Introduction to SNSeg and Examples

SNSeg supports change-points estimation for both univariate and multivariate time series (including high-dimensional time series with dimension greater than 10.) using Self-Normalization (SN) based framework. Please read Zhao, Jiang and Shao (2022) <doi.org/10.1111/rssb.12552> for details of the SN-based algorithms.

All functions contain two important input arguments: grid_size_scale and grid_size.

Users can set their own grid_size or leave it as NULL in the input arguments. If grid_size is NULL, these functions will compute the SN test statistic using the value of grid_size_scale. If grid_size is set by users, the functions will first calculate grid_size_scale by diving the length of time, and then compute the SN test statistic.

For the other input arguments: * ts: Users should enter a time series for this argument. For SNSeg_Uni, the dimension of ts should be exactly one in most cases (or two when paras_to_test = 'bivcor'); for SNSeg_Multi, the dimension must be at least two but no more than ten; for SNSeg_HD, the dimension must be greater than ten. * paras_to_test: This argument in functions SNSeg_Uni and SNSeg_Multi allows users to enter the parameter(s) they would like to test the change in. * confidence: Users should choose a confidence level among 0.9, 0.95, 0.99, 0.995 and 0.995. A smaller confidence level is easier to reject the null hypothesis and detect a change-point.

The package also offers an option to plot the time series (this only works for the univariate time series cases!) Users need to set plot_SN = TRUE to visualize the time series, and est_cp_loc = TRUE to add the estimated change-point locations in the plot.

SN Test Statistic Plot: `max_SNsweep`

To visualize the computed SN-based test statistic at each time point, users can apply the function max_SNsweep by plugging in the output object from one of the functions SNSeg_Uni, SNSeg_Multi and SNSeg_HD. The options est_cp_loc = TRUE and critical_loc = TRUE are provided to draw the estimated change-point locations and the critical value threshold inside the test statistic plot.

max_SNsweep also returns the SN-based test statistic for each time point. A large number of test statistics in the output can be messy for users who only seek to generate a plot. To hide the test statistics output, users can create an arbitrary variable name and set it to the max_SNsweep function, e.g., SN_stat <- max_SNsweep(...).

We then provide the examples of the functions SNSeg_Uni, SNSeg_Multi and SNSeg_HD.

Examples of `SNSeg_Uni`:

The function SNSeg_Uni detect change-points for a univariate time series based on the change in a single or parameters.

For a univariate time series, the input argument paras_to_test allows one or a combination of multiple parameters from “mean”, “variance”, “acf” and a numeric quantile value between 0 and 1. For instance, to test the change in autocorrelation and 80th quantile, users should set paras_to_test to c(‘acf’, 0.8).
To test the change in correlation between two univariate time series, the input time series must be two-dimensional and paras_to_test should be set to bivcor.

We then provide examples for different cases:

# Please run the following function before running examples:
mix_GauGPD <- function(u,p,trunc_r,gpd_scale,gpd_shape){
  indicator <- u<p
  rv <- rep(0, length(u))
  rv[indicator>0] <- qtruncnorm(u[indicator>0]/p,a=-Inf,b=trunc_r)
  rv[indicator<=0] <- qgpd((u[indicator<=0]-p)/(1-p), loc=trunc_r, scale=gpd_scale,shape=gpd_shape)
  return(rv)
}

Test in a single parameter

Segmentation for Mean

set.seed(7)
n <- 2000
reptime <- 2
cp_sets <- round(n*c(0,cumsum(c(0.5,0.25)),1))
mean_shift <- c(0.4,0,0.4)
rho <- -0.7
ts <- MAR(n, reptime, rho)
no_seg <- length(cp_sets)-1
for(index in 1:no_seg){ # Mean shift
  tau1 <- cp_sets[index]+1   
  tau2 <- cp_sets[index+1]
  ts[tau1:tau2,] <- ts[tau1:tau2,] + mean_shift[index]
}
ts <- ts[,2]
# grid_size undefined
result <- SNSeg_Uni(ts, paras_to_test = "mean", confidence = 0.9,
                    grid_size_scale = 0.05, grid_size = NULL, 
                    plot_SN = FALSE, est_cp_loc = FALSE)
# grid_size defined & generate time series segmentation plot
result <- SNSeg_Uni(ts, paras_to_test = "mean", confidence = 0.9,
                    grid_size_scale = 0.05, grid_size = 116, 
                    plot_SN = TRUE, est_cp_loc = TRUE)

# Estimated change-point locations
result$est_cp
#> [1] 1018 1487
# plot the SN-based test statistic
SN_stat <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)

Segmentation for Variance

set.seed(7)
ts <- MAR_Variance(2, "V1")
ts <- ts[,2]
# grid_size defined
result <- SNSeg_Uni(ts, paras_to_test = "variance", confidence = 0.9,
                    grid_size_scale = 0.05, grid_size = 67, 
                    plot_SN = TRUE, est_cp_loc = TRUE)

# Estimated change-point locations
result$est_cp
#> [1] 407 740
# plot the SN-based test statistic
SN_stat <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)

