"Forecasting" |
Index to Module Two Notes |
Fore .an ancient term of warning bearing the threat of harm at worst, and uncertainty at best, to those within potential range...+ Cast serving up a projectile to the unseen and usually unknown beneath the deceptive surface= Forecast.... a warning to those who use it... a confession of uncertainty (or deception) by those who create it... a threat of harm to those in its path
From Tom Brown in Getting the Most of Forecasting
2.1: Introduction to
Forecasting
Although the quantitative methods of
business can be studied as independent modules, I believe it is
appropriate that the text places the forecasting material right after
decision analysis. Recall in our decision analysis problems, the
states of nature generally referred to varying levels of demand or
some other unknown variable in the future. Predicting, with some
measure of accuracy or reliability, what those levels of demand will
be is our next subject.
Forecasts are more than simple extrapolations of past data into the
future using mathematical formulas, or gathering trends from
experts
. Forecasts are mechanisms of arriving at measures for
planning the future. When done correctly, they provide an audit trail
and a measure of their accuracy. When not done correctly, they remind
us of Tom Brown's clever breakdown of the term repeated at the
opening of these notes.
Not only do forecasts help us plan, they help us save money! I am
aware of one company that reduced its investment in inventory from $
28 million to $22 million by adopting a formal forecasting method
that reduced forecast error by 10%. This is an example of forecasts
helping product companies replace inventory with information, which
not only saves money but improves customer response and service.
When we use the term "forecasting" in a quantitative
methods course, we are generally referring to
quantitative time series forecasting
methods. These models are appropriate when: 1) past
information about the variable being forecast is available, 2) the
information can be quantified, and 3) it is assumed that
patterns in the historical data will continue into the
future. If the historical data is restricted to past values of the
response variable of interest, the forecasting procedure is called a
time series method.
For example, many sales forecasts rely on the classic time series
methods that we will cover in this module. When the forecast is based
on past sales, we have a time series forecast. A side note: although
I said "sales" above, whenever possible, we try to forecast sales
based on past demand rather than sales
why?
Suppose you own a T-shirt shop at the beach. You stock 100 "Spring
Break 2000" T-shirts getting ready for Spring Break. Further suppose
that 110 Spring Breakers enter your store to buy Spring Break 2000
T-shirts. What are your sales? That's right, 100. But what is your
demand? Right again, 110. You would want to use the demand figure,
rather than the sales figure, in preparing for next year as the sales
figures do not capture your stock outs. So why do many companies make
sales forecasts based on past sales and not demand? The chief reason
is cost - sales are easily captured at the check out station, but you
need some additional feature on your management information system to
capture demand.
Back to the introduction. The other major category of forecasting
methods that rely on past data are regression models,
often referred to as "causal" models as in our text.
These models base their prediction of future values of the response
variable, sales for example, on related variables such as disposable
personal income, gender, and maybe age of the consumer. You studied
regression models in the statistics course, so we will not cover them
in this course. However, I do want to say that we should use the term
"causal" with caution, as age, gender, or disposable personal income
may be highly related to sales, but age, gender or
disposable personal income may not cause sales. We can
only prove causation in an experiment.
The final major category of forecasting models includes
qualitative methods which generally involve the use of
expert judgment to develop the forecast. These methods are useful
when we do not have historical data, such as the case when we are
launching a new product line without past experience. These methods
are also useful when we are making projections into the far distant
future. We will cover one of the qualitative models in this
introduction.
First, lets examine a simple classification scheme for general
guidelines in selecting a forecasting method, and then cover some
basic principles of forecasting.
Selecting a Forecasting Method
The following table illustrates general
guidelines for selecting a forecasting method based on time span and
purpose criteria.
Table 2.1.1
Time Span Purpose Forecasting Method Long Range (3 or more years) Capital Budgets Delphi Intermediate (1 to 3 years) Capacity Planning Regression Short Range (1 year or less) Sales Forecasting Trend Projection
Product Selection
Plant Location
Expert Judgment
Sales Force Composite
Sales Planning
Time Series Decomposition
Scheduling
Inventory Control
Moving Average
Exponential Smoothing
Please understand that these are general guidelines. You may find a
company using trend projection to make reliable forecasts for product
sales 3 years into the future. It should also be noted that since
companies use computer software time series forecasting packages
rather than hand computations, they may try several different
techniques and select the technique which has the best measure of
accuracy (lowest error).
As we discuss the various techniques, and their properties,
assumptions and limitations, I hope that you will gain an
appreciation for the above classification scheme.
Forecasting Principles
Classification schemes such as the one above are useful in
helping select forecasting methods appropriate to the time span and
purpose at hand. There are also some general principles that should
be considered when we prepare and use forecasts, especially those
based on time series methods.
Oliver W. Wight in Production and Inventory Control in the
Computer Age*, and Thomas H. Fuller in Microcomputers in
Production and Inventory Management** developed a set of
principles for the production and inventory control community a while
back that I believe have universal application.
1. Unless the method is 100% accurate, it must be simple enough so people who use it know how to use it intelligently (understand it, explain it, and replicate it)*.
2. Every forecast should be accompanied by an estimate of the error (the measure of its accuracy).*
3. Long term forecasts should cover the largest possible group of items; restrict individual item forecasts to the short term.**
4. The most important element of any forecast scheme is that thing between the keyboard and the chair.**
The first principle suggests that you can get
by with treating a forecast method as a "black box," as long as it is
100% accurate. That is, if an analyst simply feeds historical data
into the computer and accepts and implements the forecast output
without any idea how the computations were made, that analyst is
treating the forecast method as a black box. This is ok as long as
the forecast error (actual observation - forecast observation) is
zero. If the forecast is not reliable (high error), the analyst
should be, at least, highly embarrassed by not being able to explain
what went wrong. There may be much worse ramifications than
embarrassment if budgets and other planning events relied heavily on
the erroneous forecast.
