Chapter-05-Exercises.Rmd

---
title: "Chapter 5 Exercises"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(fpp3)
```


## 5.11 Exercises

1. Produce forecasts for the following series using whichever of NAIVE(y), 
SNAIVE(y) or RW(y ~ drift()) is more appropriate in each case:

  * Australian Population (global_economy)
```{r}
aus_pop <- global_economy %>%
  filter(Country == 'Australia') %>%
  select(Population)

autoplot(aus_pop) # based on the graph there is no apparent seasonality but an obvious trend

aus_pop %>%
  model(RW(Population ~ drift())) %>%
  forecast(h=1) %>%
  autoplot(aus_pop)

```

  * Bricks (aus_production)
```{r}
aus_bricks <- aus_production %>%
  filter_index("1970 Q1" ~ "2004 Q4")

autoplot(aus_bricks, Bricks) #obvious seasonality here

brick_fit <- aus_bricks %>%
  model(SNAIVE(Bricks)) 

brick_fc <- brick_fit %>%
  forecast(h=4) # 1 year

brick_fc %>% autoplot(aus_bricks)
```
  
  * NSW Lambs (aus_livestock)
```{r}
nsw_lambs <- aus_livestock %>%
  filter(Animal == 'Lambs', State == 'New South Wales')

autoplot(nsw_lambs, Count)


```
  
  *  Household wealth (hh_budget).
```{r}
aus_wealth <- hh_budget %>%
  filter(Country == 'Australia') %>%
  select(Wealth)

autoplot(aus_wealth)
```
  
  
  * Australian takeaway food turnover (aus_retail).
```{r}
aus_takeaway <- aus_retail %>%
  filter(Industry == 'Takeaway food services', State == 'Australian Capital Territory') %>%
  select(Turnover)


aus_takeaway %>%
  model(RW(Turnover ~ drift())) %>%
  forecast(h=4) %>%
  autoplot(aus_takeaway)
  
```

2. Use the Facebook stock price (data set gafa_stock) to do the following:

  a. Produce a time plot of the series.
```{r}
facebook <- gafa_stock %>%
  filter(Symbol == 'FB') %>%
  mutate(day = row_number()) %>%
  update_tsibble(index = day, regular = TRUE)

autoplot(facebook, Close)
```

  b. Produce forecasts using the Drift method and plot them.
```{r}
facebook %>%
  model(RW(Close ~ drift())) %>%
  forecast(h=20) %>%
  autoplot(facebook)
```
  
  c. Show that the forecasts are identical to extending the line drawn between the first and last observations.
```{r}
facebook %>%
  model(RW(Close ~ drift())) %>%
  forecast(h=20) %>%
  autoplot(facebook) +
  geom_segment(aes(x = 1, y = 54.71, xend = 1258, yend = 131.09))

```

```{r}
facebook$Close[1258]
```
  
  d. Try using some of the other benchmark functions to forecast the same data set. Which do you think is best? Why?
```{r}
facebook %>%
  model(NAIVE(Close)) %>%
  forecast(h=20) %>%
  autoplot(facebook)
```

The distribution ranges look similar but the point forecasts are, of course,
different. I think Naive is likely best as this is stock market data.

3. Apply a Seasonal naïve method to the quarterly Australian beer production data 
from 1992. Check if the residuals look like white noise, and plot the forecasts.
What do you conclude?

```{r}
# Extract data of interest
recent_production <- aus_production %>%
  filter(year(Quarter) >= 1992)
# Define and estimate a model
fit <- recent_production %>% model(SNAIVE(Beer))
# Look at the residuals
fit %>% gg_tsresiduals()
# Look a some forecasts
fit %>% forecast() %>% autoplot(recent_production)

```

Above residuals do not look to be entirely white noise. They are heteroskedastic
weighted toward the negatives, they are a non-normal distribution, and they appear
to have autocorrelation at the quarters. Data would likely benefit from differencing.


4. Repeat the previous exercise using the Australian Exports series from `global_economy` 
and the Bricks series from `aus_production`. Use whichever of NAIVE() or SNAIVE() 
is more appropriate in each case.
```{r}
aus_exports <- global_economy %>%
  filter(Country == 'Australia') %>%
  select(Exports)

fit <- aus_exports %>% model(NAIVE(Exports))

fit %>% forecast() %>% autoplot(aus_exports)
```

No apparent seasonality above, so a Naive model was used.


