Biomedical Engineering Reference
In-Depth Information
more measured variables. It is particularly useful when dealing with continuous quantita-
tive data like the amplitude of myoelectric signal levels as a function of muscle force, for
example.
In its simplest form, for a given series of paired variables (
x
1
,
y
1
)
,(
x
2
,
y
2
),...,(
x
n
,
y
n
)
where
n
≥
2, and following a relationship known to be linear
y
=
a
+
bx
(5.62)
The best-fitting curve in a least squares sense minimizes the residual,
ε
, to determine the
coefficients
a
and
b
n
n
]
2
]
2
ε
=
[
y
i
−
f
(
x
i
)
=
[
y
i
−
(
a
+
bx
i
)
=
min
(5.63)
i
=
1
i
=
1
To obtain the least squared error, the partial derivatives of the residual with respect to
a
and
b
, respectively, must both be zero.
n
∂ε
∂
a
=
2
[
y
i
−
(
a
+
bx
i
)
]
=
0
i
=
1
(5.64)
n
∂ε
∂
b
=
x
i
[
y
i
−
(
a
+
bx
i
)
]
=
2
0
i
=
1
Expanding and solving for
a
and
b
i
=
1
y
i
i
=
1
x
i
i
=
1
x
i
i
=
1
x
i
y
i
n
n
n
n
−
a
=
(5.65)
i
=
1
x
i
2
i
=
1
x
i
n
n
n
−
i
=
1
x
i
y
i
−
n
i
=
1
x
i
n
i
=
1
y
i
n
n
b
=
(5.66)
i
=
1
x
i
2
i
=
1
x
i
n
n
n
−
The correlation coefficient,
r
, is a measure of the degree of linear association between the
two variables and is defined as
i
=
1
i
=
1
n
n
x
i
y
i
i
=
1
x
i
y
i
n
−
n
r
=
(5.67)
n
x
i
2
n
y
i
2
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
i
=
1
i
=
1
i
=
1
x
i
i
=
1
y
i
n
n
−
−
n
n
It is seldom necessary to use these equations directly, as many scientific calculators and
computer software packages are already programmed to perform regression analysis.
Consider the measured EMG signals from the biceps as a function of the applied
force, shown as the
+
markers in Figure 5-66. The coefficients
a
and
b
are obtained in
MATLAB and are used to plot the solid best-fit line
y
=
1
.
9503
+
0
.
3531
x
. The dashed
