A thermistor calibration produces paired readings ,

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

Conditions of Validity, Ch Inner products, orthogonality and the adjoint

Problem

A thermistor calibration produces nn paired readings (ti,Ri)(t_i, R_i), i=1,…,ni = 1,\ldots,n, and a model R=α+βtR = \alpha + \beta t is to be fitted by least squares. Write b=(R1,…,Rn)T∈Rn\mathbf{b} = (R_1,\ldots,R_n)^{\mathrm T} \in \mathbb{R}^n for the measured resistances and let A\mathsf{A} be the n×2n\times 2 real matrix whose columns are 1=(1,…,1)T\mathbf{1} = (1,\ldots,1)^{\mathrm T} and t=(t1,…,tn)T\mathbf{t} = (t_1,\ldots,t_n)^{\mathrm T}, so that the fitted values are Ax\mathsf{A}\mathbf{x} with x=(α,β)T\mathbf{x} = (\alpha,\beta)^{\mathrm T}. ((a)) Starting from the projection theorem for the real inner product ⟨u,v⟩=uTv\langle \mathbf{u},\mathbf{v}\rangle = \mathbf{u}^{\mathrm T}\mathbf{v} on Rn\mathbb{R}^n, derive the normal equations satisfied by the least-squares estimate x\mathbf{x}, and state which subspace the residual b−Ax\mathbf{b} - \mathsf{A}\mathbf{x} must lie in and why. ((b)) Define the centred vectors u=t−tˉ 1\mathbf{u} = \mathbf{t} - \bar{t}\,\mathbf{1} and v=b−Rˉ 1\mathbf{v} = \mathbf{b} - \bar{R}\,\mathbf{1}, where tˉ\bar{t} and Rˉ\bar{R} are the sample means of tit_i and RiR_i. Using the Cauchy-Schwarz inequality on Rn\mathbb{R}^n, show that the sample correlation coefficient r=⟨u,v⟩/(∥u∥ ∥v∥)r = \langle \mathbf{u},\mathbf{v}\rangle / (\|\mathbf{u}\|\,\|\mathbf{v}\|) satisfies −1≤r≤1-1 \le r \le 1, with equality if and only if v\mathbf{v} is a real scalar multiple of u\mathbf{u}.

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