Segmentation for Autocorrelation

set.seed(7)
ts <- MAR_Variance(2, "A3")
ts <- ts[,2]
# grid_size defined
result <- SNSeg_Uni(ts, paras_to_test = "acf", confidence = 0.9,
          grid_size_scale = 0.05, grid_size = 92, plot_SN = TRUE,
          est_cp_loc = TRUE)

# Estimated change-point locations
result$est_cp
#> [1] 491 767
# plot the SN-based test statistic
SN_stat <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)

Segmentation for bivariate correlation

library(mvtnorm)
set.seed(7)
n <- 1000
sigma_cross <- list(4*matrix(c(1,0.8,0.8,1), nrow=2),
                    matrix(c(1,0.2,0.2,1), nrow=2),
                    matrix(c(1,0.8,0.8,1), nrow=2))
cp_sets <- round(c(0,n/3,2*n/3,n))
noCP <- length(cp_sets)-2
rho_sets <- rep(0.5, noCP+1)
ts <- MAR_MTS_Covariance(n, 2, rho_sets, cp_sets, sigma_cross)
ts <- ts[1][[1]]
# grid_size defined
result <- SNSeg_Uni(ts, paras_to_test = "bivcor", confidence = 0.9,
                    grid_size_scale = 0.05, grid_size = 77, 
                    plot_SN = TRUE, est_cp_loc = TRUE)

# Estimated change-point locations
result$est_cp
#> [1] 336 667
# plot the SN-based test statistic
SN_stat <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)

Segmentation for quantile

library(truncnorm)
library(evd)
set.seed(7)
n <- 1000
cp_sets <- c(0,n/2,n)
noCP <- length(cp_sets)-2
reptime <- 2
rho <- 0.2
# AR time series with no change-point (mean, var)=(0,1)
ts <- MAR(n, reptime, rho)*sqrt(1-rho^2)
trunc_r <- 0
p <- pnorm(trunc_r)
gpd_scale <- 2
gpd_shape <- 0.125
for(ts_index in 1:reptime){
  ts[(cp_sets[2]+1):n, ts_index] <- mix_GauGPD(pnorm(ts[(cp_sets[2]+1):n, ts_index]),
p,trunc_r,gpd_scale,gpd_shape)
}
ts <- ts[,2]
# grid_size undefined
# test in 90% quantile
result <- SNSeg_Uni(ts, paras_to_test = 0.9, confidence = 0.9,
                    grid_size_scale = 0.066, grid_size = NULL,
                    plot_SN = TRUE, est_cp_loc = TRUE)

# Estimated change-point locations
result$est_cp
#> [1] 487
# plot the SN-based test statistic
SN_stat <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)

Test in multiple parameters

set.seed(7)
n <- 1000
cp_sets <- c(0,333,667,1000)
no_seg <- length(cp_sets)-1
rho <- 0
# AR time series with no change-point (mean, var)=(0,1)
ts <- MAR(n, 2, rho)*sqrt(1-rho^2)
no_seg <- length(cp_sets)-1
sd_shift <- c(1,1.6,1)
for(index in 1:no_seg){ # Mean shift
  tau1 <- cp_sets[index]+1
  tau2 <- cp_sets[index+1]
  ts[tau1:tau2,] <- ts[tau1:tau2,]*sd_shift[index]
}
d <- 2
ts <- ts[,2]

# Test in 90th and 95th quantile with grid_size undefined
result <- SNSeg_Uni(ts, paras_to_test = c(0.9, 0.95), confidence = 0.9,
                    grid_size_scale = 0.05, grid_size = NULL, 
                    plot_SN = FALSE, est_cp_loc = FALSE)

# Test in 90th quantile and the variance with grid_size undefined
result <- SNSeg_Uni(ts, paras_to_test = c(0.9, 'variance'),
                    confidence = 0.95, grid_size_scale = 0.078,
                    grid_size = NULL, plot_SN = FALSE, 
                    est_cp_loc = FALSE)

# Test in 90th quantile, variance and acf with grid_size undefined
result <- SNSeg_Uni(ts, paras_to_test = c(0.9,'variance', "acf"),
                    confidence = 0.9, grid_size_scale = 0.064,
                    grid_size = NULL, plot_SN = TRUE, 
                    est_cp_loc = TRUE)

# Estimated change-point locations
result$est_cp
#> [1] 327 659
# Test in 60th quantile, mean, variance and acf with grid_size defined
result <- SNSeg_Uni(ts, paras_to_test = c(0.6, 'mean', "variance",   
                    "acf"), confidence = 0.9, grid_size_scale = 0.05,
                    grid_size = 83, plot_SN = TRUE, est_cp_loc = TRUE)

# Estimated change-point locations
result$est_cp
#> [1] 321 660
# SN-based test statistic segmentation plot
SN_stst <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)

Examples: `SNSeg_Mulni`

The function SNSeg_Multi detects change-points for multivariate time series based on the change in multivariate means or covariances. Users can set paras_to_test to mean or covariance for change-points estimation.

