Assume data from a plane which are roughly showing a spiral. I want employ the rationale of regression to find the parameters for the best fit by some spiral.
That means, I have to estimate the (optimal) center first.
Then I can convert the rectangular data-coordinates into polar coordinates, and then the radial distances from that center should be linearly (or polynomially) dependend on the angle by which some point is displaced by some point z0. (Here multiples of 2*Pi are significant).
My example data are in fact complex values, steming from iterations of $z_{k+1} = b^{z_k}$ where $z_0$ is some complex value in the near of a complex fixpoint and b is the base, say $b = \sqrt{2} $ , and I've already done some work with this.
But what I'm primarily looking for is an idea for the general ansatz: for the case where b is given I have the fixpoint and thus I need not estimate the center and the problem is not so difficult. But if I even do not have the center, only the data, how could I begin? I couldn't translate the rectangular coordinates into polar coordinates if the center is unknown in the beginning...
Q: How could I approach this with the method of least-squares-approximation?