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Function parameter error

  1. Nov 26, 2014 #1

    ChrisVer

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    Hi, suppose that you have some function: [itex]F(x;a)[/itex]
    where [itex]x[/itex] is the variable with which you plot the function and [itex]a[/itex] is some parameter which enters the function.
    If I want to find the error coming from some uncertainty in [itex]a[/itex], computationally, I would have to plot the function for 2 different let's say values of [itex]a[/itex]: Let's say that this means to plot the functions below:
    [itex]F(x;a)[/itex]
    [itex]F(x;2a)[/itex]
    Then I believe the error then can can be computed by (their difference):

    [itex]F(x;2a)-F(x;a)[/itex]

    as well as (their fluctuation)

    [itex]\frac{F(x;2a)-F(x;a)}{F(x;a)}[/itex]

    Which of these two are best for a plotting? Is there some physical meaning behind any of these two? like they are showing something different to the reader?
     
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  3. Nov 26, 2014 #2

    Stephen Tashi

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    I think what you're after is to "portray" the "error" visually, not to compute it. For such a purpose, I'd pefer to see a shaded graph created by plotting lots of curves on top of each other The reason is that the points on the the graphs of [itex] F(x,a) [/itex] and [itex]F(x,2a) [/itex] might not show the most extreme values. For example, it's possible that that for some [itex] a < c < 2a [/itex] that [itex] F(x,c) [/itex] might be be greater than both [itex] F(x,a) [/itex] and [itex] F(x,2a) [/itex]. If you're sure that this kind of thing won't happen then then your idea of plotting only [itex] F(x,a) [/itex] and [itex] F(x,2a) [/itex] would be sufficient.

    You haven't defined what you mean by "error". If the graph is to portray a specific statistical meaning, we'd have to know the probability model for the situation.
     
  4. Nov 26, 2014 #3

    ChrisVer

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    By error I mean something like this: in general you can't determine [itex]a[/itex] exactly, but within some range [itex](a_{min}, a_{max})[/itex]... This will cause an error to the function [itex]F(x;a)[/itex] coming from [itex]a[/itex]...
    So I thought :
    I could determine it by eg saying that I can determine [itex]a[/itex] within an order of magnitude (let's say [itex] 10 \le a \le 100[/itex]), what should I do to see the error then? I would have to plot [itex]F(x;10)[/itex] and [itex]F(x;100)[/itex] and look at their differences...
     
  5. Nov 26, 2014 #4

    Stephen Tashi

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    That would be Ok if the graph of [itex] F(x,c) [/itex] always rises as [itex]c [/itex] increases or always falls as [itex] c [/itex] increases. But suppose as [itex] c [/itex] increases between 10 and 100, the point at [itex] F(5,c) [/itex] moves up and down. Then [itex] F(5,10) [/itex] and [itex] F(5,100) [/itex] might not indicate the extremes of the movement.

    What specific [itex] F(x,a) [/itex] are you dealing with?
     
  6. Nov 26, 2014 #5

    ChrisVer

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    Recombination (cosmology) and the uncertainty in determining the recombination temperature [itex]T[/itex] in which [itex] X(T_{rec})= \frac{n_{ion}}{n_e}=1/2[/itex]
    http://www.maths.qmul.ac.uk/~jel/ASTM108lecture8.pdf [Broken]
    (Eq. 8.23 with uncertainty in [itex]\eta =\frac{n_B}{n_\gamma}= 4 - 8 \times 10^{-10}[/itex] )
     
    Last edited by a moderator: May 7, 2017
  7. Nov 27, 2014 #6

    Stephen Tashi

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    [itex] \frac{ n_{ion}}{n_e} = \frac{n_\gamma}{n_B} \ exp( \frac { E_{ion}} {k_B T} ) \ \ [/itex] (Eq.8.23)

    [itex] T = \frac{E_{ion}}{k_B} \frac{1}{ \ln({\frac{n_{ion}}{n_e})} \ - \ \ln({ \frac{n_\gamma}{n_B} )} }[/itex]

    [itex] T = \frac{E_{ion}}{k_B} \frac{1}{ \ln({\frac{n_{ion}}{n_e})} \ - \ \ln({ \frac{1}{\eta} )} }[/itex]

    So you are plotting this as [itex] y = T = f(x,a) [/itex] with [itex] a = \eta [/itex]. But what variable plays the role of [itex] x [/itex] ?
     
  8. Nov 28, 2014 #7

    ChrisVer

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    ehmm.. no, I am plotting [itex]X \equiv X(T ;\eta)= \frac{1}{\eta} \exp \Big ( \frac{E_{ion}}{k_BT} \Big) [/itex] for [itex]3000<T(Kelvin)<4500[/itex]
    And [itex]\eta= 4 \times 10^{-10} [/itex] and [itex]\eta= 8 \times 10^{-10}[/itex]
    However I'd [personally] like to generalize this to an uncertainty of [itex]\eta[/itex] within an order of magnitude...
     
  9. Nov 28, 2014 #8

    Stephen Tashi

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    Then for a given value of [itex] T [/itex] , the point on the graph, as a function of [itex] \eta [/itex] has the form [itex] y = s \frac{1}{\eta} [/itex] where [itex] s [/itex] is a constant. So I plotting points given by the extreme values of [itex] \eta [/itex] will show the extremes of variation in [itex] y [/itex].
     
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