The observations are quarterly. The Bricks column has some missing values (at the end of the series).
aus_production |>autoplot(Bricks)
An upward trend is apparent until 1980, after which the number of clay bricks being produced starts to decline. A seasonal pattern is evident in this data. Some sharp drops in some quarters can also be seen.
Lynx
interval(pelt)
<interval[1]>
[1] 1Y
Observations are made once per year.
pelt |>autoplot(Lynx)
Canadian lynx trappings are cyclic, as the extent of peak trappings is unpredictable, and the spacing between the peaks is irregular but approximately 10 years.
Interval is daily. Looking closer at the data, we can see that the index is a Date variable, and that observations occur only on trading days, creating lots of implicit missing values (weekends and public holidays).
gafa_stock |>autoplot(Close)
Stock prices for these technology stocks have risen for most of the series, until mid-late 2018.
The four stocks are on different scales, so they are not directly comparable. A plot with faceting would be better.
gafa_stock |>ggplot(aes(x = Date, y = Close, group = Symbol)) +geom_line(aes(col = Symbol)) +facet_grid(Symbol ~ ., scales ='free')
The downturn in the second half of 2018 is now very clear, with Facebook taking a big drop (about 20%) in the middle of the year.
The stocks tend to move roughly together, as you would expect with companies in the same industry.
Appears to have a daily pattern, where less electricity is used overnight. Also appears to have a working day effect (less demand on weekends and holidays).
Here the annual seasonality is clear, with high volatility in summer, and peaks in summer and winter. The weekly and daily seasonality are hidden due to the compression on the horizontal axis.
fpp3 2.10, Ex 2
Use filter() to find what days corresponded to the peak closing price for each of the four stocks in gafa_stock.
# A tsibble: 4 x 3 [!]
# Key: Symbol [4]
Symbol Date Close
<chr> <date> <dbl>
1 AAPL 2018-10-03 232.
2 AMZN 2018-09-04 2040.
3 FB 2018-07-25 218.
4 GOOG 2018-07-26 1268.
fpp3 2.10, Ex 3
Download the file tute1.csv from the book website, open it in Excel (or some other spreadsheet application), and review its contents. You should find four columns of information. Columns B through D each contain a quarterly series, labelled Sales, AdBudget and GDP. Sales contains the quarterly sales for a small company over the period 1981-2005. AdBudget is the advertising budget and GDP is the gross domestic product. All series have been adjusted for inflation.
You can read the data into R with the following script:
# Without faceting:mytimeseries |>pivot_longer(-Quarter, names_to ="Key", values_to ="Value") |>ggplot(aes(x = Quarter, y = Value, colour = Key)) +geom_line()
Each series is shown in its own panel, with its own scale. If we plot the three series on the same axes instead, the scales of the series are very different, so it is harder to see the patterns in GDP.
fpp3 2.10, Ex 4
The USgas package contains data on the demand for natural gas in the US.
Install the USgas package.
Create a tsibble from us_total with year as the index and state as the key.
Plot the annual natural gas consumption by state for the New England area (comprising the states of Maine, Vermont, New Hampshire, Massachusetts, Connecticut and Rhode Island).
library(USgas)us_tsibble <- us_total |>as_tsibble(index = year, key = state)# For each stateus_tsibble |>filter( state %in%c("Maine","Vermont","New Hampshire","Massachusetts","Connecticut","Rhode Island" ) ) |>autoplot(y /1e3) +labs(y ="billion cubic feet")
fpp3 2.10, Ex 5
Download tourism.xlsx from the book website and read it into R using readxl::read_excel().
Create a tsibble which is identical to the tourism tsibble from the tsibble package.
Find what combination of Region and Purpose had the maximum number of overnight trips on average.
Create a new tsibble which combines the Purposes and Regions, and just has total trips by State.
# Download tourism.xlsx from the book website first, and save it in your working directorymy_tourism <- readxl::read_excel("tourism.xlsx") |>mutate(Quarter =yearquarter(Quarter)) |>as_tsibble(index = Quarter,key =c(Region, State, Purpose) )my_tourism
# A tsibble: 24,320 x 5 [1Q]
# Key: Region, State, Purpose [304]
Quarter Region State Purpose Trips
<qtr> <chr> <chr> <chr> <dbl>
1 1998 Q1 Adelaide South Australia Business 135.
2 1998 Q2 Adelaide South Australia Business 110.
3 1998 Q3 Adelaide South Australia Business 166.
4 1998 Q4 Adelaide South Australia Business 127.
5 1999 Q1 Adelaide South Australia Business 137.
6 1999 Q2 Adelaide South Australia Business 200.
7 1999 Q3 Adelaide South Australia Business 169.
8 1999 Q4 Adelaide South Australia Business 134.
9 2000 Q1 Adelaide South Australia Business 154.
