# 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.

- In datasets in which frequent change-points are suspected, we recommend the SDLL model selection criterion, which works particularly well in conjunction with the WBS2 solution path.

- In datasets in which a moderate number of change-points is suspected, we recommend the IC model selection criterion, which works particularly well in conjunction with the IDetect solution path.

- The thresholding model selection criterion works particularly well in conjunction with the solution path of the sequential IDetect.

- The localised pruning model selection criterion works particularly well in conjunction with the TGUH solution path.

# 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.

- Currently supported solution path methods, with the names of the corresponding functions, are: IDetect (sol.idetect), Sequential IDetect (sol.idetect_seq), NOT (sol.not), TGUH (sol.tguh), WBS (sol.wbs), and WBS2 (sol.wbs2).

- Currently supported model selection methods, with the names of the corresponding functions, are: information criterion (model.ic), localised pruning (model.lp), steepest drop to low levels (model.sdll) and thresholding (model.thresh). In the following, we briefly describe each the above listed methods.

## 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 (2021), Metrika.

#### 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 (2021). Journal of Statistical Software, 97(8), 1–42.

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

- Extensions to serially correlated, light-tailed noise
- Robust methods for heavy-tailed noise which detect changes in the piecewise-constant median
- Data windowing for extremely long signals

# 3 Simulation study

The following simulation study shows the performance of all the solution path methods when combined with the model selection methods for various signals. The results presented in this section are based upon version 2.1 of the package, where for NOT, WBS, and WBS2 the intervals are randomly drawn. In the next version of the package we will also investigate the performance of the three aforementioned solution path methods for a deterministic grid.

## 3.1 Piecewise-constant signals

### 3.1.1 Simulated signals

The signals used in the simulation study are

- (NC1) constant signal : length 3000 with no change-points. The standard deviation is \(\sigma = 1\).

- (NC2) short constant signal: length 50 with no change-points. The standard deviation is \(\sigma = 1\).

- (NC3) shorter constant signal: length 5 with no change-points. The standard deviation is \(\sigma = 1\).

- (NC4) long constant signal: length 10000 with no change-points. The standard deviation is \(\sigma = 1\).

- (M1) blocks: length 2048 with change-points at 205, 267, 308, 472, 512, 820, 902, 1332, 1557, 1598, 1659 with values between change-points 0, 14.64, -3.66, 7.32, -7.32, 10.98, -4.39, 3.29, 19.03, 7.68, 15.37, 0. The standard deviation is \(\sigma = 10\).

- (M2) fms: length 497 with change-points at 139, 226, 243, 300, 309, 333 with values between change-points -0.18, 0.08, 1.07, -0.53, 0.16, -0.69, -0.16. The standard deviation of the noise is \(\sigma = 0.3\).

- (M3) mix: length 560 with change-points at 11, 21, 41, 61, 91, 121, 161, 201, 251, 301, 361, 421, 491 with values between change-points 7, -7, 6, -6, 5, -5, 4, -4, 3, -3, 2, -2, 1, -1. The standard deviation of the noise is \(\sigma = 4\).

- (M4) teeth: length 140 with change-points at 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131 with values between change-points 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1. The standard deviation of the noise is \(\sigma = 0.4\).

- (M5) stairs: length 150 with change-points at 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141 with values between change-points 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. The standard deviation of the noise is \(\sigma = 0.3\).

- (M6) long teeth: length 10000 with 249 change-points at 40,80,\(\ldots\),9960 with values between change-points 0,1.5,0,1.5,\(\ldots\),0,1.5. The standard deviation is \(\sigma = 1\).

- (M7) short signal: length 50 with 2 change-points at 15 and 30 with values between change-points 0,2,-0.5. The standard deviation is \(\sigma = 1\).

- (M8) extremely short signal: length 10 with 2 change-points at 3 and 7 with values between change-points 0, 3, 0. The standard deviation is \(\sigma = 1\).

- (M9) strong signal: length 2500 with 4 change-points at 100, 600, 1600, 2000 with values between change-points 0, 5, 0, 10, 2. The standard deviation is \(\sigma = 0.1\).

- (M10) one spike: length 2500 with 1 spike at 1001 with the value 100. This means that there are 2 change-points in the mean at 1000 and 1001. Before and after the spike the value of the signal is 0. The standard deviation is \(\sigma = 1\).

- (M11) no noise: length 1000 with 3 change-points at 350, 550, 750 with values between change-points 0, 5, 0, -4. There is no noise, so we set the standard deviation in our simulation study to be equal to \(\sigma = 0\).

- (M12) linear: linear signal of length 501 with no changes in the slope. The signal is 0,1,2,\(\ldots\), 500. The standard deviation is \(\sigma = 1\).

- (M13) linear no noise: The same linear signal as in (M13) but without noise, meaning that \(\sigma = 0\).

