wave.multiple.cross.correlation: Wavelet routine for multiple cross-correlation
In jfdezmacho/wavemulcor: Wavelet Routines for Global and Local Multiple Regression and Correlation

Description Usage Arguments Details Value Note Author(s) References Examples

Produces an estimate of the multiscale multiple cross-correlation (as defined below).

1	wave.multiple.cross.correlation(xx, lag.max=NULL, p=.975, ymaxr=NULL)

`xx`	A list of n (multiscaled) time series, usually the outcomes of dwt or modwt, i.e. xx <- list(v1.modwt.bw, v2.modwt.bw, v3.modwt.bw)
`lag.max`	maximum lag (and lead). If not set, it defaults to half the square root of the length of the original series.
`p`	one minus the two-sided p-value for the confidence interval, i.e. the cdf value.
`ymaxr`	index number of the variable whose correlation is calculated against a linear combination of the rest, otherwise at each wavelet level wmc chooses the one maximizing the multiple correlation.

The routine calculates one single set of wavelet multiple cross-correlations out of n variables that can be plotted as one single set of graphs (one per wavelet level), as an alternative to trying to make sense out of n(n-1)/2 . J sets of wavelet cross-correlations. The code is based on the calculation, at each wavelet scale, of the square root of the coefficient of determination in a linear combination of variables that includes a lagged variable for which such coefficient of determination is a maximum.

List of two elements:

xy.mulcor: numeric matrix with as many rows as levels in the wavelet transform object. The columns provide the point estimates for the wavelet multiple cross-correlations at different lags (and leads). The central column (lag=0) replicates the wavelet multiple correlations. Columns to the right (lag>0) give wavelet multiple cross-correlations with positive lag, i.e. with y=var[Pimax] lagging behind a linear combination of the rest: x[t]hat –> y[t+j]. Columns to the left (lag<0) give wavelet multiple cross-correlations with negative lag, i.e. with y=var[Pimax] leading a linear combination of the rest: y[t-j] –> x[t]hat.
ci.mulcor: list of two elements:
- lower: numeric matrix of the same dimensions as xy.mulcor giving the lower bounds of the corresponding 100(1-2(1-p))\% confidence interval.
- upper: idem for the upper bounds.
YmaxR: numeric vector giving, at each wavelet level, the index number of the variable whose correlation is calculated against a linear combination of the rest. By default, wmcc chooses at each wavelet level the variable maximizing the multiple correlation.

Needs waveslim package to calculate dwt or modwt coefficients as inputs to the routine (also for data in the example).

Javier Fernández-Macho, Dpt. of Quantitative Methods, University of the Basque Country, Agirre Lehendakari etorb. 83, E48015 BILBAO, Spain. (email: javier.fernandezmacho at ehu.eus).

Fernández-Macho, J., 2012. Wavelet multiple correlation and cross-correlation: A multiscale analysis of Eurozone stock markets. Physica A: Statistical Mechanics and its Applications 391, 1097–1104. <DOI:10.1016/j.physa.2011.11.002>

## Based on data from Figure 7.9 in Gencay, Selcuk and Whitcher (2001)
## plus one random series.

library(wavemulcor)
data(exchange)
returns <- diff(log(exchange))
returns <- ts(returns, start=1970, freq=12)
N <- dim(returns)[1]

wf <- "d4"
J <- trunc(log2(N))-3
lmax <- 36
n <- dim(returns)[1]

demusd.modwt <- brick.wall(modwt(returns[,"DEM.USD"], wf, J), wf)
jpyusd.modwt <- brick.wall(modwt(returns[,"JPY.USD"], wf, J), wf)
rand.modwt   <- brick.wall(modwt(rnorm(length(returns[,"DEM.USD"])), wf, J), wf)

##xx <- list(demusd.modwt.bw, jpyusd.modwt.bw)
xx <- list(demusd.modwt, jpyusd.modwt, rand.modwt)

Lst <- wave.multiple.cross.correlation(xx, lmax)
returns.cross.cor <- Lst$xy.mulcor[1:J,]
returns.lower.ci <- Lst$ci.mulcor$lower[1:J,]
returns.upper.ci <- Lst$ci.mulcor$upper[1:J,]
YmaxR <- Lst$YmaxR

# ---------------------------

##Producing correlation plot

rownames(returns.cross.cor) <- rownames(returns.cross.cor, do.NULL = FALSE, prefix = "Level ")
par(mfrow=c(3,2), las=1, pty="m", mar=c(2,3,1,0)+.1, oma=c(1.2,1.2,0,0))
ymin <- -0.1
if (length(xx)<3) ymin <- -1
for(i in J:1) {
  matplot((1:(2*lmax+1)),returns.cross.cor[i,], type="l", lty=1, ylim=c(ymin,1), xaxt="n",
          xlab="", ylab="", main=rownames(returns.cross.cor)[[i]][1])
  if(i<3) {axis(side=1, at=seq(1, 2*lmax+1, by=12), labels=seq(-lmax, lmax, by=12))}
  #axis(side=2, at=c(-.2, 0, .5, 1))
  abline(h=0,v=lmax+1)              ##Add Straight horiz and vert Lines to a Plot
  lines(returns.lower.ci[i,], lty=1, col=2) ##Add Connected Line Segments to a Plot
  lines(returns.upper.ci[i,], lty=1, col=2)
  text(1,1, labels=names(xx)[YmaxR[i]], adj=0.25, cex=.8)
}
par(las=0)
mtext('Lag (months)', side=1, outer=TRUE, adj=0.5)
mtext('Wavelet Multiple Cross-Correlation', side=2, outer=TRUE, adj=0.5)