The second principle is really important. In section 2.2 we will
introduce a simple way to measure forecast error, the difference
between what actually occurs and what was predicted to occur for each
forecast time period. Here is the idea. Suppose an auto company
predicts sales of 30 cars next month using Method A. Method B also
comes up with a prediction of 30 cars. Without knowing the measure of
accuracy of the two Methods, we would be indifferent as to their
selection. However, if we knew that the composite error for Method A
is +/- 2 cars over a relevant time horizon; and the composite error
for Method B is +/- 10 cars, we would definitely select Method A over
Method B.
Why would one method have so much error compared to another? That
will be one of our learning objectives in this module. It may be
because we used a smoothing method rather than a method that
incorporates trend projection when we should not have - such as when
the data exhibits a growth trend. Smoothing methods such as
exponential smoothing, always lag trends which results in forecast
error.
The third principle might best be illustrated by an example. Suppose
you are Director of Operations for a hospital, and you are
responsible for forecasting demand for patient beds. If your forecast
was going to be for capacity planning three years from now, you might
want to forecast total patient beds for the year 2003. On the other
hand, if you were going to forecast demand for patient beds for April
2000, for scheduling purposes, then you would need to make separate
forecasts for emergency room patient beds, surgery recovery patient
beds, OB patient beds, and so forth. When much detail is required,
stick to a short term forecast horizon; aggregate your product
lines/type of patients/etc. when making long term forecasts. This
generally reduces the forecast error in both situations.
We should apply the last principle to any quantitative method. There
is always room for judgmental adjustments to our quantitative
forecasts. I like this quote from Alfred North Whitehead in An
Introduction to Mathematics, 1911:
"[T]here is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain."
Of course, judgment can be off too. How about this forecast made in 1943 by IBM Chairman Thomas Watson:
"I think there's a world market for about five computers."
How can we improve the application of judgment?
That is our next subject.
The Delphi Method of Forecasting
The Delphi Method of forecasting is a qualitative technique made
popular by the Rand Corporation. It belongs to the family of
techniques that include methods such as Grass Roots, Market Research
Panel, Historical Analogy, Expert Judgment, and Sales Force
Composite. The thing in common with these approaches is the use of
the opinions of experts, rather than historical data, to make
predictions and forecasts. The subjects of these forecasts are
typically the prediction of political, social, economic or
technological developments that might suggest new programs, products,
or responses from the organization sponsoring the Delphi study.
My first experience with expert judgment forecasting techniques was
at my last assignment during my past career in the United States Air
Force. In that assignment, I was Director of Transportation Programs
at the Pentagon. Once a year, my boss, the Director of
Transportation, would gather senior leadership (and their action
officers) at a conference to formulate transportation plans and
programs for the next five years. These programs then became the
basis for budgeting, procurement, and so forth. One of the exercises
we did was a Delphi Method to predict developments that would have
significant impact on Air Force Transportation programs. I recall one
of the developments we predicted at a conference in the early 1980's
was the accelerated movement from decentralized to centralized
strategic transportation systems in the military. As a result, we
began to posture the Air Force for the unified transportation command
several years before it became a reality.
Step 1. The Delphi Method of Forecasting, like the other judgment
techniques, begins with selecting the experts. Of course, this is
where these techniques can fail - when the experts are really not
experts at all. Maybe the boss is included as an "expert" for the
Delphi study, but while the boss is great at managing resources, he
or she may be terrible at reading the environment and predicting
developments.
Step 2. The first formal step is to obtain an anonymous
forecast on the topic of interest. This is called Round 1.
Here, the experts would be asked to provide a political, economic,
social or technological developments of interest to the organization
sponsoring the Delphi Method.
The anonymous forecasts may be gathered through a Web Site, via
e-mail or by questionnaire. They may also be gathered in a live group
setting but the "halo effect" may stifle the free flow of the
predictions. For example, it would be common for the group of experts
gathered at the Pentagon to include general officers. Several of the
generals were great leaders in the field, but not great visionaries
when it came to logistics developments. On the other hand, their
lieutenant colonel action officers were very good thinkers and knew
much about what was on the horizon for logistics and transportation
systems. However, because of the classic respect for rank, the
younger officers might not have been forthcoming if we did not use an
anonymous method to get the first round of forecasts.
Step 3. The third step in the Delphi Method involves the group
facilitator summarizing and redistributing the results of the Round
One forecasts. This is typically a "laundry list" of developments.
The experts are then asked to respond to the Round One "laundry list"
by indicating the year in which they believed the development would
occur; or to state this development will "never occur." This is
called Round 2.
Step 4. The fourth step, Round 3, involves the group
facilitator summarizing and redistributing the results of the Round
Two. This includes a simple statistical display, typically the median
and interquartile range, for the data (years a development will
occur) from Round 2. The summary would also include the percent of
experts reporting "never occur" for a particular development. In this
Round, the experts are asked to modify, if they wish, their
predictions. The experts are also given the opportunity to provide
arguments challenging or supporting the "never occur" predictions for
a particular development, and to challenge or support the years
outside the interquartile range.
