Vignette for R Breakfast Package

1 Executive summary

The breakfast package contains a broad range of methods for computationally efficient multiple change-point detection. The main function is breakfast. The current version of the package implements the Gaussian mean-shift model, in which the signal is modelled as piecewise constant and is observed in additive noise, which is assumed to be independent, identically distributed Gaussian with zero mean and unknown constant variance. More models will be added in next versions of the package.

Change-point detection in breakfast is carried out in two stages: (i) computation of a solution path, and (ii) model selection along the path. The supported solution paths methods include IDetect, NOT, TGUH, WBS and WBS2, while the supported model selection methods are based on an information criterion (IC), localised pruning (LP), steepest drop to low levels (SDLL), and thresholding. We provide a comparison between different approaches in a comprehensive simulation study. The main findings and recommendations as follows.

2 Methodologies – a brief description

2.1 Introduction

The breakfast package contains a broad range of methods for computationally efficient multiple change-point detection. The current version implements the univariate Gaussian mean-shift model, in which the data are assumed to be a piecewise-constant signal observed with i.i.d. Gaussian noise. Change-point detection in breakfast is carried out in two stages: (i) computation of a solution path, and (ii) model selection along the path. Here a solution path can be understood as a collection of estimated sub-models indexed by certain quantities such as the threshold or the number of change-points.

A variety of solution path and model selection methods are included, which can be accessed individually, or through the breakfast function.

2.2 Solution path methods

2.2.1 Isolation and Detection (IDetect)

The Isolate-Detect method and its algorithm is described in Detecting multiple generalized change-points by isolating single ones. A. Anastasiou and P. Fryzlewicz (2019), arXiv preprint.

2.2.1.1 Sequential version

The key ingredient of Isolate-Detect is the isolation of each of the true change-points within subintervals of the domain. For a positive integer \(\lambda_T\), right- and left-expanding intervals are created as follows: the \(j^{th}\) right-expanding interval is \(R_j = [1,j{\lambda_T}]\), while the \(j^{th}\) left-expanding interval is \(L_{j} = [T - j\lambda_T + 1,T]\), where \(T\) is the length of a given data sequence. We collect these intervals in the ordered set \(S_{RL} = \left\lbrace R_1, L_1, R_2, L_2, \ldots,R_K,L_K\right\rbrace\) and firstly we identify the data-point (denoted by \(\tilde{b}_1\)) of \(R_1\) with the largest CUSUM statistic. If this value exceeds a given threshold level, then the algorithm makes a new start from \(\tilde{b}_1\). If not, then the process tests the next interval in \(S_{RL}\). We stop when all the data sequence has been scanned.

2.2.1.2 Standard version

We first employ the sequential version of the previous section with a low threshold value (lower than the one used in Section 2.2.1.1). Our aim is to prune the solution path returned from the sequential version through an iterative procedure, where at each iteration the estimation that exists in the solution path and is most likely to be spurious is removed. With \(J\) being the length of the solution path given at the first step of this process, the pruning is achieved based on the CUSUM value of each candidate \(\hat{r}_j, j=1,2,\ldots, J\), when the interval used is \([\hat{r}_{j-1}+1, \hat{r}_{j+1}]\), where \(\hat{r}_0 = 0\) and \(\hat{r}_{J+1} = T\).

2.2.2 Narrowest-over-threshold (NOT)

NOT method and its algorithm is described in Narrowest‐over‐threshold detection of multiple change points and change‐point‐like features. R. Baranowski, Y. Chen and P. Fryzlewicz (2019). Journal of Royal Statistical Society Series B, 81: 649–672.

The key ingredient of NOT is its focus on the smallest local sections of the data on which the existence of a change-point is suspected. To give more details, first one randomly draws a large number of subintervals; then given a threshold level, one considers the shortest subintervals above the threshold level, estimate the change-point’s location from this subinterval, and performs this step recursively for all the observations to both the left and right of the estimated change-point.

As such, the resulting solution path consists of a collection of estimated models indexed by the threholds. Finally, we remark that NOT is a general method that can be used to detect generalized change-points beyond the mean-shifting model setup.

2.2.3 Tail greedy unbalanced Haar (TGUH)

The tail-greedy unbalanced Haar (TGUH) algorithm is described in Tail-greedy bottom-up data decompositions and fast multiple change-point detection. P. Fryzlewicz (2018). Annals of Statistics, 46: 3390–3421.

The TGUH algorithm performs bottom-up merges of the data, starting from the regions that are the least likely to contain change-points and progressively moving towards regions that are more likely to contain some. The ‘tail-greediness’ of the decomposition algorithm, whereby multiple greedy steps are taken in a single pass through the data, enables fast computation.

2.2.4 Wild binary segmentation (WBS)

WBS method and its algorithm is described in Wild binary segmentation for multiple change-point detection. P. Fryzlewicz (2014). The Annals of Statistics, 42: 2243–2281.

To give more details, first one randomly draws a large number of subintervals; then one considers the subintervals with the largest CUSUM statistic values and estimate the change-point’s location based on data from this subinterval, and performs this step recursively for all the observations to both the left and right of the estimated change-point.

Thanks to the nested structure, the resulting solution path consists of a collection of models indexed by the number of estimated change-points.

2.2.5 Wild binary segmentation 2 (WBS2)

WBS2 is described in Detecting possibly frequent change-points: Wild Binary Segmentation 2 and steepest-drop model selection. P. Fryzlewicz (2020). Journal of the Korean Statistical Society, to appear. See also this Rejoinder.

The difference between WBS and WBS2 is in the nature of the interval sampling. In WBS, all intervals to be used are sampled before the solution path procedure is ran. In WBS2, a relatively small number of intervals are sampled at the beginning, and the first change-point candidate (that corresponding to the largest CUSUM) is identified based on these intervals only. Then, next sampling and identification of change-point candidates is carried out recursively to the left and to the right of the first change-point candidate. In other words, in WBS the sampling is done once, while in WBS2 it is done recursively as the procedure progresses.

2.3 Model selection methods

2.3.1 Information criterion (IC)

This method estimates the number and locations of change-points in the piecewise-constant mean of a noisy data sequence via the strengthened Schwarz information criterion (sSIC). Here we penalise the model complexity by subtracting to the log-likelihood a term that is proportional to \(k \log^\alpha(n)\), where \(k\) is the number of change-points, \(n\) is the number of observations, and \(\alpha\) is the parameter of sSIC. We then select the model with the largest value of the penalised log-likelihood.

2.3.2 Localised pruning (LP)

LP is described in Two-stage data segmentation permitting multiscale change points, heavy tails and dependence by H. Cho and C. Kirch (2020). See also mosum: A package for moving sums in change-point analysis by A. Meier, C. Kirch and H. Cho (2020). Journal of Statistical Software, to appear.

From a set of candidate change-point estimators, it selects a subset that satisfies the followings based on the Schwarz criterion (SC):

  • Adding further change-point estimators to the subset monotonically increases the SC.

  • Among such subsets with the smallest cardinality, it has the minimum SC.

Performing the above steps locally (using the information about the interval in which each estimator is detected), it selects the final model. The LP is implemented with MOSUM-based candidate generation procedure in the R package mosum.

2.3.3 Steepest drop to low levels (SDLL)

SDLL is described in Detecting possibly frequent change-points: Wild Binary Segmentation 2 and steepest-drop model selection. P. Fryzlewicz (2020). Journal of the Korean Statistical Society, to appear. See also this Rejoinder.

The main idea is to find a location, along the solution path, at which the maximum CUSUMs stop being ‘large’ (these are believed to correspond to real change-points) and start being ‘small’ (these are believed to correspond to noise). SDLL identifies this location as that corresponding to the biggest drop in logged CUSUMs along the solution path which is followed by suitably small logged CUSUMs - hence the name SDLL (Steepest Drop to Low Levels).

2.3.4 Thresholding (THRESH)

This method estimates the number and locations of change-points in the piecewise-constant mean of a noisy data sequence via the value of a threshold. Those estimations (that are in the solution path) with a CUSUM value above a predefined threshold are selected as change-points. The threshold used is of the form \(C\sqrt{2\log T}\), where \(C\) is a positive constant.

2.4 Planned future developments

3 Simulation study

3.1 Simulated signals

The signals used in the simulation study are

3.2 Simulation process

We ran 100 replications for each one of the abovementioned signals and the frequency distribution of \(\hat{N} - N\) for each solution path method (combined with a model selection algorithm) is presented. The methods with the highest empirical frequency of \(\hat{N} - N = 0\) (or in a neighbourhood of zero, depending on the example) and those within \(10\%\) off the highest are given in bold. As a measure of the accuracy of the detected locations, we provide Monte-Carlo estimates of the mean squared error, \({\rm MSE} = T^{-1}\sum_{t=1}^{T}{\rm E}\left(\hat{f}_t - f_t\right)^2\), where \(\hat{f}_t\) is the ordinary least square approximation of \(f_t\) between two successive change-points. The scaled Hausdorff distance, \(d_H = n_s^{-1}\max\left\lbrace \max_j\min_k\left|r_j-\hat{r}_k\right|,\max_k\min_j\left|r_j-\hat{r}_k\right|\right\rbrace,\) where \(n_s\) is the length of the largest segment, is also given in all examples apart from the signals (NC1) - (NC4), which are constant-mean signals without any change-points. Now, due to the fact that we are under a misspecification scenario, we need to be careful on how to explain the results for the signals (M13) and (M14). These signals are linear, and we are using methods developed for the detection of changes in piecewise-constant signals. Therefore in the relevant (for these signals) tables, we decided to present the average number of estimated change-points that each method returns, and the MSE, which is calculated by taking as true signal a stair-like structure, where each data point is a change-point and there is a jump of magnitude equal to one. The average computational time for all methods is also provided.

3.3 Code for running the simulations

library(breakfast)
options(expressions = 500000)
mean.from.cpt <- function(x, cpt) {
  
  n <- length(x)
  len.cpt <- length(cpt)
  if (len.cpt) cpt <- sort(cpt)
  beg <- endd <- rep(0, len.cpt+1)
  beg[1] <- 1
  endd[len.cpt+1] <- n
  if (len.cpt) {
    beg[2:(len.cpt+1)] <- cpt+1
    endd[1:len.cpt] <- cpt
  }
  means <- rep(0, len.cpt+1)
  for (i in 1:(len.cpt+1)) means[i] <- mean(x[beg[i]:endd[i]])
  rep(means, endd-beg+1)
}

## The function used for the comparisons in the case of a piecewise-constant signal model.thresh
sim_thr <- function(x, const = 1.15) {
  setClass("cpt.est", representation(cpt="numeric", nocpt="numeric", mean="numeric",time="numeric"),
  prototype(cpt=numeric(0), nocpt=0, mean=numeric(0),time=numeric(0)))
  
  print("wbs")
  wbsth <- new("cpt.est")
  z <- model.thresh(sol.wbs(x), th_const = const)
  if(z$cpts == 0){wbsth@cpt = 0
  wbsth@nocpt = 0}
  else{wbsth@cpt <- z$cpts
  wbsth@nocpt <- z$no.of.cpt}
  wbsth@mean <- z$est
  wbsth@time <- system.time(model.thresh(sol.wbs(x), th_const = const))[[3]]
  
  print("not")
  notth <- new("cpt.est")
  z <- model.thresh(sol.not(x), th_const = const)
  if(z$cpts == 0){notth@cpt = 0
  notth@nocpt = 0}
  else{notth@cpt <- z$cpts
  notth@nocpt <- z$no.of.cpt}
  notth@mean <- z$est
  notth@time <- system.time(model.thresh(sol.not(x), th_const = const))[[3]]
  
  print("idetect")
  idetectth <- new("cpt.est")
  z <- model.thresh(sol.idetect_seq(x), th_const = const)
  if(z$cpts == 0){idetectth@cpt = 0
  idetectth@nocpt = 0}
  else{idetectth@cpt <- z$cpts
  idetectth@nocpt <- z$no.of.cpt}
  idetectth@mean <- z$est
  idetectth@time <- system.time(model.thresh(sol.idetect_seq(x), th_const = const))[[3]]
  
  print("tguh")
  tguhth <- new("cpt.est")
  z <- model.thresh(sol.tguh(x), th_const = const)
  if(z$cpts == 0){tguhth@cpt = 0
  tguhth@nocpt = 0}
  else{tguhth@cpt <- z$cpts
  tguhth@nocpt <- z$no.of.cpt}
  tguhth@mean <- z$est
  tguhth@time <- system.time(model.thresh(sol.tguh(x), th_const = const))[[3]]
  
