Re: A question about Covariance??

On 2006-06-02 07:24:55 -0300, "Rune Allnor" <allnor@xxxxxxxxxxxx> said:


VijaKhara skrev:
Dear members,
For two Real Random Process X(t) and Y(t), the cross-covariance is Mean
of ((X(t)-MeanX(t))(Y(t)-MeanY(t)).
I learn that people calculate the correlation of two Random Process to
determine how similar two processes are. I am wondering what is the
target of calculating the covariance? If we want to find out the
similarity of two process, correlation is enough, why we need
covariance?

I think you are wrong. If we want to find the "similarity" between two
general signals, we need to use covariance. Correlation only works if
we want to test zero-mean signals.

Take an example:

x(n) = cos(pi*n/2)+10
y(n)= sin(pi*n/2)+ 3

The cos and sin terms (with zero mean) are orthogonal.
Adding the mean terms destroy this orthiogonality, as is
easily seen:

One period, cos and sin terms only, zero mean:

x1(n) = 1 0 -1 0
y1(n) = 0 1 0 -1

<x1,y1> = 0

Adding the means:

x2(n) = 11 10 9 10
y2(n) = 3 4 3 2

<x2,y2> = 33+40+27+20 = 110

So two signal shapes that ought to be orthogonal
end up with a non-zero correlation by adding a
non-zero mean.

The "spurious" correlation <x2,y2> above can
easily mask the "true" match <y2,y2>, as can
be seen

<y2,y2> = 9 + 16 + 9 +4 = 38 << 110.

Subtracting the means is crucial to find the correct match.

One more thing, correlation is for determine the similarity of two
processes, what is the target of Auto-correlation?

One use is to test for periodic features in the signal.

Rune

The difference between correlation and covariance is scaling and not
centering. Correlation is scaled to lie between -1 and +1. The scale
factor is the product of the two standard deviations (which is of the
same physical dimension like volts) so correlation is a pure number.
Both correlation and covariance are with respect to the mean, or in other
words they are centered.

One expects that what the original poster missed was the extra word "auto"
as in autocorrelation, autocovariance and autocrosscovariance. There is
a terminology problem. The autocovariance is between time offsets of
the same time series while the scalar version is between two different
statistical events. So autocrosscovariance would be needed for time
offsets of differing time series.

In you example the sine times series is zero correlated at offset zero
with the cosine but well correlated (value 1) at an offset of 1/4 period.

The answer to the original question is that the correlation needs to
also specifiy a timing offset. That then involves the extra adjective
of auto as in autocovariance to see if a time series is like itself
but with a delay or autocrosscovariance to see if the one time series
more like the other at some particular delay.

Ask how one detect the other series being a delayed version of the first
series as a simple case where the offset is important.





.

Relevant Pages

  • Re: point me in the right direction for time series analysis?
    ... > I want to analyse each infant separately. ... For each baby two time series are generated: ... > a correlation between two time series variables which are both strongly ... > managed to get hold of a decent book on time series analysis, ...
    (sci.stat.consult)
  • Re: Hypothesis testing, two different rates
    ... The correlation between to time series can be computed. ... significance of a correlation coefficient using "standard procedures" ... In specific ARREST RATE is significantly related to OD rate in the ...
    (sci.stat.edu)
  • Re: Statistics of cross-correlation
    ... That won't happen for two "random time series" if it ... you will get some positive correlation, ... Do you think that this would be a valid test? ...
    (sci.stat.consult)
  • Re: time varying correlation
    ... > I am trying to find out relation b/w two variables, ... You are looking at time series. ... > with monthly or weekly data, there will be very few points and that may ... > not give a good estimate of correlation. ...
    (sci.stat.math)
  • Re: Montecarlo: correlation parameter value for matched pairs?
    ... ran some simulations in order to see the effect of the correlation ... ....If I use relatively-small transistors I get too high offset ... However, if I use the infamous correlation factor, with, say, cc=0.75, ... spanish (mother tongue) and italian, ...
    (comp.cad.cadence)
Loading