Step 5. The fifth step, Round 4, repeats Round 3 - the
experts receive a new statistical display with arguments - and are
requested to provide new forecasts and/or counter arguments.
Step 6. Round 4 is repeated until
consensus is formed, or at least, a relatively narrow spread of
opinions. My experience is that by Round 4, we had a good idea of the
developments we should be focusing upon.
If the original objective of the Delphi Method is to produce a number
rather than a development trend, then Round 1 simply asks the experts
for their first prediction. This might be to predict product demand
for a new product line for a consumer products company or to predict
the DJIA one year out for a mutual fund company managing a blue chip
index fund.
Let's do a "for fun" (not graded and purely volunteer) Delphi
Exercise. Suppose you are a market expert and wish to join the other
experts in our class in predicting what the DJIA will be on April 16,
2001 (as close to tax due date as possible). I will post a Conference
Topic called "DJIA Predictions" on the course Web Board, within the
Module 2 conference. Please reply to that conference topic by simply
stating what you think the DJIA will close at on April 16, 2001.
Please respond by January 27, 2001, so I can post the summary
statistics before we leave the forecasting material on February
3rd.
We will now begin our discussion of quantitative time series
forecasting methods.
2.2: Smoothing Methods
In this section we want to cover the
components of a time series; naive, moving average and exponential
smoothing methods of forecasting; and measuring forecast accuracy for
each of the methods introduced.
Pause and Reflect
Recall that there are three general classes of forecasting or prediction models. Qualitative methods, including the Delphi, rely on expert judgment and opinion, not historical data. Regression models rely on historical information about both predictor variables and the response variable of interest. Quantitative time series forecasting methods rely on historical numerical information about the variable of interest and assume patterns in the past will continue into the future. This section begins our study of the time series models, beginning with patterns or components of time series.
Components of a Time Series
The patterns that we may find in a time series of historical data
include the average, trend, seasonal, cyclical and irregular
components. The average is simply the mean of
the historical data. Trend describes real growth or
decline in average demand or other variable of interest, and
represents a shift in the average.
The seasonal component reflects a pattern that repeats
within the total time frame of interest. For example, 15 years ago in
Southwest Florida, airline traffic was much higher in January -
April, peaking in March. October was the low month. This
seasonal pattern repeated through 1988. Between 1988
and 1992, January - April continued to repeat each year as high
months, but the peaks were not as high as before, nor the off-season
valleys as low as before, much to the delight of the hotel and
tourism industries. The point is, seasonal peaks repeat within the
time frame of interest - usually monthly or quarterly seasons within
a year, although there can be daily seasonality in the stock market
(Mondays and Fridays showing higher closing averages than Tuesdays -
Thursdays) as an example.
The cyclical component shows recurring values of the
variable of interest above or below the average or long-run trend
line over a multiyear planning horizon. The length of cycles is not
constant, as with the length of seasonal peaks and valleys, making
economic cycles much tougher to predict. Since the patterns are not
constant, multiple variable models such as econometric and multiple
regression models are better suited to predict cyclical turning
points than time series models.
The last component is what's left! The irregular
component is the random variation in demand that is unexplained by
the average, trend, seasonal and/or cyclical components of a time
series. As in regression models, we try to make the random variation
as low as possible.
Quantitative models are designed to address the various components
covered above. Obviously, the trend projection technique will work
best with time series that exhibit an historical trend pattern. Time
series decomposition, which decomposes the trend and seasonal
components of a time series, works best with times series having
trend and seasonal patterns. Where does that leave our first set of
techniques, smoothing methods? Actually, smoothing methods work well
in the presence of average and irregular components. We start with
them next.
Before we start, lets get some data. This time series consists of
quarterly demand for a product. Historical data is available for 12
quarters, or three years. Table 2.2.1 provides the history.
Table 2.2.1
Figure 2.2.1 provides a graph of the time
series. This graph was prepared in Excel using the Chart Wizard's
Line Plot chart assistant. It is not important what software is used
to graph the historical time series - but it is important to "look
at" the data. Even making a pen and paper sketch is useful to get a
"feel" for the data, and see if there might be trend and/or seasonal
components in the time series.
Figure 2.2.1
Moving Average Method
A simple technique which works well with data that has no trend,
seasonality nor cyclic components is the moving average method.
Admittedly, this example data set has trend (note the overall growth
rate from period 1 to 12), and seasonality (note that every third
quarter reflects a decrease in historical demand). But let's apply
the moving average technique to this data so we will have a basis for
comparison with other methods later on.
A three period moving average forecast is a method that takes three
periods of data and creates an average. That average is the forecast
for the next period. For this data set, the first forecast we can
compute is for Period 4, using actual historical data from Periods 1,
2 and 3 (since its a three period moving average).
Then, after Period 4 occurs, we can make a forecast for Period 5,
using historical data from Periods 2, 3, and 4. Note that Period 1
dropped off, hence the term moving average. This
technique then assumes that actual historical data in the far distant
past, is not as useful as more current historical data in making
forecasts.
Before showing the formulas and illustrating this example, let me
introduce some symbols. In this module, I will be using the symbol
Ft to represent a forecast for period t. Thus, the
forecast for period 4 would be shown as F4. I will use the
symbol Yt to represent the actual historical value of the
variable of interest, such as demand, in period t. Thus, the actual
demand for period 1 would be shown as Y1.
