Environmental Engineering Reference
In-Depth Information
apply data transformations, such as those shown in
Table 10.8, until the relationship between the variables
is approximately linear, and then fit a linear relationship,
or piecewise linear relationships, to the observed data.
Based on these considerations, regression using linear
relationships are very common, and the general
approach for linear regression between two variables
will be given here.
If the relationship between two variables,
x
and
y
, are
linear, with
x
being a deterministic variable (i.e., can be
specified with certainty) and
y
being a stochastic vari-
able, then linear regression assumes that values of
x
and
y
, denoted by
x
i
and
y
i
respectively, are related by
yields the following estimates of
a
and
b
to be used in
Equation (10.125) as
S
S
xy
b
=
(10.132)
xx
a
= −
y bx
(10.133)
where
x
and
y
are the sample means of
x
and
y
, respec-
tively. The sampling distributions of the statistics
a
and
b
are given by
a
−
α
NS
xx
for a
S
S
+
(
Nx
)
2
e
xx
t
=
(10.134)
b
−
β
S
N
y
=
α β
+
x
+
ε
(10.124)
xx
i
i
i
for b
S
e
where
α
and
β
are constants, and ε
i
is a random outcome
drawn from a normal distribution with a mean of zero
and a standard deviation of
σ
ε
. Based on these assertions,
y
i
is normally distributed with a mean of
α
+
βx
i
and a
standard deviation of
σ
ε
. If
a
and
b
denote estimates of
α
and
β
estimated from measured samples, such that the
expected value of
y
i
, denoted by
E
(
y
i
) is estimated by
where
t
has a
t
-distribution with
N
− 2 degrees of
freedom. using these results gives the
α
confidence
limits of
α
and
β
as
2
S
Nx
NS
+
(
)
xx
(10.135)
α
= ±
a t S
α
/
2
e
xx
N
S
E y
(
) = +
a bx
(10.125)
i
i
β
=
b t S
(10.136)
α
/2
e
xx
then the sample estimate of the variance,
σ
2
, is denoted
by
S
e
2
, which is calculated as
EXAMPLE 10.25
N
2
1
S S
−
−
(
S
)
∑
(
[
]
=
2
xx
yy
xy
Two water-quality variables,
X
and
Y
, are measured
simultaneously, and the results are as follows:
S
2
=
y
−
a bx
+
)
(10.126)
e
i
i
N
−
2
N N
(
2
)
S
xx
i
=
1
where
S
e
is called the standard error of the estimate, and
S
xx
,
S
yy
, and
S
xy
are defined by the relations
x
y
x
y
x
y
x
y
11.21
60.08
16.91
55.22
22.95
56.09
28.90
56.73
12.14
54.09
17.98
56.69
23.96
62.74
30.02
57.64
2
N
N
−
12.79
52.24
19.14
59.34
25.22
54.80
31.12
60.85
∑
∑
2
(10.127)
S
=
N
x
x
xx
i
i
14.20
57.90
20.10
52.23
26.25
59.54
32.19
61.81
i
=
1
i
=
1
14.78
54.03
21.06
57.06
26.87
58.92
-
-
16.15
56.45
21.89
54.90
28.18
58.12
-
-
2
N
N
−
∑
∑
(10.128)
S
=
N
y
2
y
yy
i
i
Estimate the parameters of a linear equation relating
the two variables and the 95% confidence intervals of
the parameters. It can be assumed that
X
is a determin-
istic variable.
i
=
1
i
=
1
N
N
N
−
∑
∑ ∑
S
=
N
x y
x
y
(10.129)
xy
i
i
i
i
i
=
1
i
=
1
i
=
1
Solution
Minimizing
S
e
2
in Equation (10.126) relative to
a
and
b
by requiring that
From the given data,
N
= 22. A linear equation of the
form
y
=
a
+
bx
will be fit to the data. The computations
to determine
a
and
b
are as follows:
∂
∂
S
a
2
e
0
(10.130)
=
N
1
2
∂
∂
S
b
∑
e
0
(10.131)
x
=
x
i
=
21 54
.
=
N
i
=
1
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