Biomedical Engineering Reference
In-Depth Information
analyses evaluate how well a given signal is correlated with another signal
over past, present, and future points in time. We are familiar in statistics
with the Pearson product moment correlation. It is a measure of relation-
ship between two variables and allows us to determine whether a variable
x
increases or decreases as the variable
y
increases. The strength and polarity
of this relationship is given by the correlation coefficient: the higher the value
the stronger the relationship, while the sign indicates if variables
x
and
y
are
increasing and decreasing together (positive correlation) or if one is increasing
while the other is decreasing (negative correlation). The correlation coefficient
is a normalized dimensionless number varying from
−
1to
+
1.
2.1.1 Similarity to the Pearson Correlation
Consider the formula for the Pearson product moment correlation coefficient
relating two variables,
x
and
y
:
N
1
N
(x
i
−
x )(y
i
−
y)
i
=
1
r
=
(2.1)
s
x
s
y
where:
x
i
and
y
i
are the i
th
samples of x and y,
x
and
y
are the means of
x
and
y
, and
s
x
and
s
y
are the standard deviations of
x
and
y
.
The numerator of the formula is the sum of the product of the two vari-
ables after the mean value of each variable has been subtracted. It is easy to
appreciate that if
x
and
y
are random and unrelated then (
x
i
−
y
)
will be scattered in the
x
-
y
plane about zero (see Figure 2.1). These products
x
) and (
y
i
−
y
x
Figure 2.1
Scatter diagram of variable
x
against variable
y
showing no relationship
between the variables.
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