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, … ap,
and b1, b2,
… bq 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, … ap, and b1, b2,
… bq
.
|
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........- Chapter 12: Time-Series Analysis and Forecasting
- What
Is a Time-Series Process?
- Main
Phases of Time-Series Analysis
- Modeling
Methodologies
- Box–Jenkins
Approach
- What
Is ARIMA?
- The
AR Process
- The
MA Process
- ARMA
Process
- Understanding
ARIMA Using an Eyesight Measurement Analogy
- Steps
in the Box–Jenkins Approach
- Step
1: Testing Whether the Time Series Is Stationary
- Step
2: Identifying the Model
- Step
3: Estimating the Parameters
- Step
4: Forecasting Using the Model
- Case
Study: Time-Series Forecasting Using the SAS Example
- Checking
the Model Accuracy
- Conclusion
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