Forecasting: Principles and Practice

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Forecasting: Principles and Practice is a comprehensive textbook covering statistical forecasting...
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### Load fpp3 Package in R

Source: https://otexts.com/fpp3/index.html

This command is required to load the fpp3 package and its dependencies for all R examples in the book. Ensure you have the package installed.

```r
library(fpp3)
```

--------------------------------

### MA(2) Model Example

Source: https://otexts.com/fpp3/MA.html

Illustrates a data generation example for a Moving Average model of order 2.

```R
yt=εt−εt−1+0.8εt−2
```

--------------------------------

### MA(1) Model Example

Source: https://otexts.com/fpp3/MA.html

Illustrates a data generation example for a Moving Average model of order 1.

```R
yt=20+εt+0.8εt−1
```

--------------------------------

### General Workflow for Hierarchical Forecasting

Source: https://otexts.com/fpp3/single-level.html

Illustrates the general pipeline for hierarchical and grouped forecasts, starting with data aggregation, modeling, reconciliation, and finally forecasting.

```r
data |> aggregate_key() |> model() |
  reconcile() |> forecast()
```

--------------------------------

### Compare Multiple Forecasting Methods on Beer Production

Source: https://otexts.com/fpp3/simple-methods.html

This example fits Mean, Naïve, and Seasonal Naïve models to quarterly beer production data and forecasts 14 quarters ahead. It then plots the forecasts against actual future values.

```r
# Set training data from 1992 to 2006
train <- aus_production |
  filter_index("1992 Q1" ~ "2006 Q4")
# Fit the models
beer_fit <- train |
  model(
    Mean = MEAN(Beer),
    `Naïve` = NAIVE(Beer),
    `Seasonal naïve` = SNAIVE(Beer)
  )
# Generate forecasts for 14 quarters
beer_fc <- beer_fit |> forecast(h = 14)
# Plot forecasts against actual values
beer_fc |
  autoplot(train, level = NULL) +
  autolayer(
    filter_index(aus_production, "2007 Q1" ~ .),
    colour = "black"
  ) +
  labs(
    y = "Megalitres",
    title = "Forecasts for quarterly beer production"
  ) +
  guides(colour = guide_legend(title = "Forecast"))
```

--------------------------------

### Load fpp3 package and plot time series in R

Source: https://otexts.com/fpp3/appendix-using-r.html

Loads the fpp3 package and plots a specific retail turnover time series. Ensure the fpp3 package is installed and loaded.

```R
# Load required packages
library(fpp3)

# Plot one time series
aus_retail |>
  filter(`Series ID`=="A3349640L") |>
  autoplot(Turnover)
```

--------------------------------

### Apply Seasonal Naive Method to Beer Production Data

Source: https://otexts.com/fpp3/toolbox-exercises.html

This code snippet demonstrates how to apply the Seasonal Naive (SNAIVE) method to quarterly Australian beer production data starting from 1992. It includes filtering the data, defining and estimating the model, and visualizing the time series residuals to check for white noise.

```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()

```

--------------------------------

### Simulate Aggregate Forecasts using ETS

Source: https://otexts.com/fpp3/aggregates.html

Fit an ETS model to the data and simulate 10,000 future sample paths of length 12. Then, sum the simulated values for each sample path to get the aggregate forecast distribution.

```R
fit <- auscafe |
  # Fit a model to the data
  model(ETS(Turnover))utures <- fit |
  # Simulate 10000 future sample paths, each of length 12
  generate(times = 10000, h = 12) |
  # Sum the results for each sample path
  as_tibble() |
  group_by(.rep) |
  summarise(.sim = sum(.sim)) |
  # Store as a distribution
  summarise(total = distributional::dist_sample(list(.sim)))
```

--------------------------------

### Forecast Beer Production with Regression

Source: https://otexts.com/fpp3/forecasting-regression.html

Generates forecasts for Australian quarterly beer production using a linear regression model with trend and seasonality. Requires filtering data to a specific start year and plotting the forecasts with prediction intervals.

```R
recent_production <- aus_production |>
  filter(year(Quarter) >= 1992)
fit_beer <- recent_production |
  model(TSLM(Beer ~ trend() + season()))
fc_beer <- forecast(fit_beer)
fc_beer |
  autoplot(recent_production) + 
  labs(
    title = "Forecasts of beer production using regression",
    y = "megalitres"
  )
```

--------------------------------

### Two-Step-Ahead Forecasts for VAR(1)

Source: https://otexts.com/fpp3/VAR.html

Shows how to generate two-step-ahead forecasts for a 2-dimensional VAR(1) model. This involves using previously forecasted values as inputs for the next step.

```mathematica
y^1,T+2|T=^c1+^ϕ11,1^y1,T+1|T+^ϕ12,1^y2,T+1|T
y^2,T+2|T=^c2+^ϕ21,1^y1,T+1|T+^ϕ22,1^y2,T+1|T
```

