Vignette for R Breakfast Package

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.

Methodologies – a brief description

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.

Solution path methods

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.

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 λT, right- and left-expanding intervals are created as follows: the jth right-expanding interval is Rj = [1, jλT], while the jth left-expanding interval is Lj = [T − jλT + 1, T], where T is the length of a given data sequence. We collect these intervals in the ordered set SRL = {R1, L1, R2, L2, …, RK, LK} and firstly we identify the data-point (denoted by b̃1) of R1 with the largest CUSUM statistic. If this value exceeds a given threshold level, then the algorithm makes a new start from b̃1. If not, then the process tests the next interval in SRL. We stop when all the data sequence has been scanned.

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 r̂j, j = 1, 2, …, J, when the interval used is [r̂j − 1 + 1, r̂j + 1], where r̂0 = 0 and r̂J + 1 = T.

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.

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.

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.

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.

Model selection methods

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 klogα(n), where k is the number of change-points, n is the number of observations, and α is the parameter of sSIC. We then select the model with the largest value of the penalised log-likelihood.

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.

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).

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.

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

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.

Piecewise-constant signals

Simulated signals

The signals used in the simulation study are

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

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

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

  • (NC4) long constant signal: length 10000 with no change-points. The standard deviation is σ = 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 σ = 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 σ = 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 σ = 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 σ = 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 σ = 0.3.

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

  • (M12) linear: linear signal of length 501 with no changes in the slope. The signal is 0,1,2,…, 500. The standard deviation is σ = 1.

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

Simulation process

We ran 100 replications for each one of the abovementioned signals and the frequency distribution of N̂ − 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 f̂t is the ordinary least square approximation of ft between two successive change-points. The scaled Hausdorff distance, dH = ns−1max {maxjmink|rj − r̂k|, maxkminj|rj − r̂k|}, where ns 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.

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.

