np.mojo {CptNonPar}

Nonparametric Single Lag Change Point Detection

Description

For a given lagged value of the time series, performs nonparametric change point detection of a possibly multivariate time series. If lag \ell = 0, then only marginal changes are detected. If lag \ell \neq 0, then changes in the pairwise distribution of (X_t , X_{t+\ell}) are detected.

Usage

np.mojo(
  x,
  G,
  lag = 0,
  kernel.f = c("quad.exp", "gauss", "euclidean", "laplace", "sine")[1],
  kern.par = 1,
  data.driven.kern.par = TRUE,
  alpha = 0.1,
  threshold = c("bootstrap", "manual")[1],
  threshold.val = NULL,
  reps = 199,
  boot.dep = 1.5 * (nrow(as.matrix(x))^(1/3)),
  parallel = FALSE,
  boot.method = c("mean.subtract", "no.mean.subtract")[1],
  criterion = c("eta", "epsilon", "eta.and.epsilon")[3],
  eta = 0.4,
  epsilon = 0.02,
  use.mean = FALSE
)

Arguments

x

Input data (a numeric vector or an object of classes ts and timeSeries, or a numeric matrix with rows representing observations and columns representing variables).

G

An integer value for the moving sum bandwidth; G should be less than half the length of the time series.

lag

The lagged values of the time series used to detect changes. If lag \ell = 0, then only marginal changes are detected. If lag \ell \neq 0, then changes in the pairwise distribution of (X_t , X_{t+\ell}) are detected.

kernel.f

String indicating which kernel function to use when calculating the NP-MOJO detectors statistics; with kern.par = a, possible values are

• "quad.exp": kernel h_2 in McGonigle and Cho (2023), kernel 5 in Fan et al. (2017):

  h (x,y) = \prod_{i=1}^{2p} \frac{ (2a - (x_i - y_i)^2) \exp (-\frac{1}{4a} (x_i - y_i)^2 )}{2a} .

• "gauss": kernel h_1 in McGonigle and Cho (2023), the standard Gaussian kernel:

  h (x,y) = \exp ( - \frac{a^2}{2} \Vert x - y \Vert^2) .

• "euclidean": kernel h_3 in McGonigle and Cho (2023), the Euclidean distance-based kernel:

  h (x, y ) = \Vert x - y \Vert^a .

• "laplace": kernel 2 in Fan et al. (2017), based on a Laplace weight function:

  h (x, y ) = \prod_{i=1}^{2p} \left( 1+ a^2 (x_i - y_i)^2 \right)^{-1}.

• "sine": kernel 4 in Fan et al. (2017), based on a sinusoidal weight function:

  h (x, y ) = \prod_{i=1}^{2p} \frac{-2 | x_i - y_i | + | x_i - y_i - 2a| + | x_i - y_i +2a| }{4a} .

kern.par

The tuning parameter that appears in the expression for the kernel function, which acts as a scaling parameter, only to be used if data.driven.kern.par = FALSE. If kernel.f = "euclidean", then kern.par \in (0,2), otherwise kern.par > 0.

data.driven.kern.par

A logical variable, if set to TRUE, then the kernel tuning parameter is calculated using the median heuristic, if FALSE it is given by kern.par.

alpha

A numeric value for the significance level with 0 <= alpha <= 1; use iff threshold = "bootstrap".

threshold

String indicating how the threshold is computed. Possible values are

• "bootstrap": the threshold is calculated using the bootstrap method with significance level alpha.

• "manual": the threshold is set by the user and must be specified using the threshold.val parameter.

threshold.val

The value of the threshold used to declare change points, only to be used if threshold = "manual".

reps

An integer value for the number of bootstrap replications performed, if threshold = "bootstrap".

boot.dep

A positive value for the strength of dependence in the multiplier bootstrap sequence, if threshold = "bootstrap".

parallel

A logical variable, if set to TRUE, then parallel computing is used in the bootstrapping procedure if bootstrapping is performed.

boot.method

A string indicating the method for creating bootstrap replications. It is not recommended to change this. Possible choices are

• "mean.subtract": the default choice, as described in McGonigle and Cho (2023). Empirical mean subtraction is performed to the bootstrapped replicates, improving power.

• "no.mean.subtract": empirical mean subtraction is not performed, improving size control.

criterion

String indicating how to determine whether each point k at which NP-MOJO statistic exceeds the threshold is a change point; possible values are

• "epsilon": k is the maximum of its local exceeding environment, which has at least size epsilon*G.

• "eta": there is no larger exceeding in an eta*G environment of k.

• "eta.and.epsilon": the recommended default option; k satisfies both the eta and epsilon criterion. Recommended to use with the standard value of eta that would be used if criterion = "eta" (e.g. 0.4), but much smaller value of epsilon than would be used if criterion = "epsilon", e.g. 0.02.

eta

A positive numeric value for the minimal mutual distance of changes, relative to bandwidth (if criterion = "eta" or criterion = "eta.and.epsilon").

epsilon

a numeric value in (0,1] for the minimal size of exceeding environments, relative to moving sum bandwidth (if criterion = "epsilon" or criterion = "eta.and.epsilon").

use.mean

Logical variable, only to be used if data.drive.kern.par=TRUE. If set to TRUE, the mean of pairwise distances is used to set the kernel function tuning parameter, instead of the median. May be useful for binary data, not recommended to be used otherwise.

Details

The single-lag NP-MOJO algorithm for nonparametric change point detection is described in McGonigle, E. T. and Cho, H. (2023) Nonparametric data segmentation in multivariate time series via joint characteristic functions. arXiv preprint arXiv:2305.07581.

Value

A list object that contains the following fields:

x

Input data

G

Moving window bandwidth

lag

Lag used to detect changes

kernel.f, data.driven.kern.par, use.mean

Input parameters

kern.par

The value of the kernel tuning parameter

threshold, alpha, reps, boot.dep, boot.method, parallel

Input parameters

threshold.val

Threshold value for declaring change points

criterion, eta, epsilon

Input parameters

test.stat

A vector containing the NP-MOJO detector statistics computed from the input data

cpts

A vector containing the estimated change point locations

p.vals

The corresponding p values of the change points, if the bootstrap method was used

References

McGonigle, E.T., Cho, H. (2023). Nonparametric data segmentation in multivariate time series via joint characteristic functions. arXiv preprint arXiv:2305.07581.

Fan, Y., de Micheaux, P.L., Penev, S. and Salopek, D. (2017). Multivariate nonparametric test of independence. Journal of Multivariate Analysis, 153, pp.189-210.

See Also

np.mojo.multilag

Examples

set.seed(1)
n <- 500
noise <- c(rep(1, 300), rep(0.4, 200)) * stats::arima.sim(model = list(ar = 0.3), n = n)
signal <- c(rep(0, 100), rep(2, 400))
x <- signal + noise
x.c <- np.mojo(x, G = 83, lag = 0)
x.c$cpts
x.c$p.vals