<s>
In	O
statistics	O
,	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
scaled	I-General_Concept
error	I-General_Concept
(	O
MASE	O
)	O
is	O
a	O
measure	O
of	O
the	O
accuracy	O
of	O
forecasts	O
.	O
</s>
<s>
It	O
is	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
error	I-General_Concept
of	O
the	O
forecast	O
values	O
,	O
divided	O
by	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
error	I-General_Concept
of	O
the	O
in-sample	O
one-step	O
naive	O
forecast	O
.	O
</s>
<s>
The	O
mean	B-General_Concept
absolute	I-General_Concept
scaled	I-General_Concept
error	I-General_Concept
has	O
favorable	O
properties	O
when	O
compared	O
to	O
other	O
methods	O
for	O
calculating	O
forecast	O
errors	O
,	O
such	O
as	O
root-mean-square-deviation	B-General_Concept
,	O
and	O
is	O
therefore	O
recommended	O
for	O
determining	O
comparative	O
accuracy	O
of	O
forecasts	O
.	O
</s>
<s>
The	O
mean	B-General_Concept
absolute	I-General_Concept
scaled	I-General_Concept
error	I-General_Concept
has	O
the	O
following	O
desirable	O
properties	O
:	O
</s>
<s>
Scale	O
invariance	O
:	O
The	O
mean	B-General_Concept
absolute	I-General_Concept
scaled	I-General_Concept
error	I-General_Concept
is	O
independent	O
of	O
the	O
scale	O
of	O
the	O
data	O
,	O
so	O
can	O
be	O
used	O
to	O
compare	O
forecasts	O
across	O
data	O
sets	O
with	O
different	O
scales	O
.	O
</s>
<s>
Predictable	O
behavior	O
as	O
:	O
Percentage	O
forecast	O
accuracy	O
measures	O
such	O
as	O
the	O
Mean	B-General_Concept
absolute	I-General_Concept
percentage	I-General_Concept
error	I-General_Concept
(	O
MAPE	B-General_Concept
)	O
rely	O
on	O
division	O
of	O
,	O
skewing	O
the	O
distribution	O
of	O
the	O
MAPE	B-General_Concept
for	O
values	O
of	O
near	O
or	O
equal	O
to	O
0	O
.	O
</s>
<s>
Symmetry	O
:	O
The	O
mean	B-General_Concept
absolute	I-General_Concept
scaled	I-General_Concept
error	I-General_Concept
penalizes	O
positive	O
and	O
negative	O
forecast	O
errors	O
equally	O
,	O
and	O
penalizes	O
errors	O
in	O
large	O
forecasts	O
and	O
small	O
forecasts	O
equally	O
.	O
</s>
<s>
In	O
contrast	O
,	O
the	O
MAPE	B-General_Concept
and	O
median	O
absolute	O
percentage	O
error	O
(	O
MdAPE	O
)	O
fail	O
both	O
of	O
these	O
criteria	O
,	O
while	O
the	O
"	O
symmetric	O
"	O
sMAPE	O
and	O
sMdAPE	O
fail	O
the	O
second	O
criterion	O
.	O
</s>
<s>
Interpretability	O
:	O
The	O
mean	B-General_Concept
absolute	I-General_Concept
scaled	I-General_Concept
error	I-General_Concept
can	O
be	O
easily	O
interpreted	O
,	O
as	O
values	O
greater	O
than	O
one	O
indicate	O
that	O
in-sample	O
one-step	O
forecasts	O
from	O
the	O
naïve	O
method	O
perform	O
better	O
than	O
the	O
forecast	O
values	O
under	O
consideration	O
.	O
</s>
<s>
The	O
DM	O
statistic	O
for	O
the	O
MASE	O
has	O
been	O
empirically	O
shown	O
to	O
approximate	O
this	O
distribution	O
,	O
while	O
the	O
mean	O
relative	O
absolute	O
error	O
(	O
MRAE	O
)	O
,	O
MAPE	B-General_Concept
and	O
sMAPE	O
do	O
not	O
.	O
</s>
<s>
For	O
a	O
seasonal	O
time	O
series	O
,	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
scaled	I-General_Concept
error	I-General_Concept
is	O
estimated	O
in	O
a	O
manner	O
similar	O
to	O
the	O
method	O
for	O
non-seasonal	O
time	O
series	O
:	O
</s>
<s>
The	O
main	O
difference	O
with	O
the	O
method	O
for	O
non-seasonal	O
time	O
series	O
,	O
is	O
that	O
the	O
denominator	O
is	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
error	I-General_Concept
of	O
the	O
one-step	O
"	O
seasonal	O
naive	O
forecast	O
method	O
"	O
on	O
the	O
training	O
set	O
,	O
which	O
uses	O
the	O
actual	O
value	O
from	O
the	O
prior	O
season	O
as	O
the	O
forecast	O
:	O
Ft	O
=	O
Yt−m	O
,	O
where	O
m	O
is	O
the	O
seasonal	O
period	O
.	O
</s>
<s>
In	O
this	O
case	O
the	O
MASE	O
is	O
the	O
Mean	B-General_Concept
absolute	I-General_Concept
error	I-General_Concept
divided	O
by	O
the	O
Mean	B-General_Concept
Absolute	I-General_Concept
Deviation	I-General_Concept
.	O
</s>
