Thread: Regarding the Autocovariance Function (Time Series Analysis)

1. Part of an assignment I'm working on involves a demonstration and explanation of the Wiener-Khintchine Theorem, which simply states that the autocovariance function and the power spectrum of a time series form a Fourier-transform pair. The problem is, I'm a bit confused about the definition of the autocovariance of a series.

Consider the series x(t) = 2, 3, 1. One definition I have states that:

ACVF(τ) = E[(x(t) - μ)(x(t + τ) - μ)], μ = E[x] = 2

Thus,

ACVF(0) = [(x(1) - 2)(x(1) - 2) + (x(2) - 2)(x(2) - 2) + (x(3) - 2)(x(3) - 2)] / 3

ACVF(1) = [(x(1) - 2)(x(2) - 2) + (x(2) - 2)(x(3) - 2) + (x(3) - 2)(x(4) - 2)] / 3

ACVF(-1) = [(x(1) - 2)(x(0) - 2) + (x(2) - 2)(x(1) - 2) + (x(3) - 2)(x(2) - 2)] / 3

ACVF(2) = [(x(1) - 2)(x(3) - 2) + (x(2) - 2)(x(4) - 2) + (x(3) - 2)(x(5) - 2)] / 3

ACVF(-2) = [(x(1) - 2)(x(-1) - 2) + (x(2) - 2)(x(0) - 2) + (x(3) - 2)(x(1) - 2)] / 3

My questions is, if x(1) = 2, x(2) = 3, x(3) = 1, what do x(-1), x(0), x(4) and x(5) represent in the expressions above? Do I simply assume the series is periodic with a period of 3 (ie. x(4) = x(1), x(5) = x(2), etc)? Do I assume all other values of x are 0? Or do I simply not include the terms where t ≠ 1, 2, 3 in the expressions at all?

I hope my question is clear, but even I get confused thinking about this stuff.

2. The autocovariance of a series doesn't make sense in your example. Usually you would need an infinite series.

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