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Efficient computation of N-point correlation functions in D dimensions.

Oliver H E Philcox1,2, Zachary Slepian3,4

  • 1Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08540.

Proceedings of the National Academy of Sciences of the United States of America
|August 8, 2022
PubMed
Summary

We developed efficient algorithms to compute N-point correlation functions (NPCFs) for random fields, significantly reducing computational complexity. This advancement makes higher-order correlation analysis feasible across various scientific disciplines.

Keywords:
clustering statisticscomputational physicscorrelation functionscosmologyspherical harmonics

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Area of Science:

  • Physics
  • Cosmology
  • Fluid Dynamics
  • Data Analysis

Background:

  • N-point correlation functions (NPCFs) are crucial for describing random fields in physical sciences.
  • Existing algorithms for NPCFs have high computational complexity, limiting their application to small N.
  • Efficient computation of NPCFs is needed for advanced analysis of stochastic processes.

Purpose of the Study:

  • To develop efficient algorithms for computing NPCFs in D-dimensional spaces.
  • To reduce the computational complexity of NPCF calculations.
  • To enable the use of higher-order correlation functions as a standard analytical tool.

Main Methods:

  • Projecting NPCFs onto a D-dimensional hyperspherical harmonic basis.
  • Utilizing a separable form for estimators.
  • Employing Fast Fourier Transform for grid-based evaluation.
  • Implementing algorithms in a Julia package.

Main Results:

  • Achieved computational complexity of O(n^2 log n) or O(n log n) using FFT.
  • Demonstrated significant dimensionality reduction for isotropic correlation functions.
  • Developed a practical Julia package for NPCF estimation.

Conclusions:

  • The new algorithms offer a computationally feasible method for calculating NPCFs.
  • This work facilitates the use of higher-order correlation functions in diverse scientific fields.
  • The developed methods will advance the analysis of random fields in cosmology and fluid dynamics.