Biomedical Engineering Reference
In-Depth Information
shifting one of the series forward and backward in time, the value of this
shifting will be evident later. The magnitude (
ve
or -
ve
)ofthisshiftingis
decided by the user and the time series of correlations is a function of the
phase shift,
τ
. The formula for the autocorrelation of
x (t )
is
R
xx
(τ )
:
+
T
1
T
x (t )x (t
+
τ)dt
0
R
xx
(τ )
=
(2.2)
R
xx
(
0
)
Where: x(t) has zero mean.
The formula for the cross-correlation of
x (t )
and
y(t )
is
R
xy
(τ )
:
T
1
T
x (t )y(t
+
τ)dt
0
R
xy
(τ )
=
R
xx
(
0
)R
yy
(
0
)
(2.3)
where:
x (t )
and
y(t )
have zero means.
It is easy to see the similarities between these formulae and the formula for
the Pearson product moment coefficient. The summation sign is replaced by
the integral sign, and to get the mean we now divide by
T
rather than
N
.The
denominator in these two equations, as in the Pearson equation, normalizes
the correlation to be dimensionless from
1. Also the two time series
must have a zero mean, as was the case in the Pearson formula, when the
means of
x
and
y
were subtracted. Note that the Pearson correlation is a
single
coefficient, while these auto- and cross-correlations are a series of correlation
scores over time at each value of
τ
.
−
1to
+
2.1.3 Four Properties of the Autocorrelation Function
Property #1.
The maximum value of
R
xx
(τ )
is
R
xx
(
0
)
which, in effect, is
the mean square of
x (t )
. For all values of the phase shift,
τ
, either
+
ve
or
−
ve R
xx
(τ )
is less than
R
xx
(
0
)
, which can be seen from the following proof.
From basic mathematics we know:
T
(x (t )
−
x (t
−
τ))
2
dt
≥
0
0
Expanding, we get:
T
(x (t )
2
−
τ)
2
+
x (t
−
2
x (t )x (t
−
τ))dt
≥
0
0
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