Ensemble Methods for Multiple Change-Point Detection

Description

Implements a segmentation algorithm for multiple change-point detection in univariate time series using the Ensemble Binary Segmentation of Korkas (2022).

Details

We propose a new technique for consistent estimation of the number and locations of the change-points in the structure of an irregularly spaced time series. The core of the segmentation procedure is the Ensemble Binary Segmentation method (EBS), a technique in which a large number of multiple change-point detection tasks using the Binary Segmentation (BS) method are applied on sub-samples of the data of differing lengths, and then the results are combined to create an overall answer. This methodology is applied to irregularly time series models such as the time-varying Autoregressive Conditional Duration model or the time-varying Hawkes process.

Author(s)

Karolos K. Korkas <kkorkas@yahoo.co.uk>.

Maintainer: Karolos K. Korkas <kkorkas@yahoo.co.uk>

References

Korkas, K.K., 2022. Ensemble binary segmentation for irregularly spaced data with change-points. Journal of the Korean Statistical Society, 51(1), pp.65-86.

Examples

  ## Not run: 
 pw.acd.obj <- new("simACD")
 pw.acd.obj@cp.loc <- seq(0.1,0.95,by=0.025)
 pw.acd.obj@lambda_0 <- rep(c(0.5,2),1+length(pw.acd.obj@cp.loc)/2)
 pw.acd.obj@alpha <- rep(0.2,1+length(pw.acd.obj@cp.loc))
 pw.acd.obj@beta <- rep(0.4,1+length(pw.acd.obj@cp.loc))
 pw.acd.obj@N <- 5000
 pw.acd.obj <- pc_acdsim(pw.acd.obj)
 ts.plot(pw.acd.obj@x,main="Ensemble BS");abline(v=EnBinSeg(pw.acd.obj@x)[[1]],col="red")
 #real change-points in grey
 abline(v=floor(pw.acd.obj@cp.loc*pw.acd.obj@N),col="grey",lty=2)
 ts.plot(pw.acd.obj@x,main="Standard BS");abline(v=BinSeg(pw.acd.obj@x)[[1]],col="blue")
 #real change-points in grey
 abline(v=floor(pw.acd.obj@cp.loc*pw.acd.obj@N),col="grey",lty=2)

  
## End(Not run)

An S4 method to detect the change-points in an irregularly spaced time series using Binary Segmentation.

Description

An S4 method to detect the change-points in an irregularly spaced time series using the Binary Segmentation methodology described in Korkas (2022).

Usage

BinSeg(
  H,
  thresh = "universal",
  q = 0.99,
  p = 1,
  z = NULL,
  start.values = c(0.9, 0.6),
  dampen.factor = "auto",
  epsilon = 1e-05,
  LOG = TRUE,
  process = "acd",
  acd_p = 0,
  acd_q = 1,
  do.parallel = 2
)

## S4 method for signature 'ANY'
BinSeg(
  H,
  thresh = "universal",
  q = 0.99,
  p = 1,
  z = NULL,
  start.values = c(0.9, 0.6),
  dampen.factor = "auto",
  epsilon = 1e-05,
  LOG = TRUE,
  process = "acd",
  acd_p = 0,
  acd_q = 1,
  do.parallel = 2
)

Arguments

Value

Returns a list with the detected change-points and the transformed series.

References

Korkas, K.K., 2022. Ensemble binary segmentation for irregularly spaced data with change-points. Journal of the Korean Statistical Society, 51(1), pp.65-86.

Examples

pw.acd.obj <- new("simACD")
pw.acd.obj@cp.loc <- seq(0.1,0.95,by=0.025)
pw.acd.obj@lambda_0 <- rep(c(0.5,2),1+length(pw.acd.obj@cp.loc)/2)
pw.acd.obj@alpha <- rep(0.2,1+length(pw.acd.obj@cp.loc))
pw.acd.obj@beta <- rep(0.4,1+length(pw.acd.obj@cp.loc))
pw.acd.obj@N <- 5000
pw.acd.obj <- pc_acdsim(pw.acd.obj)
ts.plot(pw.acd.obj@x,main="Standard BS");abline(v=BinSeg(pw.acd.obj@x)[[1]],col="blue")
#real change-points in grey
abline(v=floor(pw.acd.obj@cp.loc*pw.acd.obj@N),col="grey",lty=2)

An S4 method to detect the change-points in an irregularly spaced time series using Ensemble Binary Segmentation.

