Finding value of parameters to fit some data

[imsolost]

The problem is the following :

I have some measured data's obtained when measuring a physical process. Let's call these : y_E,L where E and L are 2 physical parameters of the experiment (an energy and a length).

I also know that :

$$\frac{y_{E,L}}{\sum_{k=1} ^{k=100} {\epsilon_{E,L} (k * v) * f(k*v|a,b,c) }} = constant$$ for all E, L.

where f(x|a,b,c) is a known, non-linear, parametrized function with 3 parameters that needs a fit : a, b, c. whose I know the expression of (I don't write it here because its quite long with some exponentials but I hope you get the idea).

I have no analytical expression for $\epsilon_{E,L}(x)$ but I can calculate separately each of the 100 different $\epsilon_{E,L}(k*v)$ values so I know the value of all $\epsilon_{E,L}(k*v)$ above for all E and L.

What algorithm or calculation method should I use to get a best-fit for a,b,c ?edit : trying to get the latex code working but smthing's wrong -_-' <Moderator's note: fixed>

 

[Stephen Tashi]

so I know the value of all $\epsilon_{E,L}(k*v)$ above for all E and L.

It isn't clear what you data is. For example, do $E$ and $L$ have any definite relation to "$v$"?

In a simple experimental scenario, data has the form of ordered pairs (x,y). One controls the value of x and measures the value of y. Do you have more than one "control" variable?

 

[imsolost]

It isn't clear what you data is. For example, do $E$ and $L$ have any definite relation to "$v$"?

In a simple experimental scenario, data has the form of ordered pairs (x,y). One controls the value of x and measures the value of y. Do you have more than one "control" variable?

Using your notation (x,y), then your "y" is my $y_{E,L}$. Your "x" is my couple of variables (E, L). Basically, during my experiment, I changed L (I used 3 different values of L) and E (i used 5 different values of E) for a total of 3x15 data points $y_{E,L}$. I guess this should be enough to find 3 parameters a, b, c.

"v" is just an interval : the denominator of my expression comes from a simple rectangle numerical integration (I forgot a *v in the expression inside the sum btw). So basically, I had something in the denominator like $\int_{0}^{100*v}{\epsilon_{E,L} (r) * f(r|a,b,c)*dr}$.

I hope this clarifies my problem. Any help would be really appreciated.

 

[Stephen Tashi]

Is the value of "constant" known?

 

[imsolost]

No, I don't know its value.

 

[Stephen Tashi]

(I used 3 different values of L) and E (i used 5 different values of E) for a total of 3x15 data points yE,L

I assume you mean 3x5 = 15 data points.

You could formulate the problem as an optimization problem - to find values of a,b,c that minimize a certain function subject to certain constraints on a,b,c (if there are any - e.g. perhaps some of the parameters need to be positive or within known bounds).

There are various numerical techniques for solving such optimization problems, such as "simulated annealing" and "conjugate gradient".

let the function that is supposed to be some common constant for a given value of $E,L$ be $g(E,L,a,b,c)$.

Let $h(a,b,c) = \sum_{i=1}^5 \sum_{j=1}^3 g(E_i,L_j,a,b,c)$ and $\bar{h} = \frac{ h(a,b,c)} {15}$

Let the function to minimize be $s(a,b,c) = \sum_{i=1}^5 \sum_{j=1}^3 ( g(E_i,L_j,a,b,c) - \bar{h})^2$ which is proportional to the sample variance of the functions $g(E_i,L_j,a,b,c)$. If each of the functions $g(E_i,L_j,a,b,c)$ had the same common value, the variance would be zero.