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Root mean square error (RMSE) or mean absolute error (MAE)?

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Abstract

Both the root mean square error (RMSE) and the mean absolute error (MAE) are regularly employed in model evaluation studies. Willmott and Matsuura (2005) have suggested that the RMSE is not a good indicator of average model performance and might be a misleading indicator of average error and thus the MAE would be a better metric for that purpose. Their paper has been widely cited and may have influenced many researchers in choosing MAE when presenting their model evaluation statistics. However, we contend that the proposed avoidance of RMSE and the use of MAE is not the solution to the problem. In this technical note, we demonstrate that the RMSE is not ambiguous in its meaning, contrary to what was claimed by Willmott et al. (2009). The RMSE is more appropriate to represent model performance than the MAE when the error distribution is expected to be Gaussian. In addition, we show that the RMSE satisfies the triangle inequality requirement for a distance metric.
Geosci. Model Dev., 7, 1247–1250, 2014
www.geosci-model-dev.net/7/1247/2014/
doi:10.5194/gmd-7-1247-2014
© Author(s) 2014. CC Attribution 3.0 License.
Root mean square error (RMSE) or mean absolute error (MAE)? –
Arguments against avoiding RMSE in the literature
T. Chai1,2 and R. R. Draxler1
1NOAA Air Resources Laboratory (ARL), NOAA Center for Weather and Climate Prediction,
5830 University Research Court, College Park, MD 20740, USA
2Cooperative Institute for Climate and Satellites, University of Maryland, College Park, MD 20740, USA
Correspondence to: T. Chai (tianfeng.chai@noaa.gov)
Received: 10 February 2014 – Published in Geosci. Model Dev. Discuss.: 28 February 2014
Revised: 27 May 2014 – Accepted: 2 June 2014 – Published: 30 June 2014
Abstract. Both the root mean square error (RMSE) and the
mean absolute error (MAE) are regularly employed in model
evaluation studies. Willmott and Matsuura (2005) have sug-
gested that the RMSE is not a good indicator of average
model performance and might be a misleading indicator of
average error,and thus the MAE would be a better metric for
that purpose. While some concerns over using RMSE raised
by Willmott and Matsuura (2005) and Willmott et al. (2009)
are valid, the proposed avoidance of RMSE in favor of MAE
is not the solution. Citing the aforementioned papers, many
researchers chose MAE over RMSE to present their model
evaluation statistics when presenting or adding the RMSE
measures could be more beneficial. In this technical note, we
demonstrate that the RMSE is not ambiguous in its mean-
ing, contrary to what was claimed by Willmott et al. (2009).
The RMSE is more appropriate to represent model perfor-
mance than the MAE when the error distribution is expected
to be Gaussian. In addition, we show that the RMSE satis-
fies the triangle inequality requirement for a distance metric,
whereas Willmott et al. (2009) indicated that the sums-of-
squares-based statistics do not satisfy this rule. In the end, we
discussed some circumstances where using the RMSE will be
more beneficial. However, we do not contend that the RMSE
is superior over the MAE. Instead, a combination of metrics,
including but certainly not limited to RMSEs and MAEs, are
often required to assess model performance.
1 Introduction
The root mean square error (RMSE) has been used as a stan-
dard statistical metric to measure model performance in me-
teorology, air quality, and climate research studies. The mean
absolute error (MAE) is another useful measure widely used
in model evaluations. While they have both been used to
assess model performance for many years, there is no con-
sensus on the most appropriate metric for model errors. In
the field of geosciences, many present the RMSE as a stan-
dard metric for model errors (e.g., McKeen et al., 2005;
Savage et al., 2013; Chai et al., 2013), while a few others
choose to avoid the RMSE and present only the MAE, cit-
ing the ambiguity of the RMSE claimed by Willmott and
Matsuura (2005) and Willmott et al. (2009) (e.g., Taylor
et al., 2013; Chatterjee et al., 2013; Jerez et al., 2013). While
the MAE gives the same weight to all errors, the RMSE pe-
nalizes variance as it gives errors with larger absolute values
more weight than errors with smaller absolute values. When
both metrics are calculated, the RMSE is by definition never
smaller than the MAE. For instance, Chai et al. (2009) pre-
sented both the mean errors (MAEs) and the rms errors (RM-
SEs) of model NO2column predictions compared to SCIA-
MACHY satellite observations. The ratio of RMSE to MAE
ranged from 1.63 to 2.29 (see Table 1 of Chai et al., 2009).
