A straight line is fitted by unweighted least squares to five points at

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

Error, Chance and the Machine, Ch Estimators and the information in data

Problem

A straight line y=a+bxy = a + bx is fitted by unweighted least squares to five points at x=1,2,3,4,5x = 1, 2, 3, 4, 5, each with the same standard uncertainty σy=1\sigma_y = 1. The design matrix is A\mathsf{A} with rows (1,xi)(1, x_i). ((a)) Using the QR factorisation of A\mathsf{A} (rather than forming the normal equations ATA\mathsf{A}^{\rm T}\mathsf{A} directly, which is the less stable route), find the upper-triangular factor R\mathsf{R} and hence the covariance matrix V=R−1R−T\mathsf{V} = \mathsf{R}^{-1}\mathsf{R}^{-\rm T} of (a^,b^)(\hat a,\hat b). ((b)) State the correlation coefficient between a^\hat a and b^\hat b. ((c)) Propagate V\mathsf{V} to find the standard deviation of the extrapolated prediction y^(x=8)=a^+8b^\hat y(x=8) = \hat a + 8\hat b.

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