Many measurements produce angle-dependent data. For example, we measure the (ferromagnetic) resonance fields when the sample is positioned at various angles with respect to the external field (see attached figure). This way we obtain a series of pairs (anglej, resultj), j=1, ... N, with angles not necessarily sampled uniformly, or sometimes not even covering one full turn (cycle), as in the attached figure. The task is to find angle0 such that a graph of resultj = resultj (anglej - angle0) has symmetry axis coinciding with line \phi=0 in radial coordinate system (or the line angle=0 is the mirror symmetry axis in Cartesian coordinates). There is no closed form formula describing the dependence result(angle) and the measurements are not exact, as usually. The angles may be assumed exact. How do you deal with problems like that?
Hello Marek,
I usually try one or several of the following, especially if I know that mirror symmetry is a physical condition my data 'have to' obey to.
1) calculate smoothened numerical derivative by convolution with first derivative of a Gaussian. Play with the width of that Gaussian to adapt to the characteristic curvature/width of the data near the extremum. Determine the position of the extremum from these procedures.
2) fit with polynomials in (alpha-alpha0)2n, i.e. only even terms allowed and get alpha0 from that
3) same with other linear combinations of even functions in (alpha-alpha0), depending on what the shape/physics of the data looks like (or depending on physical background and what I think would therefore make sense...)
Can your method help people who want to plot the direction-specific measured redshifts of objects in the sky to thereby generate a "local-angle-dependent 2-dimensional Hubble law"?
Otto, you seem to present slightly different problem, I think. You are most likely looking for any 2D non-uniformity of Hubble law. If indeed detected it would be very interesting in next(?) step to see what is its symmetry - if any.
Yes,dear Marek, precisely; and as usual, an impish or revolutionary hunch is behind it:
I believe I could prove that the Hubble redshift law is interactively generated in gravitation as conjectured by Zwicky. I published proofs to this effect, but of course no one is believing me.
So this would be an empirical proof - IF the variability is, (i) very strong across the sky and, (ii) the lowest-sloped patches are behind several voids in a row.
Then the burden of proof would no longer lie with cryodynamics and c-global.
Hello Kai,
Usually I follow your 3rd approach, using (shifted) Fourier cosine series with incomplete set of harmonics. Incomplete, as I try to decide whether the observed symmetry is of C2, C3, C4 or C6 type.- expected in crystal-related measurements. The shift is gradually adjusted to get the best approximation of available data. This method is advantageous to your 2nd approach as 1) it uses orthogonal functions, and 2) it operates on the entire data set, not just on those located nearby extremum, thus it eliminates possible personal bias in data selection. I'm not sure whether numerical differentiation, even if smoothed, is always applicable (large separations/steps, non-uniform distribution, etc.)
I asked this question because my own approach recently failed, after many years of decent behavior. And the reason appeared somewhat unexpected: my data seem to obey S2 symmetry! Hmm, my procedure certainly needs serious revision and I'm looking for more valuable hints.
Dear Marek,
Yours is an interesting problem, and I like the approach related to incomplete Fourier series that you use.
Since you are primarily interested in estimating the azimuth, alpha, would it be profitable to compare projections of your data onto the pairs x_k = cos(k phi) and y_k = sin(k phi), with k = 1, 2, 3, 6 and possibly higher harmonics of those? (I include k = 1 because the symmetry S_2 also appears to be of interest.)
An analysis of those (x, y) pairs might at least give you good starting points for the azimuth in your "shifted" fits, in an automated way and with no personal bias.
There might have to be a correction applied having to do with the incomplete and nonuniform sampling around the circle, but it seems to me that pairwise comparisons of the sin and cos components focuses the analysis on the azimuthal orientation.
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