  print("wbs2")
  wbs2th <- new("cpt.est")
  z <- model.thresh(sol.wbs2(x), th_const = const)
  if(z$cpts == 0){wbs2th@cpt = 0
  wbs2th@nocpt = 0}
  else{wbs2th@cpt <- z$cpts
  wbs2th@nocpt <- z$no.of.cpt}
  wbs2th@mean <- z$est
  wbs2th@time <- system.time(model.thresh(sol.wbs2(x), th_const = const))[[3]]
  

  list(wbsth=wbsth, notth = notth, idetectth = idetectth, tguhth = tguhth, wbs2th = wbs2th)
}


sim.study.thr <- function(signal, true.cpt=NULL,sigma, m = 100, seed = NULL, gen_const = 1.15) {
  
  setClass("est.eval", representation(avg.signal="numeric",mean="list",cpt="list",diff="matrix",
  dh="numeric",cptall="numeric",dnc="numeric", mse="numeric", time= "numeric"), 
  prototype(dnc=numeric(m), mse=numeric(m),time=numeric(m),dh=numeric(m)))
  
  wbsth <- new("est.eval")
  wbs2th <- new("est.eval")
  notth <- new("est.eval")
  idetectth <- new("est.eval")
  tguhth <- new("est.eval")
  
  no.of.cpt <- sum(abs(diff(signal)) > 0)
  n <- length(signal)
  ns <- max(c(diff(true.cpt), length(signal)))
  
  if (!is.null(seed)) set.seed(seed)
  
  for (i in 1:m) {
    
    print(i)
    x <- signal + sigma * rnorm(n)
    
    est <- sim_thr(x, const = gen_const)
    
    wbsth@dnc[i] <- est$wbsth@nocpt - no.of.cpt
    wbsth@mean[[i]] <- est$wbsth@mean
    wbsth@mse[i] <- mean((est$wbsth@mean - signal)^2)
    wbsth@cpt[[i]] <- est$wbsth@cpt
    wbsth@diff <- abs(matrix(est$wbsth@cpt,nrow=no.of.cpt,ncol=length(est$wbsth@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$wbsth@cpt),byr=F))
    wbsth@dh[i] <- max(apply(wbsth@diff,1,min),apply(wbsth@diff,2,min))/ns
    wbsth@time[i] <- est$wbsth@time

    notth@dnc[i] <- est$notth@nocpt - no.of.cpt
    notth@mean[[i]] <- est$notth@mean
    notth@mse[i] <- mean((est$notth@mean - signal)^2)
    notth@cpt[[i]] <- est$notth@cpt
    notth@diff <- abs(matrix(est$notth@cpt,nrow=no.of.cpt,ncol=length(est$notth@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$notth@cpt),byr=F))
    notth@dh[i] <- max(apply(notth@diff,1,min),apply(notth@diff,2,min))/ns
    notth@time[i] <- est$notth@time
    
    idetectth@dnc[i] <- est$idetectth@nocpt - no.of.cpt
    idetectth@mean[[i]] <- est$idetectth@mean
    idetectth@mse[i] <- mean((est$idetectth@mean - signal)^2)
    idetectth@cpt[[i]] <- est$idetectth@cpt
    idetectth@diff <- abs(matrix(est$idetectth@cpt,nrow=no.of.cpt,
    ncol=length(est$idetectth@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$idetectth@cpt),byr=F))
    idetectth@dh[i] <- max(apply(idetectth@diff,1,min),apply(idetectth@diff,2,min))/ns
    idetectth@time[i] <- est$idetectth@time

    wbs2th@dnc[i] <- est$wbs2th@nocpt - no.of.cpt
    wbs2th@mean[[i]] <- est$wbs2th@mean
    wbs2th@mse[i] <- mean((est$wbs2th@mean - signal)^2)
    wbs2th@cpt[[i]] <- est$wbs2th@cpt
    wbs2th@diff <- abs(matrix(est$wbs2th@cpt,nrow=no.of.cpt,ncol=length(est$wbs2th@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$wbs2th@cpt),byr=F))
    wbs2th@dh[i] <- max(apply(wbs2th@diff,1,min),apply(wbs2th@diff,2,min))/ns
    wbs2th@time[i] <- est$wbs2th@time
    
    tguhth@dnc[i] <- est$tguhth@nocpt - no.of.cpt
    tguhth@mean[[i]] <- est$tguhth@mean
    tguhth@mse[i] <- mean((est$tguhth@mean - signal)^2)
    tguhth@cpt[[i]] <- est$tguhth@cpt
    tguhth@diff <- abs(matrix(est$tguhth@cpt,nrow=no.of.cpt,ncol=length(est$tguhth@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$tguhth@cpt),byr=F))
    tguhth@dh[i] <- max(apply(tguhth@diff,1,min),apply(tguhth@diff,2,min))/ns
    tguhth@time[i] <- est$tguhth@time
    gc()
  }

list(wbsth=wbsth, notth=notth, idetectth = idetectth, #idetectth2 = idetectth2,
     tguhth = tguhth, wbs2th = wbs2th)
}

seed.temp=12

justnoise = rep(0,3000)
NC.small_thr1.15 = sim.study.thr(justnoise,sigma=1,true.cpt=c(0),seed=seed.temp, gen_const = 1.15)

justnoise2 = rep(0,50)
NC2.small_thr1.15 = sim.study.thr(justnoise2,true.cpt=c(0),sigma=1,seed=seed.temp, gen_const = 1.15)

justnoise3 = rep(0,5)
NC3.small_thr1.15 = sim.study.thr(justnoise3,true.cpt=c(0),sigma=1,seed=seed.temp, gen_const = 1.15)

justnoise4 = rep(0,10000)
NC4.small_thr1.15 = sim.study.thr(justnoise4,sigma=1,true.cpt=c(0),seed=seed.temp, gen_const = 1.15)

blocks = c(rep(0,205),rep(14.64,62),rep(-3.66,41),rep(7.32,164),rep(-7.32,40),rep(10.98,308),
           rep(-4.39,82),rep(3.29,430),rep(19.03,225),rep(7.68,41),rep(15.37,61),rep(0,389))

SIMR1.small_thr1.15 = sim.study.thr(blocks,true.cpt = c(205,267,308,472,512,820,902,1332,1557,1598,1659),
sigma=10,seed=seed.temp, gen_const = 1.15)

fms = c(rep(-0.18,139),rep(0.08,87),rep(1.07,17),rep(-0.53,57),rep(0.16,9),rep(-0.69,24),rep(-0.16,164))
SIMR2.small_thr1.15 = sim.study.thr(fms,true.cpt = c(139,226,243,300,309,333),sigma=0.3,seed=seed.temp,
gen_const = 1.15)

mix = c(rep(7,11),rep(-7,10),rep(6,20),rep(-6,20),rep(5,30),rep(-5,30),rep(4,40),rep(-4,40),
        rep(3,50),rep(-3,50),rep(2,60),rep(-2,60),rep(1,70),rep(-1,69))
SIMR3.small_thr1.15 = sim.study.thr(mix,true.cpt=c(11,21,41,61,91,121,161,201,251,301,361,421,491),
sigma=4,seed=seed.temp, gen_const = 1.15)

teeth10 = c(rep(0,11),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,10),rep(0,10),
rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,9))
SIMR4.small_thr1.15 = sim.study.thr(teeth10,true.cpt=seq(11,131,10),0.4,seed=seed.temp, gen_const = 1.15)

stairs10 = c(rep(1,11),rep(2,10),rep(3,10),rep(4,10),rep(5,10),rep(6,10),rep(7,10),rep(8,10),rep(9,10),
rep(10,10),rep(11,10),rep(12,10),rep(13,10),rep(14,10),rep(15,9))
SIMR5.small_thr1.15 = sim.study.thr(stairs10,true.cpt=seq(11,141,10),sigma=0.3,seed=seed.temp, 
gen_const = 1.15)

myteeth20 = rep(c(rep(0,40),rep(1.5,40)),125)
SIMR6.small_thr1.15 = sim.study.thr(myteeth20,true.cpt = seq(40,9960,40),sigma=1,seed=seed.temp,
m=100, gen_const = 1.15)

extr_short_2_cpt <- c(rep(0,15), rep(2,15), rep(-0.5,20))
SIMR8.small_thr1.15 = sim.study.thr(extr_short_2_cpt,true.cpt=c(15,30),sigma=1,seed=seed.temp,
m=100, gen_const = 1.15)

extr_short_2_cpt2 <- c(rep(0,3), rep(3,4), rep(0,3))
SIMR9.small_thr1.15 = sim.study.thr(extr_short_2_cpt2,true.cpt=c(3,7),sigma=1,seed=seed.temp,
m=100, gen_const = 1.15)

strong_sig_low_noise <- c(rep(0,100), rep(5,500), rep(0, 1000), rep(10, 400), rep(2, 500))
SIMR10.small_thr1.15 <- sim.study.thr(strong_sig_low_noise,true.cpt=c(100, 600, 1600, 2000),
sigma=0.1,seed=seed.temp, m=100, gen_const = 1.15)

one_spike_in_middle <- c(rep(0,1000),100, rep(0,999))
SIMR11.small_thr1.15 <- sim.study.thr(one_spike_in_middle,true.cpt=c(1000, 1001),sigma=1,
seed=seed.temp, m=100, gen_const = 1.15)

no_noise <- c(rep(0,250), rep(5,300), rep(0,200), rep(-4,250))
SIMR12.small_thr1.15 <- sim.study.thr(no_noise,true.cpt=c(250, 550, 750),sigma=0,seed=seed.temp,
m=1, gen_const = 1.15) ## no need for 100 replications here.

linear <- seq(0,500)
SIMR13.small_thr1.15 <- sim.study.thr(linear,true.cpt=seq(1,500,1),sigma=1,seed=seed.temp,
m=100, gen_const = 1.15)

SIMR14.small_thr1.15 <- sim.study.thr(linear,true.cpt=seq(1,500,1),sigma=0,seed=seed.temp,
m=1, gen_const = 1.15) ## no need for 100 replications here.


## The function used for the comparisons in the case of a piecewise-constant signal 
## information criterion
sim_ic <- function(x, par.max = 25) {
  setClass("cpt.est", representation(cpt="numeric", nocpt="numeric", mean="numeric",time="numeric"),
  prototype(cpt=numeric(0), nocpt=0, mean=numeric(0),time=numeric(0)))
  
  print("wbs")
  wbsic <- new("cpt.est")
  z <- model.ic(sol.wbs(x), q.max = par.max)
  if(z$cpt == 0){wbsic@cpt = 0
   wbsic@nocpt = 0}
 else{wbsic@cpt <- z$cpt
  wbsic@nocpt <- z$no.of.cpt}
  wbsic@mean <- z$est
  wbsic@time <- system.time(model.ic(sol.wbs(x), q.max = par.max))[[3]]
  
  print("tguh")
  tguhic <- new("cpt.est")
  z <- model.ic(sol.tguh(x), q.max = par.max)
  if(z$cpt == 0){tguhic@cpt = 0
  tguhic@nocpt = 0}
  else{tguhic@cpt <- z$cpt
  tguhic@nocpt <- z$no.of.cpt}
  tguhic@mean <- z$est
  tguhic@time <- system.time(model.ic(sol.tguh(x), q.max = par.max))[[3]]
  
  print("not")
  notic <- new("cpt.est")
  z <- model.ic(sol.not(x), q.max = par.max)
  if(z$cpt == 0){notic@cpt = 0
  notic@nocpt = 0}
  else{notic@cpt <- z$cpt
  notic@nocpt <- z$no.of.cpt}
  notic@mean <- z$est
  notic@time <- system.time(model.ic(sol.not(x), q.max = par.max))[[3]]
  
  print("idetect")
  idetectic <- new("cpt.est")
  z <- model.ic(sol.idetect(x), q.max = par.max)
  if(z$cpt == 0){idetectic@cpt = 0
  idetectic@nocpt = 0}
  else{idetectic@cpt <- z$cpt
  idetectic@nocpt <- z$no.of.cpt}
  idetectic@mean <- z$est
  idetectic@time <- system.time(model.ic(sol.idetect(x), q.max = par.max))[[3]]
  