### 3.1.2 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. 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.

### 3.1.3 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.

### 3.1.4 Detailed results

#### 3.1.4.1 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) | \(\checkmark\) |  |  | \(\checkmark\) |

#### 3.1.4.2 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) | \(\checkmark\) |  |  | \(\checkmark\) |

#### 3.1.4.3 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 |  | 89 | 3 | 3 | 5 | 14 \(\times 10^{-5}\) | 0.10
TGUH | (NC1) | 97 | 2 | 1 | 0 | 4 \(\times 10^{-5}\) | 0.19
WBS |  | 94 | 4 | 1 | 1 | 6 \(\times 10^{-5}\) | 0.04
WBS2 |  | 93 | 3 | 4 | 0 | 7 \(\times 10^{-5}\) | 1.22
ID |  | 86 | 2 | 7 | 5 | 62 \(\times 10^{-3}\) | 0.002
TGUH | (NC2) | 96 | 3 | 0 | 1 | 24 \(\times 10^{-3}\) | 0.010
WBS |  | 85 | 3 | 4 | 8 | 79 \(\times 10^{-3}\) | 0.002
WBS2 |  | 89 | 1 | 4 | 6 | 68 \(\times 10^{-3}\) | 0.014
ID |  | 84 | 11 | 5 | 0 | 240 \(\times 10^{-3}\) | 0.001
TGUH | (NC3) | 85 | 10 | 1 | 4 | 239 \(\times 10^{-3}\) | 0.001
WBS |  | 84 | 8 | 3 | 5 | 252 \(\times 10^{-3}\) | 0.001
WBS2 |  | 84 | 8 | 3 | 5 | 252 \(\times 10^{-3}\) | 0.001
ID |  | 94 | 3 | 2 | 1 | 19 \(\times 10^{-5}\) | 0.62
TGUH | (NC4) | 99 | 1 | 0 | 0 | 11 \(\times 10^{-5}\) | 0.46
WBS |  | 98 | 2 | 0 | 0 | 11 \(\times 10^{-5}\) | 0.14
WBS2 |  | 94 | 5 | 1 | 0 | 12 \(\times 10^{-5}\) | 4.04

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 |  | 0 | 5 | 56 | 29 | 6 | 3 | 1 | 2.992 | 223 \(\times10^{-4}\) | 0.03
TGUH | (M1) | 3 | 12 | 49 | 33 | 3 | 0 | 0 | 3.716 | 244 \(\times10^{-4}\) | 0.13
WBS |  | 1 | 11 | 64 | 17 | 2 | 3 | 2 | 2.997 | 223 \(\times10^{-4}\) | 0.03
WBS2 |  | 0 | 3 | 54 | 33 | 8 | 0 | 2 | 2.786 | 196 \(\times10^{-4}\) | 0.76
ID |  | 0 | 0 | 0 | 91 | 3 | 4 | 2 | 43 \(\times 10^{-4}\) | 23 \(\times 10^{-3}\) | 0.010
TGUH | (M2) | 0 | 2 | 63 | 29 | 5 | 0 | 1 | 99 \(\times 10^{-4}\) | 36 \(\times 10^{-3}\) | 0.059
WBS |  | 0 | 0 | 5 | 83 | 7 | 2 | 3 | 43 \(\times 10^{-4}\) | 29 \(\times 10^{-3}\) | 0.012
WBS2 |  | 0 | 0 | 1 | 91 | 4 | 3 | 1 | 42 \(\times 10^{-4}\) | 18 \(\times 10^{-3}\) | 0.176
ID |  | 12 | 24 | 30 | 30 | 4 | 0 | 0 | 1.642 | 146 \(\times 10^{-3}\) | 0.013
TGUH | (M3) | 34 | 23 | 25 | 9 | 4 | 3 | 2 | 2.489 | 196 \(\times 10^{-3}\) | 0.060
WBS |  | 10 | 21 | 39 | 23 | 5 | 1 | 1 | 1.520 | 140 \(\times 10^{-3}\) | 0.010
WBS2 |  | 12 | 24 | 36 | 18 | 7 | 1 | 2 | 1.562 | 152 \(\times 10^{-3}\) | 0.198
ID |  | 2 | 4 | 4 | 77 | 6 | 4 | 3 | 61 \(\times10^{-3}\) | 30 \(\times 10^{-3}\) | 0.009
TGUH | (M4) | 29 | 12 | 16 | 20 | 10 | 4 | 9 | 110 \(\times10^{-3}\) | 101 \(\times 10^{-3}\) | 0.025
WBS |  | 0 | 3 | 8 | 76 | 9 | 2 | 2 | 57 \(\times10^{-3}\) | 27 \(\times 10^{-3}\) | 0.006
WBS2 |  | 0 | 3 | 9 | 75 | 9 | 2 | 2 | 57 \(\times10^{-3}\) | 27 \(\times 10^{-3}\) | 0.041
ID |  | 0 | 0 | 2 | 97 | 1 | 0 | 0 | 23 \(\times10^{-3}\) | 10 \(\times 10^{-3}\) | 0.008
TGUH | (M5) | 0 | 0 | 2 | 95 | 3 | 0 | 0 | 26 \(\times10^{-3}\) | 11 \(\times 10^{-3}\) | 0.023
WBS |  | 0 | 0 | 2 | 82 | 14 | 1 | 1 | 26 \(\times10^{-3}\) | 14 \(\times 10^{-3}\) | 0.007
WBS2 |  | 0 | 0 | 2 | 82 | 15 | 0 | 1 | 26 \(\times10^{-3}\) | 13 \(\times 10^{-3}\) | 0.040

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 | 0 | 0 | 100 | 0 | 0 | 0.113 | 36 \(\times10^{-4}\) | 0.56
TGUH | 0 | 31 | 58 | 7 | 4 | 0 | 0.270 | 56 \(\times10^{-4}\) | 0.43
WBS | 1 | 99 | 0 | 0 | 0 | 0 | 0.509 | 64 \(\times10^{-4}\) | 0.14
WBS2 | 0 | 0 | 1 | 99 | 0 | 0 | 0.109 | 34 \(\times10^{-4}\) | 3.87

