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 $y$.