PCA4TS (version 0.1)

segmentTS: Segment Multivariate Time Series

Description

Calculate linear transformation of the $p$-variate time series {y_t} such that the transformed series {x_t}=B{y_t} is segmented into several lower-dimensional subseries, and those subseries are uncorrelated with each other both contemporaneously and serially.

Usage

segmentTS(Y, k0, thresh = FALSE, tuning.vec = 2, K = 5)

Arguments

Y

a data matrix with $n$ rows and $p$ columns, where $n$ is the sample size and $p$ is the dimension of the time series.

k0

a positive integer specified to calculate Wy. See (2.5) in Chang et al. (2014).

thresh

logical. If FALSE (the default), no thresholding will be applied. If TRUE, a thresholding method will be applied first to estimate Wy, see (3.4) and (3.5) in Chang et al. (2014).

tuning.vec

the value of thresholding parameter $\lambda$. The thresholding level is specified by $$u = \lambda {(log p/n)^(1/2)}.$$ Default value is 2. If tuning.vec is a vector, then a cross validation method propo

K

the number of folders used in the cross validation, the default is 5. It is required when thresh is TRUE.

Value

• An object of class "segmentTS" is a list containing the following components:
• B
  the $p$ by $p$ transformation matrix such that {x_t}=B{y_t}
• X
  the transformed series with $n$ rows and $p$ columns

Details

When $p$ is small, thresholding is not required. However, when $p$ is large, it is necessary to use the thresholding method, see more information in Chang et al. (2014).

References

Chang, J., Guo, B. and Yao, Q. (2014). Segmenting Multiple Time Series by Contemporaneous Linear Transformation: PCA for Time Series. Available at http://arxiv.org/abs/1410.2323. Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation. Journal of the American Statistical Association 106: 672-684.

See Also

permutationMax, permutationFDR.

Examples

## Example 1 (Example 5 of Chang et al.(2014)).
## p=6, x_t consists of 3 independent subseries with 3, 2 and 1 components.

p=6;n=1500
# Generate x_t
X=mat.or.vec(p,n)
x=arima.sim(model=list(ar=c(0.5, 0.3), ma=c(-0.9, 0.3, 1.2,1.3)),n=n+2,sd=1)
for(i in 1:3) X[i,]=x[i:(n+i-1)]
x=arima.sim(model=list(ar=c(0.8,-0.5),ma=c(1,0.8,1.8) ),n=n+1,sd=1)
for(i in 4:5) X[i,]=x[(i-3):(n+i-4)]
x=arima.sim(model=list(ar=c(-0.7, -0.5), ma=c(-1, -0.8)),n=n,sd=1)
X[6,]=x
# Generate y_t
A=matrix(runif(p*p, -3, 3), ncol=p)
Y=A%*%X
Y=t(Y)
Trans=segmentTS(Y, k0=5)
# The transformed series z_t
Z=Trans$X
# Plot the cross correlogram of z_t and y_t
Y=data.frame(Y);Z=data.frame(Z)
names(Y)=c("Y1","Y2","Y3","Y4","Y5","Y6")
names(Z)=c("Z1","Z2","Z3","Z4","Z5","Z6")
# The cross correlogram of y_t shows no block pattern
acfY=acf(Y)
# The cross correlogram of z_t shows 3-2-1 block pattern
acfZ=acf(Z)

## Example 2 (Example 6 of Chang et al.(2014)).
## p=20, x_t consists of 5 independent subseries with 6, 5, 4, 3 and 2 components.

p=20;n=3000
# Generate x_t
X=mat.or.vec(p,n)
x=arima.sim(model=list(ar=c(0.5, 0.3), ma=c(-0.9, 0.3, 1.2,1.3)),n.start=500,n=n+5,sd=1)
for(i in 1:6) X[i,]=x[i:(n+i-1)]
x=arima.sim(model=list(ar=c(-0.4,0.5),ma=c(1,0.8,1.5,1.8)),n.start=500,n=n+4,sd=1)
for(i in 7:11) X[i,]=x[(i-6):(n+i-7)]
x=arima.sim(model=list(ar=c(0.85,-0.3),ma=c(1,0.5,1.2)), n.start=500,n=n+3,sd=1)
for(i in 12:15) X[i,]=x[(i-11):(n+i-12)]
x=arima.sim(model=list(ar=c(0.8,-0.5),ma=c(1,0.8,1.8)),n.start=500,n=n+2,sd=1)
for(i in 16:18) X[i,]=x[(i-15):(n+i-16)]
x=arima.sim(model=list(ar=c(-0.7, -0.5), ma=c(-1, -0.8)),n.start=500,n=n+1,sd=1)
for(i in 19:20) X[i,]=x[(i-18):(n+i-19)]
# Generate y_t
A=matrix(runif(p*p, -3, 3), ncol=p)
Y=A%*%X
Y=t(Y)
Trans=segmentTS(Y, k0=5)
# The transformed series z_t
Z=Trans$X
# Plot the cross correlogram of x_t and y_t
Y=data.frame(Y);Z=data.frame(Z)
namesY=NULL;namesZ=NULL
for(i in 1:p)
{
   namesY=c(namesY,paste0("Y",i))
   namesZ=c(namesZ,paste0("Z",i))
}
names(Y)=namesY;names(Z)=namesZ
# The cross correlogram of y_t shows no block pattern
acfY=acf(Y, plot=FALSE)
plot(acfY, max.mfrow=6, xlab='', ylab='',  mar=c(1.8,1.3,1.6,0.5),
     oma=c(1,1.2,1.2,1), mgp=c(0.8,0.4,0),cex.main=1)
# The cross correlogram of z_t shows 6-5-4-3-2 block pattern
acfZ=acf(Z, plot=FALSE)
plot(acfZ, max.mfrow=6, xlab='', ylab='',  mar=c(1.8,1.3,1.6,0.5),
     oma=c(1,1.2,1.2,1), mgp=c(0.8,0.4,0),cex.main=1)
# Identify the permutation mechanism
permutation=permutationMax(Z)
permutation$Groups