A thermistor's resistance is measured at known temperatures (in )

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

Scientific Computing for Physicists, Ch Learning from physical data: regression, classification and principal components

Problem

A thermistor's resistance RR is measured at N=6N=6 known temperatures TT (in ∘C^\circ\mathrm{C}), with each measurement subject to independent Gaussian noise of known standard deviation σ=0.8 Ω\sigma = 0.8\,\Omega. The candidate model is the straight line R(T)=θ0+θ1TR(T) = \theta_0 + \theta_1 T. Data: for T=0,20,40,60,80,100 ∘CT = 0, 20, 40, 60, 80, 100\,^\circ\mathrm{C} the measured resistances are R=10.2,19.8,30.5,39.6,51.0,60.3 ΩR = 10.2, 19.8, 30.5, 39.6, 51.0, 60.3\,\Omega respectively. ((a)) State the supervised learning problem: identify the design matrix, the parameter vector θ=(θ0,θ1)\boldsymbol{\theta}=(\theta_0,\theta_1), and write the loss as the negative log-likelihood of the Gaussian noise model. ((b)) Show that minimising this negative log-likelihood over θ\boldsymbol{\theta} is equivalent to ordinary least squares. ((c)) Evaluate the negative log-likelihood, in nats, for the trial parameters θ0=10 Ω\theta_0 = 10\,\Omega, θ1=0.5 Ω/∘C\theta_1 = 0.5\,\Omega/^\circ\mathrm{C}, using the tabulated data.

Browse the problem library