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 |  | 0 | 4 | 89 | 4 | 3 | 0.232 | 55 \(\times10^{-3}\) | 0.003
TGUH | (M7) | 7 | 2 | 85 | 5 | 1 | 0.281 | 79 \(\times10^{-3}\) | 0.012
WBS |  | 0 | 4 | 86 | 5 | 5 | 0.231 | 53 \(\times10^{-3}\) | 0.003
WBS2 |  | 0 | 3 | 86 | 5 | 6 | 0.230 | 53 \(\times10^{-3}\) | 0.015
ID |  | 65 | 2 | 31 | 2 | 0 | 1.680 | 474 \(\times10^{-3}\) | 0.003
TGUH | (M8) | 72 | 2 | 24 | 2 | 0 | 1.812 | 522 \(\times10^{-3}\) | 0.002
WBS |  | 65 | 1 | 31 | 3 | 0 | 1.667 | 470 \(\times10^{-3}\) | 0.001
WBS2 |  | 65 | 1 | 31 | 3 | 0 | 1.667 | 470 \(\times10^{-3}\) | 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 | 91 | 7 | 1 | 1 | 25 \(\times 10^{-6}\) | 8 \(\times10^{-3}\) | 0.043
TGUH | 0 | 0 | 0 | 97 | 2 | 1 | 0 | 21 \(\times 10^{-6}\) | 3 \(\times10^{-3}\) | 0.163
WBS | 0 | 0 | 0 | 93 | 4 | 2 | 1 | 26 \(\times 10^{-6}\) | 6 \(\times10^{-3}\) | 0.038
WBS2 | 0 | 0 | 0 | 90 | 4 | 4 | 2 | 25 \(\times 10^{-6}\) | 11 \(\times10^{-3}\) | 1.008

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 | 87 | 6 | 7 | 0.003 | 37 \(\times 10^{-3}\) | 0.039
TGUH | 0 | 96 | 3 | 1 | 0 | 4.946 | 12 \(\times10^{-3}\) | 0.145
WBS | 0 | 92 | 7 | 1 | 0 | 4.996 | 20 \(\times10^{-3}\) | 0.033
WBS2 | 0 | 0 | 94 | 4 | 2 | 0.002 | 17 \(\times10^{-3}\) | 0.814

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 | 129.97 | 1.890 | 0.116
TGUH | 119.72 | 2.039 | 0.062
WBS | 98.11 | 3.314 | 0.012
WBS2 | 122.13 | 1.928 | 0.165

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) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\)
(NC2) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\)
(NC3) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\)
(NC4) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\)
(M1) |  | \(\checkmark\) |  | \(\checkmark\)
(M2) | \(\checkmark\) |  | \(\checkmark\) | \(\checkmark\)
(M3) | \(\checkmark\) |  |  |
(M4) | \(\checkmark\) |  | \(\checkmark\) | \(\checkmark\)
(M5) | \(\checkmark\) | \(\checkmark\) |  |
(M6) | \(\checkmark\) |  |  | \(\checkmark\)
(M7) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\)
(M8) | \(\checkmark\) |  | \(\checkmark\) | \(\checkmark\)
(M9) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\)
(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) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | \(\checkmark\)
(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) |  | \(\checkmark\) | \(\checkmark\) |

#### 3.1.4.4 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) |  | \(\checkmark\) |  | \(\checkmark\)

## 3.2 Continuous, piecewise-linear signals

### 3.2.1 Simulated signals

The signals used in the simulation study are

- (NC1) No change-points: length 300 with slope equal to 1 and no change-points. The standard deviation is \(\sigma = 1\).

- (NC2) short signal, no change-points: length 30 with slope equal to 1 and no change-points. The standard deviation is \(\sigma = 1\).

- (NC3) long signal, no change-points: length 3000 with slope equal to 1 and no change-points. The standard deviation is \(\sigma = 1\).

- (M1) one change-point: length 200 with one change-point in the middle were the slope goes from \(0.5\) to \(0\). The standard deviation is \(\sigma = 1\).

- (M2) Trapezoid: length 300 with 2 change-points at 100 and 200 with the slope taking the values \(0.5, 0, -0.5\) in the three segments created. The standard deviation is \(\sigma = 1\).

- (M3) Trapezoid with larger variance: length 300 with 2 change-points at 100 and 200 with the slope taking the values \(0.5, 0, -0.5\) in the three segments created. The standard deviation is \(\sigma = 10\).

- (M4) trapezoid repetitive: length 1500 with 14 change-points at 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400 with the values of the slope between change-points being equal to 0.25, 0, -0.25, 0.25, 0, -0.25, 0.25, 0, -0.25, 0.25, 0, -0.25, 0.25, 0, -0.25. The standard deviation of the noise is \(\sigma = 6\).

- (M5) gradual changes: length 240 with 5 change-points at 40, 80, 120, 160, 200 with slope values between change-points equal to 0.05, 0.1, 0.2, 0.4, 0.8, 1.6. The standard deviation of the noise is \(\sigma = 0.3\).

- (M6) weak, long signal: length 3000 with 2 change-point at 1000 and 2000 with slope values between change-points equal to 1, 0.95, 1. The standard deviation of the noise is \(\sigma = 2\).

### 3.2.2 Simulation process

We ran 100 replications for each one of the abovementioned signals and the frequency distribution of \(\hat{N} - N\) for the ID and NOT solution path methods (combined with a model selection algorithm) is presented. 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) - (NC3), which are signals without any change-points. The average computational time for all methods is also provided.

