Covariance and Correlation
28 Jul 2017 The reverse is not true in general: if the covariance of two random variables is 0 they can still be dependent! Page 2. –2–. Properties of ...
covariance
Scatterplots and Correlation
Measuring Linear Association: Correlation Calculate and interpret correlation. ... and motivation scores in this example range from 0 to 100.
scatterplots and correlation notes
Reminder No. 1: Uncorrelated vs. Independent
27 Feb 2013 If ρ(XY) = 0
uncorrelated vs independent
Pearson's correlation
We can categorise the type of correlation by considering as one variable increases The first three represent the “extreme” correlation values of -1 0.
pearsons
New Automatic Search Tool for Impossible Differentials and Zero
Abstract. Impossible differential and zero-correlation linear cryptanalysis are two of the most powerful cryptanalysis methods in the field of symmetric key
Links among Impossible Differential Integral and Zero Correlation
Secondly by establishing some boolean equations
The Bivariate Normal Distribution
Zero Correlation Implies Independence. If two random variables X and Y are jointly normal and are uncorrelated then they are independent.
Bivariate Normal
Zero Correlation Independence
https://www.tandfonline.com/doi/pdf/10.1080/00031305.1986.10475412
Correlation coefficient and p-values: what they are and why you
The p-value is a number between 0 and 1 representing the probability that this data would have arisen if the null hypothesis were true. In medical trials the
p values
1.10.5 Covariance and Correlation
2. If random variables X1 and X2 are independent then cov(X1X2)=0. 3. var(aX1 + bX2) =
MS NotesWeek
Will Monroe
CS 109Lecture Notes #15
July 28, 2017
Covariance and CorrelationBased on a chapter by Chris PiechCovariance and Correlation
Consider the two plots shown below. In both images I have plotted one thousand samples drawnfrom an underlying joint distribution. Clearly the two distributions are different. However, the mean
and variance are the same in both thexand theydimension. What is different?Covarianceis a quantitative measure of the extent to which the deviation of one variable from its
mean matches the deviation of the other from its mean. It is a mathematical relationship that is defined as:Cov(X;Y)=E[(XE[X])(YE[Y])]
Themeaningofthismathematicaldefinitionmaynotbeobviousatafirstglance.IfXandYareboth above their respective means,orifXandYare both below their respective means, the expression inside the outer expectation will be positive. If one is above its mean and the other is below, the term is negative. If this expression is positive on average, the two random variables will have a positive correlation. We can rewrite the above equation to get an equivalent equation:Cov(X;Y)=E[XY]E[Y]E[X]
Using this equation (and the fact that the expectation of the product of two independent randomvariables is equal to the product of the expectations) is it easy to see that if two random variables
are independent their covariance is 0. The reverse isnottrue in general: if the covariance of two random variables is 0, they can still be dependent! - 2 -Properties of Covariance
Say thatXandYare arbitrary random variables:
Cov(X;Y)=Cov(Y;X)
Cov(X;X)=E[X2]E[X]E[X]=Var(X)
Cov(aX+b;Y)=aCov(X;Y)
LetX=X1+X2++Xnand letY=Y1+Y2++Ym. The covariance ofXandYis:Cov(X;Y)=n
X i=1m X j=1Cov(Xi;Yj)Cov(X;X)=Var(X)=n
X i=1n X j=1Cov(Xi;Xj) That last property gives us a third way to calculate variance.Correlation
Covariance is interesting because it is a quantitative measurement of the relationship between two variables. Correlation between two random variables,(X;Y)is the covariance of the two variables normalized by the variance of each variable. This normalization cancels the units out and normalizes the measure so that it is always in the range [0, 1]: (X;Y)=Cov(X;Y)pVar(X)Var(Y)Correlation measures linearity betweenXandY.
(X;Y)=1Y=aX+bwherea=y=x (X;Y)=1Y=aX+bwherea=y=x (X;Y)=0absence of linear relationship If(X;Y)=0we say thatXandYare "uncorrelated." If two variables are independent, then theircorrelation will be 0. However, like with covariance. it doesn"t go the other way. A correlation of 0
does not imply independence. When people use the term correlation, they are actually referring to a specific type of correlation called "Pearson" correlation. It measures the degree to which there is a linear relationship between the two variables. An alternative measure is "Spearman" correlation, which has a formula almost identical to the correlation defined above, with the exception that the underlying random variables are first transformed into their rank. Spearman correlation is outside the scope of CS109. - 1 -Will Monroe
CS 109Lecture Notes #15
July 28, 2017
Covariance and CorrelationBased on a chapter by Chris PiechCovariance and Correlation
Consider the two plots shown below. In both images I have plotted one thousand samples drawnfrom an underlying joint distribution. Clearly the two distributions are different. However, the mean
and variance are the same in both thexand theydimension. What is different?Covarianceis a quantitative measure of the extent to which the deviation of one variable from its
mean matches the deviation of the other from its mean. It is a mathematical relationship that is defined as:Cov(X;Y)=E[(XE[X])(YE[Y])]
Themeaningofthismathematicaldefinitionmaynotbeobviousatafirstglance.IfXandYareboth above their respective means,orifXandYare both below their respective means, the expression inside the outer expectation will be positive. If one is above its mean and the other is below, the term is negative. If this expression is positive on average, the two random variables will have a positive correlation. We can rewrite the above equation to get an equivalent equation:Cov(X;Y)=E[XY]E[Y]E[X]
Using this equation (and the fact that the expectation of the product of two independent randomvariables is equal to the product of the expectations) is it easy to see that if two random variables
are independent their covariance is 0. The reverse isnottrue in general: if the covariance of two random variables is 0, they can still be dependent! - 2 -Properties of Covariance
Say thatXandYare arbitrary random variables:
Cov(X;Y)=Cov(Y;X)
Cov(X;X)=E[X2]E[X]E[X]=Var(X)
Cov(aX+b;Y)=aCov(X;Y)
LetX=X1+X2++Xnand letY=Y1+Y2++Ym. The covariance ofXandYis:Cov(X;Y)=n
X i=1m X j=1Cov(Xi;Yj)Cov(X;X)=Var(X)=n
X i=1n X j=1Cov(Xi;Xj) That last property gives us a third way to calculate variance.Correlation
Covariance is interesting because it is a quantitative measurement of the relationship between two variables. Correlation between two random variables,(X;Y)is the covariance of the two variables normalized by the variance of each variable. This normalization cancels the units out and normalizes the measure so that it is always in the range [0, 1]: (X;Y)=Cov(X;Y)pVar(X)Var(Y)Correlation measures linearity betweenXandY.
(X;Y)=1Y=aX+bwherea=y=x (X;Y)=1Y=aX+bwherea=y=x (X;Y)=0absence of linear relationship If(X;Y)=0we say thatXandYare "uncorrelated." If two variables are independent, then theircorrelation will be 0. However, like with covariance. it doesn"t go the other way. A correlation of 0
does not imply independence. When people use the term correlation, they are actually referring to a specific type of correlation called "Pearson" correlation. It measures the degree to which there is a linear relationship between the two variables. An alternative measure is "Spearman" correlation, which has a formula almost identical to the correlation defined above, with the exception that the underlying random variables are first transformed into their rank. Spearman correlation is outside the scope of CS109.