All three autocorrelation plots are consistent with white noise. The three figures all show autocorrelation plots for the random number sets that fluctuate around zero with no pattern. Each bar represents the correlation between the linear relationship of lagged values.
In each set less than 5% of the lags exceed the confidence interval. This is an acceptable amount of values that exceed that threshold and still have it be considered white noise. As the sample size increases, the autocorrelations fluctuate less around zero and the bounds become narrower.
b.
As the size of the sample sets gets larger, the 95% confidence threshold gets smaller because the larger sample size provides more precise estimates. The bounds are set by dividing the z-value of 1.96 by the square root of the length of the time series. The more samples in the set, the smaller the interval.
The autocorrelations are different in each figure because they are collections of random samples. Within each sample set there is no indication of autocorrelation, and since all sets are different the autocorrelations all differ. The larger sample size ensures less variation.
Stationary data will not have a trend or seasonality. While the Amazon closing stock price does not have seasonality, it does have a trend, indicating it is non-stationary.
The Amazon autocorrelation is plotted below.
amzn |>ACF(Close) |>autoplot()
Warning: Provided data has an irregular interval, results should be treated
with caution. Computing ACF by observation.
The Amazon closing stock price is non-stationary. For a stationary time series the ACF will drop to zero relatively quickly, the Amazon closing stock price decreases slowly, indicating non-stationary data.
The amazon partial-autocorrelation is plotted below.
amzn |>PACF(Close) |>autoplot()
Warning: Provided data has an irregular interval, results should be treated
with caution. Computing ACF by observation.
The PACF for the Amazon closing stock price shows a single spike at the first lag and then many much smaller spikes, this is consistent with a random-walk like process which is a type of non-stationary data.
The Turkish GDP is Box-Cox transformed with a Lambda of 0.157. This transformation has eliminated much of the inconsistencies of the variation in the GDP. There is still a trend but no other discernible pattern.
# A tibble: 1 × 3
Country kpss_stat kpss_pvalue
<fct> <dbl> <dbl>
1 Turkey 0.0889 0.1
The transformed data indicates the same as the un-transformed data. The p-value above 0.05 indicates that the null hypothesis should not be rejected and that the data is stationary.
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
The Box-Cox and differenced GDP data, when plotted, do not appear to have any pattern, trend, or seasonality any longer.
turk_diffs |>ACF(gdp_boxcox_diff) |>autoplot()
turk_diffs |>PACF(gdp_boxcox_diff) |>autoplot()
The ACF and PACF both show no correlation for the transformed and differrenced data.
The Turkish GDP data underwent a Box-Cox transformation, no seasonal differencing, and an ordinary differencing of 1.
b.
The State of Tasmania accommodation takings are plotted below.
take <- aus_accommodation |>filter(State =='Tasmania')take |>autoplot(Takings)
The takings data has a consistent upward trend and what appears to be a seasonal variation.
lambda <- take |>features(Takings, features = guerrero) |>pull(lambda_guerrero)take_transformed <- take |>mutate(take_boxcox =box_cox(Takings, lambda))take_transformed |>autoplot(take_boxcox) +labs(y ="Takings",x ="Quarter",title =paste0("Box-Cox transformed Tasmanian Accommodations' Takings (Lambda = ",round(lambda, 3),")" ) )
The Tasmanian Accommodations’ Takings have had their variation made consistent by the Box-Cox transformation with a lambda of 0.002. The data still has a trend and seasonal variation.
# A tibble: 1 × 2
State ndiffs
<chr> <int>
1 Tasmania 0
After seasonal differencing, there is still an indication that ordinary differencing is needed.
take_diffs |>autoplot(both_diff) +labs(y ="Box-Cox Transformed and Differenced Aus Accommodations Takings Data",x ="Year",title =paste0("Box-Cox Transformed anmd Differenced Aus Accommodations Takings GDP",")" ) )
Warning: Removed 5 rows containing missing values or values outside the scale range
(`geom_line()`).
The plot of the Takings data after Box-Cox transformation and both seasonal and ordinary differences are applied shows no more trend or seasonal variation.
