How to solve this numerical technique problem?

Question

Well, in a numerical technique test we were given the following problem:

A physical phenomenon is modeled such that,

$F(f,d) = A(f)/d^2 + L$;

Where, $F$ is a function of frequency $f$ and distance $d$, and $L$ is random error. Function $A(f)$ is unknown.

A function is given, to produce data set $(F,f,d)$. The function is:

function F = get_data(d,f)
    for i= 1:length(f)
        for j = 1:length(d)
            F(i,j) = 2*f(i)^2 / d(j)^2 + 0.3*rand()*(1-rand());
        end
    end
end

here d, f are arrays. F is a 2D array. rand is a function that gives some random numbers less than 1. It is assumed that, we cannot poke into this function for an answer. We can't just look at the function and say A(f) = 2*f^2. We have to find this or a very close approximation to this by some numerical method....

Now, for d = 1:1:10; f = 30:30:300;

we have to produce a data set for $F$. From, this data set, we have to calculate, (1) mean of $L$. (2) an expression for $A(f)$.

Now, what I understand is that, 2nd if we have $A(f)$ then we can produce the values of $L$, and can easily evaluate mean of $L$. And, for $A(f)$ we could assume $L$ to be too low that we can ignore......

But, how could I actually do it?

I'll be very happy if someone helped me.

Answer 1

I am not giving the full answer here but as an idea you could think about your data as follows.

You have homoscedastic data, which is good. But a non-normal noise term. Now there a several techniques for defining a statistical model on these data. One could do a B-spline regression on the model or one could choose an appropriate basis on the domain of $f$ for an approximation on $A(f)\approx \hat{A}(f) = \sum_i c_i b_i(f)$, with basis functions $b_i(f)$.

Either way, your fit model needs to find some coefficients and reads: $F(f,d) = \hat{A}(f)/d^2 +L$ and $\hat{A}(f)$ is the respective approximation to $A(f)$. Once you have obtained the fit, the residuals represent $L$. A hypothesis test should reveal if $E(L)=0$ in your model, so you catch the features of $A(f)$.

Likely, the fit matrix will be ill-conditioned, so when performing the fit you need to apply some regularization technique, e.g. truncated svd.