Detailed results

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 N̂ − N over 100 simulated data sequences from (NC1)-(NC4). Also the average MSE and computational times for each method are given.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID 99 1 0 0 49 ×10−5 0.12
NOT 100 0 0 0 43 ×10−5 0.08
TGUH (NC1) 100 0 0 0 43 ×10−5 0.21
WBS 99 1 0 0 49 ×10−5 0.07
WBS2 99 1 0 0 49 ×10−5 1.13
ID 88 7 5 0 45 ×10−3 0.005
NOT 88 5 5 2 52 ×10−3 0.010
TGUH (NC2) 89 4 5 2 48 ×10−3 0.017
WBS 88 5 5 2 52 ×10−3 0.009
WBS2 87 5 6 2 54 ×10−3 0.022
ID 60 30 10 0 373 ×10−3 0.002
NOT 0 0 0 100 903 ×10−3 0.002
TGUH (NC3) 0 0 0 100 903 ×10−3 0.003
WBS 0 0 0 100 903 ×10−3 0.003
WBS2 0 0 0 100 903 ×10−3 0.001
ID 99 0 1 0 14 ×10−5 0.84
NOT 99 0 1 0 14 ×10−5 0.17
TGUH (NC4) 100 0 0 0 10 ×10−5 0.54
WBS 100 0 0 0 10 ×10−5 0.15
WBS2 100 0 0 0 10 ×10−5 4.09
Table 2: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signals (M1)-(M5). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID 0 3 40 51 5 1 0 2.886 171 ×10−4 0.04
NOT 0 5 58 35 2 0 0 2.733 175 ×10−4 0.09
TGUH (M1) 0 5 54 36 4 1 0 3.356 212 ×10−4 0.16
WBS 0 4 49 47 0 0 0 2.680 145 ×10−4 0.06
WBS2 0 5 47 45 1 2 0 2.743 153 ×10−4 0.84
ID 0 2 0 89 7 2 0 45 ×10−4 20 ×10−3 0.013
NOT 0 2 0 95 3 0 0 40 ×10−4 17 ×10−3 0.049
TGUH (M2) 2 13 6 62 13 4 0 66 ×10−4 46 ×10−3 0.064
WBS 0 2 0 95 3 0 0 44 ×10−4 17 ×10−3 0.017
WBS2 0 1 0 96 3 0 0 42 ×10−4 15 ×10−3 0.181
ID 7 33 25 28 7 0 0 1.626 147 ×10−3 0.015
NOT 9 30 31 22 6 0 2 1.729 144 ×10−3 0.048
TGUH (M3) 12 29 32 16 4 2 5 1.985 181 ×10−3 0.059
WBS 11 24 31 29 4 0 1 1.670 141 ×10−3 0.034
WBS2 9 26 32 30 3 0 0 1.578 146 ×10−3 0.176
ID 3 5 2 66 21 2 1 61 ×10−3 44 ×10−3 0.011
NOT 6 10 0 69 12 3 0 64 ×10−3 43 ×10−3 0.021
TGUH (M4) 22 2 2 35 14 13 12 102 ×10−3 215 ×10−3 0.027
WBS 1 4 0 82 12 0 1 54 ×10−3 31 ×10−3 0.012
WBS2 2 5 0 75 17 0 1 58 ×10−3 40 ×10−3 0.039
ID 0 0 0 89 9 1 1 22 ×10−3 10 ×10−3 0.012
NOT 0 0 0 86 8 4 2 21 ×10−3 10 ×10−3 0.100
TGUH (M5) 0 0 0 74 19 5 2 24 ×10−3 13 ×10−3 0.031
WBS 0 0 0 60 34 3 3 24 ×10−3 14 ×10−3 0.032
WBS2 0 0 0 60 30 8 2 24 ×10−3 14 ×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 N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M6). The average MSE, dH and computational time are also given.
