The Beylkin-Cramer Summation Rule and a New Fast Algorithm of Cosmic Statistics for Large Data Sets

Based on the Beylkin-Cramer summation rule, we introduce a new fast algorithm that enables us to explore the high-order statistics efficiently in large data sets. Central to this technique is to make decompositions both of fields and operators within the framework of multiresolution analysis and to realize their discrete representations. Accordingly, a homogeneous point process could be equivalently described by the operation of a Toeplitz matrix on a vector, which is accomplished by making use of the fast Fourier transformation. The algorithm could be applied widely in cosmic statistics to tackle large data sets. We demonstrate this novel technique using the spherical, cubic, and cylindrical counts in cells. The numerical test shows that the algorithm produces an excellent agreement with the expected results. Moreover, the algorithm naturally introduces a sharp filter, which is capable of suppressing shot noise in weak signals. In the numerical procedures, the algorithm is somewhat similar to particle-mesh methods in N-body simulations. Since it is scaled with O(N log N), it is significantly faster than the current particle-based methods, and its computational cost does not rely on the shape or size of the sampling cells. In addition, based on this technique, we further propose a simple fast scheme to compute the second statistics for cosmic density fields and justify it using simulation samples. Hopefully, the technique developed here allows us to make a comprehensive study of non-Gaussianity of the cosmic fields in high-precision cosmology. A specific implementation of the algorithm is publicly available on request to the author.