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main.R
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main.R
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#' WCM.gSA
#'
#' WCM.gSA procedure for multiple change point detection in the mean of serially correlated time series
#'
#' @param x input data (a \code{numeric} vector)
#' @param R number of deterministic intervals to be drawn at each iteration of WBS2
#' @param min.len minimum distance between candidate change point estimators;
#' if \code{min.len = NULL}, it is set to be \code{max(20, p.max + ceiling(log(length(x))^1.1)}
#' @param Q maximum number of allowable change points
#' @param max.iter maximum number of nested change point models considered
#' @param p.max maximum AR order
#' @param pen penalty used for the Schwarz criterion
#' @return
#' \item{cp}{change point location estimators; \code{integer(0)} is returned if there is no change point}
#' \item{rcp}{refined change point estimators}
#' \item{cp.info}{matrix containing information about the change point estimators, such as their order of detection,
#' the intervals \code{(s, e)} in which each estimator (\code{b}) is detected, and the associated max-CUSUMs}
#' @references H. Cho and P. Fryzlewicz (2021) Multiple change point detection under serial dependence:
#' wild contrast maximisation and gappy Schwarz algorithm. arXiv:2011.13884.
#' @examples
#' set.seed(111)
#' f <- rep(c(0, 5, 2, 8, 1, -2), c(100, 200, 200, 50, 200, 250))
#' x <- f + arima.sim(list(ar = c(.75, -.5), ma = c(.8, .7, .6, .5, .4, .3)), n = length(f), sd = 1)
#' wcm.gsa(x)
#' @export
wcm.gsa <- function(x, R = 100, min.len = NULL,
gappy = TRUE, Q = floor(log(length(x))^1.9), max.iter = 5,
p.max = 10, pen = log(length(x))^1.01) {
n <- length(x)
if(is.null(min.len)) min.len <- max(20, p.max + ceiling(log(n)^1.1))
res <- t(wbs2(x, R, min.len, 1))
res <- res[order(- res[, 4]), ]
colnames(res) <- c('s', 'b', 'e', 'cusum')
Q <- min(Q, dim(res)[1])
z <- log(res[1:Q, 4])
cs <- cumsum(z[1:Q])
ecp.seq <- res[1:Q, 2]
ecp.list <- list()
current <- c()
if(gappy){
niter <- ii <- 0
for(k in sort(sort(abs(diff(z)), decreasing = TRUE, index.return = TRUE)$ix[1:max.iter])){
new.cp <- sort(ecp.seq[(ii + 1):k])
current <- sort(c(current, new.cp))
ecp.list <- c(ecp.list, list(current))
ii <- k
}
} else{
for(ii in 1:(Q - 1)){
current <- sort(c(current, ecp.seq[ii]))
ecp.list <- c(ecp.list, list(current))
}
}
ll <- length(ecp.list)
while(ll >= 1){
if(ll == 1) current <- numeric(0) else current <- ecp.list[[ll - 1]]
new.cp <- setdiff(ecp.list[[ll]], current)
out <- rep(0, length(current) + 1)
brks <- c(0, current, n)
for(jj in 1:(length(current) + 1)){
int <- (brks[jj] + 1):brks[jj + 1]
ecp <- new.cp[new.cp %in% int]
if(length(ecp) > 0){
loc.brks <- c(brks[jj], ecp, brks[jj + 1])
if(length(int) > p.max + log(n) + length(ecp) + 1 & max(diff(loc.brks)) > p.max + 1 + log(n)){
ms <- ms.sc(x, s = brks[jj], e = brks[jj + 1], ecp = ecp, p.max = p.max, pen = pen)
if(ms$cp.sc >= ms$ncp.sc) out[jj] <- -1 else out[jj] <- 1
} else out[jj] <- -1
}
}
if(any(out < 0)) flag <- FALSE else{
rcm <- refine.chp.model(ecp.list[[ll]], x, c = .5)
ms <- ms.sc(x, s = 0, e = n, ecp = rcm$rcp, p.max = p.max, pen = pen)
if(ms$cp.sc >= ms$ncp.sc) flag <- FALSE else flag <- TRUE
}
if(flag) break
ll <- ll - 1
}
if(ll == 0){
rcp <- cp <- integer(0)
cp.info <- matrix(0, nrow = 0, ncol = 4)
colnames(cp.info) <- colnames(res)
} else{
cp <- ecp.list[[ll]]
rcp <- refine.chp.model(cp, x, c = .5)$rcp
cp.info <- res[1:length(cp),, drop = FALSE]
}
return(list(cp = cp, rcp = rcp, cp.info = cp.info))
}
## internal functions -- not to be called directly by the user
#' WBS2
#' @keywords internal
wbs2 <- function(x, R = 100, min.len = 0, top.scale = 0){
n <- length(x)
if (n <= 2 * min.len + 1) return(matrix(NA, 4, 0)) else {
if(top.scale == 1){