N̂ − N
Method  ≤ −150 (−150, −50] (−50, −10) [−10, 10] (10, 50]  > 50 MSE dH Time (s)
ID 0 0 0 100 0 0 0.110 29 ×10−4 0.31
NOT 13 87 0 0 0 0 0.483 167 ×10−4 0.13
TGUH 0 100 0 0 0 0 0.376 84 ×10−4 0.39
WBS 0 100 0 0 0 0 0.453 168 ×10−4 0.12
WBS2 0 0 0 100 0 0 0.108 34 ×10−4 2.97
Table 4: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signals (M7) and (M8). The average MSE, dH and computational time are also given.
N̂ − N
Method Model −2 −1 0 1  ≥ 2 MSE dH Time (s)
ID 0 0 78 19 3 0.203 47 ×10−3 0.008
NOT 0 0 79 15 6 0.209 50 ×10−3 0.010
TGUH (M7) 0 0 71 24 5 0.257 58 ×10−3 0.016
WBS 0 0 81 12 7 0.207 49 ×10−3 0.008
WBS2 0 0 80 14 6 0.200 47 ×10−3 0.019
ID 14 6 68 11 1 0.884 164 ×10−3 0.003
NOT 0 0 0 0 100 0.920 200 ×10−3 0.003
TGUH (M8) 0 0 0 0 100 0.920 200 ×10−3 0.005
WBS 0 0 0 0 100 0.920 200 ×10−3 0.003
WBS2 0 0 0 0 100 0.920 200 ×10−3 0.004
Table 5: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M9). The average MSE, dH and computational time are also given.
N̂ − N
Method  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID 0 0 0 98 2 0 0 23 ×10−6 29 ×10−3 0.038
NOT 0 0 0 98 2 0 0 23 ×10−6 26 ×10−3 0.071
TGUH 0 0 0 98 2 0 0 23 ×10−6 29 ×10−3 0.132
WBS 0 0 0 98 2 0 0 23 ×10−6 26 ×10−3 0.051
WBS2 0 0 0 98 2 0 0 23 ×10−6 26 ×10−3 0.712
Table 6: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M10). The average MSE, dH and computational time are also given.
N̂ − N
Method −2 −1 0 1  ≥ 2 MSE dH Time (s)
ID 0 0 100 0 0 0.001 0 0.039
NOT 1 0 98 1 0 0.051 66 ×10−4 0.062
TGUH 22 0 7 10 61 1.293 3981 ×10−4 0.114
WBS 1 0 97 2 0 0.052 115 ×10−4 0.050
WBS2 0 0 98 2 0 0.002 66 ×10−4 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 N̂, MSE, and computational times over 100 simulated data sequences from the linear signal (M12).
Method N̂ 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) ✓ ✓ ✓ ✓ ✓
(NC2) ✓ ✓ ✓ ✓ ✓
(NC3) ✓
(NC4) ✓ ✓ ✓ ✓ ✓
(M1) ✓ ✓
(M2) ✓ ✓ ✓ ✓
(M3) ✓ ✓ ✓
(M4) ✓ ✓
(M5) ✓ ✓
(M6) ✓ ✓
(M7) ✓ ✓ ✓ ✓
(M8) ✓
(M9) ✓ ✓ ✓ ✓ ✓
(M10) ✓ ✓ ✓ ✓
(M11) ✓ ✓ ✓ ✓ ✓
(M12) ✓ ✓
(M13) ✓ ✓
Criterion: The top two methods in terms of speed
(NC1) ✓ ✓
(NC2) ✓ ✓
(NC3) ✓ ✓ ✓
(NC4) ✓ ✓
(M1) ✓ ✓
(M2) ✓ ✓
(M3) ✓ ✓
(M4) ✓ ✓
(M5) ✓ ✓
(M6) ✓ ✓
(M7) ✓ ✓
(M8) ✓ ✓ ✓
(M9) ✓ ✓
(M10) ✓ ✓
(M12) ✓ ✓

