onlineCOV (version 1.3)

stopping.rule: Online change-point detection by the stopping rule.

Description

Function to determine whether a process with continually arrving data should be terminated, based on the proposed stopping rule.

Usage

stopping.rule(ARL, H, mu, M, cor, old.data, new.data)

Arguments

ARL

The expected value of the stopping time when there is no change, eg. ARL = 5000.

H

The window size so that the stopping rule only considers H observations from the current time, eg. H=100.

mu

The mean vector of the observation with dimension 1 by p, can be estimated from a training sample through the function "nuisance.est".

M

M dependence, can be estimated from a training sample through the function "nuisance.est", eg. M=0 means data are temporally independent.

cor

A value used to obtain the standard deviation of the test statistic in the stopping rule, can be estimated from a training sample through the function "nuisance.est".

old.data

The observed sequence of data. The dataset has dimension H by p, where H is the window size, or the number of observed data (row), and p is the number of components (column).

new.data

A newly arrived observation with dimension 1 by p.

Value

Returns a list with items "decision" and "old.updated". See below for more detail.

decision

returns 1 if the stopping rule detects a change point, and returns 0 otherwise.

old.updated

The updated observed dataset in this step, with dimension H by p. The Hth observation is the newly arrived observation, and the rest H-1 observations come from the previous dataset.

References

Li, L. and Li, J. (2020) "Online Change-Point Detection in High-Dimensional Covariance Structure with Application to Dynamic Networks." arXiv:1911.07762.

Examples

# NOT RUN {
# The following is an example to detect covariance structure change 
# in a real-time manner, in the sense that we pretend that 
# the observations in the dataset continually arrive in time. 
# At each time, we determine whether the process should be 
# terminated through the proposed stopping rule.
# there is an immediate change point at n0=200


p<-200;n<-10000;n0<-200 #n0 is traing sample size
rho<-0.6;M<-2
H<-100;ARL<-5000

Gam1<-diag(1,p,p)
times<-1:p
d<-abs(outer(times, times, "-"))
sigma<-rho^d
Gam2<-eigen(sigma,symmetric=TRUE)$vectors%*%diag(sqrt(eigen(sigma,symmetric=TRUE)$values),p)
Gam<-cbind(Gam1,Gam2)

data_Mat<-matrix(0,n0,p)
L<-M+1
Z<-matrix(rnorm(p*(n+L-1)),p*(n+L-1),1)
vec.coef<-1/rep(c(L:1),each=p)

 	for(j in 1:n0){
 	Gam.m<-Gam[,1:p]	
 	Gam.mat<-t(apply(Gam.m,1,rep,L))*matrix(vec.coef,ncol=L*p,nrow=p,byrow=TRUE)	
 	data_Mat[j,]<-matrix((Gam.mat%*%Z[((j-1)*p+1):((j+L-1)*p),]),1,p,byrow=FALSE)	
 	}

old.data<-data_Mat 
nuisance.results<-nuisance.est(old.data)
mu<-nuisance.results$mu.hat
M<-nuisance.results$M.hat
cor<-nuisance.results$cor.hat

 j<-n0+1;decision = 0

 	while(decision==0){
	
 	Gam.m<-Gam[,(p+1):(2*p)]	
 	Gam.mat<-t(apply(Gam.m,1,rep,L))*matrix(vec.coef,ncol=L*p,nrow=p,byrow=TRUE)	
 	new.data<-matrix((Gam.mat%*%Z[((j-1)*p+1):((j+L-1)*p),]),1,p,byrow=FALSE)    	
      
 	result<-stopping.rule(ARL,H,mu,M,cor,old.data,new.data)
 	decision<-result$decision
 	old.data<-result$old.updated
 	cpt.est<-j-n0

 	j<-j+1
 			}

 print(cpt.est) #The point where the detection procedure terminates.
# }