# Correlograms

Also covariances

August 8, 2018 — September 22, 2019

dynamical systems
Hilbert space
linear algebra
signal processing
statistics
time series

This material is revised and expanded from the appendix of draft versions of a recent conference submission, for my own reference. I used (deterministic) correlograms a lot in that, and it was startlingly hard to find a decent summary of their properties anywhere. Nothing new here, but… see the matrial about doing this in a probabilistic way via Wiener-Khintchine representation and covariance kernels which lead to a natural probabilistic spectral analysis.

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Consider an $$L_2$$ signal $$f: \bb{R}\to\bb{R}.$$ We frequently overload notation and refer to as signal with free argument $$t$$, so that $$f(rt-\xi),$$ for example, refers to the signal $$t\mapsto f(rt-\xi).$$ We write the inner product between signals $$t\mapsto f(t)$$ and $$t\mapsto f'(t)$$ as $$\inner{f(t)}{f'(t)}$$. Where it is not clear that the free argument is, e.g. $$t$$, we annotate it $$\finner{f(t)}{f'(t)}{t}$$.

The correlogram $$\cc{A}:L_2(\bb{R}) \to L_2(\bb{R})$$ maps signals to signals. Specifically, $$\mathcal{A}\{f\}$$ is a signal $$\bb{R}\to\bb{R}$$ such that

$\mathcal{A}\{f\}:=\xi \mapsto \finner{ f(t) }{ f(t-\xi) }{t}$ This is the covariance between $$f(t)$$ and $$f(t-\xi).$$ (Note that we here discuss the covariance between given deterministic signals, not between two stochastic sources; covariance of stochastic processes is a broader, let alone inferring the covariance of stochastic processes.) Note also that this is what I would call an autocovariance not an auto-correlation, since it’s not normalized, but I’ll stick with the latter for now since for reasons of convention.

We derive the properties of this transform.

Multiplication by a constant. Consider a constant $$c\in \bb{R}.$$

\begin{aligned}\mathcal{A}\{cf\}(\xi)&= \inner{ cf(t) }{ cf(t-\xi) }\\ &= c^2\finner{ f(t) }{ f(t-\xi) }{t}\\ &= c^2\mathcal{A}\{f\}(\xi).\\ \end{aligned}

Time scaling:

\begin{aligned}\mathcal{A}\{f(r t)\}(\xi) &=\finner{ f(r t) }{ f(r t-\xi) }{t}\\ &= \int f(r t)f(r t-\xi)\dd t\\ &= \frac{1}{r }\int f(t)f(t-\frac{\xi}{r})\dd t\\ &= \frac{1}{r} \mathcal{A}\{f\}\left(\frac{\xi}{r}\right)\\ \end{aligned}

\begin{aligned}\mathcal{A}\{f+f'\}(\xi) &=\finner{ f(t)+f'(t) }{ f(t-\xi)+f'(t-\xi) }{t}\\ &=\finner{ f(t) }{ f(t-\xi)\rangle+\langle f(t),f'(t-\xi) }{t} +\finner{ f'(t) }{ f(t-\xi)\rangle+\langle f'(t),f'(t-\xi) }{t}\\ &= \mathcal{A}\{f\}(\xi)+ \finner{ f'(t) }{ f(t-\xi)}{t} +\finner{f(t)}{f'(t-\xi) }{t} +\mathcal{A}\{f'\}(\xi).\\ &= \mathcal{A}\{f\}(\xi)+ \finner{ f'(t) }{ f(t-\xi)}{t} +\finner{f(t+\xi)}{f'(t) }{t} +\mathcal{A}\{f'\}(\xi).\\ &= \mathcal{A}\{f\}(\xi)+ \finner{ f'(t) }{ f(t-\xi)}{t} +\finner{f'(t) }{f(t+\xi)}{t} +\mathcal{A}\{f'\}(\xi).\\ \end{aligned}

We can say little about the term $$\finner{ f'(t) }{ f(t-\xi)}+\finner{f'(t) }{f(t+\xi)}{t}$$ without more information about the signals in question. However, we can solve a randomized version. Suppose $$S_i, \, i \in\bb{N}$$ are i.i.d. Rademacher variables, i.e. that they assume a value in $$\{+1,-1\}$$ with equal probability. Then, we can introduce the following property:

\begin{aligned} \bb{E}[ \mathcal{A}\{S_1f + S_2f'\}(\xi) &=\bb{E}[ \mathcal{A}\{S_1f\}(\xi) + \finner{ S_2 f'(t) }{ S_1 f(t-\xi)}{t} +\finner{S_2f'(t) }{S_1 f(t+\xi)}{t} +\mathcal{A}\{S_2f'\}(\xi)]\\ &=\bb{E}[ \mathcal{A}\{S_1f\}(\xi)] + \bb{E}\finner{ S_2 f'(t) }{ S_1 f(t-\xi)}{t} + \bb{E}\finner{S_2f'(t) }{S_1 f(t+\xi)}{t} +\bb{E}[ \mathcal{A}\{S_2f'\}(\xi)]\\ &=\mathcal{A}\{f\}(\xi)+ \bb{E}[ S_1S_2]\finner{ f'(t) }{ f(t-\xi) }{t} + \bb{E}[ S_1S_2]\finner{ f'(t) }{ f(t+\xi) }{t}+\mathcal{A}\{f'\}(\xi)\\ &=\mathcal{A}\{f\}(\xi)+ \mathcal{A}\{f'\}(\xi)\\ \end{aligned}

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