<s>
In	O
statistics	O
,	O
the	O
explained	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
ESS	O
)	O
,	O
alternatively	O
known	O
as	O
the	O
model	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
or	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
regression	O
(	O
SSR	O
–	O
not	O
to	O
be	O
confused	O
with	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
RSS	O
)	O
or	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
of	O
errors	O
)	O
,	O
is	O
a	O
quantity	O
used	O
in	O
describing	O
how	O
well	O
a	O
model	O
,	O
often	O
a	O
regression	O
model	O
,	O
represents	O
the	O
data	O
being	O
modelled	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
explained	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
measures	O
how	O
much	O
variation	O
there	O
is	O
in	O
the	O
modelled	O
values	O
and	O
this	O
is	O
compared	O
to	O
the	O
total	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
TSS	O
)	O
,	O
which	O
measures	O
how	O
much	O
variation	O
there	O
is	O
in	O
the	O
observed	O
data	O
,	O
and	O
to	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
,	O
which	O
measures	O
the	O
variation	O
in	O
the	O
error	O
between	O
the	O
observed	O
data	O
and	O
modelled	O
values	O
.	O
</s>
<s>
The	O
explained	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
ESS	O
)	O
is	O
the	O
sum	O
of	O
the	O
squares	O
of	O
the	O
deviations	O
of	O
the	O
predicted	O
values	O
from	O
the	O
mean	O
value	O
of	O
a	O
response	O
variable	O
,	O
in	O
a	O
standard	O
regression	O
model	O
—	O
for	O
example	O
,	O
,	O
where	O
yi	O
is	O
the	O
i	O
th	O
observation	O
of	O
the	O
response	O
variable	O
,	O
xji	O
is	O
the	O
i	O
th	O
observation	O
of	O
the	O
j	O
th	O
explanatory	O
variable	O
,	O
a	O
and	O
bj	O
are	O
coefficients	O
,	O
i	O
indexes	O
the	O
observations	O
from	O
1	O
to	O
n	O
,	O
and	O
εi	O
is	O
the	O
ith	O
value	O
of	O
the	O
error	O
term	O
.	O
</s>
<s>
The	O
following	O
equality	O
,	O
stating	O
that	O
the	O
total	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
TSS	O
)	O
equals	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(=	O
SSE	O
:	O
the	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
errors	I-Algorithm
of	I-Algorithm
prediction	I-Algorithm
)	O
plus	O
the	O
explained	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
SSR	O
:the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
regression	O
or	O
explained	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
)	O
,	O
is	O
generally	O
true	O
in	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
:	O
</s>
<s>
The	O
residual	O
vector	O
is	O
,	O
so	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
is	O
,	O
after	O
simplification	O
,	O
</s>
