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Correlation and Covariance

Module by: Nick Kingsbury

Summary: This module introduces correlation and covariance.

Correlation and covariance are techniques for measuring the similarity of one signal to another. For a random process Xtα X t α they are defined as follows.
  • Auto-correlation function:
    r X X t1t2=EXt1αXt2α=∫∫x1x2fx1x2dx1dx2 r X X t1 t2 X t1 α X t2 α x2 x1 x1 x2 f x1 x2 (1)
    where the expectation is performed over all α∈ α (i.e. the whole ensemble), and fx1x2 f x1 x2 is the joint pdf when x1 x1 and x2 x2 are samples taken at times t1 t1 and t2 t2 from the same random event αα of the random process XX.
  • Auto-covariance function:
    c X X t1t2=EXt1α-Xt1¯Xt2α-Xt2¯=∫x1∫x2x1-Xt1¯x2-Xt2¯fx1x2dx1dx2= r X X t1t2-2Xt1¯Xt2¯+Xt1¯Xt2¯= r X X t1t2-Xt1¯Xt2¯ c X X t1 t2 X t1 α X t1 X t2 α X t2 x2 x1 x1 x2 x1 X t1 x2 X t2 f x1 x2 r X X t1 t2 2 X t1 X t2 X t1 X t2 r X X t1 t2 X t1 X t2 (2)
    where the same conditions apply as for auto-correlation and the means Xt1¯ X t1 and Xt2¯ X t2 are taken over all α∈ α . Covariances are similar to correlations except that the effects of the means are removed.
  • Cross-correlation function: If we have two different processes, Xtα X t α and Ytα Y t α , both arising as a result of the same random event αα, then cross-correlation is defined as
    r X Y t1t2=EXt1αYt2α=∫∫x1y2fx1y2dx1dy2 r X Y t1 t2 X t1 α Y t2 α y2 x1 x1 y2 f x1 y2 (3)
    where fx1y2 f x1 y2 is the joint pdf when x1 x1 and y2 y2 are samples of XX and YY taken at times t1 t1 and t2 t2 as a result of the same random event αα. Again the expectation is performed over all α∈ α .
  • Cross-covariance function:
    c X Y t1t2=EXt1α-Xt1¯Yt2α-Yt2¯=∫x1∫y2x1-Xt1¯y2-Yt2¯fx1y2dx1dy2= r X Y t1t2-Xt1¯Yt2¯ c X Y t1 t2 X t1 α X t1 Y t2 α Y t2 y2 x1 x1 y2 x1 X t1 y2 Y t2 f x1 y2 r X Y t1 t2 X t1 Y t2 (4)
For Deterministic Random Processes which depend deterministically on the random variable αα (or some function of it), we can simplify the above integrals by expressing the joint pdf in that space. E.g. for auto-correlation:
r X X t1t2=EXt1αXt2α=∫xt1αxt2αfαdα r X X t1 t2 X t1 α X t2 α α x t1 α x t2 α f α (5)

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