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Math Help - How does autocorrelation for statisticians relate to the same but for engineers?

  1. #1
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    How does autocorrelation for statisticians relate to the same but for engineers?

    Hi, I was wondering if you could help me. I'm a little stuck with marrying up the ideas of autocorrelation from a statisticians point of view and an engineers point of view. I am neither of these but need to understand how they relate to one another.

    The statisticians point of view:
    Chris Chatfield defines in 'The Analysis of Time Series' p23 the autocovariance coefficients at lag k for a discrete time series \left\{x_t\right\}_{t=1}^N,

    c_k=\frac{1}{N}\sum_{t=1}^{N}(x_t-\overline{x})(x_{[t+k]mod N}-\overline{x})

    I'm taking my time series to have period N here. Then the autocorrelation at lag k is defined as,

    r_k=\frac{c_k}{c_0}

    The autocorrelation is normalised with respect to the autocorrelation at lag 0 and centred about the mean. I understand that the method looks to identify any linear relationships present in the data at differing lag times.

    Another definition in the same book is the autocorrelation function,

    \frac{Cov[X(t),X(t+\tau)]}{Var[X(t)]}

    Is this the form of the autocorrelation at lag k, but for lag \tau with an infinite time series? Can I use both forms for unrealised random variables?

    The engineers define the autocorrelation in 'Digital Signal Processing' by Sanjit Mitra p105, at lag k to be,

    r_k(x)=\frac{1}{N}\sum_{t=1}^Nx_tx_{[t-k]modN}

    This form isn't centred about the mean or normalised and is defined with the lag backwards in time. I believe the engineers are trying to check for patterns in the time series just the same, but only wish to check whether the result is associated with white noise or not. Does anyone know if this is really what is being done and I'm not missing something?

    Thanks :-)
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  2. #2
    MHF Contributor
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    Re: How does autocorrelation for statisticians relate to the same but for engineers?

    Hey alanaj5.

    It's hard to say what the case is (with the mean being assumed to be zero).

    As a guess, you might want to check the book if it's referring to a distribution with mean 0 and variance 1 (which fits a residual model for normal distributions especially in the context of errors).

    Can you give a bit more context and information that surrounds the buildup to this definition?
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