What is the best method for plotting function parameter errors?

[ChrisVer]

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

F(x;2a)-F(x;a)

as well as (their fluctuation)

\frac{F(x;2a)-F(x;a)}{F(x;a)}

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?

 

[Stephen Tashi]

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 F(x,a) and F(x,2a) might not show the most extreme values. For example, it's possible that that for some a < c < 2a that F(x,c) might be be greater than both F(x,a) and F(x,2a). If you're sure that this kind of thing won't happen then then your idea of plotting only F(x,a) and F(x,2a) 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.

 

[ChrisVer]

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

 

[Stephen Tashi]

I would have to plot F(x;10) and F(x;100) and look at their differences...

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

What specific F(x,a) are you dealing with?

 

[ChrisVer]

Recombination (cosmology) and the uncertainty in determining the recombination temperature T in which X(T_{rec})= \frac{n_{ion}}{n_e}=1/2
http://www.maths.qmul.ac.uk/~jel/ASTM108lecture8.pdf
(Eq. 8.23 with uncertainty in \eta =\frac{n_B}{n_\gamma}= 4 - 8 \times 10^{-10} )

 

[Stephen Tashi]

(Eq. 8.23 with uncertainty in \eta =\frac{n_B}{n_\gamma}= 4 - 8 \times 10^{-10} )

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

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

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

So you are plotting this as y = T = f(x,a) with a = \eta. But what variable plays the role of x ?

 

[ChrisVer]

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

 

[Stephen Tashi]

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