### 3.2.3 Code for running the simulations

    library(breakfast)

    ## For thresholding
    sim_thr_lin <- function(x, const = 1.4) {
      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("not")
      notth <- new("cpt.est")
      z <- model.thresh(sol.not(x, type = "lin.cont"), 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, type = "lin.cont"), th.const = const))[[3]]

      print("idetect")
      idetectth <- new("cpt.est")
      z <- model.thresh(sol.idetect_seq(x, type = "lin.cont"), 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, type = "lin.cont"), th.const = const))[[3]]

      list(notth = notth, idetectth = idetectth)
    }

    sim.study.thr_lin <- function(signal, true.cpt=NULL,sigma, m = 100, seed = NULL, gen_const = 1.4) {

      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)))

      notth <- new("est.eval")
      idetectth <- new("est.eval")

      no.of.cpt <- length(true.cpt)
      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_lin(x, const = gen_const)

        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

        gc()
      }

      list(notth=notth, idetectth = idetectth)
    }

    ## The function used for the comparisons in the case of 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("not")
      notic <- new("cpt.est")
      z <- model.ic(sol.not(x, type = "lin.cont"), q.max = par.max)
      if(z$cpts == 0){notic@cpt = 0
      notic@nocpt = 0}
      else{notic@cpt <- z$cpts
      notic@nocpt <- z$no.of.cpt}
      notic@mean <- z$est
      notic@time <- system.time(model.ic(sol.not(x, type = "lin.cont"), q.max = par.max))[[3]]

      print("idetect")
      idetectic <- new("cpt.est")
      z <- model.ic(sol.idetect(x, type = "lin.cont"), q.max = par.max)
      if(z$cpts == 0){idetectic@cpt = 0
      idetectic@nocpt = 0}
      else{idetectic@cpt <- z$cpts
      idetectic@nocpt <- z$no.of.cpt}
      idetectic@mean <- z$est
      idetectic@time <- system.time(model.ic(sol.idetect(x, type = "lin.cont"), q.max = par.max))[[3]]

      list(notic = notic, idetectic = idetectic)
    }

    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)))

      notic <- new("est.eval")
      idetectic <- new("est.eval")

      no.of.cpt <- length(true.cpt)
      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)

        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

      }

      list(notic=notic, idetectic = idetectic)
    }

    ## The function used for the comparisons in the case of 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("idetect")
      idetectsdll <- new("cpt.est")
      z <- model.sdll(sol.idetect(x, type = "lin.cont", thr_ic_lin = 1))
      if(length(z$cpts) == 0){idetectsdll@cpt = 0
      idetectsdll@nocpt = 0}
      else{idetectsdll@cpt <- z$cpts
      idetectsdll@nocpt <- z$no.of.cpt}
      idetectsdll@mean <- z$est
      idetectsdll@time <- system.time(model.sdll(sol.idetect(x, type = "lin.cont", thr_ic_lin = 1)))[[3]]

      list(idetectsdll = idetectsdll)
    }

    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)))

      idetectsdll <- new("est.eval")

      no.of.cpt <- length(true.cpt)
      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)

        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

      }

      list(idetectsdll = idetectsdll)
    }

    ## Signals for simulation study

    seed.temp <- 12
    no_cpt <- seq(1,300,1)

    lin.SIMR0_thr <- sim.study.thr_lin(no_cpt, sigma = 1, seed = seed.temp, true.cpt = integer(0))
    lin.SIMR0_ic <- sim.study.ic(no_cpt, sigma = 1, seed = seed.temp, true.cpt = integer(0))
    lin.SIMR0_sdll <- sim.study.sdll(no_cpt, sigma = 1, seed = seed.temp, true.cpt = integer(0))

    ####
    no_cpt_2 <- seq(1,30,1)

    lin.SIMR0_2_thr <- sim.study.thr_lin(no_cpt_2, sigma = 1, seed = seed.temp, true.cpt = integer(0))
    lin.SIMR0_2_ic <- sim.study.ic(no_cpt_2, sigma = 1, seed = seed.temp, true.cpt = integer(0))
    lin.SIMR0_2_sdll <- sim.study.sdll(no_cpt_2, sigma = 1, seed = seed.temp, true.cpt = integer(0))

    ####

    no_cpt_3 <- seq(1,3000,1)

    lin.SIMR0_3_thr <- sim.study.thr_lin(no_cpt_3, sigma = 1, seed = seed.temp, true.cpt = integer(0))
    lin.SIMR0_3_ic <- sim.study.ic(no_cpt_3, sigma = 1, seed = seed.temp, true.cpt = integer(0))
    lin.SIMR0_3_sdll <- sim.study.sdll(no_cpt_3, sigma = 1, seed = seed.temp, true.cpt = integer(0))

    ####

    one_cpt <- c(seq(0,49.5,0.5), rep(49.5,100))
    lin.SIMR1_thr <- sim.study.thr_lin(one_cpt, sigma = 1, seed = seed.temp, true.cpt = c(100))
    lin.SIMR1_ic <- sim.study.ic(one_cpt, sigma = 1, seed = seed.temp, true.cpt = c(100))
    lin.SIMR1_sdll <- sim.study.sdll(one_cpt, sigma = 1, seed = seed.temp, true.cpt = c(100))

    ####

    trapezoid <- c(seq(0,49.5,0.5), rep(49.5, 100), seq(49,-0.5,-0.5))
    plot.ts(trapezoid + rnorm(300, sd = 10))
    lin.SIMR2_thr <- sim.study.thr_lin(trapezoid, sigma = 1, seed = seed.temp, true.cpt = c(100,200))
    lin.SIMR2_ic <- sim.study.ic(trapezoid, sigma = 1, seed = seed.temp, true.cpt = c(100,200))
    lin.SIMR2_sdll <- sim.study.sdll(trapezoid, sigma = 1, seed = seed.temp, true.cpt = c(100,200))

    lin.SIMR2_10_thr <- sim.study.thr_lin(trapezoid, sigma = 10, seed = seed.temp, true.cpt = c(100,200))
    lin.SIMR2_10_ic <- sim.study.ic(trapezoid, sigma = 10, seed = seed.temp, true.cpt = c(100,200))
    lin.SIMR2_10_sdll <- sim.study.sdll(trapezoid, sigma = 10, seed = seed.temp, true.cpt = c(100,200))