10 2000 Q2 Adelaide South Australia Business 169.
# ℹ 24,310 more rows
tourism
# A tsibble: 24,320 x 5 [1Q]
# Key: Region, State, Purpose [304]
Quarter Region State Purpose Trips
<qtr> <chr> <chr> <chr> <dbl>
1 1998 Q1 Adelaide South Australia Business 135.
2 1998 Q2 Adelaide South Australia Business 110.
3 1998 Q3 Adelaide South Australia Business 166.
4 1998 Q4 Adelaide South Australia Business 127.
5 1999 Q1 Adelaide South Australia Business 137.
6 1999 Q2 Adelaide South Australia Business 200.
7 1999 Q3 Adelaide South Australia Business 169.
8 1999 Q4 Adelaide South Australia Business 134.
9 2000 Q1 Adelaide South Australia Business 154.
10 2000 Q2 Adelaide South Australia Business 169.
# ℹ 24,310 more rows
The aus_arrivals data set comprises quarterly international arrivals to Australia from Japan, New Zealand, UK and the US.
Use autoplot(), gg_season() and gg_subseries() to compare the differences between the arrivals from these four countries.
Can you identify any unusual observations?
aus_arrivals |>autoplot(Arrivals)
Generally the number of arrivals to Australia is increasing over the entire series, with the exception of Japanese visitors which begin to decline after 1995. The series appear to have a seasonal pattern which varies proportionately to the number of arrivals. Interestingly, the number of visitors from NZ peaks sharply in 1988. The seasonal pattern from Japan appears to change substantially.
The seasonal pattern of arrivals varies between countries. Arrivals from the UK and the US peak in Q4 and Q1, which include the Australian summer and the Christmas and New Year holiday period, with Q2 and Q3 being the troughs. Arrivals from Japan are lowest in Q2, and usually highest in Q1 or Q4, although in some years Q3 is the peak; Q3 arrivals possibly correspond to the winter ski season or to visits to northern Australia during the dry season. New Zealand is noticeably different: arrivals are lowest in Q1 (the summer quarter) in every year, and usually highest in Q3.
aus_arrivals |>gg_subseries(Arrivals)
The subseries plot reveals more interesting features. It is evident that whilst the UK arrivals is increasing, most of this increase is seasonal. More arrivals are coming during Q1 and Q4, whilst the increase in Q2 and Q3 is less extreme. The growth in arrivals from NZ and US appears fairly similar across all quarters. There exists an unusual spike in arrivals from the US in 1991 Q3.
Unusual observations:
2000 Q3: Spikes from the US (Sydney Olympics arrivals)
2001 Q3-Q4 are unusual for US (9/11 effect)
1991 Q3 is unusually high for the US
fpp3 2.10, Ex 7
Monthly Australian retail data is provided in aus_retail. Select one of the time series as follows (but choose your own seed value):
With this seed, we have turnover for clothing, footwear and personal accessory retailing in the Northern Territory. The data features a non-linear upward trend and a strong seasonal pattern. The variability in the data appears proportional to the amount of turnover (level of the series) over the time period.
Strong seasonality is evident in the season plot. Large increases in clothing retailing can be observed in December (probably a Christmas effect). There is also a peak in July that appears to be getting stronger over time. 2016 had an unusual pattern in the first half of the year.
myseries |>gg_subseries(Turnover) +labs(y ="Turnover (million $AUD)", x ="")
There is a strong trend in all months, with the largest trend in December and a larger increase in July and August than most other months.
The lag plots and ACF show strong positive autocorrelation that decays slowly (due to the trend), with peaks at lags 12, 24, 36 and 48 (due to the seasonality).
fpp3 2.10, Ex 8
Use the following graphics functions: autoplot(), gg_season(), gg_subseries(), gg_lag(), ACF() and explore features from the following time series: “Total Private” Employed from us_employment, Bricks from aus_production, Hare from pelt, “H02” Cost from PBS, and Barrels from us_gasoline.
Can you spot any seasonality, cyclicity and trend?
In all of these plots, the trend is so dominant that it is hard to see anything else. We need to remove the trend so we can explore the other features of the data.
Brick production in Australia
aus_production |>autoplot(Bricks)
A positive trend in the first 20 years, and a negative trend in the next 25 years. Clear quarterly seasonality, although it is small compared to the trend, with some cyclicity – note the recessions in the 1970s and 1980s, which show up as unusually low production in 1975 Q1, and in 1982 Q4 and 1983 Q1. Production was also unusually low in early 2001.
aus_production |>gg_season(Bricks)
Brick production tends to be lowest in the first quarter and peak in either quarter 2 or quarter 3.
aus_production |>gg_subseries(Bricks)
The decrease in the last 25 years has been weakest in Q1.
aus_production |>gg_lag(Bricks, geom ='point')
aus_production |>ACF(Bricks) |>autoplot()
The strong positive relationships in the lag plots are dominated by the trend. The seasonality shows up as small peaks at lags 4, 8, 12, 16, 20, …. The trend is seen with the slow decline on the positive side.