Using hypothetical sets of four errors, Willmott and
Matsuura (2005) demonstrated that while keeping the MAE
as a constant of 2.0, the RMSE varies from 2.0 to 4.0.
They concluded that the RMSE varies with the variability
of the the error magnitudes and the total-error or average-
error magnitude (MAE), and the sample size n. They further
Published by Copernicus Publications on behalf of the European Geosciences Union.
1248 T. Chai and R. R. Draxler: RMSE or MAE
demonstrated an inconsistency between MAEs and RMSEs
using 10 combinations of 5 pairs of global precipitation data.
They summarized that the RMSE tends to become increas-
ingly larger than the MAE (but not necessarily in a mono-
tonic fashion) as the distribution of error magnitudes be-
comes more variable. The RMSE tends to grow larger than
the MAE with n1
2since its lower limit is fixed at the MAE
and its upper limit (n1
2·MAE) increases with n1
2. Further,
Willmott et al. (2009) concluded that the sums-of-squares-
based error statistics such as the RMSE and the standard er-
ror have inherent ambiguities and recommended the use of
alternates such as the MAE.
As every statistical measure condenses a large number of
data into a single value, it only provides one projection of the
model errors emphasizing a certain aspect of the error char-
acteristics of the model performance. Willmott and Matsuura
(2005) have simply proved that the RMSE is not equivalent
to the MAE, and one cannot easily derive the MAE value
from the RMSE (and vice versa). Similarly, one can readily
show that, for several sets of errors with the same RMSE, the
MAE would vary from set to set.
Since statistics are just a collection of tools, researchers
must select the most appropriate tool for the question being
addressed. Because the RMSE and the MAE are defined dif-
ferently, we should expect the results to be different. Some-
times multiple metrics are required to provide a complete
picture of error distribution. When the error distribution is
expected to be Gaussian and there are enough samples, the
RMSE has an advantage over the MAE to illustrate the error
distribution.
The objective of this note is to clarify the interpretation of
the RMSE and the MAE. In addition, we demonstrate that
the RMSE satisfies the triangle inequality requirement for a
distance metric, whereas Willmott and Matsuura (2005) and
Willmott et al. (2009) have claimed otherwise.
2 Interpretation of RMSE and MAE
To simplify, we assume that we already have nsamples of
model errors calculated as (ei,i=1,2,...,n). The un-
certainties brought in by observation errors or the method
used to compare model and observations are not considered
here. We also assume the error sample set is unbiased. The
RMSE and the MAE are calculated for the data set as
MAE =1
n
n
X
i=1|ei|(1)
RMSE =v
u
u
t
1
n
n
X
i=1
e2
i.(2)
The underlying assumption when presenting the RMSE is
that the errors are unbiased and follow a normal distribution.
Table 1. RMSEs and MAEs of randomly generated pseudo-errors
with a zero mean and unit variance Gaussian distribution. Five sets
of errors of size nare generated with different random seeds.
nRMSEs MAEs
4 0.92, 0.65, 1.48, 1.02, 0.79 0.70, 0.57, 1.33, 1.16, 0.76
10 0.81, 1.10, 0.83, 0.95, 1.01 0.65, 0.89, 0.72, 0.84, 0.78
100 1.05, 1.03, 1.03, 1.00, 1.04 0.82, 0.81, 0.79, 0.78, 0.78
1000 1.04, 0.98, 1.01, 1.00, 1.00 0.82, 0.78, 0.80, 0.80, 0.81
10 000 1.00, 0.98, 1.01, 1.00, 1.00 0.79, 0.79, 0.79, 0.81, 0.80
100 000 1.00, 1.00, 1.00, 1.00, 1.00 0.80, 0.80, 0.80, 0.80, 0.80
1000 000 1.00, 1.00, 1.00, 1.00, 1.00 0.80, 0.80, 0.80, 0.80, 0.80
Thus, using the RMSE or the standard error (SE)1helps to
provide a complete picture of the error distribution.