    ####
    trapezoid2 <- c(seq(0,24.75,0.25), rep(24.75, 100), seq(24.5,-0.25,-0.25))
    small.trapezoid.teeth <- rep(trapezoid2, 5)
    lin.SIMR3_thr <- sim.study.thr_lin(small.trapezoid.teeth, sigma = 6, seed = seed.temp, true.cpt = seq(100,1400,100))
    lin.SIMR3_ic <- sim.study.ic(small.trapezoid.teeth, sigma = 6, seed = seed.temp, true.cpt = seq(100,1400,100))
    lin.SIMR3_sdll <- sim.study.sdll(small.trapezoid.teeth, sigma = 6, seed = seed.temp, true.cpt = seq(100,1400,100))

    ####
    five_cpts_difficult <- c(seq(0,1.95,0.05), seq(2.05,5.95,0.1), seq(6.15,13.95,0.2), seq(14.35, 29.95, 0.4), seq(30.75, 61.95,0.8), seq(63.55, 125.95, 1.6))
    lin.SIMR4_thr <- sim.study.thr_lin(five_cpts_difficult, sigma = 0.3, seed = seed.temp, true.cpt = seq(40,200,40))
    lin.SIMR4_ic <- sim.study.ic(five_cpts_difficult, sigma = 0.3, seed = seed.temp, true.cpt = seq(40,200,40))
    lin.SIMR4_sdll <- sim.study.sdll(five_cpts_difficult, sigma = 0.3, seed = seed.temp, true.cpt = seq(40,200,40))

    ####
    weaker_signal_1 <- c(seq(1,1000,1), seq(1000.95,1950,0.95), seq(1951,2950,1))
    lin.SIMR0_8_thr <- sim.study.thr_lin(weaker_signal_1, sigma = 2, seed = seed.temp, true.cpt = c(1000,2000))
    lin.SIMR0_8_ic <- sim.study.ic(weaker_signal_1, sigma = 2, seed = seed.temp, true.cpt = c(1000,2000))
    lin.SIMR0_8_sdll <- sim.study.sdll(weaker_signal_1, sigma = 2, seed = seed.temp, true.cpt = c(1000,2000))

### 3.2.4 Detailed results

#### 3.2.4.1 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 33 and 34 below. 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 33: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1)-(NC3) of the continuous piecewise-linear case. 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 | (NC1) | 100 | 0 | 0 | 0 | 6 \(\times 10^{-3}\) | 0.011
NOT |  | 99 | 1 | 0 | 0 | 6 \(\times 10^{-3}\) | 0.043
ID | (NC2) | 98 | 1 | 1 | 0 | 50 \(\times 10^{-3}\) | 0.003
NOT |  | 95 | 4 | 1 | 0 | 55 \(\times 10^{-3}\) | 0.017
ID | (NC3) | 100 | 0 | 0 | 0 | 1 \(\times 10^{-3}\) | 0.382
NOT |  | 100 | 0 | 0 | 0 | 1 \(\times 10^{-3}\) | 0.178

Table 34: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the continuous, piecewise-linear signals (M1)-(M6). 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 | (M1) | 0 | 0 | 0 | 97 | 3 | 0 | 0 | 0.062 | 9 \(\times10^{-3}\) | 0.011
NOT |  | 0 | 0 | 0 | 97 | 3 | 0 | 0 | 0.020 | 7 \(\times10^{-3}\) | 0.054
ID | (M2) | 0 | 0 | 0 | 94 | 6 | 0 | 0 | 94 \(\times 10^{-3}\) | 11 \(\times 10^{-3}\) | 0.013
NOT |  | 0 | 0 | 0 | 100 | 0 | 0 | 0 | 21 \(\times 10^{-3}\) | 2 \(\times 10^{-3}\) | 0.137
ID | (M3) | 0 | 0 | 0 | 97 | 3 | 0 | 0 | 3.639 | 48 \(\times 10^{-3}\) | 0.012
NOT |  | 0 | 0 | 0 | 100 | 0 | 0 | 0 | 2.086 | 25 \(\times 10^{-3}\) | 0.077
ID | (M4) | 0 | 1 | 7 | 89 | 3 | 0 | 0 | 1.642 | 20 \(\times 10^{-3}\) | 0.044
NOT |  | 0 | 0 | 11 | 85 | 3 | 1 | 0 | 1.270 | 18 \(\times 10^{-3}\) | 0.368
ID | (M5) | 0 | 0 | 0 | 92 | 8 | 0 | 0 | 11 \(\times10^{-3}\) | 21 \(\times 10^{-3}\) | 0.014
NOT |  | 0 | 0 | 5 | 87 | 7 | 1 | 0 | 10 \(\times10^{-3}\) | 23 \(\times 10^{-3}\) | 0.129
ID | (M6) | 0 | 0 | 0 | 96 | 4 | 0 | 0 | 45 \(\times10^{-3}\) | 6 \(\times 10^{-3}\) | 0.124
NOT |  | 0 | 0 | 0 | 100 | 0 | 0 | 0 | 8 \(\times10^{-3}\) | 1 \(\times 10^{-3}\) | 0.435

#### 3.2.4.2 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 35 and 36 below. 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 35: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1)-(NC3) of the continuous piecewise-linear case. 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 | (NC1) | 99 | 1 | 0 | 0 | 6 \(\times 10^{-3}\) | 0.018
NOT |  | 95 | 5 | 0 | 0 | 6 \(\times 10^{-3}\) | 0.023
ID | (NC2) | 91 | 6 | 3 | 0 | 53 \(\times 10^{-3}\) | 0.004
NOT |  | 81 | 11 | 6 | 2 | 59 \(\times 10^{-3}\) | 0.003
ID | (NC3) | 100 | 0 | 0 | 0 | 1 \(\times 10^{-3}\) | 0.529
NOT |  | 100 | 0 | 0 | 0 | 1 \(\times 10^{-3}\) | 0.095