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 N̂ − N over 100 simulated data sequences from (NC1)-(NC4). Also the average MSE and computational times for each method are given.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID 86 2 9 3 159 ×10−5 0.245
NOT 82 13 2 3 93 ×10−5 0.063
TGUH (NC1) 93 6 1 0 45 ×10−5 0.147
WBS 79 9 10 2 135 ×10−5 0.058
WBS2 79 17 2 2 91 ×10−5 0.934
ID 73 9 12 6 76 ×10−3 0.002
NOT 59 16 14 11 86 ×10−3 0.003
TGUH (NC2) 81 16 1 2 30 ×10−3 0.011
WBS 48 20 15 17 121 ×10−3 0.002
WBS2 59 15 13 13 106 ×10−3 0.017
ID 55 29 14 2 390 ×10−3 0.002
NOT 67 24 1 8 317 ×10−3 0.003
TGUH (NC3) 64 25 3 8 333 ×10−3 0.002
WBS 50 34 8 8 410 ×10−3 0.001
WBS2 50 34 8 8 410 ×10−3 0.001
ID 89 2 8 1 37 ×10−5 1.614
NOT 89 8 2 1 21 ×10−5 0.119
TGUH (NC4) 94 6 0 0 12 ×10−5 0.351
WBS 89 7 4 0 22 ×10−5 0.112
WBS2 88 10 2 0 15 ×10−5 2.911
Table 10: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signals (M1)-(M5). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID 0 2 56 40 2 0 0 2.807 19 ×10−3 0.025
NOT 0 15 62 22 1 0 0 3.217 28 ×10−3 0.063
TGUH (M1) 1 14 64 19 2 0 0 3.680 23 ×10−3 0.109
WBS 0 3 60 34 3 0 0 2.812 27 ×10−3 0.049
WBS2 0 3 49 38 6 4 0 2.914 26 ×10−3 0.582
ID 0 0 2 87 8 3 0 45 ×10−4 28 ×10−3 0.008
NOT 0 2 21 64 11 1 1 60 ×10−4 54 ×10−3 0.040
TGUH (M2) 0 1 61 36 2 0 0 96 ×10−4 32 ×10−3 0.056
WBS 0 0 2 75 17 4 2 47 ×10−4 50 ×10−3 0.029
WBS2 0 0 4 84 8 4 0 45 ×10−4 29 ×10−3 0.150
ID 1 20 42 31 6 0 0 1.617 120 ×10−3 0.009
NOT 5 20 44 26 5 0 0 1.909 123 ×10−3 0.042
TGUH (M3) 47 33 17 3 0 0 0 2.706 228 ×10−3 0.061
WBS 0 10 51 27 10 2 0 1.649 102 ×10−3 0.031
WBS2 2 17 41 27 12 1 0 1.548 119 ×10−3 0.168
ID 10 7 17 63 2 1 0 66 ×10−3 47 ×10−3 0.006
NOT 5 14 26 48 6 1 0 75 ×10−3 49 ×10−3 0.014
TGUH (M4) 84 10 3 3 0 0 0 168 ×10−3 129 ×10−3 0.025
WBS 3 4 21 65 6 1 0 60 ×10−3 37 ×10−3 0.005
WBS2 5 4 21 63 6 1 0 62 ×10−3 39 ×10−3 0.038
ID 0 0 0 97 2 1 0 22 ×10−3 9 ×10−3 0.006
NOT 0 0 34 59 6 0 1 32 ×10−3 28 ×10−3 0.054
TGUH (M5) 0 0 0 99 1 0 0 23 ×10−3 9 ×10−3 0.029
WBS 0 0 0 90 7 2 1 25 ×10−3 11 ×10−3 0.025
WBS2 0 0 0 84 15 1 0 25 ×10−3 13 ×10−3 0.042
Table 11: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M6). The average MSE, dH and computational time are also given.
N̂ − N
Method  ≤ −150 (−150, −50] (−50, −10) [−10, 10] (10, 50]  > 50 MSE dH Time (s)
ID 0 0 24 76 0 0 0.124 58 ×10−4 0.195
NOT 100 0 0 0 0 0 0.504 240 ×10−4 0.123
TGUH 4 96 0 0 0 0 0.480 166 ×10−4 0.361
WBS 66 34 0 0 0 0 0.486 230 ×10−4 0.114
WBS2 0 0 34 66 0 0 0.131 52 ×10−4 2.810
Table 12: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signals (M7) and (M8). The average MSE, dH and computational time are also given.
N̂ − N
Method Model −2 −1 0 1  ≥ 2 MSE dH Time (s)
ID 0 1 85 8 6 0.203 54 ×10−3 0.003
NOT 0 37 44 12 7 0.408 166 ×10−3 0.002
TGUH (M7) 0 2 87 9 2 0.247 62 ×10−3 0.009
WBS 0 0 71 18 11 0.215 75 ×10−3 0.001
WBS2 0 0 71 18 11 0.212 74 ×10−3 0.014
ID 22 11 61 4 2 1.082 231 ×10−3 0.002
NOT 24 41 27 6 2 1.509 354 ×10−3 0.001
TGUH (M8) 27 24 43 4 2 1.318 301 ×10−3 0.003
WBS 16 12 64 6 2 1.026 199 ×10−3 0.001
WBS2 16 12 64 6 2 1.026 199 ×10−3 0.002
Table 13: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M9). The average MSE, dH and computational time are also given.
N̂ − N
Method  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID 0 0 0 90 2 8 0 29 ×10−6 10 ×10−3 0.038
NOT 0 0 0 86 10 4 0 25 ×10−6 15 ×10−3 0.064
TGUH 0 0 0 94 6 0 0 21 ×10−6 7 ×10−3 0.125
WBS 0 0 0 76 15 9 0 31 ×10−6 26 ×10−3 0.048
WBS2 0 0 0 80 16 4 0 26 ×10−6 15 ×10−3 0.706
Table 14: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M10). The average MSE, dH and computational time are also given.
N̂ − N
Method −2 −1 0 1  ≥ 2 MSE dH Time (s)
ID 0 0 91 4 5 0.002 24 ×10−3 0.038
NOT 0 1 86 10 3 0.151 44 ×10−3 0.052
TGUH 0 95 4 1 0 4.996 17 ×10−3 0.110
WBS 0 1 79 13 7 0.052 58 ×10−3 0.045
WBS2 0 0 88 7 5 0.002 38 ×10−3 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 N̂, MSE, and computational times over 100 simulated data sequences from the linear signal (M12).
Method N̂ 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) ✓ ✓
(NC2) ✓ ✓
(NC3) ✓ ✓
(NC4) ✓ ✓ ✓ ✓ ✓
(M1) ✓ ✓
(M2) ✓ ✓
(M3) ✓ ✓ ✓
(M4) ✓ ✓ ✓
(M5) ✓ ✓ ✓
(M6) ✓
(M7) ✓ ✓
(M8) ✓ ✓ ✓
(M9) ✓ ✓ ✓
(M10) ✓ ✓ ✓
(M11) ✓ ✓ ✓ ✓ ✓
(M12) ✓ ✓ ✓
(M13) ✓ ✓ ✓ ✓ ✓
Criterion: The top two methods in terms of speed
(NC1) ✓ ✓
(NC2) ✓ ✓
(NC3) ✓ ✓
(NC4) ✓ ✓
(M1) ✓ ✓
(M2) ✓ ✓
(M3) ✓ ✓
(M4) ✓ ✓
(M5) ✓ ✓
(M6) ✓ ✓
(M7) ✓ ✓
(M8) ✓ ✓
(M9) ✓ ✓
(M10) ✓ ✓
(M12) ✓ ✓

