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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 behavesin general. This proves to be a very important observation. However, suppose we have several random processes measuringdifferent aspects of a system. The relationship between these different processes will also be an important observation. Thecovariance and correlation are two important tools in finding these relationships. Below we will go into more details as towhat these words mean and how these tools are helpful. Note that much of the following discussions refer to just randomvariables, 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 havea 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.

Covariance
A measure of how much the deviations of two or more variables or processes match.
For two processes, X and Y , if they are not closely related then the covariance will be small, and if they are similar then the covariancewill be large. Let us clarify this statement by describing what we mean by "related" and "similar." Two processes are"closely related" if their distribution spreads are almost equal and they are around the same, or a very slightlydifferent, mean.

Mathematically, covariance is often written as σ x y and is defined as

cov X Y σ x y X X Y Y
This can also be reduced and rewritten in the following twoforms:
σ x y x y x y
σ x y y x X X Y Y f x y

Useful properties

  • If X and Y are independent and uncorrelated or one of them has zero mean value, then σ x y 0
  • If X and Y are orthogonal, then σ 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/independenceamong 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.

Correlation
A measure of how much one random variable depends upon the other.
This measure of association between the variables will provideus with a clue as to how well the value of one variable can be predicted from the value of the other. The correlation isequal to the average of the product of two random variables and is defined as
cor X Y X Y y x x y f x y

Correlation coefficient

It is often useful to express the correlation of random variables with a range of numbers, like a percentage. For agiven set of variables, we use the correlation coefficient to give us the linear relationship between our variables. The correlation coefficient of twovariables is defined in terms of their covariance and standard deviations , denoted by σ x , as seen below

ρ cov X Y σ x σ y
where we will always have -1 ρ 1 This provides us with a quick and easy way to view the correlation between our variables. If there is norelationship between the variables then the correlation coefficient will be zero and if there is a perfect positivematch it will be one. If there is a perfect inverse relationship, where one set of variables increases while theother decreases, then the correlation coefficient will be negative one. This type of correlation is often referred tomore specifically as the Pearson's Correlation Coefficient ,or Pearson's Product Moment Correlation.

Positive Correlation
Negative Correlation
Uncorrelated (No Correlation)
Types of Correlation

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 oftime, 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 twosets of random numbers. We are given the following data sets: x 3 1 6 3 4 y 1 5 3 4 3 To begin with, for the covariance we will need to find the expected value , or mean, of x and y . x 1 5 3 1 6 3 4 3.4 y 1 5 1 5 3 4 3 3.2 x y 1 5 3 5 18 12 12 10 Next we will solve for the standard deviations of our two setsusing the formula below (for a review click here ). σ X X 2 σ x 1 5 0.16 5.76 6.76 0.16 0.36 1.625 σ y 1 6 4.84 3.24 0.04 0.64 0.04 1.327 Now we can finally calculate the covariance using one of the two formulas found above. Since we calculated the threemeans, we will use that formula since it will be much simpler. σ x y 10 3.4 3.2 -0.88 And for our last calculation, we will solve for thecorrelation coefficient, ρ . ρ -0.88 1.625 1.327 -0.408

Matlab code for example

The above example can be easily calculated using Matlab. Below I have included the code to find all of the valuesabove.

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)

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
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Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
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Harshika Reply
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Harshika
find the subring of gaussian integers?
Rofiqul
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Shirley Reply
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Mark
I need quadratic equation link to Alpa Beta
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find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
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use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
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Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
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How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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