Table 36: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the continuous, piecewise-linear signals (M1)-(M6). 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 | (M1) | 0 | 0 | 0 | 95 | 5 | 0 | 0 | 0.066 | 12 \(\times10^{-3}\) | 0.003
NOT |  | 0 | 0 | 3 | 83 | 13 | 0 | 1 | 1.649 | 44 \(\times10^{-3}\) | 0.017
ID | (M2) | 0 | 0 | 0 | 96 | 4 | 0 | 0 | 84 \(\times 10^{-3}\) | 9 \(\times 10^{-3}\) | 0.002
NOT |  | 0 | 0 | 0 | 94 | 6 | 0 | 0 | 136\(\times 10^{-3}\) | 14 \(\times 10^{-3}\) | 0.027
ID | (M3) | 0 | 0 | 0 | 98 | 2 | 0 | 0 | 3.355 | 43 \(\times 10^{-3}\) | 0.005
NOT |  | 0 | 0 | 0 | 92 | 8 | 0 | 0 | 4.471 | 59 \(\times 10^{-3}\) | 0.021
ID | (M4) | 0 | 0 | 7 | 92 | 1 | 0 | 0 | 1.459 | 19 \(\times 10^{-3}\) | 0.019
NOT |  | 0 | 3 | 24 | 73 | 0 | 0 | 0 | 2.042 | 26 \(\times 10^{-3}\) | 0.061
ID | (M5) | 0 | 0 | 0 | 97 | 3 | 0 | 0 | 12 \(\times10^{-3}\) | 18 \(\times 10^{-3}\) | 0.004
NOT |  | 0 | 0 | 1 | 94 | 4 | 1 | 0 | 23 \(\times10^{-3}\) | 27 \(\times 10^{-3}\) | 0.030
ID | (M6) | 0 | 0 | 0 | 98 | 2 | 0 | 0 | 41 \(\times10^{-3}\) | 5 \(\times 10^{-3}\) | 0.076
NOT |  | 0 | 0 | 0 | 98 | 2 | 0 | 0 | 55 \(\times10^{-3}\) | 6 \(\times 10^{-3}\) | 0.105

#### 3.2.4.3 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 37 and 38. 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 37: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1)-(NC3) of the continuous piecewise-linear case. Also the average MSE and computational times are given.
|  | \(\hat{N} - N\) |  |  |  |  |
--- | --- | --- | --- | --- | --- | --- | ---
Method | Model | \(0\) | \(1\) | \(2\) | \(\geq 3\) | MSE | Time (s)
ID | (NC1) | 95 | 3 | 0 | 2 | 6 \(\times 10^{-3}\) | 0.006
ID | (NC2) | 87 | 6 | 2 | 5 | 67 \(\times 10^{-3}\) | 0.003
ID | (NC3) | 94 | 5 | 1 | 0 | 1 \(\times 10^{-3}\) | 0.307

Table 38: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the continuous, piecewise-linear signals (M1)-(M6). 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 | (M1) | 0 | 0 | 0 | 94 | 3 | 1 | 2 | 87 \(\times 10^{-3}\) | 23 \(\times10^{-3}\) | 0.004
ID | (M2) | 0 | 0 | 0 | 93 | 5 | 1 | 1 | 103\(\times 10^{-3}\) | 19 \(\times 10^{-3}\) | 0.004
ID | (M3) | 0 | 0 | 0 | 94 | 2 | 3 | 1 | 3.813 | 55 \(\times 10^{-3}\) | 0.005
ID | (M4) | 0 | 0 | 9 | 88 | 2 | 1 | 0 | 1.890 | 22 \(\times 10^{-3}\) | 0.025
ID | (M5) | 0 | 0 | 0 | 91 | 7 | 0 | 2 | 13 \(\times10^{-3}\) | 26 \(\times 10^{-3}\) | 0.006
ID | (M6) | 0 | 0 | 0 | 93 | 5 | 2 | 0 | 71 \(\times10^{-3}\) | 14 \(\times 10^{-3}\) | 0.100

## 3.3 Discontinuous, piecewise-linear signals

### 3.3.1 Simulated signals

The signals used in the simulation study are

- (NC1) No change-points: length 300 with slope equal to 1 and no change-points. The standard deviation is \(\sigma = 1\).

- (M1) one change-point: length 200 with one change-point in the middle were the slope goes from \(0.5\) to \(0\) and there is also a jump of magnitude 10.5. The standard deviation is \(\sigma = 1\).

- (M2) Trapezoid: length 250 with 2 change-points at 50 and 200 with the slope taking the values \(1, 0, -1\) in the three segments created. At both change-points we also introduce a jump of magnitude 50. The standard deviation is \(\sigma = 1\).

- (M3) Trapezoid with larger variance: length 250 with 2 change-points at 50 and 200 with the slope taking the values \(1, 0, -1\) in the three segments created. At both change-points we also introduce a jump of magnitude 50. The standard deviation is \(\sigma = 15\).

- (M4) trapezoid repetitive: length 1250 with 14 change-points at 50, 200, 250, 300, 450, 500, 550, 700, 750, 800, 950, 1000, 1050, 1200 with the values of the slope between change-points being equal to 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1. At 50, 200, 300, 450, 550, 700, 800, 950, 1050, and 1200 there are also jumps of magnitudes equal to 50. The standard deviation of the noise is \(\sigma = 15\).