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 N̂ − N over 100 simulated data sequences from (NC1)-(NC4). Also the average MSE and computational times for each method are given.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID 89 3 3 5 14 ×10−5 0.10
TGUH (NC1) 97 2 1 0 4 ×10−5 0.19
WBS 94 4 1 1 6 ×10−5 0.04
WBS2 93 3 4 0 7 ×10−5 1.22
ID 86 2 7 5 62 ×10−3 0.002
TGUH (NC2) 96 3 0 1 24 ×10−3 0.010
WBS 85 3 4 8 79 ×10−3 0.002
WBS2 89 1 4 6 68 ×10−3 0.014
ID 84 11 5 0 240 ×10−3 0.001
TGUH (NC3) 85 10 1 4 239 ×10−3 0.001
WBS 84 8 3 5 252 ×10−3 0.001
WBS2 84 8 3 5 252 ×10−3 0.001
ID 94 3 2 1 19 ×10−5 0.62
TGUH (NC4) 99 1 0 0 11 ×10−5 0.46
WBS 98 2 0 0 11 ×10−5 0.14
WBS2 94 5 1 0 12 ×10−5 4.04
Table 18: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signals (M1)-(M5). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID 0 5 56 29 6 3 1 2.992 223 ×10−4 0.03
TGUH (M1) 3 12 49 33 3 0 0 3.716 244 ×10−4 0.13
WBS 1 11 64 17 2 3 2 2.997 223 ×10−4 0.03
WBS2 0 3 54 33 8 0 2 2.786 196 ×10−4 0.76
ID 0 0 0 91 3 4 2 43 ×10−4 23 ×10−3 0.010
TGUH (M2) 0 2 63 29 5 0 1 99 ×10−4 36 ×10−3 0.059
WBS 0 0 5 83 7 2 3 43 ×10−4 29 ×10−3 0.012
WBS2 0 0 1 91 4 3 1 42 ×10−4 18 ×10−3 0.176
ID 12 24 30 30 4 0 0 1.642 146 ×10−3 0.013
TGUH (M3) 34 23 25 9 4 3 2 2.489 196 ×10−3 0.060
WBS 10 21 39 23 5 1 1 1.520 140 ×10−3 0.010
WBS2 12 24 36 18 7 1 2 1.562 152 ×10−3 0.198
ID 2 4 4 77 6 4 3 61 ×10−3 30 ×10−3 0.009
TGUH (M4) 29 12 16 20 10 4 9 110 ×10−3 101 ×10−3 0.025
WBS 0 3 8 76 9 2 2 57 ×10−3 27 ×10−3 0.006
WBS2 0 3 9 75 9 2 2 57 ×10−3 27 ×10−3 0.041
ID 0 0 2 97 1 0 0 23 ×10−3 10 ×10−3 0.008
TGUH (M5) 0 0 2 95 3 0 0 26 ×10−3 11 ×10−3 0.023
WBS 0 0 2 82 14 1 1 26 ×10−3 14 ×10−3 0.007
WBS2 0 0 2 82 15 0 1 26 ×10−3 13 ×10−3 0.040
Table 19: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M6). The average MSE, dH and computational time are also given.
N̂ − N
Method  ≤ −150 (−150, −50] (−50, −10) [−10, 10] (10, 50]  > 50 MSE dH Time (s)
ID 0 0 0 100 0 0 0.113 36 ×10−4 0.56
TGUH 0 31 58 7 4 0 0.270 56 ×10−4 0.43
WBS 1 99 0 0 0 0 0.509 64 ×10−4 0.14
WBS2 0 0 1 99 0 0 0.109 34 ×10−4 3.87
Table 20: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signals (M7) and (M8). The average MSE, dH and computational time are also given.
N̂ − N
Method Model −2 −1 0 1  ≥ 2 MSE dH Time (s)
ID 0 4 89 4 3 0.232 55 ×10−3 0.003
TGUH (M7) 7 2 85 5 1 0.281 79 ×10−3 0.012
WBS 0 4 86 5 5 0.231 53 ×10−3 0.003
WBS2 0 3 86 5 6 0.230 53 ×10−3 0.015
ID 65 2 31 2 0 1.680 474 ×10−3 0.003
TGUH (M8) 72 2 24 2 0 1.812 522 ×10−3 0.002
WBS 65 1 31 3 0 1.667 470 ×10−3 0.001
WBS2 65 1 31 3 0 1.667 470 ×10−3 0.002
Table 21: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M9). The average MSE, dH and computational time are also given.
N̂ − N
Method  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID 0 0 0 91 7 1 1 25 ×10−6 8 ×10−3 0.043
TGUH 0 0 0 97 2 1 0 21 ×10−6 3 ×10−3 0.163
WBS 0 0 0 93 4 2 1 26 ×10−6 6 ×10−3 0.038
WBS2 0 0 0 90 4 4 2 25 ×10−6 11 ×10−3 1.008
Table 22: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M10). The average MSE, dH and computational time are also given.
N̂ − N
Method −2 −1 0 1  ≥ 2 MSE dH Time (s)
ID 0 0 87 6 7 0.003 37 ×10−3 0.039
TGUH 0 96 3 1 0 4.946 12 ×10−3 0.145
WBS 0 92 7 1 0 4.996 20 ×10−3 0.033
WBS2 0 0 94 4 2 0.002 17 ×10−3 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 N̂, MSE, and computational times over 100 simulated data sequences from the linear signal (M12).
Method N̂ 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) ✓ ✓ ✓ ✓
(NC2) ✓ ✓ ✓ ✓
(NC3) ✓ ✓ ✓ ✓
(NC4) ✓ ✓ ✓ ✓
(M1) ✓ ✓
(M2) ✓ ✓ ✓
(M3) ✓
(M4) ✓ ✓ ✓
(M5) ✓ ✓
(M6) ✓ ✓
(M7) ✓ ✓ ✓ ✓
(M8) ✓ ✓ ✓
(M9) ✓ ✓ ✓ ✓
(M10) ✓ ✓
(M11) ✓ ✓ ✓ ✓
(M12) ✓ ✓ ✓
(M13) ✓ ✓ ✓ ✓
Criterion: The top two methods in terms of speed
(NC1) ✓ ✓
(NC2) ✓ ✓
(NC3) ✓ ✓ ✓ ✓
(NC4) ✓ ✓
(M1) ✓ ✓
(M2) ✓ ✓
(M3) ✓ ✓
(M4) ✓ ✓
(M5) ✓ ✓
(M6) ✓ ✓
(M7) ✓ ✓
(M8) ✓ ✓ ✓
(M9) ✓ ✓
(M10) ✓ ✓
(M12) ✓ ✓