  print("wbs2")
  wbs2ic <- new("cpt.est")
  z <- model.ic(sol.wbs2(x), q.max = par.max)
  if(z$cpt == 0){wbs2ic@cpt = 0
  wbs2ic@nocpt = 0}
  else{wbs2ic@cpt <- z$cpt
  wbs2ic@nocpt <- z$no.of.cpt}
  wbs2ic@mean <- z$est
  wbs2ic@time <- system.time(model.ic(sol.wbs2(x), q.max = par.max))[[3]]
  
  
  list(wbsic=wbsic, notic = notic,
    idetectic = idetectic, tguhic = tguhic, wbs2ic = wbs2ic)
}


sim.study.ic <- function(signal, true.cpt=NULL,sigma, m = 100, seed = NULL, max.par = 25) {
  
  setClass("est.eval", representation(avg.signal="numeric",mean="list",cpt="list",diff="matrix",
  dh="numeric",cptall="numeric",dnc="numeric", mse="numeric", time= "numeric"),
  prototype(dnc=numeric(m), mse=numeric(m),time=numeric(m),dh=numeric(m)))
  
  wbsic <- new("est.eval")
  wbs2ic <- new("est.eval")
  notic <- new("est.eval")
  idetectic <- new("est.eval")
  tguhic <- new("est.eval")
  
  no.of.cpt <- sum(abs(diff(signal)) > 0)
  n <- length(signal)
  ns <- max(c(diff(true.cpt), length(signal)))
  
  if (!is.null(seed)) set.seed(seed)
  
  for (i in 1:m) {
    
    print(i)
    x <- signal + sigma * rnorm(n)
    
    est <- sim_ic(x, par.max = max.par)
    
    wbsic@dnc[i] <- est$wbsic@nocpt - no.of.cpt
    wbsic@mean[[i]] <- est$wbsic@mean
    wbsic@mse[i] <- mean((est$wbsic@mean - signal)^2)
    wbsic@cpt[[i]] <- est$wbsic@cpt
    wbsic@diff <- abs(matrix(est$wbsic@cpt,nrow=no.of.cpt,ncol=length(est$wbsic@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$wbsic@cpt),byr=F))
    wbsic@dh[i] <- max(apply(wbsic@diff,1,min),apply(wbsic@diff,2,min))/ns
    wbsic@time[i] <- est$wbsic@time

    wbs2ic@dnc[i] <- est$wbs2ic@nocpt - no.of.cpt
    wbs2ic@mean[[i]] <- est$wbs2ic@mean
    wbs2ic@mse[i] <- mean((est$wbs2ic@mean - signal)^2)
    wbs2ic@cpt[[i]] <- est$wbs2ic@cpt
    wbs2ic@diff <- abs(matrix(est$wbs2ic@cpt,nrow=no.of.cpt,ncol=length(est$wbs2ic@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$wbs2ic@cpt),byr=F))
    wbs2ic@dh[i] <- max(apply(wbs2ic@diff,1,min),apply(wbs2ic@diff,2,min))/ns
    wbs2ic@time[i] <- est$wbs2ic@time

    notic@dnc[i] <- est$notic@nocpt - no.of.cpt
    notic@mean[[i]] <- est$notic@mean
    notic@mse[i] <- mean((est$notic@mean - signal)^2)
    notic@cpt[[i]] <- est$notic@cpt
    notic@diff <- abs(matrix(est$notic@cpt,nrow=no.of.cpt,ncol=length(est$notic@cpt),
    byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$notic@cpt),byr=F))
    notic@dh[i] <- max(apply(notic@diff,1,min),apply(notic@diff,2,min))/ns
    notic@time[i] <- est$notic@time
    
    idetectic@dnc[i] <- est$idetectic@nocpt - no.of.cpt
    idetectic@mean[[i]] <- est$idetectic@mean
    idetectic@mse[i] <- mean((est$idetectic@mean - signal)^2)
    idetectic@cpt[[i]] <- est$idetectic@cpt
    idetectic@diff <- abs(matrix(est$idetectic@cpt,nrow=no.of.cpt,ncol=length(est$idetectic@cpt),
    byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$idetectic@cpt),byr=F))
    idetectic@dh[i] <- max(apply(idetectic@diff,1,min),apply(idetectic@diff,2,min))/ns
    idetectic@time[i] <- est$idetectic@time
    
    tguhic@dnc[i] <- est$tguhic@nocpt - no.of.cpt
    tguhic@mean[[i]] <- est$tguhic@mean
    tguhic@mse[i] <- mean((est$tguhic@mean - signal)^2)
    tguhic@cpt[[i]] <- est$tguhic@cpt
    tguhic@diff <- abs(matrix(est$tguhic@cpt,nrow=no.of.cpt,ncol=length(est$tguhic@cpt),
    byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$tguhic@cpt),byr=F))
    tguhic@dh[i] <- max(apply(tguhic@diff,1,min),apply(tguhic@diff,2,min))/ns
    tguhic@time[i] <- est$tguhic@time
    
  }
  
  list(wbsic=wbsic, notic=notic, idetectic = idetectic, tguhic = tguhic, wbs2ic = wbs2ic)
}

seed.temp=12

justnoise = rep(0,3000)
NC.small_ic = sim.study.ic(justnoise,sigma=1,true.cpt=c(0),seed=seed.temp)

justnoise2 = rep(0,50)
NC2.small_ic = sim.study.ic(justnoise2,true.cpt=c(0),sigma=1,seed=seed.temp)

justnoise3 = rep(0,5)
NC3.small_ic = sim.study.ic(justnoise3,true.cpt=c(0),sigma=1,seed=seed.temp)

justnoise4 = rep(0,10000)
NC4.small_ic = sim.study.ic(justnoise4,sigma=1,true.cpt=c(0),seed=seed.temp)

blocks = c(rep(0,205),rep(14.64,62),rep(-3.66,41),rep(7.32,164),rep(-7.32,40),rep(10.98,308),
           rep(-4.39,82),rep(3.29,430),rep(19.03,225),rep(7.68,41),rep(15.37,61),rep(0,389))


SIMR1.small_ic = sim.study.ic(blocks,true.cpt = c(205,267,308,472,512,820,902,1332,1557,1598,1659),
sigma=10,seed=seed.temp)

fms = c(rep(-0.18,139),rep(0.08,87),rep(1.07,17),rep(-0.53,57),rep(0.16,9),rep(-0.69,24),
rep(-0.16,164))
SIMR2.small_ic = sim.study.ic(fms,true.cpt = c(139,226,243,300,309,333),sigma=0.3,seed=seed.temp)

mix = c(rep(7,11),rep(-7,10),rep(6,20),rep(-6,20),rep(5,30),rep(-5,30),rep(4,40),rep(-4,40),
        rep(3,50),rep(-3,50),rep(2,60),rep(-2,60),rep(1,70),rep(-1,69))
SIMR3.small_ic = sim.study.ic(mix,true.cpt=c(11,21,41,61,91,121,161,201,251,301,361,421,491), 
sigma=4,seed=seed.temp)

teeth10 = c(rep(0,11),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,10),
rep(0,10),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,9))
SIMR4.small_ic = sim.study.ic(teeth10,true.cpt=seq(11,131,10),0.4,seed=seed.temp)

stairs10 = c(rep(1,11),rep(2,10),rep(3,10),rep(4,10),rep(5,10),rep(6,10),rep(7,10),rep(8,10),
rep(9,10),rep(10,10),rep(11,10),rep(12,10),rep(13,10),rep(14,10),rep(15,9))
SIMR5.small_ic = sim.study.ic(stairs10,true.cpt=seq(11,141,10),sigma=0.3,seed=seed.temp)

myteeth20 = rep(c(rep(0,40),rep(1.5,40)),125)
SIMR6.small_ic = sim.study.ic(myteeth20,true.cpt = seq(40,9960,40),sigma=1,seed=seed.temp, m=100)

long_teeth = rep(c(rep(0,40),rep(1.5,40)),1250)
SIMR7.small_ic = sim.study.ic(long_teeth,true.cpt = seq(40,99960,40),sigma=1,seed=seed.temp, m=100)

extr_short_2_cpt <- c(rep(0,15), rep(2,15), rep(-0.5,20))
SIMR8.small_ic = sim.study.ic(extr_short_2_cpt,true.cpt=c(15,30),sigma=1,seed=seed.temp, m=100)

extr_short_2_cpt2 <- c(rep(0,3), rep(3,4), rep(0,3))
SIMR9.small_ic = sim.study.ic(extr_short_2_cpt2,true.cpt=c(3,7),sigma=1,seed=seed.temp, m=100)

strong_sig_low_noise <- c(rep(0,100), rep(5,500), rep(0, 1000), rep(10, 400), rep(2, 500))
SIMR10.small_ic <- sim.study.ic(strong_sig_low_noise,true.cpt=c(100, 600, 1600, 2000),sigma=0.1,
seed=seed.temp, m=100)

one_spike_in_middle <- c(rep(0,1000),100, rep(0,999))
SIMR11.small_ic <- sim.study.ic(one_spike_in_middle,true.cpt=c(1000, 1001),sigma=1,
seed=seed.temp, m=100)

no_noise <- c(rep(0,250), rep(5,300), rep(0,200), rep(-4,250))
SIMR12.small_ic <- sim.study.ic(no_noise,true.cpt=c(250, 550, 750),sigma=0,seed=seed.temp, m=1) 
## no need for 100 replications here.

linear <- seq(0,500)
SIMR13.small_ic <- sim.study.ic(linear,true.cpt=seq(1,500,1),sigma=1,seed=seed.temp, m=100, 
max.par = 550)

SIMR14.small_ic <- sim.study.ic(linear,true.cpt=seq(1,500,1),sigma=0,seed=seed.temp, m=1, 
max.par = 550) ## no need for 100 replications here.


## The function used for the comparisons in the case of a piecewise-constant signal sdll
sim_sdll <- function(x) {
  setClass("cpt.est", representation(cpt="numeric", nocpt="numeric", mean="numeric",time="numeric"), 
  prototype(cpt=numeric(0), nocpt=0, mean=numeric(0),time=numeric(0)))
  
  print("wbs")
  wbssdll <- new("cpt.est")
  z <- model.sdll(sol.wbs(x))
  if(length(z$cpts) == 0){wbssdll@cpt = 0
  wbssdll@nocpt = 0}
  else{wbssdll@cpt <- z$cpts
  wbssdll@nocpt <- z$no.of.cpt}
  wbssdll@mean <- z$est
  wbssdll@time <- system.time(model.sdll(sol.wbs(x)))[[3]]
  

  print("tguh")
  tguhsdll <- new("cpt.est")
  z <- model.sdll(sol.tguh(x))
  if(length(z$cpts) == 0){tguhsdll@cpt = 0
  tguhsdll@nocpt = 0}
  else{tguhsdll@cpt <- z$cpt
  tguhsdll@nocpt <- z$no.of.cpt}
  tguhsdll@mean <- z$est
  tguhsdll@time <- system.time(model.sdll(sol.tguh(x)))[[3]]
  
  print("idetect")
  idetectsdll <- new("cpt.est")
  z <- model.sdll(sol.idetect(x))
  if(length(z$cpts) == 0){idetectsdll@cpt = 0
  idetectsdll@nocpt = 0}
  else{idetectsdll@cpt <- z$cpt
  idetectsdll@nocpt <- z$no.of.cpt}
  idetectsdll@mean <- z$est
  idetectsdll@time <- system.time(model.sdll(sol.idetect(x)))[[3]]
  
  print("wbs2")
  wbs2sdll <- new("cpt.est")
  z <- model.sdll(sol.wbs2(x))
  if(length(z$cpts) == 0){wbs2sdll@cpt = 0
  wbs2sdll@nocpt = 0}
  else{wbs2sdll@cpt <- z$cpt
  wbs2sdll@nocpt <- z$no.of.cpt}
  wbs2sdll@mean <- z$est
  wbs2sdll@time <- system.time(model.sdll(sol.wbs2(x)))[[3]]
  
  
  list(wbssdll=wbssdll, idetectsdll = idetectsdll, tguhsdll = tguhsdll, wbs2sdll = wbs2sdll)
}


sim.study.sdll <- function(signal, true.cpt=NULL,sigma, m = 100, seed = NULL) {
  
  setClass("est.eval", representation(avg.signal="numeric",mean="list",cpt="list",diff="matrix",
  dh="numeric",cptall="numeric",dnc="numeric", mse="numeric", time= "numeric"), 
  prototype(dnc=numeric(m), mse=numeric(m),time=numeric(m),dh=numeric(m)))
  
