Environmental Engineering Reference
In-Depth Information
is given by
y
i
−
y
i
,
calcd
2
bx
i
)
2
.
W
=
=
(y
i
−
a
−
i
i
To obtain the value of
a
and
b
, one minimizes the function
W
with respect to both
a
and
b
, that is, differentiate the function with respect to
a
and
b
and set them equal to
zero. Thus,
2
i
∂W
∂a
=−
( y
i
−
a
−
bx
i
)
=
0
and
2
i
∂W
∂b
=−
( y
i
−
−
bx
i
) x
i
=
a
0.
The above equations give us the following two equations:
b
i
na
+
x
i
=
y
i
,
i
a
i
b
i
x
i
=
x
i
+
x
i
y
i
,
i
where
n
is the total number of paired
(x
,
y)
data points.
Solving the above equations gives us the values of
a
and
b
as follows:
i
x
i
i
y
i
−
i
x
i
i
x
i
y
i
n
i
x
i
−
i
x
i
2
a
=
,
n
i
x
i
y
i
−
i
x
i
i
y
i
n
i
x
i
−
i
x
i
2
b
=
.
The respective
standard deviations
in the slope and intercept are then given by the
following equations:
i
x
i
n
i
x
i
−
i
x
i
2
1
/
2
σ
a
= σ
y
,
n
n
i
x
i
−
i
x
i
2
1
/
2
σ
b
= σ
y
,
where
i
(y
i
−
a
−
bx
i
)
2
(n
−
2
σ
y
=
.
2
)
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