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 N̂ − N over 100 simulated data sequences from (NC1)-(NC4). Also the average MSE and computational times for each method are given.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
NOT 37 19 17 27 373 ×10−5 0.392
TGUH (NC1) 88 9 2 1 78 ×10−5 4.511
WBS 45 20 20 15 332 ×10−5 0.243
WBS2 48 17 15 20 285 ×10−5 1.745
NOT 97 3 0 0 26 ×10−3 0.002
TGUH (NC2) 93 6 1 0 32 ×10−3 0.009
WBS 91 6 3 0 37 ×10−3 0.002
WBS2 91 5 4 0 38 ×10−3 0.011
NOT 99 1 0 0 174 ×10−3 0.002
TGUH (NC3) 67 33 0 0 325 ×10−3 0.010
WBS 85 15 0 0 249 ×10−3 0.002
WBS2 85 15 0 0 249 ×10−3 0.003
NOT 51 16 23 10 91 ×10−5 3.065
TGUH (NC4) 89 7 4 0 22 ×10−5 11.738
WBS 57 14 13 16 91 ×10−5 1.884
WBS2 36 25 20 19 99 ×10−5 13.128
Table 26: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signals (M1)-(M5). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
NOT 4 0 17 44 22 10 3 4.084 571 ×10−4 0.143
TGUH (M1) 2 2 15 56 17 8 0 3.717 320 ×10−4 0.181
WBS 8 1 17 43 16 14 1 5.500 745 ×10−4 0.056
WBS2 5 1 14 45 15 14 6 4.043 561 ×10−4 0.623
NOT 0 12 6 56 16 7 3 59 ×10−4 53 ×10−3 0.034
TGUH (M2) 0 4 2 69 14 9 2 50 ×10−4 36 ×10−3 0.049
WBS 0 2 0 77 8 10 3 47 ×10−4 43 ×10−3 0.028
WBS2 0 3 0 76 12 7 2 49 ×10−4 47 ×10−3 0.130
NOT 12 21 34 28 5 0 0 2.241 103 ×10−3 0.036
TGUH (M3) 1 12 27 43 11 5 1 1.738 98 ×10−3 0.298
WBS 4 26 42 26 2 0 0 2.127 98 ×10−3 0.029
WBS2 0 12 38 42 8 0 0 1.506 94 ×10−3 0.144
NOT 60 17 7 16 0 0 0 113 ×10−3 145 ×10−3 0.013
TGUH (M4) 19 24 2 55 0 0 0 75 ×10−3 67 ×10−3 0.021
WBS 31 23 16 30 0 0 0 86 ×10−3 92 ×10−3 0.008
WBS2 44 21 8 27 0 0 0 100 ×10−3 122 ×10−3 0.031
NOT 0 0 23 70 5 2 0 27 ×10−3 23 ×10−3 0.170
TGUH (M5) 0 0 0 99 1 0 0 23 ×10−3 9 ×10−3 0.024
WBS 0 3 27 70 0 0 0 36 ×10−3 26 ×10−3 0.021
WBS2 0 1 26 73 0 0 0 35 ×10−3 23 ×10−3 0.032
Table 27: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M6). The average MSE, dH and computational time are also given.
N̂ − N
Method  ≤ −150 (−150, −50] (−50, −10) [−10, 10] (10, 50]  > 50 MSE dH Time (s)
NOT 68 32 0 0 0 0 0.437 580 ×10−4 0.21
TGUH 2 8 71 19 0 0 0.192 147 ×10−4 4.63
WBS 98 2 0 0 0 0 0.472 774 ×10−4 0.18
WBS2 0 0 0 100 0 0 0.105 36 ×10−4 3.85
Table 28: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signals (M7) and (M8). The average MSE, dH and computational time are also given.
N̂ − N
Method Model −2 −1 0 1  ≥ 2 MSE dH Time (s)
NOT 33 38 26 3 0 0.692 321 ×10−3 0.004
TGUH (M7) 0 0 92 8 0 0.212 36 ×10−3 0.010
WBS 0 3 97 0 0 0.181 30 ×10−3 0.001
WBS2 0 2 96 2 0 0.172 29 ×10−3 0.013
NOT 95 3 2 0 0 2.210 678 ×10−3 0.002
TGUH (M8) 16 5 79 0 0 0.761 147 ×10−3 0.003
WBS 39 13 48 0 0 1.313 335 ×10−3 0.002
WBS2 39 13 48 0 0 1.313 335 ×10−3 0.002
Table 29: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M9). The average MSE, dH and computational time are also given.
N̂ − N
Method  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
NOT 0 0 2 45 37 10 6 182 ×10−3 54 ×10−3 0.071
TGUH 0 0 0 72 18 6 4 34 ×10−6 19 ×10−3 1.642
WBS 3 1 8 45 14 20 9 610 ×10−3 97 ×10−3 0.470
WBS2 2 2 5 44 16 16 15 750 ×10−3 104 ×10−3 2.812
Table 30: Distribution of N̂ − N over 100 simulated data sequences of the piecewise-constant signal (M10). The average MSE, dH and computational time are also given.
N̂ − N
Method −2 −1 0 1  ≥ 2 MSE dH Time (s)
NOT 73 2 25 0 0 4.972 371 ×10−4 0.069
TGUH 35 9 56 0 0 4.925 185 ×10−4 0.863
WBS 80 4 16 0 0 4.981 404 ×10−4 0.048
WBS2 46 8 46 0 0 4.925 240 ×10−4 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 N̂, MSE, and computational times over 100 simulated data sequences from the linear signal (M12).
Method N̂ 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) ✓
(NC2) ✓ ✓ ✓ ✓
(NC3) ✓
(NC4) ✓
(M1) ✓
(M2) ✓ ✓ ✓
(M3) ✓ ✓
(M4) ✓
(M5) ✓
(M6) ✓
(M7) ✓ ✓ ✓
(M8) ✓
(M9) ✓
(M10) ✓
(M11) ✓
(M12) ✓ ✓ ✓
(M13) ✓
Criterion: The top two methods in terms of speed
(NC1) ✓ ✓
(NC2) ✓ ✓
(NC3) ✓ ✓
(NC4) ✓ ✓
(M1) ✓ ✓
(M2) ✓ ✓
(M3) ✓ ✓
(M4) ✓ ✓
(M5) ✓ ✓
(M6) ✓ ✓
(M7) ✓ ✓
(M8) ✓ ✓ ✓
(M9) ✓ ✓
(M10) ✓ ✓
(M12) ✓ ✓