  wbssdll <- new("est.eval")
  wbs2sdll <- new("est.eval")
  idetectsdll <- new("est.eval")
  tguhsdll <- new("est.eval")
  
  no.of.cpt <- sum(abs(diff(signal)) > 0)
  n <- length(signal)
  ns <- max(c(diff(true.cpt), length(signal)))
  
  if (!is.null(seed)) set.seed(seed)
  
  for (i in 1:m) {
    
    print(i)
    x <- signal + sigma * rnorm(n)
    
    est <- sim_sdll(x)
    
    wbssdll@dnc[i] <- est$wbssdll@nocpt - no.of.cpt
    wbssdll@mean[[i]] <- est$wbssdll@mean
    wbssdll@mse[i] <- mean((est$wbssdll@mean - signal)^2)
    wbssdll@cpt[[i]] <- est$wbssdll@cpt
    wbssdll@diff <- abs(matrix(est$wbssdll@cpt,nrow=no.of.cpt,ncol=length(est$wbssdll@cpt),
    byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$wbssdll@cpt),byr=F))
    wbssdll@dh[i] <- max(apply(wbssdll@diff,1,min),apply(wbssdll@diff,2,min))/ns
    wbssdll@time[i] <- est$wbssdll@time
    
    wbs2sdll@dnc[i] <- est$wbs2sdll@nocpt - no.of.cpt
    wbs2sdll@mean[[i]] <- est$wbs2sdll@mean
    wbs2sdll@mse[i] <- mean((est$wbs2sdll@mean - signal)^2)
    wbs2sdll@cpt[[i]] <- est$wbs2sdll@cpt
    wbs2sdll@diff <- abs(matrix(est$wbs2sdll@cpt,nrow=no.of.cpt,ncol=length(est$wbs2sdll@cpt),
    byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$wbs2sdll@cpt),byr=F))
    wbs2sdll@dh[i] <- max(apply(wbs2sdll@diff,1,min),apply(wbs2sdll@diff,2,min))/ns
    wbs2sdll@time[i] <- est$wbs2sdll@time
    
    idetectsdll@dnc[i] <- est$idetectsdll@nocpt - no.of.cpt
    idetectsdll@mean[[i]] <- est$idetectsdll@mean
    idetectsdll@mse[i] <- mean((est$idetectsdll@mean - signal)^2)
    idetectsdll@cpt[[i]] <- est$idetectsdll@cpt
    idetectsdll@diff <- abs(matrix(est$idetectsdll@cpt,nrow=no.of.cpt,
    ncol=length(est$idetectsdll@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$idetectsdll@cpt),byr=F))
    idetectsdll@dh[i] <- max(apply(idetectsdll@diff,1,min),apply(idetectsdll@diff,2,min))/ns
    idetectsdll@time[i] <- est$idetectsdll@time
    
    tguhsdll@dnc[i] <- est$tguhsdll@nocpt - no.of.cpt
    tguhsdll@mean[[i]] <- est$tguhsdll@mean
    tguhsdll@mse[i] <- mean((est$tguhsdll@mean - signal)^2)
    tguhsdll@cpt[[i]] <- est$tguhsdll@cpt
    tguhsdll@diff <- abs(matrix(est$tguhsdll@cpt,nrow=no.of.cpt,ncol=length(est$tguhsdll@cpt),
    byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$tguhsdll@cpt),byr=F))
    tguhsdll@dh[i] <- max(apply(tguhsdll@diff,1,min),apply(tguhsdll@diff,2,min))/ns
    tguhsdll@time[i] <- est$tguhsdll@time
    
  }
  
  list(wbssdll=wbssdll, idetectsdll = idetectsdll, tguhsdll = tguhsdll, wbs2sdll = wbs2sdll)
}

seed.temp=12

justnoise = rep(0,3000)
NC.small_sdll = sim.study.sdll(justnoise,sigma=1,true.cpt=c(0),seed=seed.temp)

justnoise2 = rep(0,50)
NC2.small_sdll = sim.study.sdll(justnoise2,true.cpt=c(0),sigma=1,seed=seed.temp)

justnoise3 = rep(0,5)
NC3.small_sdll = sim.study.sdll(justnoise3,true.cpt=c(0),sigma=1,seed=seed.temp)

justnoise4 = rep(0,10000)
NC4.small_sdll = sim.study.sdll(justnoise4,sigma=1,true.cpt=c(0),seed=seed.temp)

blocks = c(rep(0,205),rep(14.64,62),rep(-3.66,41),rep(7.32,164),rep(-7.32,40),rep(10.98,308),
           rep(-4.39,82),rep(3.29,430),rep(19.03,225),rep(7.68,41),rep(15.37,61),rep(0,389))

SIMR1.small_sdll = sim.study.sdll(blocks,true.cpt = c(205,267,308,472,512,820,902,1332,1557,
1598,1659), sigma=10,seed=seed.temp)

fms = c(rep(-0.18,139),rep(0.08,87),rep(1.07,17),rep(-0.53,57),rep(0.16,9),rep(-0.69,24),
rep(-0.16,164))
SIMR2.small_sdll = sim.study.sdll(fms,true.cpt = c(139,226,243,300,309,333),sigma=0.3,seed=seed.temp)

mix = c(rep(7,11),rep(-7,10),rep(6,20),rep(-6,20),rep(5,30),rep(-5,30),rep(4,40),rep(-4,40),
        rep(3,50),rep(-3,50),rep(2,60),rep(-2,60),rep(1,70),rep(-1,69))
SIMR3.small_sdll = sim.study.sdll(mix,true.cpt=c(11,21,41,61,91,121,161,201,251,301,361,421,491), 
sigma=4,seed=seed.temp)

teeth10 = c(rep(0,11),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,10),
rep(0,10),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,9))
SIMR4.small_sdll = sim.study.sdll(teeth10,true.cpt=seq(11,131,10),0.4,seed=seed.temp)

stairs10 = c(rep(1,11),rep(2,10),rep(3,10),rep(4,10),rep(5,10),rep(6,10),rep(7,10),rep(8,10),
rep(9,10),rep(10,10),rep(11,10),rep(12,10),rep(13,10),rep(14,10),rep(15,9))
SIMR5.small_sdll = sim.study.sdll(stairs10,true.cpt=seq(11,141,10),sigma=0.3,seed=seed.temp)

myteeth20 = rep(c(rep(0,40),rep(1.5,40)),125)
SIMR6.small_sdll = sim.study.sdll(myteeth20,true.cpt = seq(40,9960,40),sigma=1,seed=seed.temp,
m=100)

long_teeth = rep(c(rep(0,40),rep(1.5,40)),1250)
SIMR7.small_sdll = sim.study.sdll(long_teeth,true.cpt = seq(40,99960,40),sigma=1,seed=seed.temp,
m=100)

extr_short_2_cpt <- c(rep(0,15), rep(2,15), rep(-0.5,20))
SIMR8.small_sdll = sim.study.sdll(extr_short_2_cpt,true.cpt=c(15,30),sigma=1,seed=seed.temp,
m=100, gen_const = 1.1)

extr_short_2_cpt2 <- c(rep(0,3), rep(3,4), rep(0,3))
SIMR9.small_sdll = sim.study.sdll(extr_short_2_cpt2,true.cpt=c(3,7),sigma=1,seed=seed.temp,
m=100, gen_const = 1.1)

strong_sig_low_noise <- c(rep(0,100), rep(5,500), rep(0, 1000), rep(10, 400), rep(2, 500))
SIMR10.small_sdll <- sim.study.sdll(strong_sig_low_noise,true.cpt=c(100, 600, 1600, 2000),
sigma=0.1, seed=seed.temp, m=100, gen_const = 1.1)

one_spike_in_middle <- c(rep(0,1000),100, rep(0,999))
SIMR11.small_sdll <- sim.study.sdll(one_spike_in_middle,true.cpt=c(1000, 1001),sigma=1,seed=seed.temp, 
m=100, gen_const = 1.1)

no_noise <- c(rep(0,250), rep(5,300), rep(0,200), rep(-4,250))
SIMR12.small_sdll <- sim.study.sdll(no_noise,true.cpt=c(250, 550, 750),sigma=0,seed=seed.temp, m=1) 
## no need for 100 replications here.

linear <- seq(0,500)
SIMR13.small_sdll <- sim.study.sdll(linear,true.cpt=seq(1,500,1),sigma=1,seed=seed.temp, m=100)

SIMR14.small_sdll <- sim.study.sdll(linear,true.cpt=seq(1,500,1),sigma=0,seed=seed.temp, m=1) 
## no need for 100 replications here.


## The function used for the comparisons in the case of a piecewise-constant signal 
## with localised pruning
sim_lp <- function(x) {
  setClass("cpt.est", representation(cpt="numeric", nocpt="numeric", mean="numeric",time="numeric"),
  prototype(cpt=numeric(0), nocpt=0, mean=numeric(0),time=numeric(0)))
  
  print("wbs")
  wbsloc.prune <- new("cpt.est")
  z <- model.lp(sol.wbs(x))
  if(length(z$cpts) == 0){wbsloc.prune@cpt = 0
  wbsloc.prune@nocpt = 0}
  else{wbsloc.prune@cpt <- z$cpts
  wbsloc.prune@nocpt <- z$no.of.cpt}
  wbsloc.prune@mean <- z$est
  wbsloc.prune@time <- system.time(model.lp(sol.wbs(x)))[[3]]
  
  print("not")
  notloc.prune <- new("cpt.est")
  z <- model.lp(sol.not(x))
  if(length(z$cpts) == 0){notloc.prune@cpt = 0
  notloc.prune@nocpt = 0}
  else{notloc.prune@cpt <- z$cpts
  notloc.prune@nocpt <- z$no.of.cpt}
  notloc.prune@mean <- z$est
  notloc.prune@time <- system.time(model.lp(sol.not(x)))[[3]]
  
  print("wbs2")
  wbs2loc.prune <- new("cpt.est")
  z <- model.lp(sol.wbs2(x))
  if(length(z$cpts) == 0){wbs2loc.prune@cpt = 0
  wbs2loc.prune@nocpt = 0}
  else{wbs2loc.prune@cpt <- z$cpts
  wbs2loc.prune@nocpt <- z$no.of.cpt}
  wbs2loc.prune@mean <- z$est
  wbs2loc.prune@time <- system.time(model.lp(sol.wbs2(x)))[[3]]
  
  print("tguh")
  tguhloc.prune <- new("cpt.est")
  z <- model.lp(sol.tguh(x))
  if(length(z$cpts) == 0){tguhloc.prune@cpt = 0
  tguhloc.prune@nocpt = 0}
  else{tguhloc.prune@cpt <- z$cpts
  tguhloc.prune@nocpt <- z$no.of.cpt}
  tguhloc.prune@mean <- z$est
  tguhloc.prune@time <- system.time(model.lp(sol.tguh(x)))[[3]]
  
  list(wbsloc.prune=wbsloc.prune, notloc.prune = notloc.prune, tguhloc.prune = tguhloc.prune, 
  wbs2loc.prune = wbs2loc.prune)
}


sim.study.lp <- function(signal, true.cpt=NULL,sigma, m = 100, seed = NULL) {
  
  setClass("est.eval", representation(avg.signal="numeric",mean="list",cpt="list",
  diff="matrix", dh="numeric",cptall="numeric",dnc="numeric", mse="numeric", time= "numeric"),
  prototype(dnc=numeric(m), mse=numeric(m),time=numeric(m),dh=numeric(m)))
  
  wbsloc.prune <- new("est.eval")
  wbs2loc.prune <- new("est.eval")
  notloc.prune <- new("est.eval")
  tguhloc.prune <- new("est.eval")
  
  no.of.cpt <- sum(abs(diff(signal)) > 0)
  n <- length(signal)
  ns <- max(c(diff(true.cpt), length(signal)))
  
  if (!is.null(seed)) set.seed(seed)
  
  for (i in 1:m) {
    
    print(i)
    x <- signal + sigma * rnorm(n)
    
    est <- sim_lp(x)
    
