What Is Time Series Forecasting and How Does the ARIMA Method Work?

Time series forecasting is one of the most widely used quality tools for predicting future values based on historical data. Whether you are monitoring production output, customer demand, or equipment performance, understanding how patterns repeat over time helps you make better decisions.

This article explains the core concepts of time series analysis, introduces the Box-Jenkins ARIMA approach, and shows how you can apply it with a simple illustrative example.

What It Is

A time series is a sequence of data points recorded at successive, equally spaced points in time—for example, daily sales, weekly defect counts, or monthly energy usage. Time series forecasting uses statistical models to identify underlying patterns and project them into the future.

The two most important patterns are:

• Trend: a long-term upward or downward movement in the data.
• Seasonality: a regular, repeating pattern within a fixed period (e.g., weekly, monthly, or yearly).

The Box-Jenkins method, named after statisticians George Box and Gwilym Jenkins, is a systematic approach to building autoregressive integrated moving average (ARIMA) models. It is considered a standard reference for univariate time series forecasting.

How It Works: The ARIMA Model

ARIMA is denoted as ARIMA(p, d, q), where:

• p = the number of autoregressive (AR) terms—how much past values influence the current value.
• d = the number of differences needed to make the series stationary (i.e., remove trend and seasonality).
• q = the number of moving average (MA) terms—how much past forecast errors influence the current value.

The general ARIMA equation for a stationary series can be written as:

\[
Y_t = c + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \theta_1 \varepsilon_{t-1} + \dots + \theta_q \varepsilon_{t-q} + \varepsilon_t
\]

Where:

• \(Y_t\) is the current value,
• \(\phi\) are the autoregressive coefficients,
• \(\theta\) are the moving average coefficients,
• \(\varepsilon_t\) is the current random error (white noise),
• \(c\) is a constant.

Steps in the Box-Jenkins approach:

1. Identify: Plot the data and examine autocorrelation (ACF) and partial autocorrelation (PACF) plots to choose tentative values of p, d, and q.
2. Estimate: Fit the model parameters using statistical software or a dedicated tool.
3. Diagnose: Check residuals to confirm they behave like white noise (no remaining autocorrelation).
4. Forecast: Use the validated model to generate future predictions.

A Worked Illustrative Example

Example data (illustrative only)

Suppose you have monthly defect counts for a production line over 24 months, and the data shows a clear upward trend with no strong seasonality. You decide to use ARIMA(1,1,1):

• d = 1: you difference the series once to remove the trend.
• p = 1: the current value depends on the previous differenced value.
• q = 1: the current value depends on the previous forecast error.

After fitting the model, you obtain the following estimated equation (illustrative):

\[
\Delta Y_t = 0.65 \Delta Y_{t-1} + \varepsilon_t - 0.30 \varepsilon_{t-1}
\]

Where \(\Delta Y_t = Y_t - Y_{t-1}\).

If the last observed defect count was 120 and the last forecast error was +5, the next forecast is:

\[
\Delta Y_{t+1} = 0.65 \times (120 - 115) + 0 - 0.30 \times 5 = 0.65 \times 5 - 1.5 = 1.75
\]

So the predicted value for the next month is \(120 + 1.75 = 121.75\), which you would round to 122 defects.

This simple example shows how ARIMA combines historical levels and past errors to generate a forecast.

Common Pitfalls

• Ignoring stationarity: Applying ARIMA to non-stationary data without differencing leads to unreliable results.
• Overfitting: Choosing too many parameters (large p and q) may fit noise rather than the true pattern.
• Ignoring seasonality: If your data has a regular seasonal pattern, use a seasonal ARIMA model (e.g., SARIMA) instead of plain ARIMA.
• Forgetting to validate: Always check residuals for autocorrelation before trusting the forecast.

Try It Yourself

Time series forecasting is a powerful addition to your quality toolbox. To apply the ARIMA method without writing code, you can use the free online tool at:

https://www.6sq.com/tools/timeseries/

Upload your historical data, let the tool guide you through identification, estimation, and forecasting, and turn your time-based data into actionable insight.