The Tasmanian accommodations takings data underwent a Box-Cox transformation, a seasonally differencing of 1, and an ordinary differencing of 1.
c.
The souvenirs Sales data is plotted below.
souv <- souvenirssouv |>autoplot(Sales)
The souvenirs Sales data shows an upward trend, a seasonal variation which peaks in December, and progressively increasing variation.
The Box-Cox transformation has made the variation consistent using a lambda of 0.002. However the data still appears to have a trend and seasonal variation.
Warning: Removed 12 rows containing missing values or values outside the scale range
(`geom_line()`).
The plot of the Box-Cox transformed and seasonally differenced data no longer has a trend and has a cyclical variation but one that does not strictly appear seasonal.
Unit root indicates that there is no need for an ordinary difference application.
The souvenir sales data underwent a Box-Cox transformation, as seasonal diferencing of 1, and no ordinary differencing.
9.5
# set the seedset.seed(47)# select the retail datamyseries <- aus_retail |>filter(`Series ID`==sample(aus_retail$`Series ID`, 1))# plot the training data and the entire dataautoplot(myseries, Turnover)
The retail data has a trend, increasing variation over time, and a possible seasonal trend.
Warning: Removed 13 rows containing missing values or values outside the scale range
(`geom_line()`).
The retail turnover data now appears to have no trend or seasonality. After a Box-Cox transformation, a seasonal differencing, and an ordinary differencing the data appears to now be stationary.
9.6
a.
y <-numeric(100)e <-rnorm(100)for (i in2:100) { y[i] <-0.6* y[i -1] + e[i]}sim <-tsibble(idx =seq_len(100), y = y, index = idx)autoplot(sim, y)
for (i in2:100) { y[i] <--0.9* y[i -1] + e[i]}sim1 <-tsibble(idx =seq_len(100), y = y, index = idx)autoplot(sim1, y)
for (i in2:100) { y[i] <-0.9* y[i -1] + e[i]}sim2 <-tsibble(idx =seq_len(100), y = y, index = idx)autoplot(sim2, y)
As \(\phi\) decreases the variation in the time series increases in its extremeness, as \(\phi\) increases the variation decreases.
c.
y <-numeric(100)e <-rnorm(100)for (i in2:100) { y[i] <- e[i] +0.6* e[i-1]}tim <-tsibble(idx =seq_len(100), y = y, index = idx)
d.
autoplot(tim, y)
for (i in2:100) { y[i] <- e[i] +-0.9* e[i-1]}tim1 <-tsibble(idx =seq_len(100), y = y, index = idx)autoplot(tim1, y)
for (i in2:100) { y[i] <- e[i] +0.9* e[i-1]}tim2 <-tsibble(idx =seq_len(100), y = y, index = idx)autoplot(tim2, y)
As \(\theta\) decreases, the extremeness of the variation increases. As \(\theta\) increases the extremeness of the variation decreases.
e.
y <-numeric(100)e <-rnorm(100)for (i in2:100) { y[i] <-0.6* y[i -1] +0.6* e[i-1] + e[i]}jim <-tsibble(idx =seq_len(100), y = y, index = idx)autoplot(jim, y)
f.
y <-numeric(100)e <-rnorm(100)for (i in3:100) { y[i] <- (-0.8* y[i -1]) +0.3* y[i -2] + e[i]}gim <-tsibble(idx =seq_len(100), y = y, index = idx)autoplot(gim, y)
g.
The plot of the ARMA model in letter e. produces stationary data. There is no pattern of trend or seasonality.
The plot of AR2 shows a non-stationary pattern of progressively and dramatically increasing variation around zero.
9.7
a.
aus_airpassengers |>autoplot(Passengers)
The plot of Australian air passengers shows a steady increasing trend over time with no signs seasonality or of variation changes, therefore a Box-Cox trasnformation will not be performed.
Warning: Model specification induces a quadratic or higher order polynomial trend. This
is generally discouraged, consider removing the constant or reducing the number
of differences.
