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I'm trying to understand the different aspects of a wavelet transform. Wavelet power has made enough sense to me as an analogy of the covariance. However, the wavelet coherence does not make sense to me. The notes in the R package 'WaveletComp' states that coherence is analogous to the coefficient of correlation. In my intuition, that would mean coherence between independent time series should be reasonably low. However, this is not the case, as is shown by the R code below.

library(tidyverse)
library(WaveletComp)

complete_noise <- tibble(x = seq(1:1024)) %>% 
      mutate(
        sample_1 = rnorm(x),
        sample_2 = rnorm(x)
  )

wc_noise <- analyze.coherency(complete_noise, c("sample_1", 
    "sample_2"), loess.span = 0, make.pval = F)

wc.image(wc_noise, which.image = "wc", color.key = "interval")

enter image description here

The plot shows the majority of the area as red, which is up near a coherence of 1.0. Why do independent time series have a high coherence in most of these points on the period-time plot?

Edit: I turned off smoothing (it is on by default with WaveletComp) and now the plot shows a coherence of 1 everywhere. I'm taking this to mean the drops in coherence are due to smoothing, which also doesn't make sense to me.

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1 Answer 1

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While I don't have an answer to the question as titled, I did find an answer to the underlying question (how can I make the analysis work).

First, the fact that without smoothing, wavelet transforms are . In page 7 of the WaveletComp documentation, it says

The concept of Fourier coherency measures the cross-correlation between two time series as a function of frequency; an analogous concept in wavelet theory is the notion of wavelet coherency, which, however, requires smoothing of both the cross-wavelet spectrum and the normalizing individual wavelet power spectra (without smoothing, its absolute value would be identically 1; see Liu [8])

This aligns perfectly with what I found - without smoothing, all coherence values are 1. I still don't understand why, but that is not necessary for the work I am doing right now.

The real solution to my problem was that the smoothing was too small of a window - by increasing the smoothing some (about a factor of 5), I got a similar value for coherence to what I was expecting.

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