```{r}
aus_bricks <- aus_production %>%
  select(Bricks)

fit <- aus_production %>%
  model(SNAIVE(Bricks))

fc <- fit %>% forecast()

fc %>%
  autoplot(aus_production)
```

Model above should definitely use `SNAIVE()`, but for some reason the forecast
won't plot claiming missing values.


5. Produce forecasts for the 7 Victorian series in `aus_livestock` using `SNAIVE()`. 
Plot the resulting forecasts including the historical data. Is this a reasonable 
benchmark for these series?
```{r}
victorian_livestock <- aus_livestock %>%
  filter(State == 'Victoria')

#########################
victorian_bulls <- victorian_livestock %>%
  filter(Animal == 'Bulls, bullocks and steers')

fit <- victorian_bulls %>%
  model(SNAIVE(Count))

fit %>% forecast() %>%
  autoplot(victorian_bulls)

########################
victorian_calves <- victorian_livestock %>%
  filter(Animal == 'Calves')

fit <- victorian_calves %>%
  model(SNAIVE(Count))

fit %>% forecast() %>%
  autoplot(victorian_calves)

###########################
victorian_cattle <- victorian_livestock %>%
  filter(Animal == 'Cattle (excl. calves)')

fit <- victorian_cattle %>%
  model(SNAIVE(Count))

fit %>% forecast() %>%
  autoplot(victorian_cattle)

##########################
victorian_cows <- victorian_livestock %>%
  filter(Animal == 'Cows and heifers')

fit <- victorian_cows %>%
  model(SNAIVE(Count))

fit %>% forecast() %>%
  autoplot(victorian_cows)

##########################
victorian_lambs <- victorian_livestock %>%
  filter(Animal == 'Lambs')

fit <- victorian_lambs %>%
  model(SNAIVE(Count))

fit %>% forecast() %>%
  autoplot(victorian_lambs)

###########################
victorian_pigs <- victorian_livestock %>%
  filter(Animal == 'Pigs')

fit <- victorian_pigs %>%
  model(SNAIVE(Count))

fit %>% forecast() %>%
  autoplot(victorian_pigs)

############################
victorian_sheep <- victorian_livestock %>%
  filter(Animal == 'Sheep')

fit <- victorian_sheep %>%
  model(SNAIVE(Count))

fit %>% forecast() %>%
  autoplot(victorian_sheep)
  

###########################
```

Models seem to capture the seasonality in the data. I think this would be a 
reasonable benchmark.



6. Are the following statements true or false? Explain your answer.

  a. Good forecast methods should have normally distributed residuals.
    - True. If the residuals are not normally distributed, there is unaccounted 
    for bias
  b. A model with small residuals will give good forecasts.
    - False. A model will small residuals could be overfit to the training data.
  c. The best measure of forecast accuracy is MAPE.
    - False. There are multiple different ways to measure depending on what is
    being measured.
  d. If your model doesn’t forecast well, you should make it more complicated.
    - False. This should not be a tactic for improvement.
  e. Always choose the model with the best forecast accuracy as measured on the test set.
    - False. While this may be a good tactic, there are other things to consider.


7. For your retail time series (from Exercise 8 in Section 2.10):

  a. Create a training dataset consisting of observations before 2011 using

```{r}
set.seed(222)
myseries <- aus_retail %>%
  filter(`Series ID` == sample(aus_retail$`Series ID`,1))

myseries_train <- myseries %>%
  filter(year(Month) < 2011)
```

  b. Check that your data have been split appropriately by producing the following plot.

```{r}
autoplot(myseries, Turnover) +
  autolayer(myseries_train, Turnover, colour = "red")
```

  c. Fit a Seasonal naïve model using SNAIVE() applied to your training data (myseries_train).\
```{r}
fit <- myseries_train %>%
  model(SNAIVE(Turnover))
```

  d. Check the residuals. Do the residuals appear to be uncorrelated and normally distributed?
```{r}
fit %>% gg_tsresiduals()
```

  e. Produce forecasts for the test data
```{r}
fc <- fit %>%
  forecast(new_data = anti_join(myseries, myseries_train))
fc %>% autoplot(myseries)
```

  f. Compare the accuracy of your forecasts against the actual values.
```{r}
fit %>% accuracy()
fc %>% accuracy(myseries)
```

  g. How sensitive are the accuracy measures to the amount of training data used?

8. Consider the number of pigs slaughtered in New South Wales (data set aus_livestock).

  a. Produce some plots of the data in order to become familiar with it.
```{r}
nsw_pigs <- aus_livestock %>%
  filter(State == 'New South Wales', Animal == 'Pigs')

autoplot(nsw_pigs)
gg_season(nsw_pigs)
gg_subseries(nsw_pigs)
```
  
  b. Create a training set of 486 observations, withholding a test set of 72 
  observations (6 years).
```{r}
train <- nsw_pigs %>% filter(year(Month) < 2013) 
test <- nsw_pigs %>% filter(year(Month) >= 2013)

```
  
  c. Try using various benchmark methods to forecast the training set and compare 
  the results on the test set. Which method did best?
```{r}

nsw_pig_fit <- train %>%
  model(
    Mean = MEAN(Count),
    Naive = NAIVE(Count),
    `Seasonal Naive` = SNAIVE(Count),
    Drift = RW(Count ~ drift())
    
