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3 Determining the Correlation Function

The two-point angular autocorrelation function is defined by the joint probability tex2html_wrap_inline451P of finding two sources in each of the elements of solid angle tex2html_wrap_inline525 and tex2html_wrap_inline527 separated by angle tex2html_wrap_inline529:
where N is the mean surface density. Many derivations for estimators of tex2html_wrap_inline535 have been given before (see, for example, Peebles 1980). For those unfamiliar with the field, we outline a basic derivation given by Sicotte (1995).

It follows from the definition that for a catalog of n data points covering a solid angle tex2html_wrap_inline539, the mean number of sources at a distance tex2html_wrap_inline541 from a randomly picked data point is tex2html_wrap_inline543 where tex2html_wrap_inline545 is the mean solid angle about a randomly chosen data point. The total number of pairs with separations in the interval tex2html_wrap_inline541 is then given by tex2html_wrap_inline549. tex2html_wrap_inline551 can be measured directly from the catalog, and when combined with an estimate of tex2html_wrap_inline553, can give an estimate of tex2html_wrap_inline535. For an all-sky catalog tex2html_wrap_inline557 and tex2html_wrap_inline559. For surveys with complicated geometries, it is more practical to calculate tex2html_wrap_inline553 using a field of randomly distributed points covering the same area as the survey. Assuming there are the same number of random points as there are data points, the number of pairs of random points with separations between tex2html_wrap_inline541 is given by tex2html_wrap_inline565. One can clearly measure this quantity and obtain a value for tex2html_wrap_inline567 which is approximately equal to tex2html_wrap_inline553. A simple estimator of tex2html_wrap_inline535 is thus given by DD/RR-1. Strictly, one needs to have more random points than data points so that the tex2html_wrap_inline535 estimate is not limited by statistical errors in the random points. An improvement on this estimate of tex2html_wrap_inline553 can be obtained using the quantity DR(tex2html_wrap_inline529): the number of pairs of points separated by angle tex2html_wrap_inline529 where one point is taken from the data field and one is from the random field. Using DR instead of RR enables one to measure the mean solid angle around data points [tex2html_wrap_inline553] as opposed to the mean solid angle around random points [tex2html_wrap_inline567].

Sicotte (1995) gives a detailed comparison of the various estimators of tex2html_wrap_inline535. We have chosen to quote results using the the standard (S) estimator [tex2html_wrap_inline589] as well as the estimator suggested by Landy and Szalay (1993, hereafter LS) [tex2html_wrap_inline591]. The LS estimator has been shown to have smaller uncertainties on larger scales and Sicotte claims it will remove the effects of large scale fluctuations in density for estimates of tex2html_wrap_inline535 at small tex2html_wrap_inline529 . The method introduced by Hamilton (1993) [tex2html_wrap_inline597] gives results very similar to the LS estimator.

To determine the uncertainties associated with each estimate, a `bootstrap' analysis of the errors was performed. This method of estimating uncertainties is described in detail in Ling, Frenk and Barrow (1986) and in Fisher et al. (1994). A set of `bootstrap' catalogs, each the same size as the data catalog, are generated using the following procedure. A source is picked at random from the data catalog and inserted into the first bootstrap catalog. Random sources from the data catalog continue to be included in the bootstrap catalog until it has the same number of sources as the data catalog. Some of the sources in the original data set will appear more than once in the bootstrap catalog, some will not appear at all. A set of these bootstrap catalogs can then be generated and the correlation function can be calculated for each one. The result is that at each tex2html_wrap_inline529 we produce a set of normally distributed estimates of the correlation function. The variance around the mean can then be used as an estimate of the uncertainty in the measurement of w at each tex2html_wrap_inline529. This process established that both the S estimator and the LS estimator of tex2html_wrap_inline535 had similar uncertainties associated with them out to about tex2html_wrap_inline607. The Poissonian estimate of the uncertainties given by tex2html_wrap_inline609 is less than the bootstrap estimate by a factor of 2 on small scales (tex2html_wrap_inline611) and by more than an order of magnitude on larger scales (tex2html_wrap_inline613).

As explained above, the random field measurements RR and DR correct for the `edge effects'; that is, they estimate the mean solid angle about points in the area of the survey for a given tex2html_wrap_inline529. Therefore, the random field should contain all the biases that the data field contains. We consider the following:

In addition to the angular autocorrelation function for the catalog as a whole and various source subsamples, we have determined the angular cross correlation function for the radio sources and Abell clusters. We use the estimator tex2html_wrap_inline631 where Dd is the number of radio sources separated by tex2html_wrap_inline529 degrees from an Abell cluster center and Rd is the number of random field points tex2html_wrap_inline529 degrees from an Abell cluster center.

next up previous
Next: 4 Results Up: The Angular Two-Point Correlation Previous: 2 The Catalog

Richard L. White,
FIRST Home Page
1996 Dec 20