Required Packages

Let’s start by loading the necessary packages for our analysis:

# Load required packages
library(fpp3)      # For forecasting using tidyverts
library(tidyverse) # For data manipulation and visualization
library(lubridate) # For date handling

Additional Useful Packages

# Additional useful packages
library(seasonal)  # For seasonal adjustment
library(forecast)  # Traditional forecasting methods
library(tsibble)   # For time series tibbles

What is Time Series?

Definition and Key Features

• Time series: A sequence of observations taken sequentially in time
• Each observation is associated with a particular time
• Examples: stock prices, temperature readings, sales data, population counts
• Key characteristic: temporal ordering matters

Components of Time Series

• Trend: Long-term increase or decrease in data
• Seasonality: Pattern that repeats over fixed periods
• Cycles: Pattern that repeats over varying lengths
• Irregular: Random fluctuations

Basic Time Series in R

Creating Time Series Objects

# Using ts() function
# ts_data <- ts(data, start = c(2020, 1), frequency = 12)

# Using tsibble for modern approach
library(tsibble)
tsbl_data <- tsibble(
  Month = yearmonth("2020 Jan") + 0:23,
  Sales = rnorm(24, 100, 10),
  index = Month
 )

Example: Australian GDP

# Load built-in dataset
data(global_economy)
# Filter for Australia
aus_economy <- global_economy |>   filter(Country == "Australia")

# View the data structure
(aus_economy) |> slice_sample(n = 8)

# A tsibble: 8 x 9 [1Y]
# Key:       Country [1]
  Country   Code   Year     GDP Growth    CPI Imports Exports Population
  <fct>     <fct> <dbl>   <dbl>  <dbl>  <dbl>   <dbl>   <dbl>      <dbl>
1 Australia AUS    1986 1.82e11   4.06  44.5     18.1    15.0   16018400
2 Australia AUS    1964 2.38e10   6.98   8.40    13.8    14.9   11167000
3 Australia AUS    1981 1.77e11   3.34  30.0     16.7    14.9   14927000
4 Australia AUS    1990 3.11e11   3.56  59.8     17.1    15.1   17065100
5 Australia AUS    1996 4.00e11   3.88  69.4     19.4    18.9   18311000
6 Australia AUS    1973 6.37e10   2.61  12.5     11.0    14.2   13380000
7 Australia AUS    2016 1.21e12   2.83 113.      21.5    19.3   24210809
8 Australia AUS    1994 3.22e11   3.98  64.6     18.5    18.0   17855000

What is Forecasting?

Definition and Purpose

• Forecasting: Predicting future values based on historical data
• Goal: Make informed decisions about the future
• Applications in public health:
  • Disease outbreak prediction
  • Healthcare resource planning
  • Mortality and morbidity projections

Types of Forecasting

• Point forecasts: Single predicted value
• Interval forecasts: Range of possible values
• Probability forecasts: Probability distributions

Forecasting Steps

1. Problem definition
2. Gather information
3. Preliminary analysis
4. Choose and fit models
5. Use and evaluate forecasting model

Time Series Graphics

Basic Time Plots

# Simple time plot
aus_economy |>
  autoplot(GDP) +
  labs(title = "Australian GDP",
       y = "GDP (billions USD)")

Seasonal Plots

# Seasonal plot for monthly data
aus_retail <- aus_retail |>
  filter(State == "Victoria", Industry == "Food retailing")

aus_retail |>
  gg_season(Turnover, labels = "both") +
  labs(y = "Turnover (millions AUD)",
       title = "Seasonal plot: Food retail sales")

Seasonal Subseries Plots

aus_retail |>
  gg_subseries(Turnover) +
  labs(y = "Turnover (millions AUD)",
       title = "Seasonal subseries plot")

Time Series Decomposition

Classical Decomposition

Additive model: \(y_t = T_t + S_t + E_t\)

Multiplicative model: \(y_t = T_t \times S_t \times E_t\)

Where:

• \(y_t\) = observed value at time t
• \(T_t\) = trend component
• \(S_t\) = seasonal component
• \(E_t\) = error/irregular component

STL Decomposition

# STL decomposition
aus_retail |>
  model(stl = STL(Turnover)) |>
  components() |>
  autoplot() +
  labs(title = "STL decomposition of retail sales")

Time Series Features

Measures of Strength

# Calculate features
aus_retail |>
  features(Turnover, list(
    trend_strength = feat_stl,
    seasonal_strength = feat_stl
  ))

# A tibble: 1 × 20
  State    Industry       trend_strength_trend_strength trend_strength_seasona…¹
  <chr>    <chr>                                  <dbl>                    <dbl>
1 Victoria Food retailing                         0.999                    0.959
# ℹ abbreviated name: ¹​trend_strength_seasonal_strength_year
# ℹ 16 more variables: trend_strength_seasonal_peak_year <dbl>,
#   trend_strength_seasonal_trough_year <dbl>, trend_strength_spikiness <dbl>,
#   trend_strength_linearity <dbl>, trend_strength_curvature <dbl>,
#   trend_strength_stl_e_acf1 <dbl>, trend_strength_stl_e_acf10 <dbl>,
#   seasonal_strength_trend_strength <dbl>,
#   seasonal_strength_seasonal_strength_year <dbl>, …

