Fourier Analysis of Stochastic Sampling Strategies for Assessing Bias and Variance in Integration

Each pixel in a photorealistic, computer generated picture is calculated by approximately integrating all the light arriving at the pixel, from the virtual scene. A common strategy to calculate these high-dimensional integrals is to average the estimates at stochastically sampled locations. The strategy with which the sampled locations are chosen is of utmost importance in deciding the quality of the approximation, and hence rendered image. We derive connections between the spectral properties of stochastic sampling patterns and the first and second order statistics of estimates of integration using the samples. Our equations provide insight into the assessment of stochastic sampling strategies for integration. We show that the amplitude of the expected Fourier spectrum of sampling patterns is a useful indicator of the bias when used in numerical integration. We deduce that estimator variance is directly dependent on the variance of the sampling spectrum over multiple realizations of the sampling pattern. We then analyse Gaussian jittered sampling, a simple variant of jittered sampling, that allows a smooth trade-off of bias for variance in uniform (regular grid) sampling. We verify our predictions using spectral measurement, quantitative integration experiments and qualitative comparisons of rendered images.</jats:p

Spectrum of the Laplace-Beltrami Operator and the Phase Structure of Causal Dynamical Triangulation

We propose a new method to characterize the different phases observed in the non-perturbative numerical approach to quantum gravity known as Causal Dynamical Triangulation. The method is based on the analysis of the eigenvalues and the eigenvectors of the Laplace-Beltrami operator computed on the triangulations: it generalizes previous works based on the analysis of diffusive processes and proves capable of providing more detailed information on the geometric properties of the triangulations. In particular, we apply the method to the analysis of spatial slices, showing that the different phases can be characterized by a new order parameter related to the presence or absence of a gap in the spectrum of the Laplace-Beltrami operator, and deriving an effective dimensionality of the slices at the different scales. We also propose quantities derived from the spectrum that could be used to monitor the running to the continuum limit around a suitable critical point in the phase diagram, if any is found.Comment: 21 pages, 26 figures, 2 table

Comment of Global dynamics of biological systems

In a recent study, (Grigorov, 2006) analyzed temporal gene expression profiles (Arbeitman et al., 2002) generated in a Drosophila experiment using SSA in conjunction with Monte-Carlo SSA. The author (Grigorov, 2006) makes three important claims in his article, namely: Claim1: A new method based on the theory of nonlinear time series analysis is used to capture the global dynamics of the fruit-fly cycle temporal gene expression profiles. Claim 2: Flattening of a significant part of the eigen-spectrum confirms the hypothesis about an underly-ing high-dimensional chaotic generating process. Claim 3: Monte-Carlo SSA can be used to establish whether a given time series is distinguishable from any well-defined process including deterministic chaos. In this report we present fundamental concerns with respect to the above claims (Grigorov, 2006) in a systematic manner with simple examples. The discussion provided especially discourages the choice of SSA for inferring nonlinear dynamical structure form time series obtained in any biological paradigm.Comment: 6 pages, 2 figure