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# Percentage Error: What Denominator?

## Abstract

This is the authors' second survey on the measurement of forecast error. They reported the results of their first survey in the Summer 2008 issue of Foresight (Green & Tashman, 2008). The question they asked in that survey was whether to define forecast error as Actual minus Forecast (A-F) or Forecast minus Actual (F-A). Respondents made good arguments for both of the alternatives. In the current survey, they asked how percentage forecast error should be measured. In particular, what should the denominator be when calculating percentage error? The resulting answers and comments are presented here. Copyright International Institute of Forecasters, 2009
| 27 | Winter 2008 Issue 12 FORESIGHT
Percentage Error Metrics: What Denominator?
FINDINGS OF A SURVEY CONDUCTED BY KESTEN GREEN AND LEN TASHMAN
FORECAST ACCURACY MEASUREMENT
THE ISSUE
This is our second survey on the measurement of forecast
error. We reported the results of our first survey in the
Summer 2008 issue of Foresight (Green & Tashman,
2008). The question we asked in that survey was whether
to define forecast error as Actual minus Forecast (A-
F) or Forecast minus Actual (F-A). Respondents made
good arguments for both of the alternatives.
In the current survey, we asked how percentage forecast
error should be measured. In particular: What should the
denominator be when calculating percentage error?
We posed the question to the International Institute
of Forecasters discussion list as well as to Foresight
subscribers, in the following way:
To calculate a percentage error, it is better to use…
(Check or
write in)
1. The actual value (A) as the denominator [ ]
2. The forecast (F) as the denominator [ ]
3. Neither (A) nor (F) but some other value [ ]
I recommend my choice of denominator, because:
The first two options in the questionnaire have each been
used when calculating the mean absolute percentage
error (MAPE) for multiple forecast periods. The first
option is the more traditional form.
One popular alternative to using either A or F as the
denominator is to take an average of the two: (A+F)/2.
Calculated over multiple forecast periods, this measure
is most commonly called the symmetric MAPE
(sMAPE) and has been used in recent forecasting
competitions to compare the accuracy of forecasts
from different methods. See, for example, www.
neural-forecasting-competition.com/index.htm.
SURVEY RESULTS
We received 61 usable responses. 34 of these, (a
majority of 56%) preferred option 1: using the
Actual as the denominator for the percentage error.
15% preferred option 2, using the Forecast as the
denominator, while 29% chose option 3, something
other than the actual or the forecast.
One respondent wrote: For our company, this issue
led to a very heated debate with many strong points of
view. I would imagine that many other organizations
will go through the same experience.
Option 1
Percentage Error = Error / Actual * 100
Of the 34 proponents of using the Actual value for
the denominator, 31 gave us their reasons. We have
organized their responses by theme.
A. The Actual is the forecaster’s target.
Actual value is the forecast target and therefore should
represent the baseline for measurement.
The measure of our success must be how close we
came to “the truth.”
Actual is the “stake in the ground” against which we
should measure variance.
Since forecasting what actually happened is always
our goal, we should be comparing how well we did to
the actual value.
We should measure performance against reality.
B. The Actual is the only consistent basis for
comparing forecast accuracy against a
benchmark or for judging improvement
over time.
| 28 | FORESIGHT Issue 12 Winter 2008
Actual is the only acceptable denominator because
it represents the only objective benchmark for
comparison.
Without a fixed point of reference quantity in the
denominator, you will have trouble comparing the
errors of one forecast to another.
You want to compare the forecast to actuals and not the
other way around. The actuals are the most important
factor. It drives safety stock calculations that are based
on standard deviation of forecast error calculations
that use actuals as the denominator.
Forecast error is measured here as (actual-forecast)/
actual, for comparability to other studies.
C. The Actuals serve as the weights for a weighted
MAPE.
Using the Actuals is more consistent for calculating a
weighted average percentage error (WAPE) for a group
of SKUs or even for the full product portfolio. Using
actual value as denominator is providing the weight
for the different SKUs, which is more understandable
– one is weighting different SKUs based on their actual
contribution. If we use F (forecast), this means we will
weigh them based on the forecast – but this can be
challenged as subjective. Someone may calculate the
single SKU accuracy based on F as denominator, and
then weigh according to Actual sales of each SKU, but
this unnecessarily complicates the formula.
D. The Actual is the customary and expected
denominator of the MAPE.
I would argue that the standard definition of “percent
error” uses the Actual. The Actual is used without any
discussion of alternatives in the first three textbooks I
opened, it is used in most forecasting software, and it is
used on Wikipedia (at least until someone changes it).
If you are creating a display that reads “percent
error” or “MAPE” for others to read without
further explanation, you should use Actual – this is
what is expected.
Actual is the generally used and accepted formula; if
you use an alternative, such as the Forecast, you might
need to give it a new name in order to avoid confusion.
E. Use of the Actual gives a more intuitive
interpretation.
