A degree-5 polynomial

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

t(x) = \sum_{n=0}^{5} a_n x^n

, with six free coefficients, is fitted by least squares to exactly six training measurements of seismic travel time

t

(in seconds) against epicentral distance

x

(in km). Training data: for

x = 50, 100, 150, 200, 250, 300\,\mathrm{km}

the measured times are

t = 9.1, 17.8, 26.5, 35.0, 43.9, 52.6\,\mathrm{s}

respectively. A held-out test set of four further measurements, never used in the fit, gives the following. At

x = 75, 125, 175, 225\,\mathrm{km}

the measured times are

13.6, 22.3, 30.9, 39.7\,\mathrm{s}

and the model-predicted times are

13.1, 22.9, 29.5, 41.0\,\mathrm{s}

respectively. ((a)) Explain, by counting equations and unknowns, why the least-squares fit reproduces every training value exactly. ((b)) Compute the training RMSE and the test RMSE. ((c)) State which of the two, if either, is a fair estimate of the model's error on future measurements, and why.

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