Practical Business Analytics Using SAS: A Hands-on Guide: Understanding ARIMA Using an Eyesight Measurement Analogy

I’ll use an analogy related to eyesight measurement and prescription eyeglasses as an example to explain a serious concept. Nowadays there is a lot complicated equipment to measure human eyesight. It is relatively easy to accurately measure eyesight and get the right prescription for eyeglasses. But a few decades back, before today’s sophisticated and computerized eye testing machines, doctors accomplished this task manually. Today, almost everyone who has visited an eye doctor may easily recognize the picture presented in the Figure 12-12. It is known as a Snellen chart (though this name is not as common as the chart itself).

Figure 12-12. A Snellen chart (courtesy: Wikipedia)

To test eye sight and prescribe eyeglasses, doctors perform a small test. Instead of using special equipment, in the past doctors had a box full of lenses (of different powers). The patient was asked to sit in a chair and was given an empty frame to put on her eyes. The doctor used to put differently powered lenses, on by one, in the frame and asked the patient to read from the Snellen chart. Some patients, for example, read the top seven rows and struggled with the lower ones. The doctor then removed the first lens and put another. After much such iteration, the doctor used to finalize on the exact lenses to be used in the patient’s glass. Some patients got diagnosed as nearsighted and some with farsightedness. The basic assumption in this process was that all the patients are literate. But what if the patient was illiterate and couldn’t read any letters?

So, the major steps in the process of eyesight determination and prescription can be listed as follows:

1. Assume that patient is literate.

2. Based on some tests, identify nearsightedness or farsightedness and get a rough estimate of eyesight.

3. Estimate the exact eyesight by trying various lenses.

4. Use the test results to give the prescription.

A similar analogy can be used with Box–Jenkins approach. Say you have a time series in your hands and you want to forecast some future values in this series. You first need to identify whether the time-series process is an AR process or an MA process or an ARMA process. You also need to identify the orders, p and q, of AR(p) or MA(q) or ARMA(p,q) processes as applicable. Once you identify the type of series and the orders, you can attempt to write the series equation. You are already familiar with the AR(p), MA(q), and ARMA(p,q) equations. The next step is to find parameters such as a₁, a₂, … a_p, and b₁, b₂, … b_q as applicable. Before you move on to identifying the model, there is an assumption to be made; the series has to be stationary (in the coming sections, I explain more about stationary time series). This is to simplify the overall model identification process. Table 12-2 uses the eyeglasses prescription analogy to illustrate the Box–Jenkins approach. After that I discuss various steps involved in the Box–Jenkins approach.

Table 12-2. An Analogy Between a Vision Test and the Box–Jenkins Approach

Vision Test	Box–Jenkins Approach
Assume the patient is literate.	Assume that the time series is stationary; otherwise, make it stationary.
Based on some tests, identify nearsightedness or farsightedness and get a rough estimate of eyesight.	Based on plots (ACF and PACF functions, explained later in this chapter), identify whether the model is an AR or MA or ARMA process.
Estimate the exact eyesight by trying various lenses.	Estimate the parameters such as a₁, a₂, … a_p, and b₁, b₂, … b_q .
Use the test results to give the prescription.	Use the final model for forecasting.

Steps in the Box–Jenkins Approach

Once again I’ll show a time-series forecasting problem. Consider some time-series data of a premium stock over a period of time. Let’s assume you have the stock price data for the past year. Also assume that you want to predict the stock prices for the next week. You will use the Box–Jenkins approach for this forecasting. First you need to make sure that the stock price time-series process is stationary. Then you need to identify the type of process, which approximates the pattern followed by the stock price data. Is it an AR process or an MA process or an ARMA process? Once you have the basic model equation in place, you can estimate the parameters. Successfully completing all these steps concludes the model-building process. You now have the final equation that can be used for forecasting the future values of the stock under consideration. You also need to take a look at the model accuracy or the error rate before you can continue with the final forecasting and model deployment. What follows is a detailed explanation of each step.

Step 1: Testing Whether the Time Series Is Stationary

If a time-series process is stationary, it’s much easier to build the model using the Box–Jenkins methodology.

What Is a Stationary Time Series?

“A time series is said to be stationary if there is no systematic change in mean (no trend), if there is no systematic change in variance, and if strictly periodic variations have been removed,” according to Dr. Chris Chatfield in The Analysis of Time Series: An Introduction (Chapman and Hall, 2003). A stationary time series is in a state of statistical equilibrium........