A least-squares fit to a damped oscillator's decay gives best-fit amplitude…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Data analysis and model fitting in the laboratory · Data analysis and model fitting in the laboratory

Error, Chance and the Machine, Ch Monte Carlo integration and variance reduction

Problem

A least-squares fit to a damped oscillator's decay gives best-fit amplitude A0=2.00A_0 = 2.00 (arbitrary units) and decay rate Γ0=0.50 s−1\Gamma_0 = 0.50\ \mathrm{s^{-1}}, with covariance matrix (in units of [A]2[A]^2, [A][Γ][A][\Gamma] and [Γ]2[\Gamma]^2 respectively) V=(0.00250.00120.00120.0016).\mathsf{V} = \begin{pmatrix} 0.0025 & 0.0012 \\ 0.0012 & 0.0016 \end{pmatrix}. ((a)) Find the correlation coefficient between AA and Γ\Gamma. ((b)) Find the lower-triangular Cholesky factor L\mathsf{L} with V=LLT\mathsf{V} = \mathsf{L}\mathsf{L}^{\rm T}, that is L=RT\mathsf L=\mathsf R^{\rm T} for the upper-triangular R\mathsf R with V=RTR\mathsf V=\mathsf R^{\rm T}\mathsf R. ((c)) Using independent standard normal draws z1=0.80z_1 = 0.80, z2=−0.50z_2 = -0.50, form the correlated sample (δA,δΓ)=Lz(\delta A,\delta\Gamma) = \mathsf{L}\mathbf{z} and state the resulting sample (A,Γ)(A,\Gamma).

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