    wbsloc.prune@dnc[i] <- est$wbsloc.prune@nocpt - no.of.cpt
    wbsloc.prune@mean[[i]] <- est$wbsloc.prune@mean
    wbsloc.prune@mse[i] <- mean((est$wbsloc.prune@mean - signal)^2)
    wbsloc.prune@cpt[[i]] <- est$wbsloc.prune@cpt
    wbsloc.prune@diff <- abs(matrix(est$wbsloc.prune@cpt,nrow=no.of.cpt,
    ncol=length(est$wbsloc.prune@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$wbsloc.prune@cpt),byr=F))
    wbsloc.prune@dh[i] <- max(apply(wbsloc.prune@diff,1,min),apply(wbsloc.prune@diff,2,min))/ns
    wbsloc.prune@time[i] <- est$wbsloc.prune@time
    
    wbs2loc.prune@dnc[i] <- est$wbs2loc.prune@nocpt - no.of.cpt
    wbs2loc.prune@mean[[i]] <- est$wbs2loc.prune@mean
    wbs2loc.prune@mse[i] <- mean((est$wbs2loc.prune@mean - signal)^2)
    wbs2loc.prune@cpt[[i]] <- est$wbs2loc.prune@cpt
    wbs2loc.prune@diff <- abs(matrix(est$wbs2loc.prune@cpt,nrow=no.of.cpt,
    ncol=length(est$wbs2loc.prune@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$wbs2loc.prune@cpt),byr=F))
    wbs2loc.prune@dh[i] <- max(apply(wbs2loc.prune@diff,1,min),apply(wbs2loc.prune@diff,2,min))/ns
    wbs2loc.prune@time[i] <- est$wbs2loc.prune@time
    
    notloc.prune@dnc[i] <- est$notloc.prune@nocpt - no.of.cpt
    notloc.prune@mean[[i]] <- est$notloc.prune@mean
    notloc.prune@mse[i] <- mean((est$notloc.prune@mean - signal)^2)
    notloc.prune@cpt[[i]] <- est$notloc.prune@cpt
    notloc.prune@diff <- abs(matrix(est$notloc.prune@cpt,nrow=no.of.cpt,
    ncol=length(est$notloc.prune@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$notloc.prune@cpt),byr=F))
    notloc.prune@dh[i] <- max(apply(notloc.prune@diff,1,min),apply(notloc.prune@diff,2,min))/ns
    notloc.prune@time[i] <- est$notloc.prune@time
    
    tguhloc.prune@dnc[i] <- est$tguhloc.prune@nocpt - no.of.cpt
    tguhloc.prune@mean[[i]] <- est$tguhloc.prune@mean
    tguhloc.prune@mse[i] <- mean((est$tguhloc.prune@mean - signal)^2)
    tguhloc.prune@cpt[[i]] <- est$tguhloc.prune@cpt
    tguhloc.prune@diff <- abs(matrix(est$tguhloc.prune@cpt,nrow=no.of.cpt,
    ncol=length(est$tguhloc.prune@cpt),byr=T)
    -matrix(true.cpt,nrow=no.of.cpt,ncol=length(est$tguhloc.prune@cpt),byr=F))
    tguhloc.prune@dh[i] <- max(apply(tguhloc.prune@diff,1,min),apply(tguhloc.prune@diff,2,min))/ns
    tguhloc.prune@time[i] <- est$tguhloc.prune@time
    
  }
  
  list(wbsloc.prune=wbsloc.prune, notloc.prune=notloc.prune, tguhloc.prune = tguhloc.prune, 
  wbs2loc.prune = wbs2loc.prune)
}

seed.temp=12

justnoise = rep(0,3000)
NC.small_lp = sim.study.lp(justnoise,sigma=1,true.cpt=c(0),seed=seed.temp)

justnoise2 = rep(0,50)
NC2.small_lp = sim.study.lp(justnoise2,true.cpt=c(0),sigma=1,seed=seed.temp)

justnoise3 = rep(0,5)
NC3.small_lp = sim.study.lp(justnoise3,true.cpt=c(0),sigma=1,seed=seed.temp)

justnoise4 = rep(0,10000)
NC4.small_lp = sim.study.lp(justnoise4,sigma=1,true.cpt=c(0),seed=seed.temp)

blocks = c(rep(0,205),rep(14.64,62),rep(-3.66,41),rep(7.32,164),rep(-7.32,40),rep(10.98,308),
           rep(-4.39,82),rep(3.29,430),rep(19.03,225),rep(7.68,41),rep(15.37,61),rep(0,389))

SIMR1.small_lp = sim.study.lp(blocks,true.cpt = c(205,267,308,472,512,820,902,1332,1557,1598,1659),
sigma=10,seed=seed.temp)

fms = c(rep(-0.18,139),rep(0.08,87),rep(1.07,17),rep(-0.53,57),rep(0.16,9),rep(-0.69,24),
rep(-0.16,164))
SIMR2.small_lp = sim.study.lp(fms,true.cpt = c(139,226,243,300,309,333),sigma=0.3,seed=seed.temp)

mix = c(rep(7,11),rep(-7,10),rep(6,20),rep(-6,20),rep(5,30),rep(-5,30),rep(4,40),rep(-4,40),
        rep(3,50),rep(-3,50),rep(2,60),rep(-2,60),rep(1,70),rep(-1,69))
SIMR3.small_lp = sim.study.lp(mix,true.cpt=c(11,21,41,61,91,121,161,201,251,301,361,421,491), 
sigma=4,seed=seed.temp)

teeth10 = c(rep(0,11),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,10),
rep(0,10),rep(1,10),rep(0,10),rep(1,10),rep(0,10),rep(1,9))

SIMR4.small_lp = sim.study.lp(teeth10,true.cpt=seq(11,131,10),0.4,seed=seed.temp)

stairs10 = c(rep(1,11),rep(2,10),rep(3,10),rep(4,10),rep(5,10),rep(6,10),rep(7,10),rep(8,10),
rep(9,10),rep(10,10),rep(11,10),rep(12,10),rep(13,10),rep(14,10),rep(15,9))

SIMR5.small_lp = sim.study.lp(stairs10,true.cpt=seq(11,141,10),sigma=0.3,seed=seed.temp)

myteeth20 = rep(c(rep(0,40),rep(1.5,40)),125)
SIMR6.small_lp = sim.study.lp(myteeth20,true.cpt = seq(40,9960,40),sigma=1,seed=seed.temp,
m=100)

extr_short_2_cpt <- c(rep(0,15), rep(2,15), rep(-0.5,20))
SIMR8.small_lp = sim.study.lp(extr_short_2_cpt,true.cpt=c(15,30),sigma=1,seed=seed.temp,
m=100)

extr_short_2_cpt2 <- c(rep(0,3), rep(3,4), rep(0,3))
SIMR9.small_lp = sim.study.lp(extr_short_2_cpt2,true.cpt=c(3,7),sigma=1,seed=seed.temp,
m=100)

strong_sig_low_noise <- c(rep(0,100), rep(5,500), rep(0, 1000), rep(10, 400), rep(2, 500))
SIMR10.small_lp <- sim.study.lp(strong_sig_low_noise,true.cpt=c(100, 600, 1600, 2000),
sigma=0.1,seed=seed.temp, m=100)

one_spike_in_middle <- c(rep(0,1000),100, rep(0,999))
SIMR11.small_lp <- sim.study.lp(one_spike_in_middle,true.cpt=c(1000, 1001),sigma=1,seed=seed.temp, 
m=100)

no_noise <- c(rep(0,250), rep(5,300), rep(0,200), rep(-4,250))
SIMR12.small_lp <- sim.study.lp(no_noise,true.cpt=c(250, 550, 750),sigma=0,seed=seed.temp, m=1) 
## no need for 100 replications here.

linear <- seq(0,500)
SIMR13.small_lp <- sim.study.lp(linear,true.cpt=seq(1,500,1),sigma=1,seed=seed.temp, m=100)

SIMR14.small_lp <- sim.study.lp(linear,true.cpt=seq(1,500,1),sigma=0,seed=seed.temp, m=1) 
## no need for 100 replications here.

3.4 Detailed results

3.4.1 Information-criterion based results

In this section, we give the simulation results when the model.ic function from the package is employed at each of the solution path methods. The results are given in Tables 1 - 7 below and a summary of the results is given in Table 8. We note that for ID, the results are presented for the sol.idetect function (and not for the sol.idetect_seq), which is the one that should be used when Isolate-Detect is combined with the information-criterion model selection algorithm of the model.ic function.

Table 1: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1)-(NC4). Also the average MSE and computational times for each method are given.
\(\hat{N} - N\)
Method Model \(0\) \(1\) \(2\) \(\geq 3\) MSE Time (s)
ID 99 1 0 0 49 \(\times 10^{-5}\) 0.12
NOT 100 0 0 0 43 \(\times 10^{-5}\) 0.08
TGUH (NC1) 100 0 0 0 43 \(\times 10^{-5}\) 0.21
WBS 99 1 0 0 49 \(\times 10^{-5}\) 0.07
WBS2 99 1 0 0 49 \(\times 10^{-5}\) 1.13
ID 88 7 5 0 45 \(\times 10^{-3}\) 0.005
NOT 88 5 5 2 52 \(\times 10^{-3}\) 0.010
TGUH (NC2) 89 4 5 2 48 \(\times 10^{-3}\) 0.017
WBS 88 5 5 2 52 \(\times 10^{-3}\) 0.009
WBS2 87 5 6 2 54 \(\times 10^{-3}\) 0.022
ID 60 30 10 0 373 \(\times 10^{-3}\) 0.002
NOT 0 0 0 100 903 \(\times 10^{-3}\) 0.002
TGUH (NC3) 0 0 0 100 903 \(\times 10^{-3}\) 0.003
WBS 0 0 0 100 903 \(\times 10^{-3}\) 0.003
WBS2 0 0 0 100 903 \(\times 10^{-3}\) 0.001
ID 99 0 1 0 14 \(\times 10^{-5}\) 0.84
NOT 99 0 1 0 14 \(\times 10^{-5}\) 0.17
TGUH (NC4) 100 0 0 0 10 \(\times 10^{-5}\) 0.54
WBS 100 0 0 0 10 \(\times 10^{-5}\) 0.15
WBS2 100 0 0 0 10 \(\times 10^{-5}\) 4.09
Table 2: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signals (M1)-(M5). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method Model \(\leq -3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\geq 3\) MSE \(d_H\) Time (s)
ID 0 3 40 51 5 1 0 2.886 171 \(\times10^{-4}\) 0.04
NOT 0 5 58 35 2 0 0 2.733 175 \(\times10^{-4}\) 0.09
TGUH (M1) 0 5 54 36 4 1 0 3.356 212 \(\times10^{-4}\) 0.16
WBS 0 4 49 47 0 0 0 2.680 145 \(\times10^{-4}\) 0.06
WBS2 0 5 47 45 1 2 0 2.743 153 \(\times10^{-4}\) 0.84
ID 0 2 0 89 7 2 0 45 \(\times 10^{-4}\) 20 \(\times 10^{-3}\) 0.013
NOT 0 2 0 95 3 0 0 40 \(\times 10^{-4}\) 17 \(\times 10^{-3}\) 0.049
TGUH (M2) 2 13 6 62 13 4 0 66 \(\times 10^{-4}\) 46 \(\times 10^{-3}\) 0.064
WBS 0 2 0 95 3 0 0 44 \(\times 10^{-4}\) 17 \(\times 10^{-3}\) 0.017
WBS2 0 1 0 96 3 0 0 42 \(\times 10^{-4}\) 15 \(\times 10^{-3}\) 0.181
ID 7 33 25 28 7 0 0 1.626 147 \(\times 10^{-3}\) 0.015
NOT 9 30 31 22 6 0 2 1.729 144 \(\times 10^{-3}\) 0.048
TGUH (M3) 12 29 32 16 4 2 5 1.985 181 \(\times 10^{-3}\) 0.059
WBS 11 24 31 29 4 0 1 1.670 141 \(\times 10^{-3}\) 0.034
WBS2 9 26 32 30 3 0 0 1.578 146 \(\times 10^{-3}\) 0.176
ID 3 5 2 66 21 2 1 61 \(\times10^{-3}\) 44 \(\times 10^{-3}\) 0.011
NOT 6 10 0 69 12 3 0 64 \(\times10^{-3}\) 43 \(\times 10^{-3}\) 0.021
TGUH (M4) 22 2 2 35 14 13 12 102 \(\times10^{-3}\) 215 \(\times 10^{-3}\) 0.027
WBS 1 4 0 82 12 0 1 54 \(\times10^{-3}\) 31 \(\times 10^{-3}\) 0.012
WBS2 2 5 0 75 17 0 1 58 \(\times10^{-3}\) 40 \(\times 10^{-3}\) 0.039
ID 0 0 0 89 9 1 1 22 \(\times10^{-3}\) 10 \(\times 10^{-3}\) 0.012
NOT 0 0 0 86 8 4 2 21 \(\times10^{-3}\) 10 \(\times 10^{-3}\) 0.100
TGUH (M5) 0 0 0 74 19 5 2 24 \(\times10^{-3}\) 13 \(\times 10^{-3}\) 0.031
WBS 0 0 0 60 34 3 3 24 \(\times10^{-3}\) 14 \(\times 10^{-3}\) 0.032
WBS2 0 0 0 60 30 8 2 24 \(\times10^{-3}\) 14 \(\times 10^{-3}\) 0.044

By default, the maximum number of change-points (argument q.max in the model.ic function) that the information criterion approach can detect is set to be equal to 25. The signal (M6) has 249 change-points; thus, we need to increase q.max and in the results that follow in Table 3, we set this argument to be equal to 300, which is sufficiently higher than the true number of change-points.