Now to carry forward the computations for a three period moving
average. The forecast for period four is:
F4 = (Y1 + Y2 + Y3) / 3 = (398 + 395 + 361) / 3 = 384.7
To generate the forecast for period five:
F5 = (Y2 + Y3 + Y4) / 3 = (395 + 361 + 400) / 3 = 385.3
We continue through the historical data until
we get to the end of Period 12 and make our forecast for Period 13
based on actual demand from Periods 10, 11 and 12. Since Period 12 is
the last period for which we have data, this ends our computations.
If someone was interested in making a forecast for Periods 14, 15,
and 16, as well as Period 13, the best that could be done with
the moving average method would be to make the "out period"
forecasts the same as the most current forecast. This is true because
moving average methods cannot grow or respond to trend. This is the
chief reason these types of methods are limited to short term
applications, such as what is the demand for the next period.
The forecast calculations are summarized in Table 2.2.2.
Table 2.2.2
Thus finishes our first time series
forecast...but wait a minute...is it any good? To answer that
question, we need to measure the accuracy of the forecast. Then, for
all other forecasts presented, we will include that method's measure
of accuracy.
Measuring the Error: Forecast Accuracy
The criteria for selecting between forecasting models, and for
keeping tabs of how well a forecast is doing once it is implemented
is called measuring the accuracy or the error of the
forecast. To do this, we simply have to compute the average error of
a forecast over an appropriate period of time. Typically, the
appropriate period of time would be the period of time from which
data was gathered and forecasts were applied.
Forecast error in time period t (Et) is the actual value
of the time series minus the forecasted value in time period
y.
Error in time t = Et = ( Yt - Ft )
Table 2.2.3 illustrates the error computations
for the three period moving average model.
Table 2.2.3
Since we are interested in measuring the
magnitude of the error to determine forecast accuracy,
note that I square the error to remove the plus and
minus signs. Then, we simple average the squared errors. To compute
an average or a mean, first sum the squared
errors (SSE), then divide by the number of errors to
get the mean squared error (MSE), then
take the square root of the error to get the Root Mean
Square Error (RMSE).
SSE = (235.1 + 608.4 +...+ 625.0 + 455.1) = 9061.78
MSE = 9061.78 / 9 = 1006.86
RMSE = Square Root (1006.86) = 31.73
From your statistics course(s), you will
recognize the RMSE as simply the standard deviation of forecast
errors and the MSE is simply the variance of the forecast errors.
Like the standard deviation, the lower the RMSE the
more accurate the forecast. Thus, the RMSE can be very helpful in
choosing between forecast models.
We can also use the RMSE to do some probability analysis. Since the
RMSE is the standard deviation of the forecast error, we can treat
the forecast as the mean of a distribution, and apply the
important empirical rule, assuming that forecast errors
are normally distributed. I will bet that some of you remember this
rule:
68% of the observations in a bell-shaped symmetric distribution lie within the area: mean +/- 1 standard deviation
95% of the observations lie within: mean +/- 2 standard deviations
99.7% (almost all of the observations) lie within:
mean +/- 3 standard deviations
Since the mean is the forecast, and the standard deviation is the RMSE, we can express the empirical rule as follows:
68% of actual values are expected to fall within:
Forecast +/- 1 RMSE = 454.3 +/- 31.73 = 423 to 486
95% of the actual values are expected to fall within:
Forecast +/- 2 RMSE = 454.3 +/- (2*31.73) = 391 to 518
99.7% of the actual values are expected to fall within:
Forecast +/- 3 RMSE = 454.3 +/- (3*31.73) = 359 to 549
As in studying the mean and standard deviation
in descriptive statistics, this is very important and has similar
applications. One thing we can do is use the 3 RMSE values to
determine if we have any outliers in our data that need to be
replaced. Any forecast that is more than 3 RMSE's from the actual
figure (or has an error greater than the absolute value of 3 * 31.73
or 95 is an outlier. That value should be removed since it inflates
the RMSE. The simplest way to remove an outlier in a time series is
to replace it by the average of the value just before the outlier and
just after the outlier.
Another very hand use for the RMSE is in the setting of safety
stocks in inventory situations. Lets draw out the 2 RMSE
region of the empirical rule for this forecast:
| _2.5%_ | _________________95%__________________ | _2.5% _ |359 .......391...................................454 ........................................518.........549
Since the middle 95% of the observations fall
between 391 and 518, 5% of the observations fall below 391 and above
518. Assuming the distribution is bell shaped, 2.5 % of the
observations fall below 391 and 2.5% fall above 518. Another way of
stating this is that 97.5% of the observations fall below 518 (when
measuring down to negative infinity, although the actual data should
stop at 359. Bottom line: if the firm anticipates
actual demand to be 518 (2 RMSE's above the forecast), then by
stocking an inventory of 518 they will cover 97.5% of the actual
demands that theoretically could occur. That is, the are operating at
a 97.5% customer service level. In only 2.5% of the
demand cases should they expect a stock out. That's really slick,
isn't it!!!!
Following the same methodology, if the firm stocks 549 items, or 3
RMSE's above the forecast, they are virtually assured they will not
have a stock out unless something really unusual occurs
(we call that an outlier is statistics). Finally, if
the firm stocks 486 items (2 RMSE's above the forecast), they will
have a stock out in 16% of the cases, or cover 84% of the demands
that should occur (100% - 16%). In this case, they are operating at
an 84% customer service level.
| _16%_ | _________________68%__________________ | _16% _ |359 ......423...................................454 ......................................486.........549
We could compute other probabilities associated
with other areas under the curve by finding the cumulative
probability for z scores, z = (observation - forecast) / RMSE (do you
remember that from the stat course(s)?). For our purposes here, it is
only important to illustrate the application from the statistics
course.
