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5. Forecast Accuracy Evaluation 91

b y being the forecasted outputs (or predicted time series), y the observed data (or
observed or measured time series), and N the number of observations. MBE is not
a good indicator of the model reliability because the errors often compensate each
other, but it allows to see how much it overestimates or underestimates.

• MAE is appropriate for applications with linear cost functions, that is, situations
where the costs resulting from a poor forecast are proportional to the forecast
error:
N
1 X
MAE ¼ jb yðiÞ yðiÞj (3.2)
N
i¼1
• The mean square error (MSE) uses the square of the difference between observed
and predicted values. This index penalizes the highest gaps:

N
1 X 2
MSE ¼ ðb yðiÞ yðiÞÞ (3.3)
N
i¼1
MSE is usually the index minimized by training algorithms [found in Artiﬁcial
Neural Network (ANN) methods for instance].
• The root mean square error (RMSE) is more sensitive to important forecast errors
and hence is suitable for applications where small errors are more tolerable and
larger errors lead to costs that are disproportionate, as in the case of utility
applications, for example. It is probably the reliability factor that is the most
widely used:
v ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u N
p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 1 X 2
RMSE ¼ MSE ¼ t ðb yðiÞ yðiÞÞ (3.4)
N
i¼1

• The mean absolute percentage error is close to the MAE but each gap between
observed and predicted data is divided by the observed data to get the relativegap.
N
1 X b yðiÞ yðiÞ
MAPE ¼ (3.5)
N yðiÞ
i¼1
This index has the disadvantage of being unstable when y(i) is near zero and it
cannot be deﬁned for y(i) ¼ 0.
Often, these errors are normalized, which is particularly true for the RMSE; the
mean value of irradiation is generally used as reference:
v ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u N
1
u X 2
ðb yðiÞ yðiÞÞ
t
N
i¼1
nRMSE ¼ (3.6)
y

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