--------------------------------

### AR(2) Model Example

Source: https://otexts.com/fpp3/AR.html

Example of an AR(2) model with specific coefficients and error term.

```mathematica
yt = 8 + 1.3 yt-1 - 0.7 yt-2 + \epsilon_t
```

--------------------------------

### One-Step-Ahead Forecasts for VAR(1)

Source: https://otexts.com/fpp3/VAR.html

Illustrates the calculation of one-step-ahead forecasts for a 2-dimensional VAR(1) model using estimated parameters and current observations. Errors are set to zero for forecasting.

```mathematica
y^1,T+1|T=^c1+^ϕ11,1y1,T+^ϕ12,1y2,T
y^2,T+1|T=^c2+^ϕ21,1y1,T+^ϕ22,1y2,T
```

--------------------------------

### AR(1) Model Example with Negative Coefficient

Source: https://otexts.com/fpp3/AR.html

Example of an AR(1) model where the coefficient tends to cause oscillation around the mean.

```mathematica
yt = 18 - 0.8 yt-1 + \epsilon_t
```

--------------------------------

### Aggregate Key for Mixed Structures

Source: https://otexts.com/fpp3/hts.html

Use `aggregate_key()` to combine nested and crossed factors. This example aggregates tourism data by state/region and purpose of travel, summing the 'Trips'.

```r
tourism_full <- tourism |>
  aggregate_key((State/Region) * Purpose, Trips = sum(Trips))
```

--------------------------------

### Mutate tsibble to Millions of Dollars

Source: https://otexts.com/fpp3/tsibbles.html

Creates a new variable by converting the 'Cost' from dollars to millions of dollars using the mutate function. This example is incomplete as the code is truncated in the source.

```r
PBS |>

```

--------------------------------

### Grouped Summing Matrix (S) Example

Source: https://otexts.com/fpp3/reconciliation.html

Demonstrates the summing matrix (S) for a grouped structure, where specific rows define aggregation relationships for grouped series. This matrix is used in the equation y_t = S b_t.

```mathematica
⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ytyA,tyB,tyX,tyY,tyAX,tyAY,tyBX,tyBY,t⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣111111000011101001011000010000100001⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢⎣yAX,tyAY,tyBX,tyBY,t⎤⎥ ⎥ ⎥ ⎥⎦
```

--------------------------------

### Forecast Internet Usage with Holt's Method

Source: https://otexts.com/fpp3/holt.html

Generates a forecast for internet usage using a fitted Holt's method model and visualizes the results. Ensure the 'www_usage' object is available and the model 'fit' is previously computed.

```r
fit |>
  forecast(h = 10) |>
  autoplot(www_usage) +
  labs(x="Minute", y="Number of users",
       title = "Internet usage per minute")
```

--------------------------------

### Load and Prepare Prison Population Data

Source: https://otexts.com/fpp3/hts.html

Loads the Australian prison population data from a CSV file and prepares it into a tsibble object. Ensure the 'readr' and 'tsibble' packages are installed and loaded.

```r
prison <- readr::read_csv("https://OTexts.com/fpp3/extrafiles/prison_population.csv") |>
  mutate(Quarter = yearquarter(Date)) |>
  select(-Date)  |>
  as_tsibble(key = c(Gender, Legal, State, Indigenous),
             index = Quarter) |>
  relocate(Quarter)
```

--------------------------------

### Implement Top-Down Forecasts with Proportion Averages

Source: https://otexts.com/fpp3/single-level.html

Use `top_down()` with `method = "proportion_averages"` to disaggregate aggregate forecasts based on the average historical values relative to the total aggregate.

```python
top_down(y, method = "proportion_averages")
```

--------------------------------

### Implement Top-Down Forecasts with Average Proportions

Source: https://otexts.com/fpp3/single-level.html

Use `top_down()` with `method = "average_proportions"` to disaggregate aggregate forecasts based on historical average proportions.

```python
top_down(y, method = "average_proportions")
```

--------------------------------

### Hierarchical Summing Matrix (S) Example

Source: https://otexts.com/fpp3/reconciliation.html

Illustrates the summing matrix (S) for a hierarchical structure, where rows represent aggregated series and the identity matrix ensures bottom-level series are preserved. This matrix is used in the equation y_t = S b_t.

```mathematica
⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ytyA,tyB,tyAA,tyAB,tyAC,tyBA,tyBB,t⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1111111100000111000001000001000001000001⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣yAA,tyAB,tyAC,tyBA,tyBB,t⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦
```

