Problem 1

Gold: Daily morning gold prices in US dollars. 1 January 1985 - 31 March 1989.
woolyrnq: Quarterly production of woollen yarn in Australia: tonnes. Mar 1965 - Sep 1994.
Gas: Australian monthly gas production: 1956-1995.

Part A

Part B

Gas.Frequency Gold.Frequency Wool.Frequency
12 1 4

Part C

Observation: 770 value: 593.7

Problem 2

Part A

X1 Sales AdBudget GDP
Mar-81 1020.2 659.2 251.8
Jun-81 889.2 589.0 290.9
Sep-81 795.0 512.5 290.8
Dec-81 1003.9 614.1 292.4
Mar-82 1057.7 647.2 279.1
Jun-82 944.4 602.0 254.0

Part B

Class of object:

## [1] "mts"    "ts"     "matrix"

Part C

If we remove the faceting, all three series share the same axis

Problem 3

Part A

Part B

We choose the 12th column, Turnover from New South Wales department stores.

Part C

Just by eyeballing the raw plot, there is an obvious upward trend and spikes at seamingly regular intervals. The seasonality is a little tougher to eyeball, as it’s harder to see the number of intervals between spikes

The most obvious seasonal pattern is from November to December. Almost all the invididual series have the same upward slope between those two months. There are noticeables spikes in May and January, though there is some crossover amongst the lines on their path to May.

Here, we see the same seasonal pattern but also the trend pattern. All within month lines are clearly upward sloping, indicating the trend.

Lines that remain above the dotted lined indicate a peak in the seasonality against the lagged month. EX: Month 12 should stay above the dotted line because Decemeber is always greather than the month it’s compared against. The opposite is true for troughs in seasonality. The trend pattern is also pretty apparent, as each individual line follows a clear upward slope. We also see the expected peak-trough relationships and lag 12 shows strong autocorrelation.

This follows the typical pattern for trended and seasonal data, with positive autocorrelation and peaks at 12 and 24.

Problem 6

Home Sales

This data has seasonal and cyclical patterns. There is a period between 1982 and 1989 where sales are flat, other than an outlier and some seasonal pattern. There is then a period 1991 to 1996 where sales are generally rising.

From the subseries and seasonal plots, it is apparent that the spring months are higher than the rest of the year. There are spikes at multiples of the frequency (12 ad 24) in the ACF plot. The subseries plot shows the lack of trend, as the within month pattern over time seems to be random.

US deaths

The US deaths data only shows a seasonal pattern. There aren’t enough years in the data to determine if there is a cyclic pattern and there is no obvious trend pattern. There is an obvious relation between warm weather and deaths, with troughs in the colder months and peaks in the warmer months. Lag 12 is highly linear in the lag plot.

Bricks

The brick data has a strong trend from the early 1960s until the early 1980s. After the 1980s, the trend turns to a cyclic pattern, likely dependent on the economy. There is also a weak seasonal pattern.

Quaters 2 and 3 appear a bind higher on the subseries and seasonal plots. There is also a small, but noticeable peak in the ACF at multiples of 4. With cyclical patterns over many years, the lag plot becomes too busy to be useful.

Sunspots

There is either a seasonal of cyclical pattern that is apparent, but the data has a frequency of 1. We adjust the frequency to use the seasonal and subseries plots

With a frequency of 10, the data exhibits a seasonal pattern. Peaks and troughs are apparent in the seasonal, subseries, and ACF plots. Lag 10 is also the most linear in the lag plot.

Gasoline

There are trend, cyclical, and seasonal patterns evident in this time series.

An increasing trend can be seen until the mid-2000s, afterwhich the pattern turns cyclical. This is likely because of the introduction of more fuel efficient vehicles. Differentiating between trend and cyclical can be a matter of choice, but if there is an underlying reason for a trend to cease, I would not classify it as a cycle.

With a frequency of 52, the seasonal and lag plots are less useful, but the seasonal trend can be spotted by the peaks at multiples of 52 in the ACF.