Description

An S4 method to detect the change-points in an irregularly spaced time series using the Ensemble Binary Segmentation methodology described in Korkas (2022).

Usage

EnBinSeg(
  H,
  thresh = "universal",
  q = 0.99,
  p = 1,
  start.values = c(0.9, 0.6),
  dampen.factor = "auto",
  epsilon = 1e-05,
  LOG = TRUE,
  process = "acd",
  thresh2 = 0.05,
  num_ens = 500,
  min_dist = 0.005,
  pp = 1,
  do.parallel = 2,
  b = NULL,
  acd_p = 0,
  acd_q = 1
)

## S4 method for signature 'ANY'
EnBinSeg(
  H,
  thresh = "universal",
  q = 0.99,
  p = 1,
  start.values = c(0.9, 0.6),
  dampen.factor = "auto",
  epsilon = 1e-05,
  LOG = TRUE,
  process = "acd",
  thresh2 = 0.05,
  num_ens = 500,
  min_dist = 0.005,
  pp = 1,
  do.parallel = 2,
  b = NULL,
  acd_p = 0,
  acd_q = 1
)

Arguments

Value

Returns a list with the detected change-points and the frequency table of the ensembles across M applications.

References

Korkas, K.K., 2022. Ensemble binary segmentation for irregularly spaced data with change-points. Journal of the Korean Statistical Society, 51(1), pp.65-86.

Examples

pw.acd.obj <- new("simACD")
pw.acd.obj@cp.loc <- seq(0.1,0.95,by=0.025)
pw.acd.obj@lambda_0 <- rep(c(0.5,2),1+length(pw.acd.obj@cp.loc)/2)
pw.acd.obj@alpha <- rep(0.2,1+length(pw.acd.obj@cp.loc))
pw.acd.obj@beta <- rep(0.4,1+length(pw.acd.obj@cp.loc))
pw.acd.obj@N <- 5000
pw.acd.obj <- pc_acdsim(pw.acd.obj)
ts.plot(pw.acd.obj@x,main="Ensemble BS");abline(v=EnBinSeg(pw.acd.obj@x)[[1]],col="red")
#real change-points in grey
abline(v=floor(pw.acd.obj@cp.loc*pw.acd.obj@N),col="grey",lty=2)
ts.plot(pw.acd.obj@x,main="Standard BS");abline(v=BinSeg(pw.acd.obj@x)[[1]],col="blue")
#real change-points in grey
abline(v=floor(pw.acd.obj@cp.loc*pw.acd.obj@N),col="grey",lty=2)

Transformation of an irregularly spaces time series.

Description

Transformation of a irregularly spaces time series. For the tvACD model, we calculate U_t = g_0(x_t, \psi_t) = \frac{x_t}{{\psi}_t}, where {\psi}_t = C_0 + \sum_{j=1}^p C_j x_{t-j} + \sum_{k=1}^q C_{p+k} \psi_{t-k}+\epsilon x_t. where the last term \epsilon x_t is added to ensure the boundness of U_t.

Usage

Z_trans(
  H,
  start.values = c(0.9, 0.6),
  dampen.factor = "auto",
  epsilon = 1e-05,
  LOG = TRUE,
  process = "acd",
  acd_p = 0,
  acd_q = 1
)

## S4 method for signature 'ANY'
Z_trans(
  H,
  start.values = c(0.9, 0.6),
  dampen.factor = "auto",
  epsilon = 1e-05,
  LOG = TRUE,
  process = "acd",
  acd_p = 0,
  acd_q = 1
)

Arguments

Value

Returns the transformed residual series.

References

Korkas, K.K., 2022. Ensemble binary segmentation for irregularly spaced data with change-points. Journal of the Korean Statistical Society, 51(1), pp.65-86.

