points.

no_cpt The number of change-points detected.

fit A numeric vector with the estimated piecewise-linear mean signal.

solution_path A vector containing the solution path.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

ID_plm and normalise, which are functions that are used in ht_ID_plm. In addition, see ht_ID_pcm for the case of piecewise-constant mean signals.

Examples

single.cpt <- c(seq(0, 1999, 1), seq(1998, -1, -1))
single.cpt.student <- single.cpt + rt(4000, df = 5)
cpt.single <- ht_ID_plm(single.cpt.student)

three.cpt <- c(seq(0, 3998, 2), seq(3996, -2, -2), seq(0,3998,2), seq(3996,-2,-2))
three.cpt.student <- three.cpt + rt(8000, df = 5)
cpt.three <- ht_ID_plm(three.cpt.student)

Calculate the contrast function for the continuous piecewise-linear mean case at specific values

Description

This function returns, at predefined positions, the values of the contrast function for a given data sequence with under the scenario of continuous, piecewise-linear mean signals. The routine is typically not called directly by the user; its result is used in the derivation of the solution path in the case of a piecewise-linear mean signal, which is carried out in sol_path_plm.

Usage

linear_contr_one(x, s, e, b)

Arguments

Value

A numeric vector of length l_b, of which the j^{th} element is the contrast function value at b_j, when the start- and end-points are s_j and e_j, respectively.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

cumsum_lin for the calculation of the contrast function for all data points of x. Also, see cusum_one for a function that has the same purpose, but for the case of the CUSUM statistic, which is used in piecewise-constant mean signals.

Examples

noise <- rnorm(2000)
ex.lin <- IDetect:::linear_contr_one(noise, s = c(1, 5, 9), e = c(6, 56, 71), b = c(4, 40, 45))

Calculate the log-likelihood in the case of a continuous piecewise-linear mean signal

Description

This function calculates the Gaussian log-likelihood for the continuous piecewise-linear mean signal estimated using est_signal with the changepoints at cpt and for type = ``slope''.

Usage

log_lik_slope(x, cpt)

Arguments

Value

The Gaussian log-likelihood for the continuous piecewise-linear mean signal estimated using est_signal with the changepoints at cpt.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

Examples

single.cpt.plm <- c(seq(0, 999, 1), seq(998.5, 499, -0.5))
single.cpt.plm.noise <- single.cpt.plm + rnorm(2000)
cpt_detect <- ID(single.cpt.plm.noise, contrast = "slope")
loglik_cpt <- IDetect:::log_lik_slope(single.cpt.plm.noise, cpt_detect$cpt)

Transform the noise to be closer to the Gaussian distribution

Description

This function pre-processes the given data in order to obtain a noise structure that is closer to satisfying the Gaussianity assumption. See details for more information and for the relevant literature reference.

Usage

normalise(x, sc = 3)

Arguments

Details

For a given natural number sc and data x of length T, let us denote by Q = \lceil T/sc \rceil. Then, normalise calculates

\tilde{x}_q = 1/sc\sum_{t=(q-1) * sc + 1}^{q * sc}x_t,

for q=1, 2, ..., Q-1, while

\tilde{x}_Q = (T - (Q-1) * sc)^{-1}\sum_{t = (Q-1) * sc + 1}^{T}x_t.

More details can be found in the preprint “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017).

Value

The “normalised” vector \tilde{x} of length Q, as explained in Details.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

ht_ID_pcm, ht_ID_plm, and ID, which are functions that employ normalise.

Examples

t5 <- rt(n = 10000, df = 5)
n5 <- normalise(t5, sc = 3)

Multiple change-point detection in the mean via thresholding

Description

This function performs the Isolate-Detect methodology (see Details for the relevant literature reference) with the thresholding-based stopping rule in order to detect multiple change-points in the mean of a given data sequence.