Autocorrelation Function (ACF)

Formula: \(r_k = \frac{\sum_{t=k+1}^T (y_t - \bar{y})(y_{t-k} - \bar{y})}{\sum_{t=1}^T (y_t - \bar{y})^2}\)

# ACF plot
aus_retail |>
  ACF(Turnover, lag_max = 24) |>
  autoplot() +
  labs(title = "Autocorrelation function")

The Forecaster’s Toolbox

Simple Methods

Mean Method

\(\hat{y}_{T+h|T} = \bar{y}\)

Naïve Method

\(\hat{y}_{T+h|T} = y_T\)

Seasonal Naïve

\(\hat{y}_{T+h|T} = y_{T+h-m(k+1)}\)

Drift Method

\(\hat{y}_{T+h|T} = y_T + \frac{h}{T-1}\sum_{t=2}^T (y_t - y_{t-1})\)

Implementation in R

# Fit simple models
fit <- aus_retail |>
  model(
    Mean = MEAN(Turnover),
    Naive = NAIVE(Turnover),
    Seasonal_naive = SNAIVE(Turnover),
    Drift = RW(Turnover ~ drift())
  )

Exponential Smoothing

Simple Exponential Smoothing

Forecast equation: \(\hat{y}_{t+1|t} = \alpha y_t + \alpha(1-\alpha) y_{t-1} + \alpha(1-\alpha)^2 y_{t-2} + \cdots\)

Smoothing equation: \(\ell_t = \alpha y_t + (1-\alpha)\ell_{t-1}\)

Where \(0 \leq \alpha \leq 1\) is the smoothing parameter.

Holt’s Linear Trend

Forecast equation: \(\hat{y}_{t+h|t} = \ell_t + h b_t\)

Level equation: \(\ell_t = \alpha y_t + (1-\alpha)(\ell_{t-1} + b_{t-1})\)

Trend equation: \(b_t = \beta(\ell_t - \ell_{t-1}) + (1-\beta)b_{t-1}\)

Holt-Winters Seasonal

Additive seasonality: - \(\hat{y}_{t+h|t} = \ell_t + h b_t + s_{t+h-m(k+1)}\) - \(s_t = \gamma(y_t - \ell_{t-1} - b_{t-1}) + (1-\gamma)s_{t-m}\)

# Fit exponential smoothing models
fit_ets <- aus_retail |>
  model(ETS(Turnover))

ARIMA Models

Components

• AR (Autoregressive): \(y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \cdots + \phi_p y_{t-p} + \varepsilon_t\)
• I (Integrated): Differencing to achieve stationarity
• MA (Moving Average): \(y_t = c + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + \cdots + \theta_q \varepsilon_{t-q}\)

ARIMA(p,d,q) Model

\((1-\phi_1B-\cdots-\phi_pB^p)(1-B)^d y_t = c + (1+\theta_1B+\cdots+\theta_qB^q)\varepsilon_t\)

Seasonal ARIMA

ARIMA(p,d,q)(P,D,Q)\(_m\) includes: - Non-seasonal part: ARIMA(p,d,q) - Seasonal part: ARIMA(P,D,Q) with period m

Model Selection

# Compare multiple models
fit_all <- aus_retail |>
  model(
    ETS = ETS(Turnover),
    ARIMA = ARIMA(Turnover),
    SNAIVE = SNAIVE(Turnover)
  )

Part 1 Summary: Getting Started

• Time series: Sequential observations with temporal ordering
• Components: Trend, seasonality, cycles, irregular
• Forecasting: Predicting future values for decision-making
• R tools: fpp3 package with tidyverse approach

Part 2 Summary: Analysis Methods

• Graphics: Time plots, seasonal plots, decomposition plots
• Decomposition: STL method for trend and seasonal components
• Features: ACF, trend strength, seasonal strength
• Simple methods: Mean, naïve, seasonal naïve, drift

Part 3 Summary: Advanced Methods

• Exponential smoothing: Weighted averages with ETS framework
• ARIMA: Autoregressive integrated moving average models
• Model selection: AIC, cross-validation, accuracy measures

Suggested References

Primary Resource

• Hyndman, R.J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts: Melbourne, Australia. Available at: https://otexts.com/fpp3/

R Packages

• fpp3: Data and functions for “Forecasting: Principles and Practice”
• forecast: Classical forecasting methods
• seasonal: Seasonal adjustment procedures

Additional Reading

• Shumway, R.H., & Stoffer, D.S. (2017). Time series analysis and its applications: With R examples
• Brockwell, P.J., & Davis, R.A. (2016). Introduction to time series and forecasting

Thank You

Questions and Answers

Contact: Kamarul Imran Musa

Resources:

• Forecasting: Principles and Practice: https://otexts.com/fpp3/
• R for Data Science: https://r4ds.had.co.nz/
• Tidyverts: https://tidyverts.org/