Using The Management Scientist Software Package
We will be using "The Management Scientist" Forecasting Module to
do the actual forecasts and RMSE computations. To illustrate the
package for the first example, click Windows Start/Programs/The
Management Scientist/The Management Scientist Icon/Continue/Select
Module 11 Forecasting/OK/File/New and you are ready to load the
example problem. The next dialog screen asks you to enter the number
of time periods - that is how many observations to you have - 12 in
this case. Click OK, and start entering your data (numbers and
decimal points only - the dialog screen will not allow alpha
characters or commas). Next, click Solution/Solve/Moving Average
and enter 3 where it asks for number of moving periods. You
should get the following solution:
Printout 2.2.1
FORECASTING WITH MOVING
AVERAGES
********************************
THE MOVING AVERAGE USES 3 TIME
PERIODS
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR
=========== ================= ======== ==============
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR 1 398 2 395 3 361 4 400 384.67 15.33 5 410 385.33 24.67 6 402 390.33 11.67 7 378 404.00 -26.00 8 440 396.67 43.33 9 465 406.67 58.33 10 460 427.67 32.33 11 430 455.00 -25.00 12 473 451.67 21.33
THE MEAN SQUARE ERROR 1,006.86
THE FORECAST FOR PERIOD 13 454.33
Please note that the software returns the
Mean Square Error, and to get the more useful
Root Mean Square Error, you need to take the square
root of the Mean Square Error, 1006.83 in this case. Also note that
the software provides just one forecast value, recognizing the
limitation of moving average methods that limit the projection to one
time period. Finally note that I put the data into an html table only
so you can read it better - this is only necessary in going from the
OUT file to html, not to an e-mail insertion of the OUT file or
copying an OUT file into a WORD document.
As with the decision analysis module solutions, you may then select
Solution/Print Solution and either select Printer to
print, or Text File to save for inserting into an e-mail to
me, or into a Word Document.
Before we do one more moving average example, take a look at the
forecast error column. Note that most of the errors are positive.
Since error is equal to actual time series value minus the forecasted
values, positive errors mean that the actual demand is generally
greater than the forecasted demand - we are under forecasting. In
this case, we are missing a growth trend in the data. As pointed out
earlier, moving average techniques do not work well with time series
data that exhibit trends.
Figure 2.2.2 illustrates the lag that is present when using the
moving average technique with a time series that exhibits a
trend.
Figure 2.2.2
Five Period Moving Average Forecast
Here is "The Management Scientist" solution for using 5 periods
to construct the moving average forecast.
Printout 2.2.2
FORECASTING WITH MOVING AVERAGES
********************************
THE MOVING AVERAGE USES 5 TIME
PERIODS
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR
=========== ================= ======== ==============
THE MEAN SQUARE ERROR 1,349.37
TIME PERIOD
TIME SERIES VALUE
FORECAST
FORECAST ERROR
1
398
2
395
3
361
4
400
5
410
6
402
392.80
9.20
7
378
393.60
-15.60
8
440
390.20
49.80
9
465
406.00
59.00
10
460
419.00
41.00
11
430
429.00
1.00
12
473
434.60
38.40
THE FORECAST FOR PERIOD 13 453.60
The RMSE for the Five-Period Moving Average
forecast is 36.7, which is about 16% worse than the error of the
three- period model. The reason for this is that there is a growth
trend in this data. As we increase the number of periods in the
computation of the moving average, the average begins to lag the
growth trend by greater amounts. The same would be true if the
historical data exhibited a downward trend. The moving average would
lag the trend and provide forecasts that would be above
the actual.
Pause and Reflect
The moving average forecasting method is simple to use and understand, and it works well with time series that do not have trend, seasonal or cyclical components. The technique requires little data, only enough past observations to match the number of time periods in in the moving average. Forecasts are usually limited to one period ahead. The technique does not work well with data that is not stationary - data that exhibits trend, seasonality, and/or cyclic patterns.
One-Period Moving Average Forecast or the "Naive Forecast"
A naive forecast would be one where the number of periods in the
moving average is set equal to one. That is, the next forecast is
equal to the last actual demand. Don't laugh! This technique might be
useful in the case of rapid growth trend; the forecast would only lag
the actual by one quarter or by one month, whatever the time period
of interest. Of course, it would be much better to use a model that
can make a trend projection if the trend represents a
real move from a prior stationary pattern - we will get to that a bit
later. Here is The Management Scientist result for the
One-Period Moving Average Forecast.
Printout 2.2.3
FORECASTING WITH MOVING AVERAGES
********************************
THE MOVING AVERAGE USES 1 TIME
PERIODS
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR
=========== ================= ======== ==============
TIME PERIOD
TIME SERIES VALUE
FORECAST
FORECAST ERROR
1
398
2
395
398.00
-3.00
3
361
395.00
-34.00
4
400
361.00
39.00
5
410
400.00
10.00
6
402
410.00
-8.00
7
378
402.00
-24.00
8
440
378.00
62.00
9
465
440.00
25.00
10
460
465.00
-5.00
11
430
460.00
-30.00
12
473
430.00
43.00
THE MEAN SQUARE ERROR 969.91
THE FORECAST FOR PERIOD 13 473.00
This printout reflects a slightly lower
RMSE than the three period moving average. That concludes our
introduction to smoothing techniques by examining the class of
smoothing methods called moving averages. The last smoothing method
we will examine is called exponential smoothing, which
is a form of a weighted moving average method.