Table 3: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M6). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(\leq -150\) \((-150,-50]\) \((-50,-10)\) \([-10,10]\) \((10,50]\) \(>50\) MSE \(d_H\) Time (s)
ID 0 0 0 100 0 0 0.110 29 \(\times10^{-4}\) 0.31
NOT 13 87 0 0 0 0 0.483 167 \(\times10^{-4}\) 0.13
TGUH 0 100 0 0 0 0 0.376 84 \(\times10^{-4}\) 0.39
WBS 0 100 0 0 0 0 0.453 168 \(\times10^{-4}\) 0.12
WBS2 0 0 0 100 0 0 0.108 34 \(\times10^{-4}\) 2.97
Table 4: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signals (M7) and (M8). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method Model \(-2\) \(-1\) \(0\) \(1\) \(\geq 2\) MSE \(d_H\) Time (s)
ID 0 0 78 19 3 0.203 47 \(\times10^{-3}\) 0.008
NOT 0 0 79 15 6 0.209 50 \(\times10^{-3}\) 0.010
TGUH (M7) 0 0 71 24 5 0.257 58 \(\times10^{-3}\) 0.016
WBS 0 0 81 12 7 0.207 49 \(\times10^{-3}\) 0.008
WBS2 0 0 80 14 6 0.200 47 \(\times10^{-3}\) 0.019
ID 14 6 68 11 1 0.884 164 \(\times10^{-3}\) 0.003
NOT 0 0 0 0 100 0.920 200 \(\times10^{-3}\) 0.003
TGUH (M8) 0 0 0 0 100 0.920 200 \(\times10^{-3}\) 0.005
WBS 0 0 0 0 100 0.920 200 \(\times10^{-3}\) 0.003
WBS2 0 0 0 0 100 0.920 200 \(\times10^{-3}\) 0.004
Table 5: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M9). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(\leq -3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\geq 3\) MSE \(d_H\) Time (s)
ID 0 0 0 98 2 0 0 23 \(\times 10^{-6}\) 29 \(\times10^{-3}\) 0.038
NOT 0 0 0 98 2 0 0 23 \(\times 10^{-6}\) 26 \(\times10^{-3}\) 0.071
TGUH 0 0 0 98 2 0 0 23 \(\times 10^{-6}\) 29 \(\times10^{-3}\) 0.132
WBS 0 0 0 98 2 0 0 23 \(\times 10^{-6}\) 26 \(\times10^{-3}\) 0.051
WBS2 0 0 0 98 2 0 0 23 \(\times 10^{-6}\) 26 \(\times10^{-3}\) 0.712
Table 6: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M10). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(-2\) \(-1\) \(0\) \(1\) \(\geq 2\) MSE \(d_H\) Time (s)
ID 0 0 100 0 0 0.001 0 0.039
NOT 1 0 98 1 0 0.051 66 \(\times10^{-4}\) 0.062
TGUH 22 0 7 10 61 1.293 3981 \(\times10^{-4}\) 0.114
WBS 1 0 97 2 0 0.052 115 \(\times10^{-4}\) 0.050
WBS2 0 0 98 2 0 0.002 66 \(\times10^{-4}\) 0.569

For the signal (M11), since there is no noise, we do not need to repeat our simulations 100 times; the input will be the same each time. All methods, have managed to detect all change-points accurately in this noiseless scenario and therefore, the results are not presented in a table. We proceed with the results for (M12). Since there are 500 change-points (the investigation is under a piecewise-constant and not a piecewise-linear signal) we use q.max = 550, where q.max is the argument for the maximum number of change-points that we allow to be detected in a given signal. The results are in Table 7 below.

Table 7: The average \(\hat{N}\), MSE, and computational times over 100 simulated data sequences from the linear signal (M12).
Method \(\hat{N}\) MSE Time (s)
ID 99.64 2.664 0.545
NOT 67.64 6.277 1.076
TGUH 500 0.999 1.012
WBS 71.85 5.241 0.668
WBS2 500 0.999 1.081

For the noiseless (M13), we do not present the results in a table. ID, NOT, TGUH, WBS, and WBS2 detect 83, 70, 500, 64, and 500, respectively. Table 8 below gives a summary of the results for all signals used in the simulation study.

Table 8: Summary of the results when the information criterion model selection was employed for all the solution path algorithms.
Methods
ID NOT TGUH WBS WBS2
Models Criterion: Within top \(10\%\) in terms of accuracy
(NC1) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(NC2) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(NC3) \(\checkmark\)
(NC4) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M1) \(\checkmark\) \(\checkmark\)
(M2) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M3) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M4) \(\checkmark\) \(\checkmark\)
(M5) \(\checkmark\) \(\checkmark\)
(M6) \(\checkmark\) \(\checkmark\)
(M7) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M8) \(\checkmark\)
(M9) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M10) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M11) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M12) \(\checkmark\) \(\checkmark\)
(M13) \(\checkmark\) \(\checkmark\)
Criterion: The top two methods in terms of speed
(NC1) \(\checkmark\) \(\checkmark\)
(NC2) \(\checkmark\) \(\checkmark\)
(NC3) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(NC4) \(\checkmark\) \(\checkmark\)
(M1) \(\checkmark\) \(\checkmark\)
(M2) \(\checkmark\) \(\checkmark\)
(M3) \(\checkmark\) \(\checkmark\)
(M4) \(\checkmark\) \(\checkmark\)
(M5) \(\checkmark\) \(\checkmark\)
(M6) \(\checkmark\) \(\checkmark\)
(M7) \(\checkmark\) \(\checkmark\)
(M8) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M9) \(\checkmark\) \(\checkmark\)
(M10) \(\checkmark\) \(\checkmark\)
(M12) \(\checkmark\) \(\checkmark\)

3.4.2 Thresholding based results

In this section, we give the simulation results when the model.thresh function from the package is employed at each of the solution path methods. The results are given in Tables 9 - 15 below and a summary of the results is given in Table 16. We note that for ID, the results are presented for the sol.idetect_seq function (and not for the sol.idetect), which is the one that should be used when Isolate-Detect is combined with the thresholding model selection algorithm of the model.thresh function.

Table 9: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1)-(NC4). Also the average MSE and computational times for each method are given.
\(\hat{N} - N\)
Method Model 0 1 2 \(\geq 3\) MSE Time (s)
ID 86 2 9 3 159 \(\times 10^{-5}\) 0.245
NOT 82 13 2 3 93 \(\times 10^{-5}\) 0.063
TGUH (NC1) 93 6 1 0 45 \(\times 10^{-5}\) 0.147
WBS 79 9 10 2 135 \(\times 10^{-5}\) 0.058
WBS2 79 17 2 2 91 \(\times 10^{-5}\) 0.934
ID 73 9 12 6 76 \(\times 10^{-3}\) 0.002
NOT 59 16 14 11 86 \(\times 10^{-3}\) 0.003
TGUH (NC2) 81 16 1 2 30 \(\times 10^{-3}\) 0.011
WBS 48 20 15 17 121 \(\times 10^{-3}\) 0.002
WBS2 59 15 13 13 106 \(\times 10^{-3}\) 0.017
ID 55 29 14 2 390 \(\times 10^{-3}\) 0.002
NOT 67 24 1 8 317 \(\times 10^{-3}\) 0.003
TGUH (NC3) 64 25 3 8 333 \(\times 10^{-3}\) 0.002
WBS 50 34 8 8 410 \(\times 10^{-3}\) 0.001
WBS2 50 34 8 8 410 \(\times 10^{-3}\) 0.001
ID 89 2 8 1 37 \(\times 10^{-5}\) 1.614
NOT 89 8 2 1 21 \(\times 10^{-5}\) 0.119
TGUH (NC4) 94 6 0 0 12 \(\times 10^{-5}\) 0.351
WBS 89 7 4 0 22 \(\times 10^{-5}\) 0.112
WBS2 88 10 2 0 15 \(\times 10^{-5}\) 2.911
Table 10: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signals (M1)-(M5). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method Model \(\leq -3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\geq 3\) MSE \(d_H\) Time (s)
ID 0 2 56 40 2 0 0 2.807 19 \(\times10^{-3}\) 0.025
NOT 0 15 62 22 1 0 0 3.217 28 \(\times10^{-3}\) 0.063
TGUH (M1) 1 14 64 19 2 0 0 3.680 23 \(\times10^{-3}\) 0.109
WBS 0 3 60 34 3 0 0 2.812 27 \(\times10^{-3}\) 0.049
WBS2 0 3 49 38 6 4 0 2.914 26 \(\times10^{-3}\) 0.582
ID 0 0 2 87 8 3 0 45 \(\times 10^{-4}\) 28 \(\times 10^{-3}\) 0.008
NOT 0 2 21 64 11 1 1 60 \(\times 10^{-4}\) 54 \(\times 10^{-3}\) 0.040
TGUH (M2) 0 1 61 36 2 0 0 96 \(\times 10^{-4}\) 32 \(\times 10^{-3}\) 0.056
WBS 0 0 2 75 17 4 2 47 \(\times 10^{-4}\) 50 \(\times 10^{-3}\) 0.029
WBS2 0 0 4 84 8 4 0 45 \(\times 10^{-4}\) 29 \(\times 10^{-3}\) 0.150
ID 1 20 42 31 6 0 0 1.617 120 \(\times 10^{-3}\) 0.009
NOT 5 20 44 26 5 0 0 1.909 123 \(\times 10^{-3}\) 0.042
TGUH (M3) 47 33 17 3 0 0 0 2.706 228 \(\times 10^{-3}\) 0.061
WBS 0 10 51 27 10 2 0 1.649 102 \(\times 10^{-3}\) 0.031
WBS2 2 17 41 27 12 1 0 1.548 119 \(\times 10^{-3}\) 0.168
ID 10 7 17 63 2 1 0 66 \(\times10^{-3}\) 47 \(\times 10^{-3}\) 0.006
NOT 5 14 26 48 6 1 0 75 \(\times10^{-3}\) 49 \(\times 10^{-3}\) 0.014
TGUH (M4) 84 10 3 3 0 0 0 168 \(\times10^{-3}\) 129 \(\times 10^{-3}\) 0.025
WBS 3 4 21 65 6 1 0 60 \(\times10^{-3}\) 37 \(\times 10^{-3}\) 0.005
WBS2 5 4 21 63 6 1 0 62 \(\times10^{-3}\) 39 \(\times 10^{-3}\) 0.038
ID 0 0 0 97 2 1 0 22 \(\times10^{-3}\) 9 \(\times 10^{-3}\) 0.006
NOT 0 0 34 59 6 0 1 32 \(\times10^{-3}\) 28 \(\times 10^{-3}\) 0.054
TGUH (M5) 0 0 0 99 1 0 0 23 \(\times10^{-3}\) 9 \(\times 10^{-3}\) 0.029
WBS 0 0 0 90 7 2 1 25 \(\times10^{-3}\) 11 \(\times 10^{-3}\) 0.025
WBS2 0 0 0 84 15 1 0 25 \(\times10^{-3}\) 13 \(\times 10^{-3}\) 0.042
Table 11: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M6). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(\leq -150\) \((-150,-50]\) \((-50,-10)\) \([-10,10]\) \((10,50]\) \(>50\) MSE \(d_H\) Time (s)
ID 0 0 24 76 0 0 0.124 58 \(\times10^{-4}\) 0.195
NOT 100 0 0 0 0 0 0.504 240 \(\times10^{-4}\) 0.123
TGUH 4 96 0 0 0 0 0.480 166 \(\times10^{-4}\) 0.361
WBS 66 34 0 0 0 0 0.486 230 \(\times10^{-4}\) 0.114
WBS2 0 0 34 66 0 0 0.131 52 \(\times10^{-4}\) 2.810
Table 12: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signals (M7) and (M8). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method Model \(-2\) \(-1\) \(0\) \(1\) \(\geq 2\) MSE \(d_H\) Time (s)
ID 0 1 85 8 6 0.203 54 \(\times10^{-3}\) 0.003
NOT 0 37 44 12 7 0.408 166 \(\times10^{-3}\) 0.002
TGUH (M7) 0 2 87 9 2 0.247 62 \(\times10^{-3}\) 0.009
WBS 0 0 71 18 11 0.215 75 \(\times10^{-3}\) 0.001
WBS2 0 0 71 18 11 0.212 74 \(\times10^{-3}\) 0.014
ID 22 11 61 4 2 1.082 231 \(\times10^{-3}\) 0.002
NOT 24 41 27 6 2 1.509 354 \(\times10^{-3}\) 0.001
TGUH (M8) 27 24 43 4 2 1.318 301 \(\times10^{-3}\) 0.003
WBS 16 12 64 6 2 1.026 199 \(\times10^{-3}\) 0.001
WBS2 16 12 64 6 2 1.026 199 \(\times10^{-3}\) 0.002
Table 13: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M9). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(\leq -3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\geq 3\) MSE \(d_H\) Time (s)
ID 0 0 0 90 2 8 0 29 \(\times 10^{-6}\) 10 \(\times10^{-3}\) 0.038
NOT 0 0 0 86 10 4 0 25 \(\times 10^{-6}\) 15 \(\times10^{-3}\) 0.064
TGUH 0 0 0 94 6 0 0 21 \(\times 10^{-6}\) 7 \(\times10^{-3}\) 0.125
WBS 0 0 0 76 15 9 0 31 \(\times 10^{-6}\) 26 \(\times10^{-3}\) 0.048
WBS2 0 0 0 80 16 4 0 26 \(\times 10^{-6}\) 15 \(\times10^{-3}\) 0.706
Table 14: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M10). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(-2\) \(-1\) \(0\) \(1\) \(\geq 2\) MSE \(d_H\) Time (s)
ID 0 0 91 4 5 0.002 24 \(\times 10^{-3}\) 0.038
NOT 0 1 86 10 3 0.151 44 \(\times10^{-3}\) 0.052
TGUH 0 95 4 1 0 4.996 17 \(\times10^{-3}\) 0.110
WBS 0 1 79 13 7 0.052 58 \(\times10^{-3}\) 0.045
WBS2 0 0 88 7 5 0.002 38 \(\times10^{-3}\) 0.569