Snow hare trappings in Canada
pelt |>autoplot(Hare)
There is some cyclic behaviour with substantial variation in the length of the period. The peaks in 1863–1864 and 1885–1886 are unusually high. This is annual data, so there is no seasonality and we don’t need seasonal plots.
pelt |>gg_lag(Hare, geom ='point')
pelt |>ACF(Hare) |>autoplot()
The cyclic period seems to have an average of about 10 (due to the local maximum in ACF at lag 10).
H02 sales in Australia
There are four series corresponding to H02 sales, so we will add them together.
A positive trend with strong annual seasonality: costs rise through the year to a peak in December or January, and drop suddenly every February.
h02 |>gg_season(Cost)
h02 |>gg_subseries(Cost)
The trends have been greater in the higher peaking months – this leads to increasing seasonal variation.
h02 |>gg_lag(Cost, geom ='point', lags =1:16)
h02 |>ACF(Cost) |>autoplot()
The sudden drop every February splits the lag plots into two clusters: pairs of months that straddle the drop fall well below the main band of points. At lag 12 the points lie close to a straight line, and the strong seasonality is clear in the ACF plot, with a peak at lag 12.
US gasoline sales
us_gasoline |>autoplot(Barrels)
A positive trend until 2008, and then the global financial crisis led to a drop in sales until 2012. The shape of the seasonality seems to have changed over time.
us_gasoline |>gg_season(Barrels)
There is a lot of noise making it hard to see the overall seasonal pattern. However, sales tend to be highest in Q3 (the northern summer) and lowest in Q1.
us_gasoline |>gg_subseries(Barrels)
The seasonal period of weekly data is not an integer (about 52.18 weeks per year), but gg_subseries() handles this by grouping the observations by week of the year. With 52 panels, the individual subseries are too cramped to be useful, but the blue lines are helpful in seeing the average seasonal pattern.
The lag plots are not very useful, as the seasonal period (about 52 weeks) is much longer than the lags shown; they mostly reflect the trend. The seasonality is seen in the ACF if we increase the lags to at least 2 years (approx 104 weeks).
fpp3 2.10, Ex 9
The following time plots and ACF plots correspond to four different time series. Your task is to match each time plot in the first row with one of the ACF plots in the second row.
1-B, 2-A, 3-D, 4-C
fpp3 2.10, Ex 10
The aus_livestock data contains the monthly total number of pigs slaughtered in Victoria, Australia, from Jul 1972 to Dec 2018. Use filter() to extract pig slaughters in Victoria between 1990 and 1995. Use autoplot() and ACF() for this data. How do they differ from white noise? If a longer period of data is used, what difference does it make to the ACF?
# A tsibble: 72 x 4 [1M]
# Key: Animal, State [1]
Month Animal State Count
<mth> <fct> <fct> <dbl>
1 1990 Jan Pigs Victoria 76000
2 1990 Feb Pigs Victoria 78100
3 1990 Mar Pigs Victoria 77600
4 1990 Apr Pigs Victoria 84100
5 1990 May Pigs Victoria 98000
6 1990 Jun Pigs Victoria 89100
7 1990 Jul Pigs Victoria 93500
8 1990 Aug Pigs Victoria 84700
9 1990 Sep Pigs Victoria 74500
10 1990 Oct Pigs Victoria 91900
# ℹ 62 more rows
vic_pigs |>autoplot(Count)
Although the values appear to vary erratically between months, a general upward trend is evident between 1990 and 1995. In contrast, a white noise plot does not exhibit any trend.
vic_pigs |>ACF(Count) |>autoplot()
The first 15 lags are significant, as the ACF slowly decays. This suggests that the data contains a trend. A white noise ACF plot would not usually contain any significant lags. The large spike at lag 12 suggests there is some seasonality in the data.
aus_livestock |>filter(Animal =="Pigs", State =="Victoria") |>ACF(Count) |>autoplot()
The longer series has much larger autocorrelations that decay more slowly, as the trend dominates over the longer period, plus a clear seasonal peak at lag 12.
fpp3 2.10, Ex 11
Use the following code to compute the daily changes in Google closing stock prices.
The tsibble needed re-indexing as trading happens irregularly. The new index is based only on trading days, so consecutive observations are one step apart.
dgoog |>autoplot(diff)
dgoog |>ACF(diff, lag_max =100) |>autoplot()
There are only two small significant autocorrelations (at lags 8 and 24) out of 100. Given the probability of a false positive is 5%, these look similar to white noise.
