TIME SERIES MA413
Assignment 2:
(1) A zero mean stationary process Xn has the spectral density
f(λ) =
{
1, if 0 ≤ |λ| < 3π/4
0, if |λ| ≥ 3π/4.
(a) Find the ACVF γ(h) for h = ±0,±1, ....
(b) Find P (Xn|Xn?1), the best linear predictor of Xn given Xn?1. Find the MSE.
(c) Find P (Xn|Xn?1, Xn?2) the best linear predictor of Xn given Xn?1, Xn?2 (Note:
your answer should be of the form a1Xn?1+a2Xn?2, and you should explicitly solve
for a1, a2, but it is not necessary to simplify the resulting expressions).
(2) Suppose that you wish to analyze a data sequence x1, . . . , xN by putting it into a moving
average filter of the form wn = xn + axn?1, for some real constant a.
(a) What would be an appropriate choice for a if we wish to analyze the trend of the
data sequence? (Hint: trend corresponds to low Fourier frequencies)
(b) What would be an appropriate choice for a if we wish to analyze the random part of
the data sequence? (Hint: the random part corresponds to high Fourier frequencies)
(3) Let Xn be a stationary ARMA process. Suppose that Xn is not observable directly but
only with additive noise, with the observation process satisfying Yn = Xn +Wn, where
Wn is WN(0, 1), independent of Xn.
(a) Find the autocovariance function γY (h) and cross-covariance function γXY (h) in
terms of γX(h).
(b) Construct the Wiener filter for the best linear predictor of Xn based on Yn. Your
answer should be of the form aYn, where a is a constant expressed in terms of values
of γX(h).
(c) Construct the Wiener filter for the best linear predictor of Xn based on Yn, Yn?1.
Your answer should be of the form a0Yn + a1Yn?1, where a0, a1 are constants ex-
pressed in terms of values of γX(h). You do not need to fully simplify the expressions
for a0, a1.
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