If the forecast value is > the actual value, then the
percentage error with the forecast in the denominator
cannot exceed 100%, which is misleading. For
example, if the Actual is 100 and the Forecast is 1,000,
the average percentage error with Actual is 900% but
with Forecast is only 90%. (Ed. note: See Table 1a for
an illustrative calculation.)
The reason is pragmatic. If Actual is, say, 10 and
Forecast is 20, most people would say the percentage
error is 100%, not 50%. Or they would say forecast is
twice what it should have been, not that the actual is
half the forecast.
By relating the magnitude of the forecast error to an
Actual figure, the result can be easily communicated
to non specialists.
From a retail perspective, explaining “over-
forecasting” when Forecast is the denominator seems
illogical to business audiences.
F. Using the Forecast in the denominator allows
for manipulation of the forecast result.
Utilizing the Forecast as the benchmark is subjective
and creates the opportunity for the forecaster to
manipulate results.
Use of the Actual eliminates “denominator
management.”
Using Forecast encourages high forecasting.
G. Caveats: There are occasions when the Actual
can’t be used.
Use of Actual only works for non-0 values of the
Actual.
| 29 | Winter 2008 Issue 12 FORESIGHT
If you are trying to overcome difficulties related to
specific data sets (e.g., low volume, zeroes, etc.) or
biases associated with using a percentage error, then
you may want to create a statistic that uses a different
denominator than the Actual. However, once you do
so, you need to document your nonstandard definition
of “percentage error” to anyone who will be using it.
For me, the Actual is the reference value. But in my
job I deal with long-term (5-10 years+) forecasts, and
the Actual is seldom “actually” seen. And since you’re
asking this question, my suspicion tells me the issue is
more complicated than this.
Option 2
Percentage Error = Error / Forecast * 100
Eight of the 9 respondents who preferred to use the Fore-
cast value for the denominator provided their reasons for
doing so. Their responses fell into two groups.
A. Using Forecast in the denominator enables you
to measure performance against forecast or plan.
For business assessment of forecast performance, the
relevant benchmark is the plan – a forecast, whatever
the business term. The relevant error is percent
variation from plan, not from actual (nor from an
average of the two).
For revenue forecasting, using the Forecast as the
denominator is considered to be more appropriate
since the forecast is the revenue estimate determining
and constraining the state budget. Any future budget
adjustments by the governor and legislature due
to changing economic conditions are equal to the
percentage deviations from the forecasted amounts
initially used in the budget. Therefore, the error as a
percent of the forecasted level is the true measure of the
necessary adjustment, instead of the more commonly
used ratio of (actual-forecast)/actual.
It has always made more sense to me that the
forecasted value be used as the denominator, since it
is the forecasted value on which you are basing your
decisions.
The forecast is what drives manufacturing and is what
is communicated to shareholders.
You are measuring the accuracy of a forecast, so you
divide by the forecast. I thought this was a standard
approach in science and statistics.
If we were to measure a purely statistical forecast (no
qualitative adjustments), we would use Actual value (A)
as the denominator because statistically this should be
the most consistent number. However, once qualitative
input (human judgment) from sales is included, there
A F Avg A+F
Absolute Error % Error with A % Error with F % Error with
Avg of A&F
100
100
100
200
1000
10000
150
550
5050
100
900
9900
100%
900%
9900%
50%
90%
99%
0.667
164%
196%
1a. If the Forecast exceeds the Actual, the % error cannot exceed 100%.
100
50
50
100
75
75
50
50
50%
100%
100%
50%
67%
67%
1b. Illustration of the Symmetry of the sMAPE.
0
0
50
100
25
50
50
100
#DIV/0!
#DIV/0!
100%
100%
200%
200%
1c. When the Actual equals zero, use of sMAPE always yields 200%.
Table 1. Illustrative Calculations
| 30 | FORESIGHT Issue 12 Winter 2008
is an element that is not purely statistical in nature.
For this reason, we have chosen to rather divide by
forecast value (F) such that we measure performance
to our forecast.
B. The argument that the use of Forecast in the
denominator opens the opportunity for
manipulation is weak.
The politicizing argument is very weak, since the
forecast is in the numerator in any case. It also implies
being able to tamper with the forecast after the fact,
and that an unbiased forecast is not a goal of the
forecasting process.
Option 1 or 2
Percentage Error = Error / [Actual or Forecast:
It Depends] * 100
Several respondents indicated that they would choose
A or F, depending on the purpose of the forecast.
Actual, if measuring deviation of forecast from actual
values. Forecast, if measuring actual events deviated
from the forecast.
If the data are always positive and if the zero is
meaningful, then use Actual. This gives the MAPE and
is easy to understand and explain. Otherwise we need
an alternative to Actual in the denominator.
The actual value must be used as a denominator
whenever comparing forecast performance over time
and/or between groups. Evaluating performance is
an assessment of how close the forecasters come to
the actual or “true” value. If forecast is used in the
denominator, then performance assessment is sullied
by the magnitude of the forecasted quantity.
