permutationMax: Permutation Using the Maximum Cross Correlation Method

In PCA4TS: Segmenting Multiple Time Series by Contemporaneous Linear Transformation

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

permutationMax(X, Vol = FALSE, 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.
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.

Details

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

Value

An object of class "permutationMax" 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
maxcorr	maximum correlation (over lags) of p(p-1)/2 pairs in descending order
corrRatio	ratios of successive values from maxcorr
NoConnectedPairs	number of connected pairs
Xpre	the prewhitened data with n-R rows and p columns

Note

This is the second step for segmentation by grouping the transformed time series. The first step is to seek for a contemporaneous linear transformation of the original series, see segmentTS.

Author(s)

Jinyuan Chang, Bin Guo and Qiwei Yao

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

Example output

No of groups with more than one members: 2 
Nos of members in those groups: 3 2 
No of connected pairs: 4 
     [,1] [,2]
[1,]    1    2
[2,]    3    4
[3,]    6    0
[4,]    0    0
[5,]    0    0
[6,]    0    0

No of groups with more than one members: 5 
Nos of members in those groups: 6 5 4 3 2 
No of connected pairs: 35 
      [,1] [,2] [,3] [,4] [,5]
 [1,]    1    2    3    6   11
 [2,]    5    4    8   12   16
 [3,]    7    9   14   19    0
 [4,]   10   13   18    0    0
 [5,]   15   17    0    0    0
 [6,]   20    0    0    0    0
 [7,]    0    0    0    0    0
 [8,]    0    0    0    0    0
 [9,]    0    0    0    0    0
[10,]    0    0    0    0    0
[11,]    0    0    0    0    0
[12,]    0    0    0    0    0
[13,]    0    0    0    0    0
[14,]    0    0    0    0    0
[15,]    0    0    0    0    0
[16,]    0    0    0    0    0
[17,]    0    0    0    0    0
[18,]    0    0    0    0    0
[19,]    0    0    0    0    0
[20,]    0    0    0    0    0

PCA4TS documentation built on May 2, 2019, 9:42 a.m.