Noise and covariance matrix...

I'm wondering how to calculate the covariance matrix for a noise vector
made like this

c = w1 + cos(a) * w2 + sin(a) * w2

where c,w1,w2 are a Nx1 vectors, x is a scalar and w1, w2 are vectors
representing additive gaussian white noise and they are uncorrelated
between them. The covariance matrices of w1 and of w2 are known and are the
same, but I cannot see how it all would mix. No assumptions can be made on
x: it's not such that cos(x) and sin(x) can be considered orthogonal.

Even simpler, what's the covariance of the vector

w3 = w1 + w2

in terms of covariance matrices of w1 and w2? Again they are uncorrelated.

Thanks,

kl31n

On Thu, 04 Jan 2007 00:23:04 -0000, kl31n <"kl31n(get rid of this to wri=
te  =

me back)"@hotmail.com> wrote:

> I'm wondering how to calculate the covariance matrix for a noise vecto=
r
> made like this
>
> c =3D w1 + cos(a) * w2 + sin(a) * w2

Where you have "a", do you mean "x"?


> where c,w1,w2 are a Nx1 vectors, x is a scalar and w1, w2 are vectors
> representing additive gaussian white noise and they are uncorrelated
> between them. The covariance matrices of w1 and of w2 are known and ar=
e  =

> the
> same, but I cannot see how it all would mix. No assumptions can be mad=
e  =

> on
> x: it's not such that cos(x) and sin(x) can be considered orthogonal.

Is x deterministic or random?  If it's deterministic, then the answer ca=
n  =

be found from the following facts:

* The covariance matrix of a vector of uncorrelated variables is a  =

diagonal matrix, where each diagonal element is the variance of the  =

corresponding vector element.

* If A and B are uncorrelated variables, then var{A+B} =3D var{A} + var{=
B}.

* The variance of kA (where k is a deterministic scalar) is k.var{A}.

If x is random, then you will also need the following fact (and hopefull=
y  =

you mean "independent" rather than "uncorrelated", as it's much easier t=
o  =

work with!):

* If A and B are independent variables, then var{A.B} =3D var{A}.var{B}.=





-- =

Oli

Il Thu, 04 Jan 2007 00:34:51 GMT, Oli Charlesworth ha scritto:

First of all thanks for you answer.

>> I'm wondering how to calculate the covariance matrix for a noise vector
>> made like this
>>
>> c = w1 + cos(a) * w2 + sin(a) * w2
> 
> Where you have "a", do you mean "x"?

Yes, it's x, before posting I changed x is a, assuming it was more readable
but I forgot to do the same with the other occurrences. Furthermore let me
add now that I should reformulate the thing as

c = w1 + cos(x) * w2 + sin(x) * w3

where w1, w2, w3 are independent from each other.

> Is x deterministic or random?

x is random.

> * If A and B are independent variables, then var{A.B} = var{A}.var{B}.

Which operator is that "."?

Thanks again,

kl31n

On Fri, 05 Jan 2007 00:19:44 -0000, kl31n <"kl31n(get rid of this to wri=
te  =

me back)"@hotmail.com> wrote:

> Il Thu, 04 Jan 2007 00:34:51 GMT, Oli Charlesworth ha scritto:
>
>> * If A and B are independent variables, then var{A.B} =3D var{A}.var{=
B}.
>
> Which operator is that "."?

Multiply (not dot product or anything).



-- =

Oli

kl31n <"kl31n(get rid of this to write me back)"@hotmail.com> wrote in
news:q23d5zri156c.2lps8ish24sr.dlg@40tude.net: 

> Il Thu, 04 Jan 2007 00:34:51 GMT, Oli Charlesworth ha scritto:
> 
> First of all thanks for you answer.
> 
>>> I'm wondering how to calculate the covariance matrix for a noise
>>> vector made like this
>>>
>>> c = w1 + cos(a) * w2 + sin(a) * w2
>> 
>> Where you have "a", do you mean "x"?
> 
> Yes, it's x, before posting I changed x is a, assuming it was more
> readable but I forgot to do the same with the other occurrences.
> Furthermore let me add now that I should reformulate the thing as
> 
> c = w1 + cos(x) * w2 + sin(x) * w3
> 
> where w1, w2, w3 are independent from each other.
> 
>> Is x deterministic or random?
> 
> x is random.
> 
>> * If A and B are independent variables, then var{A.B} =
>> var{A}.var{B}. 
> 
> Which operator is that "."?
> 
> Thanks again,
> 
> kl31n


If J is the Jacobian, the covariance matrix is given by,

covar = (J^T J)^{-1}

The Jacobian is made up of colums of partial(c)/partial(theta), where theta 
is one parameter.

So, if you were looking for the variance/covariance matrix in w, you would 
calculate the Jacobian

J=[ dc/dw1 dc/dw2 dc/dw3]  with one row for each element in x

 

-- 
Scott
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