Continuous, piecewise-linear signals

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 σ = 1.

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

  • (NC3) long signal, no change-points: length 3000 with slope equal to 1 and no change-points. The standard deviation is σ = 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 σ = 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 σ = 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 σ = 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 σ = 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 σ = 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 σ = 2.

Simulation process

We ran 100 replications for each one of the abovementioned signals and the frequency distribution of N̂ − 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 f̂t is the ordinary least square approximation of ft between two successive change-points. The scaled Hausdorff distance, dH = ns−1max {maxjmink|rj − r̂k|, maxkminj|rj − r̂k|}, where ns 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.

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))

Detailed results

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 N̂ − 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.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID (NC1) 100 0 0 0 6 ×10−3 0.011
NOT 99 1 0 0 6 ×10−3 0.043
ID (NC2) 98 1 1 0 50 ×10−3 0.003
NOT 95 4 1 0 55 ×10−3 0.017
ID (NC3) 100 0 0 0 1 ×10−3 0.382
NOT 100 0 0 0 1 ×10−3 0.178
Table 34: Distribution of N̂ − N over 100 simulated data sequences of the continuous, piecewise-linear signals (M1)-(M6). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID (M1) 0 0 0 97 3 0 0 0.062 9 ×10−3 0.011
NOT 0 0 0 97 3 0 0 0.020 7 ×10−3 0.054
ID (M2) 0 0 0 94 6 0 0 94 ×10−3 11 ×10−3 0.013
NOT 0 0 0 100 0 0 0 21 ×10−3 2 ×10−3 0.137
ID (M3) 0 0 0 97 3 0 0 3.639 48 ×10−3 0.012
NOT 0 0 0 100 0 0 0 2.086 25 ×10−3 0.077
ID (M4) 0 1 7 89 3 0 0 1.642 20 ×10−3 0.044
NOT 0 0 11 85 3 1 0 1.270 18 ×10−3 0.368
ID (M5) 0 0 0 92 8 0 0 11 ×10−3 21 ×10−3 0.014
NOT 0 0 5 87 7 1 0 10 ×10−3 23 ×10−3 0.129
ID (M6) 0 0 0 96 4 0 0 45 ×10−3 6 ×10−3 0.124
NOT 0 0 0 100 0 0 0 8 ×10−3 1 ×10−3 0.435

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 N̂ − 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.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID (NC1) 99 1 0 0 6 ×10−3 0.018
NOT 95 5 0 0 6 ×10−3 0.023
ID (NC2) 91 6 3 0 53 ×10−3 0.004
NOT 81 11 6 2 59 ×10−3 0.003
ID (NC3) 100 0 0 0 1 ×10−3 0.529
NOT 100 0 0 0 1 ×10−3 0.095
Table 36: Distribution of N̂ − N over 100 simulated data sequences of the continuous, piecewise-linear signals (M1)-(M6). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID (M1) 0 0 0 95 5 0 0 0.066 12 ×10−3 0.003
NOT 0 0 3 83 13 0 1 1.649 44 ×10−3 0.017
ID (M2) 0 0 0 96 4 0 0 84 ×10−3 9 ×10−3 0.002
NOT 0 0 0 94 6 0 0 136×10−3 14 ×10−3 0.027
ID (M3) 0 0 0 98 2 0 0 3.355 43 ×10−3 0.005
NOT 0 0 0 92 8 0 0 4.471 59 ×10−3 0.021
ID (M4) 0 0 7 92 1 0 0 1.459 19 ×10−3 0.019
NOT 0 3 24 73 0 0 0 2.042 26 ×10−3 0.061
ID (M5) 0 0 0 97 3 0 0 12 ×10−3 18 ×10−3 0.004
NOT 0 0 1 94 4 1 0 23 ×10−3 27 ×10−3 0.030
ID (M6) 0 0 0 98 2 0 0 41 ×10−3 5 ×10−3 0.076
NOT 0 0 0 98 2 0 0 55 ×10−3 6 ×10−3 0.105