Exponential Smoothing
This smoothing model became very popular with the production and
inventory control community in the early days of computer
applications because it did not need much memory, and allowed the
manager some judgment input capability. That is, exponential
smoothing includes a smoothing parameter that is used to weight
either past forecasts (places emphasis on the average
component) or the last observation (places emphasis on a rapid growth
or decline trend component).
The exponential smoothing model is:
Ft+1 = a Yt + (1 - a) Ft
where
Ft+1 = forecast of the time series for period t + 1
Yt = actual value of the time series in period t
Ft = forecast of the time series for period t
a = smoothing constant or parameter (0 < a < 1)
The smoothing constant or parameter, a, is shown as the Greek symbol alpha in the text - I am limited to alpha characters. In any case, if the smoothing constant is set at 1, the formula becomes the naive model we already studied:
Ft+1 = Yt
If the smoothing constant is set at 0, the formula becomes a weighted average model which gives most weight to the most recent forecast, with diminishing weight the farther back in the time series.
Ft+1 = Ft
Setting a can be done by trial
and error, perhaps trying 0.1, 0.5 and 0.9, recording the RMSE for
each run, then choosing the value of a that gives
forecasts with the lowest RMSE. Some guidelines are, set a
relatively high when there is a trend and you want the model
to be responsive; set a relatively low when there is
just the irregular component so the model will not be responding to
random movements.
Let's do some exponential smoothing forecasts with a
set at 0.6, relatively high.
To get the model started, we begin by making a forecast for Period 2
simply based on the actual demand for Period 1 (first shown in Table
2.2.1, but often repeated with each demonstration).
F2 = Y1 = 398
Then the first exponential smoothing forecast is actually made for Period 3, using information from Period 2. Thus t = 2, t+1 = 3, and Ft+1 = F2+1 = F3. For this forecast, we need the actual demand for Period 2 (Yt = Y2 = 395), the forecast for Period 2 (F2 = 398. The result is:
F3 = a Y2+ (1 - a) F2 = 0.6 (395) + (1-0.6) (398) = 396.2
The next forecast is for Period 4:
F4 = a Y3+ (1 - a) F3 = 0.6 (361) + (1-0.6) (396.2) = 375.08
This continues through the data until we get to the end of Period 12 and are ready to make our last forecast for Period 13. Note that all we have to maintain in historical data is the last forecast, the last actual demand and the value of the smoothing parameter - that is why the technique was so popular since it did not take much data. However, I do not subscribe to "throwing away" data files today - they should be archived for audit trail purposes. Anyway, the forecast for Period 13:
F13 = a Y12+ (1 - a) F12 = 0.6 (473) + (1-0.6) (439.86) = 459.74
Thankfully today, we have software like The
Management Scientist to do the computations. To use The
Management Scientist, select the Forecasting Module and load the
data as previously described in the Three Period Moving Average
demonstration. Next, click Solution/Solve/Exponential Smoothing
and enter 0.6 where it asks for the value of the smoothing
constant . Printout 2.2.4 illustrates the computer output with a
smoothing constant of 0.6.
Printout 2.2.4
FORECASTING WITH EXPONENTIAL SMOOTHING
**************************************
THE SMOOTHING CONSTANT IS 0.6
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR
=========== ================= ======== ==============
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR 1 398 2 395 398.00 -3.00 3 361 396.20 -35.20 4 400 375.08 24.29 5 410 390.03 19.97 6 402 402.01 -0.01 7 378 402.01 -24.01 8 440 387.60 52.40 9 465 419.04 45.96 10 460 446.62 13.38 11 430 454.65 -24.65 12 473 439.86 33.14
THE MEAN SQUARE ERROR 871.52
THE FORECAST FOR PERIOD 13 459.74
This model provides a single forecast
since, like the moving average techniques, it does not have the
capability to address the trend component. The Root Mean Square Error
is 29.52, (square root of the mean square error), or slightly better
than the best results of the moving average and naive techniques.
However, since the time series shows trend, we should be able to do
much better with the trend projection model that is demonstrated
next.
Pause and Reflect
The exponential smoothing technique is a simple technique that requires only five to ten historical observations to set the value of the smoothing parameter, then only the most recent actual observation and forecasting values. Forecasts are usually limited to one period ahead. The technique works best for time series that are stationary, that is, do not exhibit trend, seasonality and/or cyclic components. While historical data is generally used to "fit the model" - that is set the value of a, analysts may adjust that value in light of information reflecting changes to time series patterns.
2.3: Trend Projections
When a time series reflects a shift
from a stationary pattern to real growth or decline in the time
series variable of interest (e.g., product demand or student
enrollment at the university), that time series is demonstrating the
trend component. The trend projection method of time
series forecasting is based on the simple linear regression model.
However, we generally do not require the rigid assumptions of linear
regression (normal distribution of the error component, constant
variance of the error component, and so forth), only that the past
linear trend pattern will continue into the future.
Note that is the trend pattern reflects a curve, we would have to
rely on the more sophisticated features of multiple regression.
The trend projection model is:
Tt = b0 + b1 t
where,
Tt = Trend value for variable of interest in Period t
b0 = Intercept of the trend projection line
b1 = Slope, or rate of change, for the trend projection line
While the text illustrates the computational
formulas for the trend projection model, we will use The
Management Scientist. To use The Management Scientist,
select the Forecasting Module and load the data as previously
described in the Three Period Moving Average demonstration. Next,
click Solution/Solve/Trend Projection and enter 4 where it
asks for "Number of Periods to Forecast." Note, this is the first
method that we have covered that the software asks this question, as
it is assumed that all of the smoothing methods covered in this
course are limited to forecasting just one period ahead.