Usage

pcm_th(
  x,
  sigma = stats::mad(diff(x)/sqrt(2)),
  thr_const = 1,
  thr_fin = sigma * thr_const * sqrt(2 * log(length(x))),
  s = 1,
  e = length(x),
  points = 3,
  k_l = 1,
  k_r = 1
)

Arguments

Details

The change-point detection algorithm that is used in pcm_th is the Isolate-Detect methodology described in “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint. The concept is simple and is split into two stages; firstly, isolation of each of the true changepoints in small intervals, and secondly their detection.

Value

A numeric vector with the detected change-points.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

wind_pcm_th, ID_pcm, and ID, which employ this function. In addition, see plm_th for the case of detecting changes in the slope of a piecewise-linear and continuous signal via thresholding.

Examples

single.cpt <- c(rep(4,1000),rep(0,1000))
single.cpt.noise <- single.cpt + rnorm(2000)
cpt.single.th <- pcm_th(single.cpt.noise)

three.cpt <- c(rep(4,500),rep(0,500),rep(-4,500),rep(1,500))
three.cpt.noise <- three.cpt + rnorm(2000)
cpt.three.th <- pcm_th(three.cpt.noise)

multi.cpt <- rep(c(rep(0,50),rep(3,50)),20)
multi.cpt.noise <- multi.cpt + rnorm(2000)
cpt.multi.th <- pcm_th(multi.cpt.noise)

Multiple change-point detection in the slope of a piecewise-linear mean signal via thresholding

Description

This function performs the Isolate-Detect methodology (see Details for the relevant literature reference) with the thresholding-based stopping rule in order to detect multiple change-points in the slope of a piecewise-linear mean of a given data sequence.

Usage

plm_th(
  x,
  sigma = stats::mad(diff(diff(x)))/sqrt(6),
  thr_const = 1.4,
  thr_fin = sigma * thr_const * sqrt(2 * log(length(x))),
  s = 1,
  e = length(x),
  points = 3,
  k_l = 1,
  k_r = 1
)

Arguments

Details

The change-point detection algorithm that is used in plm_th is the Isolate-Detect methodology described in “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint. The concept is simple and is split into two stages; firstly, isolation of each of the true changepoints in small intervals, and secondly their detection.

Value

A numeric vector with the detected change-points.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

wind_plm_th, ID_plm, and ID, which employ this function. In addition, see pcm_th for the case of detecting changes in the mean of a piecewise-constant signal via thresholding.

Examples

single.cpt <- c(seq(0, 999, 1), seq(998.5, 499, -0.5))
single.cpt.noise <- single.cpt + rnorm(2000)
cpt.single.th <- plm_th(single.cpt.noise)

three.cpt <- c(seq(0, 499, 1), seq(498.5, 249, -0.5), seq(251,1249,2), seq(1248,749,-1))
three.cpt.noise <- three.cpt + rnorm(2000)
cpt.three.th <- plm_th(three.cpt.noise)

multi.cpt <- rep(c(seq(0,49,1), seq(48,0,-1)),20)
multi.cpt.noise <- multi.cpt + rnorm(1980)
cpt.multi.th <- plm_th(multi.cpt.noise)

Calculate the residuals related to the estimated signal

Description

This function returns a difference between x and the estimated signal with change-points at cpt. The input in the argument type_chg will indicate the type of changes in the signal.

Usage

resid(
  x,
  cpt,
  type_chg = c("mean", "slope"),
  type_res = c("raw", "standardised")
)

Arguments

Value

If type_res = "raw", the function returns the difference between the data and the estimated signal. If type_res = "standardised", then the function returns the difference between the data and the estimated signal, divided by the estimated standard deviation.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

Examples

single.cpt.pcm <- c(rep(4,1000),rep(0,1000))
single.cpt.pcm.noise <- single.cpt.pcm + rnorm(2000)
cpt_detect <- ID(single.cpt.pcm.noise, contrast = "mean")

residuals_cpt_raw <- resid(single.cpt.pcm.noise, cpt = cpt_detect$cpt, type_chg = "mean",
type_res = "raw")

residuals_cpt_stand. <- resid(single.cpt.pcm.noise, cpt = cpt_detect$cpt, type_chg = "mean",
type_res = "standardised")

plot(residuals_cpt_raw)
plot(residuals_cpt_stand.)