For the signal (M11), since there is no noise, we do not need to repeat our simulations 100 times; the input will be the same each time. All methods, have managed to detect all change-points accurately in this noiseless scenario and therefore, the results are not presented in a table. We proceed, in Table 15 below, with the results for (M12).

Table 15: The average \(\hat{N}\), MSE, and computational times over 100 simulated data sequences from the linear signal (M12).
Method \(\hat{N}\) MSE Time (s)
ID 111.52 2.175 0.036
NOT 75.16 5.613 0.169
TGUH 104.90 2.388 0.053
WBS 81.23 4.642 0.029
WBS2 110.10 2.242 0.115

For the noiseless (M13), we do not present the results in a table because all methods detect the 500 change-points without an error. Table 16 below gives a summary of the results for all signals used in the simulation study.

Table 16: Summary of the results when the thresholding model selection was employed for all the solution path algorithms.
Methods
ID NOT TGUH WBS WBS2
Models Criterion: Within top \(10\%\) in terms of accuracy
(NC1) \(\checkmark\) \(\checkmark\)
(NC2) \(\checkmark\) \(\checkmark\)
(NC3) \(\checkmark\) \(\checkmark\)
(NC4) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M1) \(\checkmark\) \(\checkmark\)
(M2) \(\checkmark\) \(\checkmark\)
(M3) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M4) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M5) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M6) \(\checkmark\)
(M7) \(\checkmark\) \(\checkmark\)
(M8) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M9) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M10) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M11) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M12) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M13) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
Criterion: The top two methods in terms of speed
(NC1) \(\checkmark\) \(\checkmark\)
(NC2) \(\checkmark\) \(\checkmark\)
(NC3) \(\checkmark\) \(\checkmark\)
(NC4) \(\checkmark\) \(\checkmark\)
(M1) \(\checkmark\) \(\checkmark\)
(M2) \(\checkmark\) \(\checkmark\)
(M3) \(\checkmark\) \(\checkmark\)
(M4) \(\checkmark\) \(\checkmark\)
(M5) \(\checkmark\) \(\checkmark\)
(M6) \(\checkmark\) \(\checkmark\)
(M7) \(\checkmark\) \(\checkmark\)
(M8) \(\checkmark\) \(\checkmark\)
(M9) \(\checkmark\) \(\checkmark\)
(M10) \(\checkmark\) \(\checkmark\)
(M12) \(\checkmark\) \(\checkmark\)

3.4.3 SDLL based results

In this section, we give the simulation results when the model.sdll function from the package is employed at each of the solution path methods. The results are given in Tables 17 - 23 below and a summary of the results is given in Table 24. We note that for ID, the results are presented for the sol.idetect function (and not for the sol.idetect_seq), which is the one that should be used when Isolate-Detect is combined with the SDLL model selection algorithm of the model.sdll function. In addition, we do not present any results when the NOT solution path is employed due to the fact that it should not be used in combination with the model.sdll function.

Table 17: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1)-(NC4). Also the average MSE and computational times for each method are given.
\(\hat{N} - N\)
Method Model 0 1 2 \(\geq 3\) MSE Time (s)
ID 88 9 1 2 73 \(\times 10^{-5}\) 0.11
TGUH (NC1) 93 5 1 1 45 \(\times 10^{-5}\) 0.20
WBS 79 6 7 8 211 \(\times 10^{-5}\) 0.06
WBS2 79 12 3 6 149 \(\times 10^{-5}\) 1.13
ID 90 1 1 8 62 \(\times 10^{-3}\) 0.003
TGUH (NC2) 95 3 0 2 23 \(\times 10^{-3}\) 0.012
WBS 72 7 5 16 123 \(\times 10^{-3}\) 0.001
WBS2 76 5 4 15 118 \(\times 10^{-3}\) 0.017
ID 75 13 11 1 292 \(\times 10^{-3}\) 0.002
TGUH (NC3) 74 17 2 7 292 \(\times 10^{-3}\) 0.001
WBS 70 15 8 7 320 \(\times 10^{-3}\) 0.002
WBS2 70 15 8 7 320 \(\times 10^{-3}\) 0.001
ID 94 4 1 1 15 \(\times 10^{-5}\) 0.64
TGUH (NC4) 93 6 1 0 12 \(\times 10^{-5}\) 0.43
WBS 85 4 7 4 43 \(\times 10^{-5}\) 0.12
WBS2 84 11 3 3 20 \(\times 10^{-5}\) 3.39
Table 18: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signals (M1)-(M5). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method Model \(\leq -3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\geq 3\) MSE \(d_H\) Time (s)
ID 1 15 55 24 3 1 1 3.304 245 \(\times10^{-4}\) 0.03
TGUH (M1) 2 14 54 24 6 0 0 3.662 245 \(\times10^{-4}\) 0.15
WBS 0 4 63 26 5 0 2 2.868 271 \(\times10^{-4}\) 0.05
WBS2 0 5 54 32 3 3 3 2.975 246 \(\times10^{-4}\) 0.76
ID 0 6 21 69 3 0 1 66 \(\times 10^{-4}\) 26 \(\times 10^{-3}\) 0.009
TGUH (M2) 0 4 59 34 1 2 0 99 \(\times 10^{-4}\) 34 \(\times 10^{-3}\) 0.057
WBS 0 0 2 87 6 3 2 46 \(\times 10^{-4}\) 32 \(\times 10^{-3}\) 0.033
WBS2 0 1 4 89 2 2 2 45 \(\times 10^{-4}\) 23 \(\times 10^{-3}\) 0.168
ID 28 30 18 16 4 1 3 2.115 192 \(\times 10^{-3}\) 0.013
TGUH (M3) 32 26 20 10 8 2 2 2.352 195 \(\times 10^{-3}\) 0.062
WBS 3 16 40 28 6 3 4 1.700 110 \(\times 10^{-3}\) 0.034
WBS2 5 17 38 27 7 2 4 1.600 131 \(\times 10^{-3}\) 0.191
ID 3 3 5 37 18 15 19 64 \(\times10^{-3}\) 50 \(\times 10^{-3}\) 0.008
TGUH (M4) 27 11 16 20 8 5 13 105 \(\times10^{-3}\) 76 \(\times 10^{-3}\) 0.033
WBS 1 2 10 75 8 0 4 55 \(\times10^{-3}\) 26 \(\times 10^{-3}\) 0.005
WBS2 2 2 5 70 15 1 5 55 \(\times10^{-3}\) 26 \(\times 10^{-3}\) 0.044
ID 0 0 0 94 6 0 0 22 \(\times10^{-3}\) 10 \(\times 10^{-3}\) 0.008
TGUH (M5) 0 0 0 97 3 0 0 9 \(\times10^{-3}\) 13 \(\times 10^{-3}\) 0.030
WBS 0 1 0 87 10 0 2 26 \(\times10^{-3}\) 12 \(\times 10^{-3}\) 0.031
WBS2 0 0 1 82 14 2 1 25 \(\times10^{-3}\) 12 \(\times 10^{-3}\) 0.044
Table 19: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M6). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(\leq -150\) \((-150,-50]\) \((-50,-10)\) \([-10,10]\) \((10,50]\) \(>50\) MSE \(d_H\) Time (s)
ID 0 2 1 32 54 11 0.119 23 \(\times10^{-4}\) 0.50
TGUH 0 32 54 12 2 0 0.274 52 \(\times10^{-4}\) 0.44
WBS 0 100 0 0 0 0 0.409 126 \(\times10^{-4}\) 0.12
WBS2 0 0 0 98 2 0 0.110 36 \(\times10^{-4}\) 3.46
Table 20: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signals (M7) and (M8). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method Model \(-2\) \(-1\) \(0\) \(1\) \(\geq 2\) MSE \(d_H\) Time (s)
ID 1 2 80 6 11 0.240 64 \(\times10^{-3}\) 0.005
TGUH (M7) 3 3 86 5 3 0.264 70 \(\times10^{-3}\) 0.018
WBS 0 4 83 6 7 0.228 60 \(\times10^{-3}\) 0.005
WBS2 0 5 80 7 8 0.233 64 \(\times10^{-3}\) 0.023
ID 40 2 43 14 1 1.296 323 \(\times10^{-3}\) 0.001
TGUH (M8) 44 5 39 8 4 1.430 358 \(\times10^{-3}\) 0.002
WBS 31 4 57 5 3 1.175 267 \(\times10^{-3}\) 0.003
WBS2 31 4 57 5 3 1.175 267 \(\times10^{-3}\) 0.002
Table 21: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M9). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(\leq -3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\geq 3\) MSE \(d_H\) Time (s)
ID 0 0 0 47 39 9 5 31 \(\times 10^{-6}\) 23 \(\times10^{-3}\) 0.039
TGUH 0 0 0 94 4 2 0 21 \(\times 10^{-6}\) 7 \(\times10^{-3}\) 0.142
WBS 0 0 0 80 8 4 4 34 \(\times 10^{-6}\) 23 \(\times10^{-3}\) 0.050
WBS2 0 0 0 83 9 6 2 29 \(\times 10^{-6}\) 15 \(\times10^{-3}\) 0.801
Table 22: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M10). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(-2\) \(-1\) \(0\) \(1\) \(\geq 2\) MSE \(d_H\) Time (s)
ID 0 0 56 39 5 0.002 55 \(\times 10^{-3}\) 0.037
TGUH 0 96 2 1 1 4.996 16 \(\times10^{-3}\) 0.128
WBS 0 1 81 9 9 0.053 55 \(\times10^{-3}\) 0.049
WBS2 0 0 90 4 6 0.002 31 \(\times10^{-3}\) 0.672

For the signal (M11), since there is no noise, we do not need to repeat our simulations 100 times; the input will be the same each time. All methods, have managed to detect all change-points accurately in this noiseless scenario and therefore, the results are not presented in a table. We proceed, in Table 23 below, with the results for (M12).