Examples

pw.acd.obj <- new("simACD")
pw.acd.obj@cp.loc <- c(0.25,0.75)
pw.acd.obj@lambda_0 <- c(1,2,1)
pw.acd.obj@alpha <- rep(0.2,3)
pw.acd.obj@beta <- rep(0.7,3)
pw.acd.obj@N <- 1000
pw.acd.obj <- pc_acdsim(pw.acd.obj)
ts.plot(Z_trans(pw.acd.obj@x))

A bootstrap method to calculate the threshold (stopping rule) in the BS or EBS segmentation.

Description

A bootstrap method to calculate the threshold (stopping rule) in the BS or EBS segmentation described in Cho and Korkas (2022) and adapted for irregularly time series in Korkas (2022).

Usage

boot_thresh(
  H,
  q = 0.75,
  r = 100,
  p = 1,
  start.values = c(0.9, 0.6),
  process = "acd",
  do.parallel = 2,
  dampen.factor = "auto",
  epsilon = 1e-05,
  LOG = TRUE,
  acd_p = 0,
  acd_q = 1
)

## S4 method for signature 'ANY'
boot_thresh(
  H,
  q = 0.75,
  r = 100,
  p = 1,
  start.values = c(0.9, 0.6),
  process = "acd",
  do.parallel = 2,
  dampen.factor = "auto",
  epsilon = 1e-05,
  LOG = TRUE,
  acd_p = 0,
  acd_q = 1
)

Arguments

Value

Returns the threshold C.

References

Cho, H. and Korkas, K.K., 2022. High-dimensional GARCH process segmentation with an application to Value-at-Risk. Econometrics and Statistics, 23, pp.187-203.

Examples

pw.acd.obj <- new("simACD")
pw.acd.obj@cp.loc <- c(0.25,0.75)
pw.acd.obj@lambda_0 <- c(1,2,1)
pw.acd.obj@alpha <- rep(0.2,3)
pw.acd.obj@beta <- rep(0.7,3)
pw.acd.obj@N <- 3000
pw.acd.obj <- pc_acdsim(pw.acd.obj)
boot_thresh(pw.acd.obj@x,r=20)

A method to simulate nonstationary ACD models.

Description

A S4 method that takes as an input a simACD object and outputs a simulated nonstationary ACD(1,1) model. The formulation of the of the piecewise constant ACD model is given in the simACD class.

Usage

pc_acdsim(object)

## S4 method for signature 'ANY'
pc_acdsim(object)

Arguments

Value

Returns an object of simACD class containing a simulated piecewise constant ACD time series.

References

Korkas, K.K., 2022. Ensemble binary segmentation for irregularly spaced data with change-points. Journal of the Korean Statistical Society, 51(1), pp.65-86.

Examples

pw.acd.obj <- new("simACD")
pw.acd.obj@cp.loc <- c(0.25,0.75)
pw.acd.obj@lambda_0 <- c(1,2,1)
pw.acd.obj@alpha <- rep(0.2,3)
pw.acd.obj@beta <- rep(0.7,3)
pw.acd.obj@N <- 3000
pw.acd.obj <- pc_acdsim(pw.acd.obj)
ts.plot(pw.acd.obj@x)
ts.plot(pw.acd.obj@psi)

A method to simulate nonstationary Hawkes models.

Description

A S4 method that takes as an input a simHawkes object and outputs a simulated nonstationary Hawkes model. The formulation of the of the piecewise constant ACD model is given in the simHawkes class.

Usage

pc_hawkessim(object)

## S4 method for signature 'ANY'
pc_hawkessim(object)

Arguments

Value

Returns an object of simHawkes class containing a simulated piecewise constant Hawkes series.

References

Korkas, K.K., 2022. Ensemble binary segmentation for irregularly spaced data with change-points. Journal of the Korean Statistical Society, 51(1), pp.65-86.

Examples

pw.hawk.obj <- new("simHawkes")
pw.hawk.obj@cp.loc <- c(0.5)
pw.hawk.obj@lambda_0 <- c(1,2)
pw.hawk.obj@alpha <- c(0.2,0.2)
pw.hawk.obj@beta <- c(0.7,0.7)
pw.hawk.obj@horizon <- 1000
pw.hawk.obj <- pc_hawkessim(pw.hawk.obj)
ts.plot(pw.hawk.obj@H)
ts.plot(pw.hawk.obj@cH)

An S4 class for a nonstationary ACD model.

