uv min and max distance estimation.

[Bonnarel]

Section starts with this text by Mireille (first sentence), and then Mattia

uv_distance_min and uv_distance_max are evaluated by fitting an ellipse on the visibilities present in the uv plane.

To compute the ellipse's eccentricity of the UV distribution a principal component analysis (PCA) with 2 components is performed over the data points sampling the UV plane to select the main axis of data scattering. The first component is used to rotate the distribution of UV in a way that the major variation of the distribution is leaning towards the $x$ axis of a bi dimensional $xy$ Cartesian plane. The major axis length and the minor axis length of the ellipse are therefore defined as the semi distance between the most positive point along the $x$/$y$ axis and the most negative point among the $y$ axis. For instance, if the range of the rotated UV will cover on the $x \in [-10, 10]$ the major axis distance would be 10, a similar procedure is done on the y axis. This procedure allows the definition of the UV distribution eccentricity:

I understand that the absolute farthest point from the center in the uv plane and the closest are not significant for estimation of uv_distance_min and uv_distance_max. But if we fit an outer ellipse to the distribution should we use the same PCA results used in the ellipse's ellepticity estimation and retain the semi major axis for uv_distance_max (significant) ??? Alternatively should we use some criterium estimated on the visibility amplitude (or number of visibilities) versus uv distance ? This will probably solve easier the estimation of the minimal uv_distance. Thoughts ?

[matmanc]

Hi François, indeed I think the estimation of the uv_distance_min/max and the eccentricity have to be coherent.

I was wondering how is the fitting performed? I described the PCA way because for me was the most easy way to fit the ellipse.

Thanks, Mattia

[Bonnarel]

Hi François, indeed I think the estimation of the uv_distance_min/max and the eccentricity have to be coherent.

I was wondering how is the fitting performed? I described the PCA way because for me was the most easy way to fit the ellipse.

Thanks, Mattia

sure, it should be coherent for uv_distance_max (semi major ax is length ?), but how do we define uv_distance_min ? we have to estimate the semi major axis of the central hole ? In that case the amplitude/radius or density/radius plots seems to be easier ? Other thing : in your text I think "the most positive point along the $x$/$y$ axis and the most negative point among the $y$ axis. " should read "the most positive point along the $x$/$y$ axis and the most negative point among the $x$y$ axis." (second x missing, no ?) Also : Semi distance is important for the semi major (semi minor) axis estimation but not for the eccentricity itself I think.

[Bonnarel]

Looking back to your formulae for the filling factor and after rediscovering density/radius plots in Mark Kettenis presentation at EDP forum last week, I wonder if the mean number of samples per cell at a given radius is not what we could reuse to find out the uv_distance_min ? We just have to decide at which percentage of the maximum we start, no ? uv_distance_max in these plots is already connected to semi majouir axis of the ellipse I guess. Thoughts ?

By the way in your formumla for the filling factors the sigma symbola should have "i "and "j" subscripts instead of 1 I guess.

[matmanc]

So the uv_distance_min is correlated to the smallest baseline you can have. That is not useful to estimate the minor axis of the ellipse that surrounds the uv visibilities. The uv_distance_max instead it can be used to derive the major axis of the ellipse. But for coherence I estimated both rotating the UV coordinate plots for the major dispersion axis and then considering the semidistance in the x and the y. Such major minor axis estimation are used to compute the eccentricity.

Regarding your last question, yes it is possible to use the uv_distance_min as the cell size. But usually it results in a very high number of cell and then the filling factor is next to zero. At least in the tries I did with the lofar data.

I will address the typos in the new merge request.

[Bonnarel]

To all, Interop is next week. It would be good if we could make some progress on estimation of uv_distance_min and uv_distance_max

Excentricity and filling factor are OK despite a few typos to fix.