Printout 2.3.1 illustrates the trend projection printout from The
Management Scientist.
Printout 2.3.1
FORECASTING WITH LINEAR
TREND
*****************************
THE LINEAR TREND EQUATION:
T = 367.121 + 7.776 t
where T = trend value of the time series in period t
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR
=========== ================= ======== ==============
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR 1 398 374.90 23.10 2 395 382.67 12.33 3 361 390.45 -29.45 4 400 398.23 1.78 5 410 406.00 4.00 6 402 413.78 -11.78 7 378 421.55 -43.55 8 440 429.33 10.67 9 465 437.11 27.90 10 460 444.88 15.12 11 430 452.66 -22.66 12 473 460.43 12.57
THE MEAN SQUARE ERROR 449.96
THE FORECAST FOR PERIOD 13 468.21
THE FORECAST FOR PERIOD 14 475.99
THE FORECAST FOR PERIOD 15 483.76
THE FORECAST FOR PERIOD 16 491.54
Now we are getting somewhere with a
forecast! Note the mean square error is down to 449.96, giving a root
mean square error of 21.2. Compared to the three period moving
average RMSE of 31.7, we have a 33% improvement in the accuracy of
the forecast over the relevant period.
Now, if this were products such as automobiles, to achieve a customer
service level of 97.5%, we would create a safety stock of 2 times the
RMSE above the forecast. So, for Period 13, the forecast plus 2 times
the RMSE is 468.21 + (2 * 21.2) or 511 cars. With the three period
moving average method, the same customer service level inventory
position would be: 454.3 + (2 * 31.7) or 518. The safety stocks are 2
times 21 (42 for the trend projection) compared to 2 times 31.7 (63
for the three period moving average). This is a difference of 21 cars
which could represent significant inventory carrying cost that could
be avoided with the better forecasting method.
Note that the software provides the trend equation, showing the
intercept of 367.121 and the slope of 7.776. The slope is interpreted
as in simple linear regression, demand goes up 7.776 per unit
increase in time. This means that over the course of the time series,
demand is increasing about 8 units a quarter. The intercept is only
of interest in placing the trend projection line on a time series
graph. I used the Chart Wizard in Excel to produce such a graph for
the trend projection model:
Figure 2.3.1
Note in this figure that demand falls below the trend projection line
in Periods 3, 7 and 11. This is confirmed by looking at The
Management Scientist computer Printout 2.3.1, where the errors
are negative in the same periods. That is a pattern! Since our data
is quarterly, we would suspect that there is a seasonal
pattern that results in a valley in the time series in every third
quarter.
To capture that pattern, we need the time series decomposition model
that breaks down, analyzes and forecasts the seasonal as well as the
trend components. We do that in the last section of this notes
modules.
Pause and Reflect
The trend projection model is appropriate when the time series exhibits a linear trend component that is assumed to continue into the future. While rules of thumb suggest 20 observations to compute and test parameters of linear regression models, the simple trend projection model can be created with a minimum of 10 observations. The trend projection model is generally used to make multiple period forecasts for the short range, although some firms use it for the intermediate range as well.
2.4: Trend and Seasonal
Components
The last time series forecasting method
that we examine is very powerful in that it can be used to make
forecasts with time series that exhibit trend and seasonal
components. The method is most often referred to as Time Series
Decomposition, since the technique involves breaking down and
analyzing a time series to identify the seasonal component in what
are called seasonal indexes. The seasonal indexes are
used to "deseasonalize" the time series. The
deseasonalized time series is then used to identify the trend
projection line used to make a deseasonalized projection.
Lastly, seasonal indexes are used to seasonalize the
trend projection. Let's illustrate how this works. As usual, we will
use The Management Scientist to do our work after the
illustration.
The Seasonal
Component
The seasonal component may be found by
using the centered moving average approach as presented in the text,
or by using the season average to grand average approach described
here. The latter is a simpler technique to understand, and comes very
close to the centered moving average approach for most time
series.
The first step is to gather observations from the same quarter and
find their average. I will repeat Table 2.2.1 as Table 2.4.1, so we
can easily find the data:
Table 2.4.1
To compute the average demand for Quarter 1, we gather all
observations for Quarter 1 and find their average, then repeat for
Quarters 2, 3 and 4:
Quarter 1 Average = (398 + 410 + 465) / 3 = 424.3
Quarter 2 Average = (395 + 402 + 460) / 3 = 419
Quarter 3 Average = (361 + 378 + 430) / 3 = 389.7
Quarter 4 Average = (400 + 440 + 473) / 3 = 437.7
The next step is to find the seasonal indexes for each quarter. This is done by dividing the quarterly average from above, by the grand average of all observations.
Grand Average = (398+395+361+400+410+402+378+440+465+460+430+473) / 12 = 417.7
Seasonal Index, Quarter 1 = 424.3 / 417.7 = 1.016
Seasonal Index, Quarter 2 = 419 / 417.7 = 1.003
Seasonal Index, Quarter 3 = 389.7 / 417.7 = 0.933
Seasonal Index, Quarter 4 = 437.7/ 417.7 = 1.048
These indexes are interpreted as follows. The
overall demand for Quarter 4 is 4.5 percent above the average demand,
thus making Quarter 4 a "peak quarter." The overall demand for
Quarter 3 is 6.7 percent below the average demand, thus
making Quarter 3 an "off peak" quarter. This confirms our suspicion
that demand is seasonal, and we have quantified the nature of the
seasonality for planning purposes.
