Biomedical Engineering Reference
In-Depth Information
Figure 2.4-6 The reference waveform (A) is compared with the electroencephalogram signal (B) using a running correlation to determine
to what extent the electroencephalogram signal contains this pattern. The running correlation between the two waveforms varies over
time, but at maximum is only around 0.5. The running correlation operation is described and carried out in Section 2.4.3.2.
( Figure 2.4-5 ). (These numbers are all in relative units that
relate to voltage in the brain by an unknown calibration
factor.) These basic measurement numbers are not
enlightening about the EEG signal or the processes that
created it. More insight might be gained by comparing the
EEG signal with one or more reference signals, or mathe-
matical functions. For example, wemight ask, ''Howmuch
is the EEG signal like a 10-Hz sinusoid?'' Or, ''Howmuch is
it like a 12-Hz sinusoid, or a 12-Hz diamond-shaped wave,
or any other function/waveform that might shed some
light on the nature of the signal?'' Such comparisons can be
carried out using an operation known as correlation.
Another somewhat related question we might ask of
an unknown waveform such as the EEG signal is whether
the EEG signal contains anything like a brief waveform
such as that shown in Figure 2.4-6 A, or other short time
period waveform.
independent), this is not the case in mathematical
analysis, particularly if variables are related in a non-
linear manner. In the statistical sense, if two (or more)
variables are independent, they are uncorrelated, but
the reverse is not generally true. Moreover, signals that
are very much alike can still have a mathematical cor-
relation of zero. With these caveats in mind, correlation
seeks to quantify (i.e., to assign a number to) how
much one thing is like another. When comparing two
mathematical functions, we use the technique of mul-
tiplying one by the other, then averaging the results.
This average is often scaled by a normalizing factor.
This gives us what is known as the linear association
between two sets of variables. The same approach is
used when correlation is applied to two signals. Given
two functions, their average product will have the
largest possible positive value when the two functions
are identical. This process, since it is based on multi-
plication, will have the largest negative value when the
two functions are exact opposites of one another (i.e.,
one function is the negative of the other). The average
product will be zero when the two functions are, on
average, completely dissimilar, again in a mathematical
sense. Stated as an equation, the correlation between
2.4.3.1 Standard correlation
and covariance
Although it is common in everyday language to take
the word uncorrelated as meaning unrelated (and thus
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