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Correlation vs Covariance

Student project for ECE302

by Blue



Correlation and Covariance

Correlation and covariance are very similarly related. Correlation is used to identify the relationship of two random variables, X and Y. In order to determine the dependence of the two events, the correlation coefficient,$ \rho $, is calculated as:

$ \rho (X,Y) = \frac{cov(X,Y)}{ \sqrt{var(X)var(Y)} } $

Covariance is defined as: $ C_{s}(n1, n2) = E(X-E[X])(Y-E[Y])) $ [1]

Correlation is then defined as: $ R_{s}(n1, n2) = E(XY) $ [2]

If X and Y are independent of each other, that means they are uncorrelated with each other, or cov(X,Y) = 0. However, if X and Y are uncorrelated, that does not mean they are independent of each other. 1, -1, and 0 are the three extreme points $ p\rho X,Y) $ can represent. 1 represents that X and Y are linearly dependent of each other. In other words, Y-E[Y] is a positive multiple of X-E[X]. -1 represents that X and Y are inversely dependent of each other. In other words, Y-E[Y] is a negative multiple of X-E[X]. [1]

Examples

Correlation coefficient graph pxy0.png $ \rho (X,Y) = 0 $ [1]

Correlation coefficient graph pxy10.png $ \rho (X,Y) = 1 $ [1]

Correlation coefficient graph pxy-10.png $ \rho (X,Y) = -1 $ [1]

Correlation coefficient graph pxy0 2.png $ \rho (X,Y) = .2 $ [1]

Correlation coefficient graph pxy4.png $ \rho (X,Y) = .4 $ [1]

Correlation coefficient graph pxy-7.png $ \rho (X,Y) = -.7 $ [1]

Correlation coefficient graph pxy9.png $ \rho (X,Y) = .9 $ [1]


Autocorrelation and Autocovariance

Correlation and covariance are comparing two random events. Autocorrelation and autocovariance are comparing the data points of one random event.

Autocovariance is defined as: $ C_{s}(n1, n2) = E(X_{n1}X_{n2}) $ [2]

Autocorrelation is defined as: $ R_{s}(n1, n2) = E((X_{n1}-E[X_{n1}])(X_{n2}-E[X_{n2}])) $[2]


References

[1]: Ilya Pollak. General Random Variables. 2012. Retrieved from https://engineering.purdue.edu/~ipollak/ece302/SPRING12/notes/19_GeneralRVs-4_Multiple_RVs.pdf

[2]: Ilya Pollak. Random Signals. 2004. Retrieved from https://engineering.purdue.edu/~ipollak/ee438/FALL04/notes/Section2.1.pdf

Back to ECE302 Spring 2013, Prof. Boutin

Alumni Liaison

Sees the importance of signal filtering in medical imaging

Dhruv Lamba, BSEE2010