--------------------------------

### Prepare New Data for Forecasting

Source: https://otexts.com/fpp3/complexseasonality.html

Creates new data for forecasting by repeating the last two days of historical temperatures and calculating corresponding date, day of week, working day status, and cooling variables.

```R
elec_newdata <- new_data(elec, 2*48) |>
  mutate(
    Temperature = tail(elec$Temperature, 2 * 48),
    Date = lubridate::as_date(Time),
    DOW = wday(Date, label = TRUE),
    Working_Day = (Date != "2015-01-01") &
                   !(DOW %in% c("Sat", "Sun")),
    Cooling = pmax(Temperature, 18)
  )
```

--------------------------------

### Generate Lag Plots for Time Series Data

Source: https://otexts.com/fpp3/lag-plots.html

Generates lag plots for a time series to visualize autocorrelation and seasonality. This example filters data to include only recent production and then creates a lag plot of beer production.

```r
recent_production <- aus_production |> 
  filter(year(Quarter) >= 2000)
recent_production |> 
  gg_lag(Beer, geom = "point") + 
  labs(x = "lag(Beer, k)")
```

--------------------------------

### Calculate 4-MA and 2x4-MA for Beer Production

Source: https://otexts.com/fpp3/moving-averages.html

Calculates a 4-period moving average and then a 2-period moving average on the results for the Australian quarterly beer production data starting from 1992. This is used to demonstrate a symmetric moving average.

```r
beer <- aus_production |> 
  filter(year(Quarter) >= 1992) |> 
  select(Quarter, Beer)
beer_ma <- beer |> 
  mutate(
    `4-MA` = slider::slide_dbl(Beer, mean,
                .before = 1, .after = 2, .complete = TRUE),
    `2x4-MA` = slider::slide_dbl(`4-MA`, mean,
                .before = 1, .after = 0, .complete = TRUE)
  )
```

--------------------------------

### Fit and Forecast Google Stock Prices

Source: https://otexts.com/fpp3/accuracy.html

Fits Mean, Naïve, and Drift models to Google's 2015 closing stock prices and generates forecasts for January 2016. Requires pre-loaded `google_2015` and `google_jan_2016` data.

```R
google_fit <- google_2015 |>
  model(
    Mean = MEAN(Close),
    `Naïve` = NAIVE(Close),
    Drift = RW(Close ~ drift())
  )

google_fc <- google_fit |> 
  forecast(google_jan_2016)
```

--------------------------------

### Plotting Naïve Forecasts with Prediction Intervals

Source: https://otexts.com/fpp3/distaccuracy.html

Visualizes the naïve forecasts for Google stock prices, including 80% prediction intervals. This helps in understanding the spread of the forecasts.

```R
google_fc |>
  filter(.model == "Naïve") |
  autoplot(bind_rows(google_2015, google_jan_2016), level=80)+
  labs(y = "$US",
       title = "Google closing stock prices")
```

--------------------------------

### Analyze Residuals from Naïve Method Forecasting

Source: https://otexts.com/fpp3/diagnostics.html

Generates augmented data including residuals from a naïve forecast model applied to Google's closing stock prices. Use this to visualize and analyze forecast errors.

```R
aug <- google_2015 |>
  model(NAIVE(Close)) |>
  augment()
autoplot(aug, .innov) +
  labs(y = "$US",
       title = "Residuals from the naïve method")
```

--------------------------------

### Compute Tourism Features with feasts

Source: https://otexts.com/fpp3/exploring-australian-tourism-data.html

Computes a comprehensive set of time series features for the tourism dataset using the feasts package. Requires the 'feasts' package to be installed. This generates 48 features for each combination of Region, State, and Purpose.

```R
tourism_features <- tourism |>
  features(Trips, feature_set(pkgs = "feasts"))
tourism_features
```

--------------------------------

### Plot Time Series with Training Data Highlight

Source: https://otexts.com/fpp3/toolbox-exercises.html

Visualizes a time series and highlights the training data subset in red. Helps verify the data split for model training.

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

--------------------------------

### Generate One-Year Forecasts Using Seasonal Naïve Method

Source: https://otexts.com/fpp3/toolbox-exercises.html

Produces one-year-ahead forecasts using the seasonal naïve (SNAIVE) method for different training set sizes. Used to evaluate forecast performance as training data varies.

```r
gc_fc_1 <- gc_train_1 |> model(SNAIVE()) |> forecast(h = 4)
gc_fc_2 <- gc_train_2 |> model(SNAIVE()) |> forecast(h = 4)
gc_fc_3 <- gc_train_3 |> model(SNAIVE()) |> forecast(h = 4)
```
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