# Numeric methods to estimate curve parameter from 2 input data with Gaussian noise

I have:

• a function of the form

$S=M f(t, \theta)$

where $\theta$ is a variable, $t$ and $M$ are parameters.

• two observations

$(\theta_1, S_1)$ and $(\theta_2, S_2)$

in which $S_1$ and $S_2$ are observed values for $t$, and $\theta_1$ and $\theta_2$ are the observed values of $\theta$.

Furthermore, we know that my $S_1$, $S_2$ contains noise, which are normal distribution $N(0, d_1)$ $N(0,d_2)$ with unknown variance $d_1$, $d_2$.

Now I want to estimate $t$.

Mathematically, if $S_1$ and $S_2$ are observed precisely, we have

$$S_1= M f( t, \theta_1)$$ $$S_2= M f( t, \theta_2)$$

Then, by division we have $\frac{S_1}{S_2}=g(t,\theta_1,\theta_2)$ following which the parameter $t$ can be estimated by a Mathematica tool like Findroot w.r.t. the parameter $t$.

However, the problem here is the noise of normal distribution. The goal is to find a $t$ as precise as possible, in the sense of some probability-theoretic characteristics. The preferred way is to use a Matlab or Mathematica function, although I don't think they have a direct solution.

I am totally new to numeric analysis, and totally new to Matlab. My naive idea is that:

1. Should the first step be removing the parameter $M$?
2. If so, there are infinitely many ways to do that, among which we can list

• $\frac{S_1}{S_2} = \frac{f(t, \theta_1)}{f(t,\theta_2)}$
• $\frac{S_1+S_2}{S_1-S_2} = \frac{f(t, \theta_1)+f(t,\theta_2)}{f(t, \theta_1)-f(t,\theta_2)}$
• $\frac{S_1^2 }{S_2^2} = \frac{f(t, \theta_1) ^2 }{f(t,\theta_2) ^2}$

It is clear that all these variations are equivalent mathematically if $S_1$, $S_2$ are observed precisely. However, they should yield different $t$ (because of the Gaussian noise in $S_1$ and $S_2$)

My questions are

1. Do you think my first step should be to remove $M$ using the foregoing approach?
2. Is there some kind of combination of $S_1$, $S_2$ that gives better results than others?
3. What would be the best way to estimate $t$ of this problem? (Maybe in Matlab, or Mathematica)

Thank you. I hope I was clear.

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I don't want to edit your question but it is a little confusing. since you are talking to statisticians. You should say that S1 and S2 are observed values for t and theta1 and theta2 are the observed values of theta. You misuse the term precise. To a statistician precise means small variability where I am pretty sure you mean to say that S1 and S2 are observed exactly (without any noise). You also should not say that after division t and be calculated rather you should say it can be estimated. – Michael Chernick May 12 '12 at 17:19
Also English can be a confusing language with plurals, the plural of dog is dogs but the plural of noise is noise not noises. – Michael Chernick May 12 '12 at 17:20
Nice remarks. Thank you. – zell May 12 '12 at 18:02
A mistake in my comment that I am too late to edit was saying that S1 and S2 were sample values of t. – Michael Chernick May 12 '12 at 18:10