Introduction to Time Series Analysis

Introduction to Time Series Analysis

• Time series analysis is a statistical methodology that deals with time series data, which are data points indexed in time order.
• Time series data are prevalent in fields such as economics, finance, and medicine.
• Forecasting is essential to decision makers in various fields, and many forecasting techniques have been developed over the centuries.

Applications of Time Series Data

• Examples of time series data:
  • Business: weekly interest rates, daily closing stock prices, monthly price indices, yearly sales figures
  • Meteorology: daily high and low temperatures, annual precipitation and drought indices, hourly wind speeds
  • Agriculture: annual figures for crop and livestock production, soil erosion, and export sales
  • Biology: electrical activity of the heart at millisecond intervals
  • Ecology: abundance of an animal species

Purposes of Time Series Analysis

• Two main purposes of time series analysis:
  1. Understanding or modeling the stochastic mechanism that gives rise to an observed series
  2. Forecasting the future values of a series based on the history of that series and possibly other related series or factors

Types of Time Series Data

• Four types of time series data:
  1. Time series with increasing trend and seasonal behavior (e.g., Australian electricity data)
  2. Time series with decreasing trend (e.g., US Treasury bill contracts)
  3. Time series with constant trend and random spikes (e.g., Sales of product C)
  4. Time series with increasing trend, seasonal behavior, and cyclical pattern (e.g., Australian clay brick data)

Steps in a Forecasting Task

• Five basic steps in a forecasting task:
  1. Problem definition: Define the problem and determine what needs to be forecasted
  2. Gathering information: Obtain past data and other relevant information
  3. Preliminary analysis: Analyze the data to identify patterns and trends
  4. Choosing and fitting models: Choose an appropriate forecasting model and fit it to the data
  5. Using and evaluating a forecasting model: Use the model to make forecasts and evaluate its performance

Basic Forecasting Tools

• Four example data sets used to illustrate time series analysis methodology:
  1. Cross-sectional customer survey data
  2. Monthly global oil prices (in USD)
  3. Daily stock prices (in ZAR)
  4. Australian beer production (in megalitres)

Graphical Summaries

• Types of graphical summaries:
  1. Time plots and time series patterns
  2. Seasonal plots
  3. Visualizations for cross-sectional data (scatter plots, histograms, bar plots)

Typical Time Series Patterns

• Five typical time series patterns:
  1. Horizontal (H) pattern: Fluctuates horizontally around a constant mean
  2. Seasonal (S) pattern: Influenced by seasonal factors (e.g., quarterly, monthly, or daily)
  3. Cyclical (C) pattern: Exhibits rises and falls that are not of a fixed period
  4. Trend (T) pattern: Exhibits a long-term increase or decrease
  5. Irregular (E) component: No identifiable pattern### Time Series Analysis
• Time series data is a sequence of data points measured at regular time intervals.
• Characteristics of time series data:
  • Horizontal pattern: a consistent pattern over time.
  • Seasonal pattern: a pattern that repeats at fixed intervals (e.g., daily, weekly, monthly).
  • Cyclical pattern: a long-term pattern that is not seasonal.
  • Trend pattern: a persistent upward or downward movement over time.
  • Random pattern: a pattern that is unpredictable.

Autocovariance and Autocorrelation

• Autocovariance (ck) measures the covariance between a time series and its lagged values:
  • ck = Σ(yt - ȳ)(yt-k - ȳ) / n, where ȳ is the mean of the time series.
• Autocorrelation (rk) measures the correlation between a time series and its lagged values:
  • rk = ck / Σ(yt - ȳ)² / n.

Forecasting Methods

• Naïve Forecast 1 (NF1) method: uses the most recent observation as a forecast.
• Moving Average (MA) method: uses the average of past k observations as a forecast:
  • ŷt+1 = (yt + yt-1 + ... + yt-k+1) / k.

Measuring Forecasting Accuracy

• Mean Error (ME): measures the average difference between forecasts and actual values.
• Mean Absolute Error (MAE): measures the average absolute difference between forecasts and actual values.
• Mean Square Error (MSE): measures the average squared difference between forecasts and actual values.
• Mean Percentage Error (MPE): measures the average percentage difference between forecasts and actual values.
• Mean Absolute Percentage Error (MAPE): measures the average absolute percentage difference between forecasts and actual values.

Theil's U-Statistic

• Measures the goodness-of-fit of a forecasting model compared to the NF1 method:
  • U < 1: the model is better than the NF1 method.
  • U = 1: the model is as good as the NF1 method.
  • U > 1: the NF1 method is better than the model.

Key Formulas

• Autocovariance: ck = Σ(yt - ȳ)(yt-k - ȳ) / n
• Autocorrelation: rk = ck / Σ(yt - ȳ)² / n
• Moving Average: ŷt+1 = (yt + yt-1 + ... + yt-k+1) / k
• Mean Error: ME = Σ(yt - ŷt) / n
• Mean Absolute Error: MAE = Σ|yt - ŷt| / n
• Mean Square Error: MSE = Σ(yt - ŷt)² / n
• Mean Percentage Error: MPE = Σ(yt - ŷt) / yt / n
• Mean Absolute Percentage Error: MAPE = Σ|yt - ŷt| / yt / n
• Theil's U-Statistic: U = Σ(ŷt+1 - yt+1)² / Σ(yt+1 - yt)² / n

Autocorrelation and Autocovariance

• Autocorrelation measures the correlation between a time series yt and its lagged values yt-k.
• The sample autocovariance for the kth lag is calculated as ck = (1/n) * Σ[(yt - ȳ) * (yt-k - ȳ)], where yt is the observation at time t, ȳ is the mean, and n is the number of observations.
• The sample autocorrelation for the kth lag is calculated as rk = (Σ[(yt - ȳ) * (yt-k - ȳ)]) / (Σ[(yt - ȳ)2]), where rk can be interpreted as the correlation between yt and yt-k.

Autocorrelation Function (ACF)

• The autocorrelations r1, r2, ..., rK form the autocorrelation function (ACF).
• The ACF can be plotted to visualize the autocorrelations, known as a correlogram or ACF plot.
• In R, the ACF can be easily calculated using the acf() function.

Example Calculations

• For the beer data, the autocovariances and autocorrelations were calculated using R, resulting in c1 = 158.81, c2 = 21.36, r1 = 0.421, and r2 = 0.057.
• The ACF plot for the beer data is shown in Figure 8.

Interpretation of Autocorrelation

• When k = 0, the autocovariance is equal to the mean squared deviation (MSD), and the autocorrelation is equal to 1.
• The autocorrelation rk can be interpreted as the correlation between yt and yt-k.
• Autocorrelation is used to summarize time series data and plays a similar role to covariance and correlation coefficient for cross-sectional data.