Consider the white noise process wt, and a 5-point moving average process vt of it. (a) Determine the theoretical auto-covariance for vt,

Consider the white noise process wt, and a 5-point moving average process vt of it.

(a) Determine the theoretical auto-covariance for vt, and the theoretical cross-covariance function between wt and vt.

(b) Generate a realization of wt of length 1000, and compute the associated 5-point moving average vt. Plot these two time series on the same graph.

Calculate the corresponding sample versions for the ACF and CCF of part (a). Remark on how these resemble and dier from the theoretical re- sults. Then try the same thing for another realization of length 1000 and remark.

3. Oceanographic and meteorological information is recorded at  the Halifax buoy located about 20 km SE of Halifax Harbour at 44.50N 63.40W. I have posted, on the course website, 20 days of hourly summer air and sea temperature (start time is July 3, 2011). This is a .RData le with AT (Air Tempera- ture) and SST (Sea Surface Temperature) as variables. Plot and compare the series. Then compute and plot the sample ACF for both series and interpret the results in terms of the statistical character of each series (use a maximum lag of 50 hours). Next, compute and plot the sample CCF between the series and state what this result says about the relative timing of temperature changes in the ocean versus the atmosphere (i.e. which one leads or lags the other, and by how much)? Note that plotting the series one over the other may help to see the lead/lag relationship.

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