Derives a subset of integers from a given set

Description

This function finds two subsets of integers in a given interval [s,e]. The routine is typically not called directly by the user; its result is used in order to construct the expanding intervals, where the Isolate-Detect method is going to be applied.

Usage

s_e_points(r, l, s, e)

Arguments

Value

e_points A vector containing the points that will be used as end-points, in order to create the left-expanding intervals. It consists of the input e and all the elements in the input vector r that are in (s,e).

s_points A vector containing the points that will be used as start-points, in order to create the left-expanding intervals. It consists of the input s and all the elements in the input vector l that are in (s,e)

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

Examples

s_e_points(r = seq(10,1000,10), l = seq(991,1,-10), s=435, e = 786)
s_e_points(r = seq(3,100,3), l = seq(98,1,-3), s=43, e = 86)

Schwarz Information Criterion penalty

Description

This function evaluates the penalty term for the Schwarz Information Criterion. The routine is typically not called directly by the user; its name can be passed as an argument to cpt_ic_pcm and cpt_ic_plm.

Usage

sic_pen(n, n_param)

Arguments

Value

The penalty term log(n) * n_param.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

ssic_pen for the strengthened Schwarz Information Criterion penalty.

Examples

three.cpt <- c(rep(4,400),rep(0,400),rep(-4,400),rep(1,400))
three.cpt.noise <- three.cpt + rnorm(1600)
detected_cpts <- cpt_ic_pcm(three.cpt.noise, penalty = "sic_pen")

The solution path for the case of piecewise-constant mean signals

Description

This function starts by over-estimating the number of true change-points. After that, following a CUSUM-based approach, it sorts the estimated change-points in a way that the estimation, which is most-likely to be correct appears first, whereas the least likely to be correct, appears last. The routine is typically not called directly by the user; it is employed in cpt_ic_pcm.

Usage

sol_path_pcm(x, thr_ic = 0.9, points = 3)

Arguments

Value

The solution path for the case of piecewise-constant mean signals.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

Examples

three.cpt <- c(rep(4,4000),rep(0,4000),rep(-4,4000),rep(1,4000))
three.cpt.noise <- three.cpt + rnorm(16000)
solution.path <- sol_path_pcm(three.cpt.noise)

The solution path for the case of continuous piecewise-linear mean signals

Description

This function starts by over-estimating the number of true change-points. After that, following an approach based on the values of a contrast function, it sorts the estimated change-points in a way that the estimation, which is most-likely to be correct appears first, whereas the least likely to be correct, appears last. The routine is typically not called directly by the user; it is employed in cpt_ic_plm.

Usage

sol_path_plm(x, thr_ic = 1.25, points = 3)

Arguments

Value

The solution path for the case of continuous piecewise-linear mean signals.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

Examples

three.cpt <- c(seq(0, 499, 1.2), seq(498.5, 249, -0.5), seq(250.5,999,1.5), seq(998,499,-1))
three.cpt.noise <- three.cpt + rnorm(2000)
solution.path <- sol_path_plm(three.cpt.noise)

Strengthened Schwarz Information Criterion penalty

Description

This function evaluates the penalty term for the strengtened Schwarz Information Criterion proposed in Fryzlewicz (2014). The routine is typically not called directly by the user; its name can be passed as an argument to cpt_ic_pcm and cpt_ic_plm.

Usage

ssic_pen(n, n_param, alpha = 1.01)

Arguments

Details

The strengthened Schwarz Information Criterion was introduced in Fryzlewicz (2014). Taking alpha = 1 will give the known Schwarz Information Criterion of sic_pen.

