In my student-research meeting, Kate Storey-Fisher (NYU), Abby Williams (NYU), and I discussed how we could make a flexible, parameterized model of a correlation function (for large-scale structure) with a continuous spatial gradient in it. The idea is that we are going to use Storey-Fisher's continuous-function estimator for the correlation function to look for departures from statistical homogeneity. Our inspiration is the power asymmetry in the cosmic microwave background.
Our approach to building the flexible model is to take a model for the homogeneous correlation function and analytically Taylor-expanding it around a fiducial point. In this formalism, the local spatial gradient at the fiducial point turns into a linear dependence on position, and the local second derivative leads to a quadratic term, and so on. Of course an alternative approach would be to just measure the correlation funciton in spatial patches, and compare the differences to the measurement varainces. But binning is sinning, and I also believe that the continuous estimation will be higher precision.
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