A logger applies a two-point boxcar (running-mean) filter to raw independent readings…

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

A logger applies a two-point boxcar (running-mean) filter to raw independent readings εn\varepsilon_n (zero mean, variance σε2\sigma_\varepsilon^2, mutually independent), producing the smoothed series xn=εn+εn−12.x_n = \frac{\varepsilon_n + \varepsilon_{n-1}}{2}. ((a)) Derive Var(xn)\mathrm{Var}(x_n) and Cov(xn,xn+1)\mathrm{Cov}(x_n, x_{n+1}) directly from the definition of xnx_n, and hence the lag-one autocorrelation ρ(1)\rho(1). ((b)) Show that ρ(k)=0\rho(k) = 0 for k≥2k \ge 2, and use τint=12+∑k=1∞ρ(k)\tau_{\rm int} = \tfrac12 + \sum_{k=1}^\infty \rho(k) to find τint\tau_{\rm int} and the statistical inefficiency ss of the smoothed series. ((c)) For N=5000N = 5000 boxcar-smoothed readings, find the effective number of independent readings NeffN_{\rm eff}, and state in one sentence why treating all 50005000 smoothed values as independent would understate the uncertainty of their mean.

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