# Statistical Performance of Radio Interferometric Calibration

## TL;DR

This paper analyzes how calibration affects residual data in radio interferometry, deriving relationships between input and output data distributions to assess calibration performance.

## Contribution

It introduces an analytical framework linking calibration input and residual statistics, highlighting the eigenvalue of the Jacobian as a performance indicator.

## Key findings

- Eigenvalue of Jacobian indicates calibration quality
- Derived analytical relationship between input and residual data
- Simulation results validate the eigenvalue as a performance metric

## Abstract

Calibration is an essential step in radio interferometric data processing that corrects the data for systematic errors and in addition, subtracts bright foreground interference to reveal weak signals hidden in the residual. These weak and unknown signals are much sought after to reach many science goals but the effect of calibration on such signals is an ever present concern. The main reason for this is the incompleteness of the model used in calibration. Distributed calibration based on consensus optimization has been shown to mitigate the effect due to model incompleteness by calibrating data covering a wide bandwidth in a computationally efficient manner. In this paper, we study the statistical performance of direction dependent distributed calibration, i.e., the distortion caused by calibration on the residual statistics. In order to study this, we consider the mapping between the input uncalibrated data and the output residual data. We derive an analytical relationship for the influence of the input on the residual and use this to find the relationship between the input and output probability density functions. Using simulations we show that the smallest eigenvalue of the Jacobian of this mapping is a reliable indicator of the statistical performance of calibration. The analysis developed in this paper can also be applied to other data processing steps in radio interferometry such as imaging and foreground subtraction as well as to many other machine learning problems.