# Open-end Nonparametric Sequential Change-Point Detection Test for (Possibly) Multivariate Time Series Sensitive to Changes in the Distribution Function Open-end nonparametric sequential test for change-point detection based on a retrospective CUSUM statistic constructed from differences of empirical distribution functions. The observations can be univariate or multivariate (low-dimensional), and serially dependent. To carry out the test, two steps are required. The first step consists of computing a detector function. The second step consists of comparing the detector function to a suitable constant threshold function. Each of these steps corresponds to one of the functions in the usage section below. The current implementation is preliminary and not optimized for real-time monitoring (but could still be used for that). Details can be found in the first reference. ```r detOpenEndCpDist(x.learn, x, pts = NULL, r = NULL, sigma = NULL, kappa = 1.5, ...) monOpenEndCpDist(det, alpha = 0.05, plot = TRUE) ``` ## Arguments - `x.learn`: a numeric matrix representing the learning sample. - `x`: a numeric matrix representing the observations collected after the beginning of the monitoring. - `pts`: a numeric matrix whose rows represent the evaluation points; if not provided by user, chosen automatically from the learning sample using parameter `r`. - `r`: integer greater or equal than 2 representing the number of evaluation points per dimension to be chosen from the learning sample; used only if `pts = NULL`. - `sigma`: a numeric matrix representing the covariance matrix to be used; if `NULL`, estimated by `sandwich::lrvar()`. - `kappa`: constant involved in the point selection procedure; used only if the multivariate case; should be larger than 1. - `...`: optional arguments passed to `sandwich::lrvar()`. - `det`: an object of class `det.OpenEndCpDist` representing a detector function computed using `detOpenEndCpDist()`. - `alpha`: the value of the desired significance level for the sequential test. - `plot`: logical indicating whether the monitoring should be plotted. ## Details The testing procedure is described in detail in the first reference. ## Returns Both functions return lists whose components have explicit names. The function `monOpenEndCpDist()` in particular returns a list whose components are - **alarm**: a logical indicating whether the detector function has exceeded the threshold function. - **time.alarm**: an integer corresponding to the time at which the detector function has exceeded the threshold function or `NA`. - **times.max**: a vector of times at which the successive detectors have reached their maximum; this sequence of times can be used to estimate the time of change from the time of alarm. - **time.change**: an integer giving the estimated time of change if `alarm` is `TRUE`; the latter is simply the value in `times.max` which corresponds to `time.alarm`. - **statistic**: the value of `statistic` in the call of the function. - **eta**: the value of `eta` in the call of the function. - **p**: number of evaluations points of the empirical distribution functions. - **pts**: evaluation points of the empirical distribution functions. - **alpha**: the value of `alpha` in the call of the function. - **sigma**: the value of `sigma` in the call of the function. - **detector**: the successive values of the detector. - **threshold**: the value of the constant threshold for the detector. ## References M. Holmes, I. Kojadinovic and A. Verhoijsen (2022), Multi-purpose open-end monitoring procedures for multivariate observations based on the empirical distribution function, 45 pages, [https://arxiv.org/abs/2201.10311](https://arxiv.org/abs/2201.10311). M. Holmes and I. Kojadinovic (2021), Open-end nonparametric sequential change-point detection based on the retrospective CUSUM statistic, **Electronic Journal of Statistics** 15:1 , pages 2288-2335, tools:::Rd_expr_doi("10.1214/21-EJS1840") . ## See Also See `detOpenEndCpMean()` for the corresponding test sensitive to changes in the mean, `selectPoints()` for the underlying point selection procedure used in the multivariate case and `lrvar()` for information on the estimation of the underlying long-run covariance matrix. ## Examples ```r ## Not run: ## Example of open-end monitoring m <- 800 # size of the learning sample nm <- 5000 # number of collected observations after the start n <- nm + m # total number of observations set.seed(456) ## Univariate, no change in distribution r <- 5 # number of evaluation points x <- rnorm(n) ## Step 1: Compute the detector det <- detOpenEndCpDist(x.learn = matrix(x[1:m]), x = matrix(x[(m + 1):n]), r = r) ## Step 2: Monitoring mon <- monOpenEndCpDist(det = det, alpha = 0.05, plot = TRUE) ## Univariate, change in distribution k <- 2000 # m + k + 1 is the time of change x[(m + k + 1):n] <- rt(nm - k, df = 3) det <- detOpenEndCpDist(x.learn = matrix(x[1:m]), x = matrix(x[(m + 1):n]), r = r) mon <- monOpenEndCpDist(det = det, alpha = 0.05, plot = TRUE) ## Bivariate, no change d <- 2 r <- 4 # number of evaluation points per dimension x <- matrix(rnorm(n * d), nrow = n, ncol = d) det <- detOpenEndCpDist(x.learn = x[1:m, ], x = x[(m + 1):n, ], r = r) mon <- monOpenEndCpDist(det = det, alpha = 0.05, plot = TRUE) ## Bivariate, change in the mean of the first margin x[(m + k + 1):n, 1] <- x[(m + k + 1):n, 1] + 0.3 det <- detOpenEndCpDist(x.learn = x[1:m, ], x = x[(m + 1):n, ], r = r) mon <- monOpenEndCpDist(det = det, alpha = 0.05, plot = TRUE) ## Bivariate, change in the dependence structure x1 <- rnorm(n) x2 <- c(rnorm(m + k, 0.2 * x1[1:(m + k)], sqrt((1 - 0.2^2))), rnorm(nm - k, 0.7 * x1[(m + k + 1):n], sqrt((1 - 0.7^2)))) x <- cbind(x1, x2) det <- detOpenEndCpDist(x.learn = x[1:m, ], x = x[(m + 1):n, ], r = r) mon <- monOpenEndCpDist(det = det, alpha = 0.05, plot = TRUE) ## End(Not run) ```

seqOpenEndCpDist function