PCA4TS (version 0.1)

permutationFDR: Permutation Using the FDR Method

Description

The permutation is determined by grouping the components of a multivariate series X into $q$ groups, where $q$ and the cardinal numbers of those groups are also unknown.

Usage

permutationFDR(X, Vol = FALSE, Beta, m = NULL)

Arguments

X

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

Vol

logical. If FALSE (the default), then prewhiten each series by fitting a univariate AR model with the order between 0 and 5 determined by AIC. If TRUE, then prewhiten each volatility process using GARCH(1,1) model.

Beta

the error rate in FDR

m

a positive constant used to calculate the maximum cross correlation over the lags between $-m$ and $m$. If $m$ is not specified, the default constant $10*log10(n/p)$ will be used.

Value

An object of class "permutationFDR" is a list containing the following components:

NoGroups

number of groups with at least two components series

Nos_of_Members

number of members in each of groups with at least two members

Groups

indices of components in each of groups with at least two members

Pvalues

Pvalue for multiple test H_0 for each of $p(p-1)/2$ pairs in ascending order

NoConnectedPairs

number of connected pairs

Xpre

the prewhitened data with $n-R$ rows and $p$ columns

Details

See Chang et al. (2014) for the permutation step and more information.

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

See Also

segmentTS, permutationMax

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$X
# Plot the cross correlogram of x_t and y_t
Z=data.frame(Z)
names(Z)=c("Z1","Z2","Z3","Z4","Z5","Z6")
# The cross correlogram of z_t shows 3-2-1 block pattern
acfZ=acf(Z, plot=FALSE)
plot(acfZ, max.mfrow=6, xlab='', ylab='',  mar=c(1.8,1.3,1.6,0.5),
    oma=c(1,1.2,1.2,1), mgp=c(0.8,0.4,0),cex.main=1)
# Identify the permutation mechanism
permutation=permutationFDR(Z,Beta=10^(-8))
permutation$Groups


## 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$X
# Identify the permutation mechanism
permutation=permutationFDR(Z,Beta=10^(-200))
permutation$Groups