first-time poster here, wondering if anyone can advise me on a curve fitting issue.
I have a function $y(x)$ that depends non-linearly upon two scalar parameters, $p$ and $q$, that I want to estimate. (To be more specific: $x$ is the angular position of an induction coil and is $y$ the magnetic flux through the coil, both are measured; $p$ is the coil area and $q$ is the coil rotation radius, both considered unknown. This is a sensor calibration problem.)
My analytical model is $y(x;p,q)=k*log(\frac{1 + (p+q)^2 - 2(p+q)cos(x)}{ 1 + (p-q)^2 + 2(p-q)cos(x)})$ , and I can fit that to the measured curve numerically, by using iterative least-squares. No problem there, this works nicely and the results are robust and stable. My question is, how can I prove rigorously that the solution I have found is unique ?
In the linear case the answer would depend upon the conditioning of the resulting normal equation system, that is clear enough. I’m struggling however to find a simple criterion for the non-linear case.
The simplest approach I could come up with is the following. For any given value of $x$, it is always possible to find parameter increments $dp$ and $dq$ that do not change locally the value of the function, i.e. that make the following differential vanish: $$dy = \frac{\partial y}{\partial p} dp + \frac{\partial y}{\partial q} dq =0$$ The solutions, if they exist, of the differential equation $$\frac{dp}{dq} = - \frac{ \frac{\partial y}{\partial p}}{\frac{\partial y}{\partial q}} (1) $$ describe a family of curves in the $(p,q)$ plane, each representing the set of parameter pairs corresponding to the same solution $y(x)$. To give a concrete example, let us consider the two functions:
$$y_1 = sin((p+q)x)$$ $$y_2= p*sin(q x)$$
For the first one, eq. (1) gives $\frac{dp}{dq}=-1$, which can be solved to obtain the family of curves $p+q=const$. In this case, it is obvious that the waveform $y_1$ is determined by the sum of the parameters, rather than by their individual values. The solution of the fitting problem cannot be unique.
For the second, $\frac{dp}{dq}=-\frac{tan(q x)}{p x}$; the r.h.s. contains the independent variable and an analogous family of solutions does not exist. In this case, the curve fitting problem is well-posed, because the two parameters have a very different effect on the sinusoidal waveform $y_2$: $p$ controls its amplitude, while $q$ controls its frequency.
To summarize, it seems to me that a general criterion for the existence of a unique solution to the curve-fitting problem, might be that the r.h.s. of eq. (1) contains or not the independent variable.
I would be really grateful for any comments or pointers to relevant literature.
Not an answer but too long for a comment.
I have seen so bizarre things with nonlinear regression during the last sixty years that I shall not answer the question.
A first comment is that, for numerical stability (it makes the parameters more independent), I should prefer to write the model as $$y=k\,\log\left(\frac{1 + a^2 - 2a\cos(x)}{ 1 +b^2 + 2b\cos(x)}\right)\quad \quad a=(p+q)\quad b=(p-q)$$
I do not know if $k$ has to be adjusted too. But, for any value of $k$, the model can be linearized and, for it, the solution is unique.