Biomedical Engineering Reference
In-Depth Information
perfect. In practice, because of the error in data, one is not to expect a perfect match between
the data and the model. The ratio of the unexplained variation to the total variation is called
random factor
:
ðn 1Þs
2
P
i ¼ 1
ðy
i
yÞ
2
UV
TV
¼
F
R
¼
(7.25)
The complementary of the
random factor
is usually called the
coefficient of determination
and is
defined by
ðn 1Þs
2
P
i ¼ 1
ðy
i
yÞ
2
R
2
¼ 1 F
R
¼ 1
(7.26)
The square root of the coefficient of determination,
R
, is called the
correlation coefficient
, which
is sometimes called as
correlation
in short. Therefore, from parametric estimation point of
view, the higher the value of
R
, the more reliable are the model parameters. If
R
is close to
0orif
F
R
is close to unity, we say that
x
i
and
y
i
are not correlated (randomly related). If
R
is close to unity or random factor is close to zero, we say that
x
i
and
y
i
are strongly correlated
by the model.
From
Eqn (7.26)
, the correlation coefficient can be obtained in general as
t
p
1 F
R
ðn 1Þs
2
P
i ¼1
y
i
n
P
i ¼ 1
y
i
2
R ¼
¼
1
(7.27)
2
or the
unexplained variations at final correlation conditions is readily available. Therefore, compu-
tation of
R
will not require extensive computational effort. The value of
R
is frequently used
as an indicator to the quality of fit.
For the flow rate versus rotameter reading data in
Table 7.1
, the regression model is linear.
Minimizing the variance of the data around the model yield the final model as given by
y ¼ 0:0661 þ 0:05842x
2
, the computation of the minimum
If the regression is carried out by minimizing
s
s
(7.28)
and the variance of the data in
Table 7.1
around the model,
Eqn (7.28)
, is given by
s
2
¼ 0:001554
From the data in
Table 7.1
, we can also compute
X
X
n
n
y
i
¼ 218:1895;
y
i
¼ 56:75405
i ¼ 1
i ¼ 1
Therefore,
t
t
1
ðn 1Þs
2
P
i ¼ 1
y
i
n
P
i ¼ 1
y
i
2
ð19 1Þ0:001554
218:1895
1
R ¼
1
¼
¼ 0:999712
19
56:75405
2
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