The dimension of parameters to be tested must be at most ten. If the parameter is mean, the dimension is equivalent to the number of time series. If the parameter is covariance, the dimension is equivalent to the number of unknown parameters within the covariance matrix.

For examples of multivariate means and covariances, please check the following commands:

# Please run this function before running examples:
exchange_cor_matrix <- function(d, rho){
  tmp <- matrix(rho, d, d)
  diag(tmp) <- 1
  return(tmp)
}

Segmentation for Multivariate Mean

library(mvtnorm)
set.seed(10)
d <- 5
n <- 1000
cp_sets <- round(n*c(0,cumsum(c(0.075,0.3,0.05,0.1,0.05)),1))
mean_shift <- c(-3,0,3,0,-3,0)/sqrt(d)
mean_shift <- sign(mean_shift)*ceiling(abs(mean_shift)*10)/10
rho_sets <- 0.5
sigma_cross <- list(exchange_cor_matrix(d,0))
ts <- MAR_MTS_Covariance(n, 2, rho_sets, cp_sets=c(0,n), sigma_cross)
noCP <- length(cp_sets)-2
no_seg <- length(cp_sets)-1
for(rep_index in 1:2){
  for(index in 1:no_seg){ # Mean shift
    tau1 <- cp_sets[index]+1
    tau2 <- cp_sets[index+1]
    ts[[rep_index]][,tau1:tau2] <- ts[[rep_index]][,tau1:tau2] + mean_shift[index]
    }
}
ts <- ts[1][[1]]

# grid_size undefined
result <- SNSeg_Multi(ts, paras_to_test = "mean", confidence = 0.95,
                      grid_size_scale = 0.079, grid_size = NULL)
# grid_size defined
result <- SNSeg_Multi(ts, paras_to_test = "mean", confidence = 0.99,
                      grid_size_scale = 0.05, grid_size = 65)
# Estimated change-point locations
result$est_cp
#> [1] 73
# SN-based test statistic segmentation plot
SN_stst <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)

Segmentation for Multivariate Covariance

library(mvtnorm)
set.seed(10)
reptime <- 2
d <- 4
n <- 1000
sigma_cross <- list(exchange_cor_matrix(d,0.2),
                    2*exchange_cor_matrix(d,0.5),
                    4*exchange_cor_matrix(d,0.5))
rho_sets <- c(0.3,0.3,0.3)
mean_shift <- c(0,0,0) # with mean change
cp_sets <- round(c(0,n/3,2*n/3,n))
ts <- MAR_MTS_Covariance(n, reptime, rho_sets, cp_sets, sigma_cross)
noCP <- length(cp_sets)-2
no_seg <- length(cp_sets)-1
for(rep_index in 1:reptime){
  for(index in 1:no_seg){ # Mean shift
    tau1 <- cp_sets[index]+1
    tau2 <- cp_sets[index+1]
    ts[[rep_index]][,tau1:tau2] <- ts[[rep_index]][,tau1:tau2] +
      mean_shift[index]
    }
}
ts <- ts[[1]]

# grid_size undefined
result <- SNSeg_Multi(ts, paras_to_test = "covariance", 
                      confidence = 0.9, grid_size_scale = 0.05, 
                      grid_size = NULL)
# grid_size defined
result <- SNSeg_Multi(ts, paras_to_test = "covariance", 
                      confidence = 0.9, grid_size_scale = 0.05, 
                      grid_size = 81)
# Estimated change-point locations
result$est_cp
#> [1] 321 651
# SN-based test statistic segmentation plot
SN_stst <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)

Examples: `SNSeg_HD`

The function SNSeg_HD performs change-points estimation for high-dimensional time series (dimension greater than 10) using the change in high-dimensional means. For usage examples of SNSeg_HD, we provide the followig code:

n <- 600
p <- 100
nocp <- 5
cp_sets <- round(seq(0,nocp+1,1)/(nocp+1)*n)
num_entry <- 5
kappa <- sqrt(4/5) # Wang et al(2020)
mean_shift <- rep(c(0,kappa),100)[1:(length(cp_sets)-1)]
set.seed(1)
ts <- matrix(rnorm(n*p,0,1),n,p)
no_seg <- length(cp_sets)-1
for(index in 1:no_seg){ # Mean shift
  tau1 <- cp_sets[index]+1
  tau2 <- cp_sets[index+1]
  ts[tau1:tau2,1:num_entry] <- ts[tau1:tau2,1:num_entry] +
    mean_shift[index] # sparse change
}

# grid_size undefined
result <- SNSeg_HD(ts, confidence = 0.9, grid_size_scale = 0.05,
                   grid_size = NULL)
# grid_size defined
result <- SNSeg_HD(ts, confidence = 0.9, grid_size_scale  = 0.05,
                   grid_size = 52)
# Estimated change-point locations
result$est_cp
#> [1] 100 195 300 404 498
# SN-based test statistic segmentation plot
SN_stst <- max_SNsweep(result, plot_SN = TRUE, est_cp_loc = TRUE,
                       critical_loc = TRUE)