The trend upwards of the ARIMA010 model is similar to that of ARIMA021 in part A. However the trend increases at slightly lower rate. The forecast is essentially a straight line, like in part A, with little variation.
The ARIMA212 model forecast shows a similar trend and rate of level increase as the ARIMA010 model. The ARIMA 212 model include variation in the forecast that the previous ARIMA models did not.
The ARIMA021 with constant forecast is very similar to the ARIMA212 without the constant, its rate of increase and lack of significant variation appear very similar or the same. The constant combined with the second differencing allows for quadratic forecast growth.
9.8
a.
usa <- global_economy |>filter(Country =="United States") |>select(Country, Code, Year, GDP)autoplot(usa)
Plot variable not specified, automatically selected `.vars = GDP`
The plot of the United States GDP information does not show a significant change in variation during the course of the plot, therefore a Box-Cox may not be necessary. However, one is applied below:
lambda <- usa |>features(GDP, features = guerrero) |>pull(lambda_guerrero)usa_transformed <- usa |>mutate(usa_boxcox =box_cox(GDP, lambda) )usa_transformed |>autoplot(usa_boxcox) +labs(y ="Box-Cox Transformed USA GDP",x ="Year",title =paste0("Box-Cox Transformed USA GDP (Lambda = ",round(lambda, 3),")" ) )
The transformation has made the plot of the data appear closer to linear.
# A mable: 1 x 2
# Key: Country [1]
Country search
<fct> <model>
1 United States <ARIMA(1,1,0) w/ drift>
The ARIMA search function returned an ARIMA(1, 1, 0) model with drift.
usa_fit |>gg_tsresiduals()
The search model looks relatively good. However, the residuals do appear to have increasing variation over time, all ACF values are within the bounds of the interval, the histogram of the residuals approaches normal but is not normal and has a strong left skew.
c.
usa_test <- usa |>model(no_boxcox =ARIMA(GDP, stepwise =FALSE) ) usa_test
# A mable: 1 x 2
# Key: Country [1]
Country no_boxcox
<fct> <model>
1 United States <ARIMA(0,2,2)>
The data without a Box-Cox transformation yields the an ARIMA(0, 2, 2) model.
According to the AICc scores the ARIMA(1, 1, 0) with drift model is the preferred model from the group.
usa_fit_2 |>select("ARIMA112") |>gg_tsresiduals()
The ARIMA(1, 1, 2) plot of residuals is similar to the ARIMA(1, 1, 0), the ACF has no values that extend past the bounds of the interval. The histogram is similar being left skewed.
usa_fit_2 |>select("ARIMA011") |>gg_tsresiduals()
The ARIMA(0, 1, 1) plot of residuals is similar to the ARIMA(1, 1, 0), the ACF has no values that extend past the bounds of the interval. The histogram is similar being left skewed but is closer to approaching normal.
usa_fit_2 |>select("ARIMA012") |>gg_tsresiduals()
The ARIMA(0, 1, 2) plot of residuals is similar to the ARIMA(1, 1, 0), the ACF has no values that extend past the bounds of the interval. The histogram is similar being left skewed but is closer to approaching normal.
usa_fit_2 |>select("ARIMA111") |>gg_tsresiduals()
The ARIMA(1, 1, 1) plot of residuals is similar to the ARIMA(1, 1, 0), the ACF has no values that extend past the bounds of the interval. The histogram is similar being left skewed.
d.
As previously demonstrated the best model of this group based on the AICc was the ARIMA(1, 1, 0) with drift. Below is the residual plots:
usa_fit |>gg_tsresiduals()
The search model looks relatively good. However, the residuals do appear to have increasing variation over time with a possible upward trend, all ACF values are within the bounds of the interval, the histogram of the residuals approaches but is not normal and has a strong left skew.
e.
The best model of this group is likely the select model of ARIMA(1,1,0) with drift.
When comparing the ARIMA(1, 1, 0) model and the ETS model, the ARIMA model forecast increases at a higher rate, however the prediction intervals of the ETS model indicate more uncertainty in the prediction interval.