### 3.3.2 Simulation process

We ran 100 replications for each one of the abovementioned signals and the frequency distribution of \(\hat{N} - N\) for the ID and NOT solution path methods (combined with a model selection algorithm) is presented. 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 signal (NC1), which does not have any change-points. The average computational time for both methods is also provided.

### 3.3.3 Code for running the simulations

    ## For thresholding
    sim_thr_lin_dis <- function(x, const = 1.4) {
      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("not")
      notth <- new("cpt.est")
      z <- model.thresh(sol.not(x, type = "lin.discont"), 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, type = "lin.discont"), th.const = const))[[3]]

      print("idetect")
      idetectth <- new("cpt.est")
      z <- model.thresh(sol.idetect_seq(x, type = "lin.discont"), 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, type = "lin.discont"), th.const = const))[[3]]

      list(notth = notth, idetectth = idetectth)
    }

    sim.study.thr_lin_dis <- function(signal, true.cpt=NULL,sigma, m = 100, seed = NULL, gen_const = 1.4) {

      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)))

      notth <- new("est.eval")
      idetectth <- new("est.eval")

      no.of.cpt <- length(true.cpt)
      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_lin_dis(x, const = gen_const)

        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

        gc()
      }

      list(notth=notth, idetectth = idetectth)
    }

    ## The function used for the comparisons in the case of information criterion
    sim_ic_dis <- 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("not")
      notic <- new("cpt.est")
      z <- model.ic(sol.not(x, type = "lin.discont"), q.max = par.max)
      if(z$cpts == 0){notic@cpt = 0
      notic@nocpt = 0}
      else{notic@cpt <- z$cpts
      notic@nocpt <- z$no.of.cpt}
      notic@mean <- z$est
      notic@time <- system.time(model.ic(sol.not(x, type = "lin.discont"), q.max = par.max))[[3]]

      print("idetect")
      idetectic <- new("cpt.est")
      z <- model.ic(sol.idetect(x, type = "lin.discont"), q.max = par.max)
      if(z$cpts == 0){idetectic@cpt = 0
      idetectic@nocpt = 0}
      else{idetectic@cpt <- z$cpts
      idetectic@nocpt <- z$no.of.cpt}
      idetectic@mean <- z$est
      idetectic@time <- system.time(model.ic(sol.idetect(x, type = "lin.discont"), q.max = par.max))[[3]]

      list(notic = notic, idetectic = idetectic)
    }

    sim.study.ic_dis <- 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)))

      notic <- new("est.eval")
      idetectic <- new("est.eval")

      no.of.cpt <- length(true.cpt)
      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_dis(x, par.max = max.par)

        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

      }

      list(notic=notic, idetectic = idetectic)
    }

    ## The function used for the comparisons in the case of sdll
    sim_sdll_dis <- 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("idetect")
      idetectsdll <- new("cpt.est")
      z <- model.sdll(sol.idetect(x, type = "lin.discont", thr_ic_lin = 1))
      if(length(z$cpts) == 0){idetectsdll@cpt = 0
      idetectsdll@nocpt = 0}
      else{idetectsdll@cpt <- z$cpts
      idetectsdll@nocpt <- z$no.of.cpt}
      idetectsdll@mean <- z$est
      idetectsdll@time <- system.time(model.sdll(sol.idetect(x, type = "lin.discont", thr_ic_lin = 1)))[[3]]

      list(idetectsdll = idetectsdll)
    }

    sim.study.sdll_dis <- 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)))

      idetectsdll <- new("est.eval")

      no.of.cpt <- length(true.cpt)
      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_dis(x)

        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

      }

      list(idetectsdll = idetectsdll)
    }

    ## Signals for simulation study

    seed.temp <- 1
    no_cpt <- seq(1,300,1)

    lin.SIMR0_thr <- sim.study.thr_lin_dis(no_cpt, sigma = 1, seed = seed.temp, true.cpt = integer(0))
    lin.SIMR0_ic <- sim.study.ic_dis(no_cpt, sigma = 1, seed = seed.temp, true.cpt = integer(0))
    lin.SIMR0_sdll <- sim.study.sdll_dis(no_cpt, sigma = 1, seed = seed.temp, true.cpt = integer(0))

    ####

    one_cpt <- c(seq(0,49.5,0.5), rep(60,100))
    lin.SIMR1_thr <- sim.study.thr_lin_dis(one_cpt, sigma = 1, seed = seed.temp, true.cpt = c(100))
    lin.SIMR1_ic <- sim.study.ic_dis(one_cpt, sigma = 1, seed = seed.temp, true.cpt = c(100))
    lin.SIMR1_sdll <- sim.study.sdll_dis(one_cpt, sigma = 1, seed = seed.temp, true.cpt = c(100))

    ####

    trapezoid <- c(seq(0,49.5,0.5), rep(60, 100), seq(49,-0.5,-0.5))
    lin.SIMR2_thr <- sim.study.thr_lin_dis(trapezoid, sigma = 1, seed = seed.temp, true.cpt = c(100,200))
    lin.SIMR2_ic <- sim.study.ic_dis(trapezoid, sigma = 1, seed = seed.temp, true.cpt = c(100,200))
    lin.SIMR2_sdll <- sim.study.sdll_dis(trapezoid, sigma = 1, seed = seed.temp, true.cpt = c(100,200))

    lin.SIMR2_10_thr <- sim.study.thr_lin_dis(trapezoid, sigma = 10, seed = seed.temp, true.cpt = c(100,200))
    lin.SIMR2_10_ic <- sim.study.ic_dis(trapezoid, sigma = 10, seed = seed.temp, true.cpt = c(100,200))
    lin.SIMR2_10_sdll <- sim.study.sdll_dis(trapezoid, sigma = 10, seed = seed.temp, true.cpt = c(100,200))