Table 1 shows RMSEs and MAEs for randomly generated
pseudo-errors with zero mean and unit variance Gaussian
distribution. When the sample size reaches 100 or above, us-
ing the calculated RMSEs one can re-construct the error dis-
tribution close to its “truth” or “exact solution”, with its stan-
dard deviation within 5 % to its truth (i.e., SE =1). When
there are more samples, reconstructing the error distribution
using RMSEs will be even more reliable. The MAE here is
the mean of the half-normal distribution (i.e., the average of
the positive subset of a population of normally distributed
errors with zero mean). Table 1 shows that the MAEs con-
verge to 0.8, an approximation to the expectation of q2
π. It
should be noted that all statistics are less useful when there
are only a limited number of error samples. For instance, Ta-
ble 1 shows that neither the RMSEs nor the MAEs are robust
when only 4 or 10 samples are used to calculate those values.
In those cases, presenting the values of the errors themselves
(e.g., in tables) is probably more appropriate than calculat-
ing any of the statistics. Fortunately, there are often hundreds
of observations available to calculate model statistics, unlike
the examples with n=4 (Willmott and Matsuura, 2005) and
n=10 (Willmott et al., 2009).
Condensing a set of error values into a single number, ei-
ther the RMSE or the MAE, removes a lot of information.
The best statistics metrics should provide not only a perfor-
mance measure but also a representation of the error distribu-
tion. The MAE is suitable to describe uniformly distributed
errors. Because model errors are likely to have a normal dis-
tribution rather than a uniform distribution, the RMSE is a
better metric to present than the MAE for such a type of data.
1For unbiased error distributions, the standard error (SE) is
equivalent to the RMSE as the sample mean is assumed to be
zero. For an unknown error distribution, the SE of mean is the
square root of the “bias-corrected sample variance”. That is, SE =
s1
n1
n
P
i=1(ei)2, where =1
n
n
P
i=1ei.
Geosci. Model Dev., 7, 1247–1250, 2014 www.geosci-model-dev.net/7/1247/2014/
T. Chai and R. R. Draxler: RMSE or MAE 1249
3 Triangle inequality of a metric
Both Willmott and Matsuura (2005) and Willmott et al.
(2009) emphasized that sums-of-squares-based statistics do
not satisfy the triangle inequality. An example is given in
a footnote of Willmott et al. (2009). In the example, it is
given that d(a,c) =4, d (a, b) =2, and d(b, c) =3, where
d(x , y) is a distance function. The authors stated that d(x , y)
as a “metric” should satisfy the “triangle inequality” (i.e.,
d(a , c) d(a, b) +d (b, c)). However, they did not specify
what a,b, and crepresent here before arguing that the sum
of squared errors does not satisfy the “triangle inequality”
because 4 2+3, whereas 4222+32. In fact, this exam-
ple represents the mean square error (MSE), which cannot be
used as a distance metric, rather than the RMSE.
