# 5. Center determination¶

Determining the central coordinates of a SC with accuracy is of utmost importance, since this parameter directly affects the generation of the radial density profile, and thus also the estimation of the SC’s radius.

ASteCA determines the center value in two steps: a coarse approximation is done first and a Gaussian kernel density estimation (KDE) is then applied to obtain precise final coordinates.

The process is independent of the system of coordinates used and can be equally applied to positional data stored in pixels or degrees.

## 5.1. Approximate center values¶

The positional data is first binned in a 2D histogram. The bin width is either automatically calculated as 1% of the minimum range in either coordinate (x, y), or set manually via the params_input.dat file in the Structure functions parameters block, by setting the bin_width parameter (CH ID line).

Important

Unless there is a specific reason to use the manual mode to set the bin width, it is recommended to use the 1% range auto value. This figure has proven to be a very good heuristic value in a large number of situations.

This 2D histogram is processed with a Gaussian filter to smooth it out, using five different bandwidths of increasing values (see get_center.py function for more details)

The maximum density values of each Gaussian filtered version of the 2D histogram, are presented in the final output image as seen in Fig. 5.1. Fig. 5.1 Approximate center coordinate values. The standard deviation values in the bottom right corner are the bandwidth values used in the Gaussian filter. The box in the top left corner contains the averaged coordinate values and their standard deviations (shown in the plot as continuous and dashed lines respectively).

Each one of these values represents an approximation to the final center coordinates obtained in the next step (see section below).

If the semi_input.dat file is used to pass approximate center coordinates, this analysis is skipped and only the passed value is presented in the plot.

## 5.2. Precise center value¶

Once the approximate center coordinates have been obtained, ASteCA proceeds to refine this result. To this end, the code employs the approximate center coordinates obtained in the previous step using the smallest bandwidth value (St dev: 2.0 in Fig. 5.1), or $$center_{bw-min}$$.

These coordinates serve as initial values for the cluster’s center. The final precise values are determined as the maximum spatial density of a two-dimensional Gaussian kernel, generated on a reduced square area of the positional diagram of the cluster; see Fig. 5.2. Fig. 5.2 In yellow, area where the center coordinates are searched via the KDE. Blue lines are contour curves, the center found is marked with a cross.

The square area where the final center coordinates are searched is centered on the $$center_{bw-min}$$ values, with a side length that spans 50% of the minimum x,y range. The bandwidth of the fitted kernel is automatically defined via Scott’s rule (Scott 1992).

In Fig. 5.3 we can see the final center coordinates (equivalently: the maximum value of the positional KDE) plotted over the Gaussian filtered 2D histogram obtained using the minimum bandwidth, as defined in Sect. 5.1. Fig. 5.3 Cluster’s center found via the KDE analysis. The Bin value in the top right corner is the bin width of the 2D histogram mentioned in the previous section, used to obtain the approximate center coordinates.

After this analysis is completed, the final center coordinates are compared with the approximate values obtained in the previous section. Two flags are raised warning the user about large deviations that could require careful revision, and stored in the output file asteca_output.dat.

The flags must meet the following conditions in order to be raised:

• f1: The median value for the approximate center coordinates in either x or y, is more than 10% away from the final center coordinates.
• f2: The standard deviation for the approximate center coordinates in either x or y, is larger than 10% of the final center coordinates.