If Sales and Marketing are being measured and
provided incentives based on how well they forecast,
then we measure the variance of the forecast of each
from the actual value. If Sales forecast 150 and
Marketing forecast 60 and actual is 100, then Sales
forecast error is (150-100)/150=33% while Marketing
forecast error is (70-100)/70=43%. When Forecast is
the denominator, then Sales appears to be the better
forecaster – even though their forecast had a greater
difference to actual.
When assessing the impact of forecast error on
deployment and/or production, then forecast error
should be calculated with Forecast in the denominator
because inventory planning has been done assuming
the forecast is the true value.
Option 3
Percentage Error = Error / [Something Other
Than Actual or Forecast] * 100
One respondent indicated use of Actual or Forecast,
whichever had the highest value. No explanation
was given.
Three respondents use the average of the Actual
and the Forecast.
Averaging actual and forecast to get the denominator
results in a symmetrical percent-error measure. (Ed.
note: See Table 1b for an illustration, and the article by
Goodwin and Lawton (1999) for a deeper analysis of
the symmetry of the sMAPE.)
There likely is no “silver bullet” here, but it might be
worthwhile to throw into the mix using the average of
F and A – this helps solve the division-by-zero issues
and helps take out the bias. Using F alone encourages
high forecasting; using A alone does not deal with zero
actuals. (Ed. note: Unfortunately, the averaging of A
and F does not deal with the zero problem. When A is
zero, the division of the forecast error by the average
of A and F always results in a percentage error equal
to 200%, as shown in Table 1c below and discussed by
Boylan and Syntetos .)
I find the corrected sMAPE adequate for most
empirical applications without implying any cost
structure, although it is slightly downward biased.
In company scenarios, I have switched to suggesting
a weighted MAPE (by turnover, etc.) if it is used for
decision making and tracking.
| 31 | Winter 2008 Issue 12 FORESIGHT
CONTACT
Kesten@ForPrin.com
LenTashman@forecasters.org
Four respondents suggest use of some “average of
Actual values” in the denominator.
Use the mean of the series. Handles the case of
intermittent data, is symmetrical, and works for cross
section. (Ed. note: This recommendation leads to use
of the MAD/Mean, as recommended by Kolassa and
Schutz .)
My personal favorite is MAD/Mean. It is stable, even
for slow-moving items, it can be easily explained, and
it has a straightforward percentage interpretation.
A median baseline, or trimmed average, using
recent periods, provides a stable and meaningful
denominator.
I prefer a “local level” as the denominator in all the
error % calculations. (Ed. note: The local level can be
thought of as a weighted average of the historical data.)
When using Holt-Winters, I use the level directly, as it
is a highly reliable indication of the current trading
level of the time series. In addition, it isn’t affected by
outliers and seasonality. The latter factors may skew
to incorrect decisions.
With other types of forecasting – such as multivariate
there’s always some “local constant” that can be used.
Even a median of the last 6 months would do. The main
problem that arises here is what to do when this level
approaches zero. This hopefully does not happen
often in any set of data to be measured. It would rather
point, as a diagnostic, to issues other than forecasting
that need dire attention.
Two respondents recommend that the denominator
be the absolute average of the period-over-period
differences in the data, yielding a MASE (Mean
Absolute Scaled Error).
The denominator should be equal to the mean of the
absolute differences in the historical data. This is
better, for example, than the mean of the historical data,
because that mean could be close to zero. And, if the
data are nonstationary (e.g., trended), then the mean of
the historical data will change systematically as more
data are collected. However, the mean of the absolute
differences will be well behaved, even if the data are
nonstationary, and it will always be positive. It has the
added advantage of providing a neat, interpretable
statistic: the MASE. Values less than 1 mean that the
forecasts are more accurate than the in-sample, naïve,
one-step forecasts. (See Hyndman, 2006.)
Mean absolute scaled error, which uses the average
absolute error for the random walk forecast (i.e., the
absolute differences in the data).
FOLLOW-UP
We welcome your reactions to these results. Have
they clarified the issue? Have they provided new food
for thought? Have they changed your mind? See our
contact information at bottom.
REFERENCES
Boylan, J. & Syntetos, A. (2006). Accuracy and accuracy-implica-
tion metrics for intermittent demand, Foresight: The International
Journal of Applied Forecasting, Issue 4, 39-42.
Goodwin, P. & Lawton R. (1999). On the asymmetry of the sym-
metric MAPE, International Journal of Forecasting, 15, 405-408.
Green, K.C. & Tashman, L. (2008). Should we define forecast
error as e= F-A or e= A-F? Foresight: The International Journal
of Applied Forecasting, Issue 10, 38-40.
Hyndman, R. (2006). Another look at forecast-accuracy metrics
for intermittent demand, Foresight: The International Journal of
Applied Forecasting, Issue 4, 43-46.
Kolassa, S. & Schutz, W. (2007). Advantages of the MAD/MEAN
ratio over the MAPE. Foresight: The International Journal of
Applied Forecasting, Issue 6, 40-43.