Following a certain order, the errors ei,i=1,...,n can
be written into a n-dimensional vector . The L1-norm and
L2-norm are closely related to the MAE and the RMSE, re-
spectively, as shown in Eqs. (3) and (4):
||1= n
X
i=1|ei|!=n·MAE (3)
||2=v
u
u
t n
X
i=1
e2
i!=n·RMSE.(4)
All vector norms satisfy |X+Y|≤|X|+|Y|and |X| =
|X|(see, e.g., Horn and Johnson, 1990). It is trivial to
prove that the distance between two vectors measured by
Lp-norm would satisfy |XY|p≤ |X|p+|Y|p. With three
n-dimensional vectors, X,Y, and Z, we have
|XY|p= |(XZ)(YZ)|p≤ |XZ|p+|YZ|p.(5)
For n-dimensional vectors and the L2-norm, Eq. (5) can
be written as
v
u
u
t
n
X
i=1
(xiyi)2v
u
u
t
n
X
i=1
(xizi)2+v
u
u
t
n
X
i=1
(yizi)2,(6)
which is equivalent to
v
u
u
t
1
n
n
X
i=1
(xiyi)2v
u
u
t
1
n
n
X
i=1
(xizi)2
+v
u
u
t
1
n
n
X
i=1
(yizi)2.(7)
This proves that RMSE satisfies the triangle inequality re-
quired for a distance function metric.
4 Summary and discussion
We present that the RMSE is not ambiguous in its meaning,
and it is more appropriate to use than the MAE when model
errors follow a normal distribution. In addition, we demon-
strate that the RMSE satisfies the triangle inequality required
for a distance function metric.
The sensitivity of the RMSE to outliers is the most com-
mon concern with the use of this metric. In fact, the exis-
tence of outliers and their probability of occurrence is well
described by the normal distribution underlying the use of the
RMSE. Table 1 shows that with enough samples (n100),
including those outliers, one can closely re-construct the er-
ror distribution. In practice, it might be justifiable to throw
out the outliers that are several orders larger than the other
samples when calculating the RMSE, especially if the num-
ber of samples is limited. If the model biases are severe, one
may also need to remove the systematic errors before calcu-
lating the RMSEs.
One distinct advantage of RMSEs over MAEs is that RM-
SEs avoid the use of absolute value, which is highly unde-
sirable in many mathematical calculations. For instance, it
might be difficult to calculate the gradient or sensitivity of
the MAEs with respect to certain model parameters. Further-
more, in the data assimilation field, the sum of squared er-
rors is often defined as the cost function to be minimized by
adjusting model parameters. In such applications, penalizing
large errors through the defined least-square terms proves to
be very effective in improving model performance. Under the
circumstances of calculating model error sensitivities or data
assimilation applications, MAEs are definitely not preferred
over RMSEs.
An important aspect of the error metrics used for model
evaluations is their capability to discriminate among model
results. The more discriminating measure that produces
higher variations in its model performance metric among dif-
ferent sets of model results is often the more desirable. In this
regard, the MAE might be affected by a large amount of aver-
age error values without adequately reflecting some large er-
rors. Giving higher weighting to the unfavorable conditions,
the RMSE usually is better at revealing model performance
differences.
In many of the model sensitivity studies that use only
RMSE, a detailed interpretation is not critical because varia-
tions of the same model will have similar error distributions.
When evaluating different models using a single metric, dif-
ferences in the error distributions become more important.
As we stated in the note, the underlying assumption when
presenting the RMSE is that the errors are unbiased and fol-
low a normal distribution. For other kinds of distributions,
more statistical moments of model errors, such as mean, vari-
ance, skewness, and flatness, are needed to provide a com-
plete picture of the model error variation. Some approaches
that emphasize resistance to outliers or insensitivity to non-
normal distributions have been explored by other researchers
(Tukey, 1977; Huber and Ronchetti, 2009).
As stated earlier, any single metric provides only one pro-
jection of the model errors and, therefore, only emphasizes
a certain aspect of the error characteristics. A combination
www.geosci-model-dev.net/7/1247/2014/ Geosci. Model Dev., 7, 1247–1250, 2014
1250 T. Chai and R. R. Draxler: RMSE or MAE
of metrics, including but certainly not limited to RMSEs and
MAEs, are often required to assess model performance.
Acknowledgements. This study was supported by NOAA grant
NA09NES4400006 (Cooperative Institute for Climate and
Satellites – CICS) at the NOAA Air Resources Laboratory in
collaboration with the University of Maryland.
Edited by: R. Sander
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