# 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

 `1` ```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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57``` ```## 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.