Smooth maps from clumpy data
Institüt für Astrophysik und Extraterrestrische Forschung, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
Corresponding author: M. Lombardi, firstname.lastname@example.org
Accepted: 12 April 2001
We study an estimator for smoothing irregularly sampled data into a smooth map. The estimator has been widely used in astronomy, owing to its low level of noise; it involves a weight function -or smoothing kernel -. We show that this estimator is not unbiased, in the sense that the expectation value of the smoothed map is not the underlying process convolved with w, but a convolution with a modified kernel . We show how to calculate for a given kernel w and investigate its properties. In particular, it is found that (1) is normalized, (2) has a shape "similar" to the original kernel w, (3) converges to w in the limit of high number density of data points, and (4) reduces to a top-hat filter in the limit of very small number density of data points. Hence, although the estimator is biased, the bias is well understood analytically, and since has all the desired properties of a smoothing kernel, the estimator is in fact very useful. We present explicit examples for several filter functions which are commonly used, and provide a series expression valid in the limit of a large density of data points.
Key words: methods: statistical / methods: analytical / methods: data analysis / gravitational lensing
© ESO, 2001