Value

The penalty term log(n)^alpha * n_param.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

References

Fryzlewicz, P. (2014). Wild Binary Segmentation for multiple change-point detection. Annals of Statistics, Vol. 42, No. 6, 2243-2281.

See Also

sic_pen for the Schwarz Information Criterion penalty.

Examples

three.cpt <- c(rep(4,400),rep(0,400),rep(-4,400),rep(1,400))
three.cpt.noise <- three.cpt + rnorm(1600)
detected_cpts <- cpt_ic_pcm(three.cpt.noise, penalty = "ssic_pen")

A window-based approach for multiple change-point detection in the mean via thresholding

Description

This function performs the windows-based variant of the Isolate-Detect methodology with the thresholding-based stopping rule in order to detect multiple change-points in the mean of a given data sequence. It is particularly helpful for very long data sequences, as due to applying Isolate-Detect on moving windows, it reduces the computational time (see Details for the relevant literature reference).

Usage

wind_pcm_th(
  xd,
  sigma = stats::mad(diff(xd)/sqrt(2)),
  thr_con = 1,
  c_win = 3000,
  w_points = 3,
  l_win = 12000
)

Arguments

Details

The method that is implemented by this function is based on splitting the given data sequence uniformly into smaller parts (windows), to which Isolate-Detect is then applied. An idea of the computational improvement that this structure offers over the classical Isolate-Detect in the case of large data sequences is explained in the supplement of “Detecting multiple generalized change-points by isolating single ones”, Anastasiou and Fryzlewicz (2017), preprint.

Value

A numeric vector with the detected change-points.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

pcm_th, which is the function that wind_pcm_th is based on. Also, see ID_pcm and ID, which employ wind_pcm_th. In addition, see wind_plm_th for the case of detecting changes in the slope of a piecewise-linear and continuous signal via thresholding.

Examples

single.cpt <- c(rep(4,1000),rep(0,1000))
single.cpt.noise <- single.cpt + rnorm(2000)
cpt.single.th <- wind_pcm_th(single.cpt.noise)

three.cpt <- c(rep(4,4000),rep(0,4000),rep(-4,4000),rep(1,4000))
three.cpt.noise <- three.cpt + rnorm(16000)
cpt.three.th <- wind_pcm_th(three.cpt.noise)

A window-based approach for multiple change-point detection in the slope of a picewise-linear mean signal via thresholding

Description

This function performs the windows-based variant of the Isolate-Detect methodology with the thresholding-based stopping rule in order to detect multiple change-points in the slope of a given data sequence. It is particularly helpful for very long data sequences, as due to applying Isolate-Detect on moving windows, it reduces the computational time (see Details for the relevant literature reference).

Usage

wind_plm_th(
  xd,
  sigma = stats::mad(diff(diff(xd)))/sqrt(6),
  thr_con = 1.4,
  c_win = 3000,
  w_points = 3,
  l_win = 12000
)

Arguments

Details

The method that is implemented by this function is based on splitting the given data sequence uniformly into smaller parts (windows), to which Isolate-Detect is then applied.

Value

A numeric vector with the detected change-points.

Author(s)

Andreas Anastasiou, anastasiou.andreas@ucy.ac.cy

See Also

plm_th, which is the function that wind_plm_th is based on. Also, see ID_plm and ID, which employ wind_plm_th. In addition, see wind_pcm_th for the case of detecting changes in the mean of a piecewise-constant signal via thresholding.

Examples

single.cpt <- c(seq(0, 999, 1), seq(998.5, 499, -0.5))
single.cpt.noise <- single.cpt + rnorm(2000)
cpt.single.th <- wind_plm_th(single.cpt.noise)

three.cpt <- c(seq(0, 3999, 1), seq(3998.5, 1999, -0.5), seq(2001,9999,2), seq(9998,5999,-1))
three.cpt.noise <- three.cpt + rnorm(16000)
cpt.three.th <- wind_plm_th(three.cpt.noise)