A ridge of algorithms

I spend some time with the growth-curve method during last days. Its looks nice, but relative hard to control.

Some side product of the game is comparison of the original Stetson's algorithm for estimate of the arithmetical mean with new ones. To get mean, Stetson have introduced some rejection-type algorithm. I've no more detailed information about it - there is no reference in source code or a paper. Opposite with this, Munipack estimate of the mean by a method on base of minimisation of a general function. It minimises function in shape of the least-square near of minimum and suppress it to zero otherwise by method of the maximum like-hood. The chapter describing of a robust statistical methods in Numerical recipes is an ideal start point to get detailed view to this field. A source code of Munipack uses of those ideas, but it doesn't describe it. The new estimator has been included to Munipack approximately eight years ago.

I compared of the estimators on case of the ridge of magnitude on differences of following apertures. Both three-dimensional graphs shows number of the differences in cells specified by apertures and its value's ranges. The horizontal (x) axis takes magnitude differences, the horizontal (y) axis takes size of aperture and vertical (z) shows the number of values in that bin interval. The top side of the data cube shows projection of the number of values to a plane. Colors of the histogram maps number of points: dark - near zero, light - near maximum (hundred).

Muniphot's ridge

Stetson's ridge

Both images looks similar. Only one difference is a small systematic shift of the ridge to negative region on Stetson's graph. It is about 0.01 in magnitude, so I think, it is not important in real situations. Perhaps. It may means that the new method for the mean estimations works little bit better.

Ridges height strongly depends on aperture. Small values gives perfect estimation against apertures greater that fifteen (in my case) when the the histogram shows no peaks. The characteristics may be important for growth-curve fitting.

As test data for both graph, I used combined image of M67 as in previous post. However, graphs of ridges looks differently (gnuplot's pm3d feature has been used on graphs here). It is due of my mistake during data processing. The previous perfect ridge with strong peaks at large apertures comes from many zero's differences of bad magnitudes of 99.999. Keep smiling.
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