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Error propagation with dependent variables

  1. Oct 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Based on Microdosimetry theory, trying to figure out error propagation for a lot of quantities that are produced from radiation spectra. I am having trouble finding information on how to calculate and propagate errors when the quantities in my equations are not independent.

    2. Relevant equations

    I have a function called the dose-weighted lineal energy distribution:

    [itex]d(y) = \frac{yf(y)}{y_{F}} = \frac{yf(y)}{\int{yf(y)dy}}[/itex]

    I have calculated the constant [itex]y_F\pm\Delta y_F[/itex] using the measured quantity [itex]f(y)\pm\sqrt{f(y)}[/itex] but how do I find the uncertainty in the [itex]d(y)[/itex] distribution when these quantities are not independent? Note: [itex]\Delta y \approx 0[/itex] so this only concerns [itex]f(y)[/itex] and [itex]y_F[/itex].

    3. The attempt at a solution
    I had attempted doing this with the simplification method that I did in one of my 3rd year stats classes however I realised that this only applies for independent variables; don't know where to go know.

    Thanks :)
     
  2. jcsd
  3. Oct 26, 2015 #2

    mfb

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    Staff: Mentor

    How do you calculate yF based on f(y)?
     
  4. Oct 26, 2015 #3
    Edit: sorry, fixing up latex.

    Thanks for responding. Given that each channel has been measured independently of the others, the counts in each channel [itex]f(y)\pm \sqrt{f(y)}[/itex] can be used in the "simplification method". To expand on the definition:

    [itex]y_F = \int_{0}^{\infty}yf(y)dy = \Delta y \sum_{i = 1}^{n}y_{i}f(y_{i}) = \Delta y(y_{1}f(y_{1}) + y_{2}f(y_{2}) + ... + y_{n}f(y_{n})[/itex]

    where [itex]\Delta y[/itex] is the lineal energy channel width, no the error in y - forgive my lack of consistency. Anyway, [itex]\Delta \Delta y \approx 0[/itex] so we don't consider it in the error calculation except as a scaling constant.

    Treating each [itex]f(y_{i})\pm \Delta f(y_{i})[/itex] as indepdent variables we get:

    [itex]
    \begin{align*}
    \Delta y_{F} &= \sqrt{(\frac{\partial}{\partial f(y_{1})}[y_{F}]\Delta f(y_{1}))^2 + (\frac{\partial}{\partial f(y_{2})}[y_{F}]\Delta f(y_{2}))^2 + ... + (\frac{\partial}{\partial f(y_{n})}[y_{F}]\Delta f(y_{n}))^2} \\
    &= \Delta y \sqrt{(y_{1}\Delta f(y_{1}))^2 + (y_{2}\Delta f(y_{2}))^2 + ... + (y_{n}\Delta f(y_{n})])^2}
    \end{align*}
    [/itex]

    Is this valid?
     
    Last edited: Oct 26, 2015
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