An integral over the unit square is estimated two ways
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods
Error, Chance and the Machine, Ch Monte Carlo integration and variance reduction
Problem
An integral over the unit square [0,1]2 is estimated two ways, each with N points. Two point plots for N=256 are described below (imagine them plotted in the unit square). Plot A: the 256 points show visible empty patches in some corners and tight clumps of several points elsewhere. Plot B: the 256 points are spread so that every small sub-square of side 1/16 contains almost exactly one point, with no empty patches and no clumps. The magnitude of the estimate's error was also recorded for a range of sample sizes for each method, on the same smooth integrand: Data: at N=102 the errors are 4.5×10−2 (method 1) and 2.0×10−2 (method 2); at N=104 they are 4.6×10−3 and 2.1×10−4; at N=106 they are 4.5×10−4 and 2.1×10−6.
((a)) Say which plot, A or B, belongs to method 1 and which to method 2, giving the reason from the pattern of points.
((b)) Using the N=102 and N=106 rows, estimate the power-law exponent p in error ∝Np for each method (fit a straight line on log-log axes through the two points), and state which standard rate each matches.
((c)) Explain, in one or two sentences, why method 2's curve in this table cannot be turned into a confidence interval on its own, even though its errors are smaller at every N.