Covariance Matrix

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  • #1
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if X= (3, 5, 7) & Y = (2, 4, 1)

What is the 3x3 covariance matrix for X & Y?
 

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  • #2
mathman
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if X= (3, 5, 7) & Y = (2, 4, 1)

What is the 3x3 covariance matrix for X & Y?
You have two variables, so the matrix is 2x2. The elements are var(X), var(Y) along the diagonal and cov(X,Y) off diagonal (both).
 
  • #3
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Since 2x2 we need two diagonal elements and two off diagonal elements.

Are the following two elements "off diagonal elements"?

cov(X,Y) & cov(Y,X);
 
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  • #4
chiro
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Yes they are the off diagonal elements.
 
  • #5
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The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?
 
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  • #6
mathman
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The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?
In this context what do you mean by dimensional? X and Y are real valued. Do you mean the number of samples?
 
  • #8
mathman
Science Advisor
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461
The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?
1 refers to real valued (1 dimensional) random variables.
2 is a generalization to vectors (n or m dimensional) which have random variables as components.
 

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