tmp <- rbind(systematic.cusums(x[1:round(n/2)], R, min.len)$max.val,
systematic.cusums(x[round(n/4) + 1:round(n/2)], R, min.len)$max.val + c(rep(round(n/4), 3), 0),
systematic.cusums(x[(round(n/2) + 1):n], R, min.len)$max.val + c(rep(round(n/2), 3), 0))
cpt <- t(tmp[which.max(tmp[, 4]), , drop = FALSE])
} else cpt <- t(systematic.cusums(x, R, min.len)$max.val)
return(cbind(cpt, wbs2(x = x[1:cpt[2]], R, min.len, 0),
wbs2(x = x[(cpt[2] + 1):n], R, min.len, 0) + c(rep(cpt[2], 3), 0)))
}
}
systematic.cusums <- function(x, R, min.len) {
y <- c(0, cumsum(x))
n <- length(x)
R <- min(R, (n - 1)*n/2)
ind <- grid.intervals(n, R)
R <- dim(ind)[2]
res <- matrix(0, R, 4)
res[,1:2] <- t(ind)
res[,3:4] <- t(apply(ind, 2, max.cusum, y, min.len))
max.ind <- which.max(abs(res[,4]))
max.val <- res[max.ind, c(1, 3, 2, 4), drop = FALSE]
list(res = res, max.val = max.val, R.eff = R)
}
grid.intervals <- function(n, R){
if(n == 2){
ind <- matrix(c(1, 2), 2, 1)
} else if(R >= (n-1)*n/2){
ind <- all.intervals.flat(n)
} else{
k <- 1
while (k*(k-1)/2 < R) k <- k + 1
ind2 <- all.intervals.flat(k)
ind2.mx <- max(ind2)
ind <- round((ind2 - 1) * ((n - 1) / (ind2.mx - 1)) + 1)
}
ind
}
all.intervals.flat <- function(n){
if (n == 2) ind <- matrix(1:2, 2, 1) else {
R <- (n - 1)*n/2
ind <- matrix(0, 2, R)
ind[1,] <- rep(1:(n - 1), (n - 1):1)
ind[2,] <- 2:(R + 1) - rep(cumsum(c(0, (n - 2):1)), (n - 1):1)
}
ind
}
max.cusum <- function(ind, y, min.len) {
m <- ind[2] - ind[1] + 1
if(m > 2 * min.len + 1){
z <- y[(ind[1] + 1):(ind[2] + 1)] - y[ind[1]]
ip <- sqrt(((m - 1):1) / m / (1:(m - 1))) * z[1:(m - 1)] - sqrt((1:(m - 1))/m/((m - 1):1)) * (z[m] - z[1:(m - 1)])
ip.max <- which.max(abs(ip[(min.len + 1):(m - min.len - 1)])) + min.len
return(c(ip.max + ind[1] - 1, abs(ip[ip.max])))
} else{
return(c(ind[1], 0))
}
}
#' gSA: model selection using Schwarz criterion
#' @keywords internal
ms.sc <- function(x, s, e, ecp, p.max = 10, pen = NULL){
n <- length(x)
brks <- c(s, ecp, e)
cp.lik.seq <- ncp.lik.seq <- cp.sc.seq <- ncp.sc.seq <- rep(0, p.max + 1)
for(ii in 1:(p.max + 1)){
pp <- ii - 1
len <- e - max(pp, s)
y <- x[(max(pp, s) + 1):e]
R <- L <- c()
if(pp > 0) for(kk in 1:pp) L <- cbind(L, x[(max(pp, s) + 1):e - kk])
for(kk in 1:(length(brks) - 1)){
if(brks[kk + 1] <= pp) next
tmp <- rep(0, len)
tmp[max(1, brks[kk] - max(pp, s) + 1):(brks[kk + 1] - max(pp, s))] <- 1
R <- cbind(R, tmp)
}
X <- cbind(L, R)
lm.fit1 <- lm(y ~ 0 + X)
cp.lik.seq[ii] <- loglik1 <- -len/2*log(2*pi*sum(resid(lm.fit1)^2)/len) - len/2
cp.sc.seq[ii] <- (pp + length(ecp) + 1) * pen - loglik1
if(pp > 0) z <- y - L %*% coef(lm.fit1)[1:pp] else z <- y
ncp.lik.seq[ii] <- loglik2 <- -len/2*log(2*pi*sum((z - mean(z))^2)/len) - len/2
ncp.sc.seq[ii] <- (pp + 1) * pen - loglik2
}
p <- which.min(cp.sc.seq[-1])
out <- list(cp.sc = cp.sc.seq[p + 1], ncp.sc = ncp.sc.seq[p + 1], p = p,
cp.lik = cp.lik.seq, ncp.lik = ncp.lik.seq,
cp.sc.seq = cp.sc.seq, ncp.sc.seq = ncp.sc.seq)
return(out)
}
#' Refinement of change point location estimators
#' @keywords internal
refine.chp.model <- function(ecp, x, c = .5){
n <- length(x)
rcp <- ecp * 0
nres <- c()
for(i in 1:length(ecp)){
if(i == 1){
if(i < length(ecp)) int <- 1:floor(ecp[i]*(1 - c) + ecp[i + 1]*c) else int <- 1:n
} else{
if(i < length(ecp)) int <- floor(ecp[i]*(1 - c) + ecp[i - 1]*c + 1):floor(ecp[i]*(1 - c) + ecp[i + 1]*c) else int <- floor(ecp[i]*(1 - c) + ecp[i - 1]*c + 1):n
}
if(length(int) < 2){
if(i == 1) int <- 1:ecp[i + 1] else if(i == length(ecp)) int <- (ecp[i - 1] + 1):n else int <- (ecp[i - 1] + 1):ecp[i + 1]
}
csx <- cumsum(x[int])
len <- length(int)
evalx <- (len/((1:(len - 1)) * ((len - 1):1)))^.5 * (csx[1:(len - 1)] - csx[len]/len * (1:(len - 1)))
rcp[i] <- int[1] + which.max(abs(evalx)) - 1
nres <- rbind(nres, c(min(int), rcp[i], max(int), max(abs(evalx))))
}
return(list(rcp = rcp, nres = nres))
}