David Schlegel suggested using a likelihood estimator to select quasars from stars. The basic idea is to have a templates of "quasar objects" and "all other objects" we would expect to see in the Sloan data set. We then take a "test object," one which we are trying to determine if it is a quasar or not, and compute a likelihood between it and the two templates. The likelihood of a test object (i) is for a set of template data (j) with color filters (f) is defined as follows:

where x is the flux of an object in the template data and mu is the flux of your test object and sigma if the error in your test object flux measurement.

I computed this likelihood with a set of test object where I knew if they were quasars or not. Below is a plot of the above likelihood computed on the test objects with a "quasar template" and a "everything else template." I am plotting the likelihood of the test objects with everything else (on the x axis) versus with quasars (on the y axis). Because I know what these test objects are, I can color the quasars (green) and the stars (blue). You can see that I get a clear separation in likelihood-likelihood space of these populations, where the quasars fall along a higher likelihood for quasars and the stars fall along a higher likelihood for everything else:

where x is the flux of an object in the template data and mu is the flux of your test object and sigma if the error in your test object flux measurement.

I computed this likelihood with a set of test object where I knew if they were quasars or not. Below is a plot of the above likelihood computed on the test objects with a "quasar template" and a "everything else template." I am plotting the likelihood of the test objects with everything else (on the x axis) versus with quasars (on the y axis). Because I know what these test objects are, I can color the quasars (green) and the stars (blue). You can see that I get a clear separation in likelihood-likelihood space of these populations, where the quasars fall along a higher likelihood for quasars and the stars fall along a higher likelihood for everything else:

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