# A CUSUM-type test to detect changes in the correlation. Performs a CUSUM-based test on changes in Spearman's rho or Kendall's tau. ```r cor_cusum(x, version = c("tau", "rho"), method = "kernel", control = list(), fpc = TRUE, tol = 1e-08, plot = FALSE) ``` ## Arguments - `x`: time series (matrix or ts object with numeric/integer values). - `version`: version of the test. Either `rho` or `tau`. - `method`: method for estimating the long run variance. - `control`: a list of control parameters. - `fpc`: finite population correction (boolean). - `tol`: tolerance of the distribution function (numeric), which is used do compute p-values. - `plot`: should the test statistic be plotted (cf. `plot.cpStat`)? Boolean. ## Returns A list of the class "htest" containing the following components: - **statistic**: return value of the function `cor_stat`. - **p.value**: p-value (numeric). - **alternative**: alternative hypothesis (character string). - **method**: name of the performed test (character string). - **cp.location**: index of the estimated change point location (integer). - **data.name**: name of the data (character string). - **lrv**: list containing the compontents `method`, `param` and `value`. ## Details The function perform a CUSUM-type test on changes in the correlation of a time series $x$. Formally, the hypothesis pair can be written as $$ H_0: \xi_1 = ... = \xi_n $$ $$ vs. $$ $$ H_1: \exists k \in \{1, ..., n-1\}: \xi_k \neq \xi_{k+1} $$ where $\xi_i$ is a fluctuation measure (either Spearman's rho or Kendall's tau) for the $i$-th observations and $n$ is the length of the time series. $k$ is called a 'change point'. The test statistic is computed using `cor_stat` and asymptotically follows a Kolmogorov distribution. To derive the p-value, the funtion `pKSdist` is used. ## References Wied, D., Dehling, H., Van Kampen, M., and Vogel, D. (2014). A fluctuation test for constant Spearman’s rho with nuisance-free limit distribution. Computational Statistics & Data Analysis, 76, 723-736. , A. (2022+). "Finite sample correction for cusum tests", **unpublished manuscript** ## Author(s) Sheila ## See Also `cor_stat`, `lrv`, `pKSdist` ## Examples ```r ### first: generate a time series with a burn-in period of m and a change point k = n/2 require(mvtnorm) n <- 500 m <- 100 N <- n + m k <- m + floor(n * 0.5) n1 <- N - k ## Spearman's rho: rho <- c(0.4, -0.9) # serial dependence: theta1 <- 0.3 theta2 <- 0.2 theta <- cbind(c(theta1, 0), c(0, theta2)) q <- rho * sqrt( (theta1^2 + 1) * (theta2^2 + 1) / (theta1 * theta2 + 1)) # shape matrices of the innovations: S0 <- cbind(c(1, q[1]), c(q[1], 1)) S1 <- cbind(c(1, q[2]), c(q[2], 1)) e0 <- rmvt(k, S0, 5) e1 <- rmvt(n1, S1, 5) e <- rbind(e0, e1) # generate the data: x <- matrix(numeric(N * 2), ncol = 2) x[1, ] <- e[1, ] invisible(sapply(2:N, function(i) x[i, ] <<- e[i, ] + theta %*% e[i-1, ])) x <- x[-(1:m), ] cor_cusum(x, "rho") ## Kendall's tau S0 <- cbind(c(1, rho[1]), c(rho[1], 1)) S1 <- cbind(c(1, rho[2]), c(rho[2], 1)) e0 <- rmvt(k, S0, 5) e1 <- rmvt(n1, S1, 5) e <- rbind(e0, e1) x <- matrix(numeric(N * 2), ncol = 2) x[1, ] <- e[1, ] # AR(1): invisible(sapply(2:N, function(i) x[i, ] <<- 0.8 * x[i-1, ] + e[i, ])) x <- x[-(1:m), ] cor_cusum(x, version = "tau") ```

cor_cusum function