Description

A specification class to create an object of a simulated piecewise constant conditional duration model of order (1,1). x_t / \psi_t = \varepsilon_t \; \sim \mathcal{G}(\theta_2) \psi_t = \omega(t) + \sum_{j=1}^p \alpha_{j}(t)x_{t-j} + \sum_{k=1}^q \beta_{k}(t)\psi_{t-k}. where \psi_{t} = \mathcal{E} [x_t | x_t,\ldots,x_1| \theta_1] is the conditional mean duration of the t-th event with parameter vector \theta_1 and \mathcal{G}(.) is a general distribution over (0,+\infty) with mean equal to 1 and parameter vector \theta_2. In this work we assume that \varepsilon_t \; \sim \exp(1).

Value

Returns an object of simACD class.

Slots

x

The durational time series.

psi

The psi time series.

N

Sample sze of the time series.

cp.loc

The vector with the location of the changepoints. Takes values from 0 to 1 or NULL. Default is NULL.

lambda_0

The vector of the parameters lambda_0 in the ACD series as in the above formula.

alpha

The vector of the parameters alpha in the ACD series as in the above formula.

beta

The vector of the parameters beta in the ACD series as in the above formula.

BurnIn

The size of the burn-in sample. Note that this only applies at the first simulated segment. Default is 500.

References

Korkas, K.K., 2022. Ensemble binary segmentation for irregularly spaced data with change-points. Journal of the Korean Statistical Society, 51(1), pp.65-86.

Examples

pw.acd.obj <- new("simACD")
pw.acd.obj@cp.loc <- c(0.25,0.75)
pw.acd.obj@lambda_0 <- c(1,2,1)
pw.acd.obj@alpha <- rep(0.2,3)
pw.acd.obj@beta <- rep(0.7,3)
pw.acd.obj@N <- 3000
pw.acd.obj <- pc_acdsim(pw.acd.obj)
ts.plot(pw.acd.obj@x)
ts.plot(pw.acd.obj@psi)

An S4 class for a nonstationary ACD model.

Description

A specification class to create an object of a simulated piecewise constant Hawkes model of order (1,1). We consider the following time-varying piecewise constant Hawkes process (which we term tvHawkes) \lambda({\upsilon}) = \lambda_0({\upsilon}) +\sum_{{\upsilon}_t < s} \alpha({\upsilon})e^{-\beta({\upsilon}) ({\upsilon}-{\upsilon}_t)}, \ \mbox{for} \ {\upsilon} = 1, \ldots,T.

Value

Returns an object of simHawkes class.

Slots

H

The durational time series.

cH

The psi time series.

horizon

The time horizon of a Hawkes process typically expressed in seconds. Effective sample size will differ depending on the size of the parameters.

N

Effective sample size which differs depending on the size of the parameters.

cp.loc

The vector with the location of the changepoints. Takes values from 0 to 1 or NULL if none. Default is NULL.

lambda_0

The vector of the parameters lambda_0 in the Hawkes model as in the above formula.

alpha

The vector of the parameters alpha in the Hawkes model as in the above formula.

beta

The vector of the parameters beta in the Hawkes model as in the above formula.

References

Korkas, K.K., 2022. Ensemble binary segmentation for irregularly spaced data with change-points. Journal of the Korean Statistical Society, 51(1), pp.65-86.

Examples

pw.hawk.obj <- new("simHawkes")
pw.hawk.obj@cp.loc <- c(0.5)
pw.hawk.obj@lambda_0 <- c(1,2)
pw.hawk.obj@alpha <- c(0.2,0.2)
pw.hawk.obj@beta <- c(0.7,0.7)
pw.hawk.obj@horizon <- 1000
pw.hawk.obj <- pc_hawkessim(pw.hawk.obj)
ts.plot(pw.hawk.obj@H)
ts.plot(pw.hawk.obj@cH)