Table 23: The average \(\hat{N}\), MSE, and computational times over 100 simulated data sequences from the linear signal (M12).
Method \(\hat{N}\) MSE Time (s)
ID 123.09 1.862 0.075
TGUH 124.55 1.899 0.062
WBS 101.24 3.562 0.031
WBS2 122.53 1.945 0.136

For the noiseless (M13), we do not present the results in a table since there is no need to repeat the experiment 100 times due to its noiseless structure. All methods detect 500 change-points without an error. Table 24 below gives a summary of the results for all signals used in the simulation study.

Table 24: Summary of the results when the SDLL model selection was employed for all the solution path algorithms.
Methods
ID TGUH WBS WBS2
Models Criterion: Within top \(10\%\) in terms of accuracy
(NC1) \(\checkmark\) \(\checkmark\)
(NC2) \(\checkmark\) \(\checkmark\)
(NC3) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(NC4) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M1) \(\checkmark\)
(M2) \(\checkmark\) \(\checkmark\)
(M3) \(\checkmark\) \(\checkmark\)
(M4) \(\checkmark\) \(\checkmark\)
(M5) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M6) \(\checkmark\)
(M7) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M8) \(\checkmark\) \(\checkmark\)
(M9) \(\checkmark\)
(M10) \(\checkmark\) \(\checkmark\)
(M11) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M12) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M13) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
Criterion: The top two methods in terms of speed
(NC1) \(\checkmark\) \(\checkmark\)
(NC2) \(\checkmark\) \(\checkmark\)
(NC3) \(\checkmark\) \(\checkmark\)
(NC4) \(\checkmark\) \(\checkmark\)
(M1) \(\checkmark\) \(\checkmark\)
(M2) \(\checkmark\) \(\checkmark\)
(M3) \(\checkmark\) \(\checkmark\)
(M4) \(\checkmark\) \(\checkmark\)
(M5) \(\checkmark\) \(\checkmark\)
(M6) \(\checkmark\) \(\checkmark\)
(M7) \(\checkmark\) \(\checkmark\)
(M8) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M9) \(\checkmark\) \(\checkmark\)
(M10) \(\checkmark\) \(\checkmark\)
(M12) \(\checkmark\) \(\checkmark\)

3.4.4 Localised pruning based results

In this section, we give the simulation results when the model.lp function from the package is employed at each of the solution path methods. The results are given in Tables 25 - 31 below and a summary of the results is given in Table 32. We do not present any results when the ID solution path is employed due to the fact that it should not be used in combination with the model.lp function. In order to present the results for (NC3), we had to set the value of min.d equal to 2, due to the fact that the length of the (NC3) signal is equal to 5.

Table 25: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1)-(NC4). Also the average MSE and computational times for each method are given.
\(\hat{N} - N\)
Method Model \(0\) \(1\) \(2\) \(\geq 3\) MSE Time (s)
NOT 37 19 17 27 373 \(\times 10^{-5}\) 0.392
TGUH (NC1) 88 9 2 1 78 \(\times 10^{-5}\) 4.511
WBS 45 20 20 15 332 \(\times 10^{-5}\) 0.243
WBS2 48 17 15 20 285 \(\times 10^{-5}\) 1.745
NOT 97 3 0 0 26 \(\times 10^{-3}\) 0.002
TGUH (NC2) 93 6 1 0 32 \(\times 10^{-3}\) 0.009
WBS 91 6 3 0 37 \(\times 10^{-3}\) 0.002
WBS2 91 5 4 0 38 \(\times 10^{-3}\) 0.011
NOT 99 1 0 0 174 \(\times 10^{-3}\) 0.002
TGUH (NC3) 67 33 0 0 325 \(\times 10^{-3}\) 0.010
WBS 85 15 0 0 249 \(\times 10^{-3}\) 0.002
WBS2 85 15 0 0 249 \(\times 10^{-3}\) 0.003
NOT 51 16 23 10 91 \(\times 10^{-5}\) 3.065
TGUH (NC4) 89 7 4 0 22 \(\times 10^{-5}\) 11.738
WBS 57 14 13 16 91 \(\times 10^{-5}\) 1.884
WBS2 36 25 20 19 99 \(\times 10^{-5}\) 13.128
Table 26: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signals (M1)-(M5). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method Model \(\leq -3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\geq 3\) MSE \(d_H\) Time (s)
NOT 4 0 17 44 22 10 3 4.084 571 \(\times10^{-4}\) 0.143
TGUH (M1) 2 2 15 56 17 8 0 3.717 320 \(\times10^{-4}\) 0.181
WBS 8 1 17 43 16 14 1 5.500 745 \(\times10^{-4}\) 0.056
WBS2 5 1 14 45 15 14 6 4.043 561 \(\times10^{-4}\) 0.623
NOT 0 12 6 56 16 7 3 59 \(\times 10^{-4}\) 53 \(\times 10^{-3}\) 0.034
TGUH (M2) 0 4 2 69 14 9 2 50 \(\times 10^{-4}\) 36 \(\times 10^{-3}\) 0.049
WBS 0 2 0 77 8 10 3 47 \(\times 10^{-4}\) 43 \(\times 10^{-3}\) 0.028
WBS2 0 3 0 76 12 7 2 49 \(\times 10^{-4}\) 47 \(\times 10^{-3}\) 0.130
NOT 12 21 34 28 5 0 0 2.241 103 \(\times 10^{-3}\) 0.036
TGUH (M3) 1 12 27 43 11 5 1 1.738 98 \(\times 10^{-3}\) 0.298
WBS 4 26 42 26 2 0 0 2.127 98 \(\times 10^{-3}\) 0.029
WBS2 0 12 38 42 8 0 0 1.506 94 \(\times 10^{-3}\) 0.144
NOT 60 17 7 16 0 0 0 113 \(\times10^{-3}\) 145 \(\times 10^{-3}\) 0.013
TGUH (M4) 19 24 2 55 0 0 0 75 \(\times10^{-3}\) 67 \(\times 10^{-3}\) 0.021
WBS 31 23 16 30 0 0 0 86 \(\times10^{-3}\) 92 \(\times 10^{-3}\) 0.008
WBS2 44 21 8 27 0 0 0 100 \(\times10^{-3}\) 122 \(\times 10^{-3}\) 0.031
NOT 0 0 23 70 5 2 0 27 \(\times10^{-3}\) 23 \(\times 10^{-3}\) 0.170
TGUH (M5) 0 0 0 99 1 0 0 23 \(\times10^{-3}\) 9 \(\times 10^{-3}\) 0.024
WBS 0 3 27 70 0 0 0 36 \(\times10^{-3}\) 26 \(\times 10^{-3}\) 0.021
WBS2 0 1 26 73 0 0 0 35 \(\times10^{-3}\) 23 \(\times 10^{-3}\) 0.032
Table 27: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M6). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(\leq -150\) \((-150,-50]\) \((-50,-10)\) \([-10,10]\) \((10,50]\) \(>50\) MSE \(d_H\) Time (s)
NOT 68 32 0 0 0 0 0.437 580 \(\times10^{-4}\) 0.21
TGUH 2 8 71 19 0 0 0.192 147 \(\times10^{-4}\) 4.63
WBS 98 2 0 0 0 0 0.472 774 \(\times10^{-4}\) 0.18
WBS2 0 0 0 100 0 0 0.105 36 \(\times10^{-4}\) 3.85
Table 28: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signals (M7) and (M8). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method Model \(-2\) \(-1\) \(0\) \(1\) \(\geq 2\) MSE \(d_H\) Time (s)
NOT 33 38 26 3 0 0.692 321 \(\times10^{-3}\) 0.004
TGUH (M7) 0 0 92 8 0 0.212 36 \(\times10^{-3}\) 0.010
WBS 0 3 97 0 0 0.181 30 \(\times10^{-3}\) 0.001
WBS2 0 2 96 2 0 0.172 29 \(\times10^{-3}\) 0.013
NOT 95 3 2 0 0 2.210 678 \(\times10^{-3}\) 0.002
TGUH (M8) 16 5 79 0 0 0.761 147 \(\times10^{-3}\) 0.003
WBS 39 13 48 0 0 1.313 335 \(\times10^{-3}\) 0.002
WBS2 39 13 48 0 0 1.313 335 \(\times10^{-3}\) 0.002
Table 29: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M9). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(\leq -3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(\geq 3\) MSE \(d_H\) Time (s)
NOT 0 0 2 45 37 10 6 182 \(\times 10^{-3}\) 54 \(\times10^{-3}\) 0.071
TGUH 0 0 0 72 18 6 4 34 \(\times 10^{-6}\) 19 \(\times10^{-3}\) 1.642
WBS 3 1 8 45 14 20 9 610 \(\times 10^{-3}\) 97 \(\times10^{-3}\) 0.470
WBS2 2 2 5 44 16 16 15 750 \(\times 10^{-3}\) 104 \(\times10^{-3}\) 2.812
Table 30: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the piecewise-constant signal (M10). The average MSE, \(d_H\) and computational time are also given.
\(\hat{N} - N\)
Method \(-2\) \(-1\) \(0\) \(1\) \(\geq 2\) MSE \(d_H\) Time (s)
NOT 73 2 25 0 0 4.972 371 \(\times10^{-4}\) 0.069
TGUH 35 9 56 0 0 4.925 185 \(\times10^{-4}\) 0.863
WBS 80 4 16 0 0 4.981 404 \(\times10^{-4}\) 0.048
WBS2 46 8 46 0 0 4.925 240 \(\times10^{-4}\) 0.726

For the signal (M11), since there is no noise, we do not need to repeat our simulations 100 times; the input will be the same each time. NOT, WBS, and WBS2 detect only one change-point at location 550, while TGUH detects two change-points at locations 550 and 750. We proceed with the results for (M12), which are given in Table 31 below.

Table 31: The average \(\hat{N}\), MSE, and computational times over 100 simulated data sequences from the linear signal (M12).
Method \(\hat{N}\) MSE Time (s)
NOT 56.42 23.797 1.076
TGUH 55.71 7.777 0.146
WBS 45.62 12.692 0.181
WBS2 58.01 7.191 0.158

For the noiseless (M13), NOT, TGUH, WBS, and WBS2 detect 39, 51, 52, 63 change-points, respectively. Table 32 below gives a summary of the results for all signals used in the simulation study.

Table 32: Summary of the results when the localised pruning model selection was employed for all the solution path algorithms.
Methods
NOT TGUH WBS WBS2
Models Criterion: Within top \(10\%\) in terms of accuracy
(NC1) \(\checkmark\)
(NC2) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(NC3) \(\checkmark\)
(NC4) \(\checkmark\)
(M1) \(\checkmark\)
(M2) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M3) \(\checkmark\) \(\checkmark\)
(M4) \(\checkmark\)
(M5) \(\checkmark\)
(M6) \(\checkmark\)
(M7) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M8) \(\checkmark\)
(M9) \(\checkmark\)
(M10) \(\checkmark\)
(M11) \(\checkmark\)
(M12) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M13) \(\checkmark\)
Criterion: The top two methods in terms of speed
(NC1) \(\checkmark\) \(\checkmark\)
(NC2) \(\checkmark\) \(\checkmark\)
(NC3) \(\checkmark\) \(\checkmark\)
(NC4) \(\checkmark\) \(\checkmark\)
(M1) \(\checkmark\) \(\checkmark\)
(M2) \(\checkmark\) \(\checkmark\)
(M3) \(\checkmark\) \(\checkmark\)
(M4) \(\checkmark\) \(\checkmark\)
(M5) \(\checkmark\) \(\checkmark\)
(M6) \(\checkmark\) \(\checkmark\)
(M7) \(\checkmark\) \(\checkmark\)
(M8) \(\checkmark\) \(\checkmark\) \(\checkmark\)
(M9) \(\checkmark\) \(\checkmark\)
(M10) \(\checkmark\) \(\checkmark\)
(M12) \(\checkmark\) \(\checkmark\)