1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculate covariance matrix of two given numbers of events

  1. Apr 9, 2012 #1
    Hi, I am trying to follow this paper: (arXiv link).
    On page 18, Appendix A.1, the authors calculate a covariance matrix for two variables in a way I cannot understand.
    1. The problem statement, all variables and given/known data
    Variables [itex]N_1[\itex]
    and [itex]N_2[\itex], distributed on
    [itex]y \in [0, 1][\itex] as follows:
    [itex]f_1(y<y_0) = 0; f_1(y>= y_0) = \frac{1}{1-y_0}[/itex]
    [itex]f_2 = 1[\itex]
    Define: [itex]N_{<} = [\itex]amount of events falling below [itex]y_0[\itex], [itex]N_{>}[\itex] analogously...
    2. Relevant equations
    Then, we can calculate:
    [itex]N_2 = \frac{1}{y_0}N_{<}[\itex]
    [itex]N_1 = -\frac{1-y_0}{y_0}N_{<} + N_{>}[\itex]
    Covariance of two variables: [itex]Cov(a,b) = <(a·b)^2> - <a·b>^2[\itex]

    3. The attempt at a solution
    After calculating the two previous results myself (N1 and N2), I tried to calculate the covariance matrix V (please see pdf). I tried to do it from the definition of covariance, but I didn't get anywhere (or to different results). So I decided to guess if the authors were doing the usual error propagation from [itex]N_1, N_2[\itex] as a function of [itex]N_{<}, N_{>}[\itex].
    Considering the usual in physics for N large, [itex]\sigma(N_k) = \sqrt(N_k), k \in {<, >}[\itex], and then, for instance [itex]\sigma^2(N_2) =\sum_k \frac{\partial N_2}{\partial N_k} \sigma(N_k) = \frac{1}{y_0 ^2} N_{>}[\itex] (the right result).
    However, I cannot extend this to the off-diagonal terms.

    Could somebody please help me? Thanks!

    EDIT: I cannot see the latex expressions, and instead I see all my itex \itex ! How can I solve this? Thanks!
     
    Last edited: Apr 9, 2012
  2. jcsd
  3. Apr 9, 2012 #2
    Right...
    after some trial and error, I got to an expression for the right solution of the covariance matrix:

    Given the (2x2) covariance matrix V, and the variables:

    N1 = a1/c
    N2 = -(1-c)a1/c + a2

    I calculated V(m,n) = sum(i = 1 to 2) (dNm/da_i)(dNn/da_i) Error^2(Ni),
    where Error^2(Ni) = Ni, and m, n are either 1 or 2.

    This works and I get the right solution. Also, the expression makes sense... However, I couldn't find this in a book on statistics. Could somebody point me in a good direction? Do you know of any book/page were I can see the proof of this?

    Thank you!! And sorry for the wrong latex formulae in the previous post, but I don't know how to fix it. I counted the number of itex \itex and it is right :S
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Calculate covariance matrix of two given numbers of events
  1. Calculating covariance (Replies: 0)

Loading...