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I have a set of $n$ low frequency time-varying signals $y(t)$.

I would like to express the variability within this set of signals using a single number. My current idea is to:

  1. Compute the average signal $\bar y(t)$ for the set using

    $$\bar y(t) = \dfrac{y_1(t) + y_2(t) + \ldots + y_n(t)}{n} $$

  2. Compute the 'standard deviation' of each individual signal with respect to the mean signal $\bar y$ using the Pearson correlation $r$

    $$\bar r = \dfrac{ r(y_1(t), \bar y(t)) + r(y_2(t), \bar y(t)) + \ldots + r(y_n(t), \bar y(t))}{n}$$

This would then give me a sense of the variability within the set of signals.

My question is whether there are better, different, or more established ways of doing this?

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  • $\begingroup$ I'm not quite sure I can 100% visualize the math you want to do. Why not add mathematical formulas to the question, like $\bar y = \sum\ldots$, so that we can discuss this on a definite basis? $\endgroup$ Oct 15, 2015 at 9:50
  • $\begingroup$ What variations do you expect to see between the different signals? Can you model $y_k$ so that you can parameterize these variations? That might give us a better clue as to how to help. Sometimes signals that are closer to deterministic than random don't work as well as you'd like with correlations. $\endgroup$
    – Peter K.
    Oct 15, 2015 at 12:46
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    $\begingroup$ Good point. The signals are fMRI BOLD, which is usually modeled using a double gamma function. I think you are right that this the way to go. Modeling the signals would then allow me to examine the variability of many different parameters that correlation alone ignores. Thanks for the advice. $\endgroup$ Oct 15, 2015 at 13:11

2 Answers 2

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As an example of what I mean, let's have a look at a simple sinusoid (though I see you're actually assuming double gamma functions; I may rework this later to use those).

This models the signals as: $$ y_p(t) = \sin(\omega t + \phi_p) $$ so the only difference between them is the phase $\phi_p = (p-1)\pi/6$.

The top plot is the mean signal $\bar{y}(t)$ and the bottom plot shows the Pearson coefficient between $\bar{y}$ and $y_p$ for $p=1\ldots 6$ (so $n=6$).

enter image description here

As you can see, the Pearson coefficient varies quite widely even in this simple example.


R Code Below

# 26438

Nsigs <- 6
T<- 1024
omega <- 2*pi*0.00982734982
t <- 0:(T-1)

y <- matrix(,nrow = T, ncol = Nsigs)
phi <- c(0, 1*pi/6, 2*pi/6, 3*pi/6, 4*pi/6, 5*pi/6 )

for (idx in 1:Nsigs)
{
  #phi[idx] <- runif(1)*2*pi
  y[,idx] <- sin(omega*t + phi[idx])  
}

mean <- rowSums(y) / Nsigs

total <- matrix(,nrow = T, ncol = Nsigs+1)
total[,1] <- mean
total[,2:(Nsigs+1)] = y

pearson <- cor(total,use="complete.obs", method="pearson")

par(mfrow=c(2,1),pty="m")
plot(t,mean, type="l")
title("Mean value")

plot(1:6, pearson[1,2:7], pch=19, col="blue")
title("Pearson coefficient")
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Compute the matrix Mij = [ y_i(t) y_j(t) ], where 0 <= i < n and 0 <= j < n. This matrix Mij will approximate a gain times the identity matrix when the signals are perfectly uncorrelated. So I suppose you could use sum of squares of the deviation from the gain times the identity matrix as a norm.

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