  )

nsw_pig_fc <- nsw_pig_fit %>%
  forecast(h = 72)

nsw_pig_fc %>%
  autoplot(nsw_pigs %>% filter(year(Month) >=2010), level=NULL)
```

It is difficult to declare a winner here base on sight. Seasonal Naive is overfit
 to the previous year's data. Drift might actually be best potentially.

  d. Check the residuals of your preferred method. Do they resemble white noise?
```{r}
train %>%
  model(Drift = RW(Count ~ drift())) %>%
  gg_tsresiduals() 

train %>%
  model(SNAIVE(Count)) %>%
  gg_tsresiduals()

train %>%
  model(NAIVE(Count)) %>%
  gg_tsresiduals()
```
```{r}
augment(nsw_pig_fit) %>% features(.innov, ljung_box, lag=24, dof=1)
```

None of these models are perfect, of course. Seasonal Naive has a better lb_stat,
but residuals seem to be heteroskedastic compared to the Drift method, and pvalue
is significant meaning the residuals differ from white noise. 

9. a. Create a training set for household wealth (hh_budget) by withholding the last 
four years as a test set.
```{r}
hh_wealth_train <- hh_budget %>%
  filter(Year <= 2012, Country == 'Australia')

hh_wealth_test <- hh_budget %>%
  filter(Year > 2012, Country == 'Australia')
```

  b. Fit all the appropriate benchmark methods to the training set and forecast 
  the periods covered by the test set.
```{r}
hh_wealth_fit <- hh_wealth_train %>%
  model(
    Mean = MEAN(Wealth),
    Naive = NAIVE(Wealth),
    Drift = RW(Wealth ~ drift())
    
  )

hh_wealth_fc <- hh_wealth_fit %>%
  forecast(h = 4)

hh_wealth_fc %>%
  autoplot(hh_budget %>% select(Wealth), level=NULL)
```

  c. Compute the accuracy of your forecasts. Which method does best?
```{r}
accuracy(hh_wealth_fc, hh_wealth_test)
```
Unsurprisingly, Drift is the best method here although it barely fits the data.
  
  
  d. Do the residuals from the best method resemble white noise?
```{r}
hh_wealth_drift <- hh_wealth_train %>%
  model(
    Drift = RW(Wealth ~ drift()))

gg_tsresiduals(hh_wealth_drift)


augment(hh_wealth_drift) %>% features(.innov, ljung_box, lag=10, dof=1)
```

Residuals do appear to be white noise.


10. a. Create a training set for Australian takeaway food turnover (aus_retail) 
by withholding the last four years as a test set.
  b. Fit all the appropriate benchmark methods to the training set and forecast 
  the periods covered by the test set.
```{r}
aus_takeaway <- aus_retail %>%
  filter(Industry == 'Takeaway food services') %>%
  select(State, Industry, Month, Turnover) %>%
  summarize(TotalT = sum(Turnover))
#Takeaway food services

train_aus_takeaway <- aus_takeaway %>%
  filter(year(Month) < 2015)

test_aus_takeaway <- aus_takeaway %>%
  filter(year(Month) >= 2015)
```

```{r}
aus_takeaway_fit <- train_aus_takeaway %>%
  model(
    Mean = MEAN(TotalT),
    Naive = NAIVE(TotalT),
    `Seasonal Naive` = SNAIVE(TotalT),
    `Seasonal Naive Drift` = SNAIVE(TotalT ~ drift()),
    Drift = RW(TotalT ~ drift())
    
  )

aus_takeaway_fc <- aus_takeaway_fit %>%
  forecast(h = 48)

aus_takeaway_fc %>%
  autoplot(aus_takeaway %>% select(TotalT), level=NULL)
```
  
  c. Compute the accuracy of your forecasts. Which method does best?
```{r}
accuracy(aus_takeaway_fc, test_aus_takeaway)
```
  
  Seasonal Naive with drift performs best
  
  d. Do the residuals from the best method resemble white noise?
```{r}
aus_takeaway_snd <- train_aus_takeaway %>%
  model(`Seasonal Naive Drift` = SNAIVE(TotalT ~ drift()))

gg_tsresiduals(aus_takeaway_snd)

augment(aus_takeaway_snd) %>% features(.innov, ljung_box, lag=24, dof=1)
```

Residuals are not differentiated from white noise.
  

11. We will use the Bricks data from aus_production (Australian quarterly clay 
brick production 1956–2005) for this exercise.

Use an STL decomposition to calculate the trend-cycle and seasonal indices. (Experiment with having fixed or changing seasonality.)
Compute and plot the seasonally adjusted data.
Use a Naïve method to produce forecasts of the seasonally adjusted data.
Use decomposition_model() to reseasonalize the results, giving forecasts for the original data.
Do the residuals look uncorrelated?
Repeat with a robust STL decomposition. Does it make much difference?
Compare forecasts from decomposition_model() with those from SNAIVE(), using a test set comprising the last 2 years of data. Which is better?
tourism contains quarterly visitor nights (in thousands) from 1998 to 2017 for 76 regions of Australia.

Extract data from the Gold Coast region using filter() and aggregate total overnight trips (sum over Purpose) using summarise(). Call this new dataset gc_tourism.

Using slice() or filter(), create three training sets for this data excluding the last 1, 2 and 3 years. For example, gc_train_1 <- gc_tourism %>% slice(1:(n()-4)).

Compute one year of forecasts for each training set using the Seasonal naïve (SNAIVE()) method. Call these gc_fc_1, gc_fc_2 and gc_fc_3, respectively.

Use accuracy() to compare the test set forecast accuracy using MAPE. Comment on these.