Variance Analysis for Monte Carlo Integration


We propose a new spectral analysis of the variance in Monte Carlo integration, expressed in terms of the power spectra of the sampling pattern and the integrand involved. We build our framework in the Euclidean space using Fourier tools and on the sphere using spherical harmonics. We further provide a theoretical background that explains how our spherical framework can be extended to the hemispherical domain. We use our framework to estimate the variance convergence rate of different state-of-the-art sampling patterns in both the Euclidean and spherical domains, as the number of samples increases. Furthermore, we formulate design principles for constructing sampling methods that can be tailored according to available resources. We validate our theoretical framework by performing numerical integration over several integrands sampled using different sampling patterns.

ACM Transactions on Graphics (Proceedings of SIGGRAPH)

Caption: Summary of theoretical power spectra and their convergence rate for the best case and worst cases of integration. From left to right: Constant power spectrum, polynomial power spectrum with b less than 1, polynomial power spectrum with b greater than 1 and step power spectrum.

      author = {Pilleboue, Adrien and Singh, Gurprit and Coeurjolly, David and Kazhdan, Michael and Ostromoukhov, Victor},
      doi = {10.1145/2766930},
      hal_id = {hal-01150268},
      journal = {ACM Transactions on Graphics (Proceedings of SIGGRAPH)},
      keywords = {Stochastic Sampling ; Monte Carlo Integration ;
Fourier Analysis ; Spherical Harmonics ; Global
      month = {August},
      number = {4},
      pages = {14},
      pdf = {},
      title = {Variance Analysis for Monte Carlo Integration},
      url = {},
      volume = {34},
      year = {2015}