Uncertainty on an asymmetry signal

[kelly0303]

Hello! I have an experiment in which I measure the counts given some experimental parameters, call them $E$ in order to extract some physics parameter of interest, call it $X$, so I have $N(E,X)$. This $N$ (the number of counts) will have a statistical error (which goes like $\sqrt{N}$) a systematic error, from the uncertainty in some of the $E$ parameters (I am denoting with $E$ all the experimental parameters) and it can also have other source of uncertainties, for example from external electric and magnetic fields. For the purpose of this question, let's say, in the ideal case, the number of counts is proportional to $N \propto (E+X)^2$. It can be shown that if I build the following assymetry function:

$$A = \frac{N_+-N_-}{N_++N_-}$$
where $N_+ \propto (E+X)^2$ and $N_- \propto (E-X)^2$ I can significantly remove the noise from external sources (basically if I invert the experimental parameters, the external noise won't invert, so by taking the difference, I just cancel the external noise). In the above case, also assuming $X<<E$ I end up with:

$$A = \frac{N_+-N_-}{N_++N_-} = \frac{2X}{E}$$
So experimentally, by measuring $N_+$ and $N_-$, and knowing $E$ I can extract $X$. However, I am not sure how to find the best value for E, where to perform the measurement. Here is where my confusion stems from. Let's say we ignore the uncertainty on $E$ for now and the only uncertainty is the statistical one. If I denote the uncertainty on $N_+$ and $N_-$ by $d_+$ and $d_-$ respectively and I do an error propagation starting from:

$$A = \frac{N_+-N_-}{N_++N_-}$$
I get:

$$dA=\frac{2}{(N_++N_-)^2}\sqrt{N_+^2d_-^2+N_-^2d_+^2}$$
Also from:

$$A = \frac{N_+-N_-}{N_++N_-} = \frac{2X}{E}$$
I get that:

$$\frac{dA}{A}=\frac{dX}{X}$$
so I want to minimize $\frac{dA}{A}$. But if I am in the case (we can assume in general that $N_+>N_-$) where $N_-=0$ and $N_+$ is large enough (which can be achieved in practice), assuming just statistical errors, I get that $A = 1$ and $dA = \frac{2d_-}{N_+}$, which is very small (if I were to assume $d_-=\sqrt{N_-}$ that would be zero, but in this case I should assume a binomial not poisson distribution for the uncertainty, but the point is I can make $d_-<<N_+$), so $\frac{dA}{A}<<1$, which is what I want. So based on this, I want to be in a situation where one of the counts is as close as possible to zero. But this doesn't make sense to me, as if I have $N_-=0$ basically the assymmetry I am building becomes worthless, as all the info is contained just in $N_+$. Something I am doing doesn't seem right. Can someone help me figure out what is going on and how to find the best region, in terms of $N_+$ and $N_-$, where to perform the measurement? Thank you!

 

[Dale]

Sometimes figuring out an analytical answer to a question like this is difficult to impossible. My usual recommendation in such cases is to do a Monte Carlo simulation to find out instead. So simulate a few thousand or a million experiments for each experimental condition, and use that simulation to justify the experimental condition you choose for the actual experiment.

 

[kelly0303]

Sometimes figuring out an analytical answer to a question like this is difficult to impossible. My usual recommendation in such cases is to do a Monte Carlo simulation to find out instead. So simulate a few thousand or a million experiments for each experimental condition, and use that simulation to justify the experimental condition you choose for the actual experiment.

I am actually not sure how to proceed with that either. There are a few things I am confused about. For example:
1. For a given $E$ and $W$ I can generate $N$, without any noise. Then I would need to add some noise to it to simulate the experiment. What magnitude would I assume for the noise? Would it be the 3 different kind of noises, statistical ($\sqrt{N}$), systematic and external noise added in quadrature?
2. Going the other way around, if in practice I see upon measurement $N$ counts, what uncertainty would I assign to it? If I had just statistical noise, I would use $\sqrt{N}$, but now I have systematic uncertainties, too. I could combine the two, but that would seem to overestimate the overall uncertainty, no? As some of the counts are due to the systematic noise, so I shouldn't include them in the $\sqrt{N}$ part, but I have no way of knowing how much of $N$ is due to the systematic uncertainty.

 

[Dale]

What magnitude would I assume for the noise?

Add enough noise to make it look like your real data. This doesn’t have to be exact, just a reasonable approximation.

Would it be the 3 different kind of noises, statistical (N), systematic and external noise added in quadrature?

Yes, unless you have reason to believe that they are correlated.

that would seem to overestimate the overall uncertainty, no?

This is probably too fine a detail to worry about.

 

[Office_Shredder]

I'm still stuck on the part where you wrote down some formula for $A$ that assumes $X<<E$, but then you assume $N_{-}=0$. Am I missing something? It seems like an invalid sequence of approximations/