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Correlation and Covariance of a Random Signal
Summary: (Blank Abstract)
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When we take the expected value, or average, of a random process, we measure several important characteristics about how the process behaves in general. This proves to be a very important observation. However, suppose we have several random processes measuring different aspects of a system. The relationship between these different processes will also be an important observation. The covariance and correlation are two important tools in finding these relationships. Below we will go into more details as to what these words mean and how these tools are helpful. Note that much of the following discussions refer to just random variables, but keep in mind that these variables can represent random signals or random processes.
Covariance
To begin with, when dealing with more than one random process, it should be obvious that it would be nice to be able to have a number that could quickly give us an idea of how similar the processes are. To do this, we use the covariance, which is analogous to the variance of a single variable.
- Definition 1: Covariance
- A measure of how much the deviations of two or more variables or processes match.
Mathematically, covariance is often written as
Useful Properties
-
If
X Y = 0 -
If
X Y = - X Y -
The covariance is symmetric
cov X Y = cov Y X -
Basic covariance identity
cov X + Y Z = cov X Z + cov Y Z -
Covariance of equal variables
cov X X = Var X
Correlation
For anyone who has any kind of statistical background, you should be able to see that the idea of dependence/independence among variables and signals plays an important role when dealing with random processes. Because of this, the correlation of two variables provides us with a measure of how the two variables affect one another.
- Definition 2: Correlation
- A measure of how much one random variable depends upon the other.
Correlation Coefficient
It is often useful to express the correlation of random
variables with a range of numbers, like a percentage. For a
given set of variables, we use the correlation
coefficient to give us the linear relationship
between our variables. The correlation coefficient of two
variables is defined in terms of their covariance and standard
deviations, denoted by
|
note:
So far we have dealt with correlation simply as a number relating the relationship between any two variables. However, since our goal will be to relate random processes to each other, which deals with signals as a function of time, we will want to continue this study by looking at correlation functions.Example
Now let us take just a second to look at a simple example that
involves calculating the covariance and correlation of two
sets of random numbers. We are given the following data sets:
Matlab Code for Example
The above example can be easily calculated using Matlab. Below I have included the code to find all of the values above.
x = [3 1 6 3 4];
y = [1 5 3 4 3];
mx = mean(x)
my = mean(y)
mxy = mean(x.*y)
% Standard Dev. from built-in Matlab Functions
std(x,1)
std(y,1)
% Standard Dev. from Equation Above (same result as std(?,1))
sqrt( 1/5 * sum((x-mx).^2))
sqrt( 1/5 * sum((y-my).^2))
cov(x,y,1)
corrcoef(x,y)
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This work is licensed by Michael Haag under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.
Last edited by Charlet Reedstrom on Jul 18, 2005 3:42 pm GMT-5.