Please note The Management Scientist software Printout 2.4.1
provides indexes of 1.046, 1.009, 0.920, and 1.025. The peaks and off
peaks are similar to the above computations, although the specific
values are a bit different. The centered moving average approach used
by the software requires more data for computations - at least 4 or 5
repeats of the seasons, we only have 3 repeats (12 quarters gives 3
years of data).
We will let the computer program do the next steps, but I will
illustrate with a couple of examples. The next task is to
"deseasonalize" the data. We do this by dividing each actual
observation by the appropriate seasonal index. So for the first
observation, where actual demand was 398, we note that it is a first
quarter observation. The deseasonalized value for 398 is:
Deseasonalized Y1 = 398 / 1.016 = 391.7
Actual demand would have been 391.7 if there was no seasonal effects. Let's do four more:
Deseasonalized Y2 = 395 / 1.003 = 393.8
Deseasonalized Y3 = 361 / 0.933 = 386.9
Deseasonalized Y4= 400 / 1.048 = 381.7
Deseasonalized Y5 = 410 / 1.016 = 403.6
I am sure you have seen "deseasonalized"
numbers in articles in the Wall Street Journal or other
popular business press and journals. This is how those are
computed.
The next step is to find the trend line projection based on the
deseasonalized observations. This trend line is a bit
more accurate than the trend line projection based on the actual
observations since than line contains seasonal variation. The
Management Scientist gives the following trend line for this
data:
Tt = 363 + 8.4 t
This trend line a close to the line we computed
in Section 2.3, when the line was fit to the actual, rather than the
seasonal data: Tt = 367 + 7.8 t.
Once we have the trend line, making a forecast is easy. Let's say we
want to make a forecast for time period 2.
F2 = T2 * S2 = [363 + 8.4 ( 2 ) ] * 1.009 = 379.8 * 1.009 = 383.2
Of course, The Management Scientist does
all this for us. To use The Management Scientist, select the
Forecasting Module and load the data as previously described in the
Three Period Moving Average demonstration. Next, click
Solution/Solve/Trend and Seasonal, then enter 4 where it asks
for number of seasons, and 4 where it asks for number of periods to
forecast. - click OK to get the solution.
Note that number of seasons is 4 for quarterly data, 12 for monthly
data, and so forth. Here is the printout.
Printout 2.4.1
FORECASTING WITH TREND AND
SEASONAL COMPONENTS
**********************************************
SEASON SEASONAL INDEX
------ --------------
1 1.046
2 1.009
3 0.920
4 1.025
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR
=========== ================= ======== ==============
TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR 1 398 388.49 9.51 2 395 383.25 11.75 3 361 357..42 3.58 4 400 406.60 -6.60 5 410 423.81 -13.81 6 402 417.32 -15.32 7 378 388.49 -10.49 8 440 441.20 -1.20 9 465 459.12 5.88 10 460 451.38 8.62 11 430 419.57 10.43 12 473 475.80 -2.80
THE MEAN SQUARE ERROR 87.25
THE FORECAST FOR PERIOD 13 494.43
THE FORECAST FOR PERIOD 14 485.44
THE FORECAST FOR PERIOD 15 450.64
THE FORECAST FOR PERIOD 16 510.40
The Mean Square Error of 87.25, gives a
root mean square error of 9.3, a spectacular improvement over the
other techniques. A sketch of the actual and forecast data shows how
well the trend and seasonal model can do at responding to the trend
and the seasonal turn points. Note how the four period out forecast
continues the response to both
components.
Figure 2.4.1
Pause and Reflect
The trend and seasonal components method is appropriate when the time series exhibits a linear trend and seasonality. This model, compared to the others, does require significantly more historical data. It is suggested that you should have enough data to see at least four or five repetitions of the seasonal peaks and off peaks (with quarterly data, there should be 16 to 20 observations; with monthly data, there should be 48 to 60 observations).
Well, that's it to the introduction to times series forecasting
material. Texts devoted entirely to this subject go into much more
detail, of course. For example, there are exponential smoothing
models that incorporate trend; and time series decomposition models
that incorporate the cyclic component. A good reference for these is
Wilson and Keating, Business Forecasting, 2nd ed., Irwin
(1994).
Two parting thoughts. In each of the "Pause and Reflect"
paragraphs, I gave suggestions for number of observations in the
historical data base. There is always some judgment required here.
While we need a lot of data to fit the trend and trend and seasonal
models, "a lot of data" may mean going far into the past. When we go
far into the past, the patterns in the data may be different, and the
time series forecasting models assume that any patterns in the past
will continue into the future (not the values of the
past observations, but the patterns such as slope and seasonal
indexes). When worded on forecasts for airport traffic, we would love
to go back 10 years, but tourist and permanent resident business
travel is different today than 10 years ago so we must balance the
need for a lot of data with the assumption of forecasting.
The second thought is to always remember to measure the accuracy of
your models. We ended with a model that had a root mean square error
that was a 75% improvement over the 5-period moving average. I know
one company that always used a 5-period moving average
for their sales forecasts - scary, isn't it?
You should be ready to tackle the assignment for Module 2,
"Forecasting Lost Sales," in the text, pp. 210-212. The case answers
via e-mail and The Management Scientist computer output files
are due February 10, 2001. If you want "free" review of your draft
responses/output, please forward as a draft by Tuesday, February 6,
2001.
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