An AR(1) process has independent zero-mean noise and ; its stationary autocorrelation…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Scientific Computing for Physicists, Ch The uncertainty of a simulated number: inputs, sensitivity and correlated output

Problem

An AR(1) process xn+1=ϕxn+ηnx_{n+1} = \phi x_n + \eta_n has independent zero-mean noise ηn\eta_n and ∣ϕ∣<1|\phi| < 1; its stationary autocorrelation function is ρ(k)=ϕ∣k∣\rho(k) = \phi^{|k|}. Take ϕ=−0.5\phi = -0.5 (an anti-persistent, sign-alternating series). ((a)) Starting from τint=12+∑k=1∞ρ(k)\tau_{\rm int} = \tfrac12 + \sum_{k=1}^{\infty}\rho(k), derive a closed form for τint(ϕ)\tau_{\rm int}(\phi) using the geometric series, and evaluate it exactly for ϕ=−0.5\phi=-0.5. ((b)) Find the statistical inefficiency s=2τints = 2\tau_{\rm int} for this ϕ\phi. ((c)) For a run of N=104N = 10^4 samples of this series, find the effective sample size Neff=N/sN_{\rm eff} = N/s, and explain in one sentence, in terms of the series' sign-alternation, why NeffN_{\rm eff} can exceed NN.

Browse the problem library