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 N̂ − N over 100 simulated data sequences from (NC1)-(NC3) of the continuous piecewise-linear case. Also the average MSE and computational times are given.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID (NC1) 95 3 0 2 6 ×10−3 0.006
ID (NC2) 87 6 2 5 67 ×10−3 0.003
ID (NC3) 94 5 1 0 1 ×10−3 0.307
Table 38: Distribution of N̂ − N over 100 simulated data sequences of the continuous, piecewise-linear signals (M1)-(M6). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID (M1) 0 0 0 94 3 1 2 87 ×10−3 23 ×10−3 0.004
ID (M2) 0 0 0 93 5 1 1 103×10−3 19 ×10−3 0.004
ID (M3) 0 0 0 94 2 3 1 3.813 55 ×10−3 0.005
ID (M4) 0 0 9 88 2 1 0 1.890 22 ×10−3 0.025
ID (M5) 0 0 0 91 7 0 2 13 ×10−3 26 ×10−3 0.006
ID (M6) 0 0 0 93 5 2 0 71 ×10−3 14 ×10−3 0.100

Discontinuous, piecewise-linear signals

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 σ = 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 σ = 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 σ = 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 σ = 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 σ = 15.

Simulation process

We ran 100 replications for each one of the abovementioned signals and the frequency distribution of N̂ − 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 f̂t is the ordinary least square approximation of ft between two successive change-points. The scaled Hausdorff distance, dH = ns−1max {maxjmink|rj − r̂k|, maxkminj|rj − r̂k|}, where ns 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.

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))

Detailed results

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 N̂ − N over 100 simulated data sequences from (NC1). Also the average MSE and computational times for each method are given.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID (NC1) 98 2 0 0 89 ×10−3 0.176
NOT 98 2 0 0 89 ×10−3 0.043
Table 40: Distribution of N̂ − N over 100 simulated data sequences of the discontinuous, piecewise-linear signals (M1)-(M3). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID (M1) 0 0 0 99 1 0 0 0.035 4 ×10−4 0.038
NOT 0 0 0 99 1 0 0 0.030 4 ×10−4 0.050
ID (M2) 0 0 0 98 2 0 0 0.060 5 ×10−3 0.053
NOT 0 0 0 98 2 0 0 0.060 5 ×10−3 0.073
ID (M3) 0 0 0 90 10 0 0 9.592 11 ×10−3 0.048
NOT 0 0 0 98 1 1 0 8.212 5 ×10−3 0.058
ID (M4) 0 0 0 89 11 0 0 11.944 11 ×10−3 0.313
NOT 0 0 0 95 5 0 0 10.658 9 ×10−3 0.200

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 N̂ − N over 100 simulated data sequences from (NC1). Also the average MSE and computational times for each method are given.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID (NC1) 96 2 2 0 93 ×10−3 0.283
NOT 94 2 3 1 94 ×10−3 0.018
Table 42: Distribution of N̂ − N over 100 simulated data sequences of the discontinuous, piecewise-linear signals (M1)-(M3). The average MSE, dH and computational times are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID (M1) 0 0 0 98 2 0 0 0.031 5 ×10−3 0.033
NOT 0 0 0 79 21 0 0 0.044 28 ×10−3 0.019
ID (M2) 0 0 0 97 2 1 0 0.061 8 ×10−3 0.046
NOT 0 0 0 95 2 1 2 0.063 10 ×10−3 0.027
ID (M3) 0 0 0 87 12 0 1 9.946 15 ×10−3 0.049
NOT 0 0 24 34 31 8 3 106.381 175 ×10−3 0.022
ID (M4) 0 0 0 81 18 1 0 11.842 11 ×10−3 0.272
NOT 0 0 0 60 35 5 0 17.315 12 ×10−3 0.065

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 N̂ − N over 100 simulated data sequences from (NC1). Also the average MSE and computational times are given.
N̂ − N
Method Model 0 1 2  ≥ 3 MSE Time (s)
ID (NC1) 91 0 4 5 0.104 0.132
Table 44: Distribution of N̂ − N over 100 simulated data sequences of the discontinuous, piecewise-linear signals (M1)-(M4). The average MSE, dH and computational time are also given.
N̂ − N
Method Model  ≤ −3 −2 −1 0 1 2  ≥ 3 MSE dH Time (s)
ID (M1) 0 0 0 93 2 3 2 40 ×10−3 17 ×10−3 0.032
ID (M2) 0 0 0 96 1 2 1 41 ×10−3 9 ×10−3 0.035
ID (M3) 0 0 0 90 5 2 3 11.802 17 ×10−3 0.046
ID (M4) 0 0 0 87 11 0 2 12.112 11 ×10−3 0.315