computing Variance between vectors

I want to compute the variance between vectors, it's result should be a scalar number but matlab command "var" giving me a vector result. let suppose I have two vector x1= [ 2 2 2]' , x2= [4 4 4]' and X= [x1; x2]. The command "var" giving the result of a vector as var(X(1:3,1:2),0,2) = [2 2 2]. but by varriance sample formula

because the square of a vector does not mean square of all its elements. I think the square of the vector is equal to the dot product of a vector with its self which is scalar

Am I right?

Answer 1

[Bruno Luong]

No you are not right, the variance computes the square of the "dispersion" of the random variable. In case of random vector variable the variance formula applies on component by component independently and the result is the vector of the same length than the input.

The variance is actually the vector of diagonal terms of the covariance matrix, which computes also the cross correlation of the components of the variable.

For

x1= [2; 2; 2];
x2= [4; 4; 4]; 
X = [x1,x2]

the command

var(X,0,2)

works along the 2nd dimension for three identical row-vectors [2,4]. For all component

N = 2;

The expectation (mean) is

mu = (2+4)/N = 3

variance is

1/(N-1) * ((2-mu)^2 + (4-mu)^2)) =
1/1 * ((2-3)^2 + (4-3)^2) = 2

The result is [2,2,2].

The quantity 6 in this example you pretend to be variance is actually the trace of the covariance matrix, which is some time called "total variance".

[mahmood hassan]

Thanks so much for your Answer but In my question I used transpose of x1 and x2.

as x1= [ 2 2 2]' , x2= [4 4 4]'. so X contains two column vectors

[mahmood hassan]

If we take mean of two vectors it would be a vector not a scalar as mean(X). if suppose the vectors are not identical, how you justify your computation for means as a scalar.

Thnaks

[Bruno Luong]

"so X contains two column vectors "

So does mine. And your example has an error since you cocatenate vertically the vector [x1; x2].

"your computation for means as a scalar"

The mean is a scalar but for each individual row of X. The three rows is the same vector [2,4]. I started my explanation with "For all component..."

If you want a formula, the variance var(X,0,2) is simply a row vector v where

v(i) = std(X(i,:))^2

Answer 2

[mahmood hassan]

Thanks so much for your time and cooperation.

I don't think there is any difference between std(X(i,:))^2 and var(X(i,:)).

Standard deviation is the square root of variance.

Is there any difference between these two commands.