    ####
    trapezoid2 <- c(seq(0,24.75,0.25), rep(35, 100), seq(24.75,0,-0.25))
    small.trapezoid.teeth <- rep(trapezoid2, 5) ## 14 change-points with length
    lin.SIMR3_thr <- sim.study.thr_lin_dis(small.trapezoid.teeth, sigma = 6, seed = seed.temp, true.cpt = seq(100,1400,100))
    lin.SIMR3_ic <- sim.study.ic_dis(small.trapezoid.teeth, sigma = 6, seed = seed.temp, true.cpt = seq(100,1400,100))
    lin.SIMR3_sdll <- sim.study.sdll_dis(small.trapezoid.teeth, sigma = 6, seed = seed.temp, true.cpt = seq(100,1400,100))

### 3.3.4 Detailed results

#### 3.3.4.1 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 39 and 40 below. 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 39: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1). 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 | (NC1) | 98 | 2 | 0 | 0 | 89 \(\times 10^{-3}\) | 0.176
NOT |  | 98 | 2 | 0 | 0 | 89 \(\times 10^{-3}\) | 0.043

Table 40: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the discontinuous, piecewise-linear signals (M1)-(M3). 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 | (M1) | 0 | 0 | 0 | 99 | 1 | 0 | 0 | 0.035 | 4 \(\times10^{-4}\) | 0.038
NOT |  | 0 | 0 | 0 | 99 | 1 | 0 | 0 | 0.030 | 4 \(\times10^{-4}\) | 0.050
ID | (M2) | 0 | 0 | 0 | 98 | 2 | 0 | 0 | 0.060 | 5 \(\times 10^{-3}\) | 0.053
NOT |  | 0 | 0 | 0 | 98 | 2 | 0 | 0 | 0.060 | 5 \(\times 10^{-3}\) | 0.073
ID | (M3) | 0 | 0 | 0 | 90 | 10 | 0 | 0 | 9.592 | 11 \(\times 10^{-3}\) | 0.048
NOT |  | 0 | 0 | 0 | 98 | 1 | 1 | 0 | 8.212 | 5 \(\times 10^{-3}\) | 0.058
ID | (M4) | 0 | 0 | 0 | 89 | 11 | 0 | 0 | 11.944 | 11 \(\times 10^{-3}\) | 0.313
NOT |  | 0 | 0 | 0 | 95 | 5 | 0 | 0 | 10.658 | 9 \(\times 10^{-3}\) | 0.200

#### 3.3.4.2 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 41 and 42 below. 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 41: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1). 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 | (NC1) | 96 | 2 | 2 | 0 | 93 \(\times 10^{-3}\) | 0.283
NOT |  | 94 | 2 | 3 | 1 | 94 \(\times 10^{-3}\) | 0.018

Table 42: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the discontinuous, piecewise-linear signals (M1)-(M3). The average MSE, \(d_H\) and computational times are also given.
|  |  |  |  | \(\hat{N} - N\) |  |  |  |  |  |
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
Method | Model | \(\leq -3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(\geq 3\) | MSE | \(d_H\) | Time (s)
ID | (M1) | 0 | 0 | 0 | 98 | 2 | 0 | 0 | 0.031 | 5 \(\times10^{-3}\) | 0.033
NOT |  | 0 | 0 | 0 | 79 | 21 | 0 | 0 | 0.044 | 28 \(\times10^{-3}\) | 0.019
ID | (M2) | 0 | 0 | 0 | 97 | 2 | 1 | 0 | 0.061 | 8 \(\times 10^{-3}\) | 0.046
NOT |  | 0 | 0 | 0 | 95 | 2 | 1 | 2 | 0.063 | 10 \(\times 10^{-3}\) | 0.027
ID | (M3) | 0 | 0 | 0 | 87 | 12 | 0 | 1 | 9.946 | 15 \(\times 10^{-3}\) | 0.049
NOT |  | 0 | 0 | 24 | 34 | 31 | 8 | 3 | 106.381 | 175 \(\times 10^{-3}\) | 0.022
ID | (M4) | 0 | 0 | 0 | 81 | 18 | 1 | 0 | 11.842 | 11 \(\times 10^{-3}\) | 0.272
NOT |  | 0 | 0 | 0 | 60 | 35 | 5 | 0 | 17.315 | 12 \(\times 10^{-3}\) | 0.065

#### 3.3.4.3 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 43 and 44. 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 43: Distribution of \(\hat{N} - N\) over 100 simulated data sequences from (NC1). Also the average MSE and computational times are given.
|  | \(\hat{N} - N\) |  |  |  |  |
--- | --- | --- | --- | --- | --- | --- | ---
Method | Model | \(0\) | \(1\) | \(2\) | \(\geq 3\) | MSE | Time (s)
ID | (NC1) | 91 | 0 | 4 | 5 | 0.104 | 0.132

Table 44: Distribution of \(\hat{N} - N\) over 100 simulated data sequences of the discontinuous, piecewise-linear signals (M1)-(M4). 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 | (M1) | 0 | 0 | 0 | 93 | 2 | 3 | 2 | 40 \(\times 10^{-3}\) | 17 \(\times10^{-3}\) | 0.032
ID | (M2) | 0 | 0 | 0 | 96 | 1 | 2 | 1 | 41 \(\times 10^{-3}\) | 9 \(\times 10^{-3}\) | 0.035
ID | (M3) | 0 | 0 | 0 | 90 | 5 | 2 | 3 | 11.802 | 17 \(\times 10^{-3}\) | 0.046
ID | (M4) | 0 | 0 | 0 | 87 | 11 | 0 | 2 | 12.112 | 11 \(\times 10^{-3}\) | 0.315