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22-Agric-A5 Principles of Instrumentation · Undated paper

Question 5 of 7: Non-Linear Calibration and Error Structure

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 04-Agric-A5 Principles of Instrumentation, National Exam (printed exam date May 2019) — a three-hour open-book exam; any non-communicating calculator is permitted. Questions 1 and 2 are compulsory (20 marks each); candidates then choose any three (3) of Questions 3-7 (20 marks each) for a 100-mark paper. All seven questions are worked here.

Reference texts. E.O. Doebelin, Measurement Systems: Application and Design, 5th ed. (calibration, noise, dynamic sensor response, sampling and ADCs); J.P. Bentley, Principles of Measurement Systems, 4th ed. (error propagation, signal conditioning, bridge circuits); P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (op-amp circuits, precision rectifiers, instrumentation amplifiers, shot/Johnson noise); J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 5th ed. (photodetectors, gas sensors, Hall-effect and thermal sensors); F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer (forced-convection/King's-Law correlations for the hot-wire bridge).

Check — question wording used below. Question 1 includes the sub-part “What is the most common source of instrument drift?”, and part 1(j)'s converter is a 13-bit device (Intersil 7109). Question 6 has six sub-parts (a–f); part (b) asks “What ions in the solution are detected?” Question 2(i) asks “Why are optical sensors sometimes cooled to liquid nitrogen temperatures?” Page 2 carries one schematic, the three-op-amp instrumentation amplifier. Question 3 has four sub-parts (a–d).

Question 5: Non-Linear Calibration and Error Structure (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

a) Absolute error is the difference between the measured and true value expressed in the same units as the measurand itself (e.g. $\pm0.05$ signal units, regardless of the signal's own magnitude). Relative error is that same difference expressed as a fraction of the true (or measured) value — dimensionless, usually quoted as a percentage — and describes the error's size relative to the signal magnitude rather than its raw size.

b) A least-squares fit minimizes the sum of squared residuals in whichever variable is actually being regressed — $S$ itself for a direct (untransformed) fit, or $\log S$ for the log-transformed fit — and this implicitly assumes the error added to that regressed variable has constant variance (homoscedastic) across the fitted range. Whether that assumption is actually true depends on the measurement's real error structure (constant absolute vs. constant relative error in the original units of $S$), so the choice of which variable to fit in must be matched to the error model, not made by default.

c) No — a constant absolute error does not justify the logarithmic transform. Given. $S=aM^b$ with $a=2$, $b=1.5$, evaluated at a low signal $M{=}1$ ($S{=}2$) and a high signal $M{=}100$ ($S{=}2000$), each corrupted by the same constant absolute error $\varepsilon_{abs}=0.05$. Find. How the resulting log-space error compares at the two signal levels. The log-space perturbation is $$\Delta(\log S)=\log(S+\varepsilon_{abs})-\log S\approx\frac{\varepsilon_{abs}}{S},$$ which shrinks as $S$ grows. Numerically, $\Delta(\log S)\approx0.0247$ at $S{=}2$ but only $\approx2.5\times10^{-5}$ at $S{=}2000$ — a thousand-fold difference for the same absolute error. The log-transformed residuals are therefore strongly heteroscedastic (much noisier at low signal), violating the constant-variance assumption behind ordinary least squares on $\log S$, so a constant-absolute-error dataset should be fitted directly to $S=aM^b$ (e.g. by non-linear least squares), not via the log transform.

d) Yes — a constant relative error does justify the log transform. With relative error $\varepsilon_{rel}$, the corrupted signal is $S(1+\varepsilon_{rel})$, so $$\Delta(\log S)=\log\!\big(S(1+\varepsilon_{rel})\big)-\log S=\log(1+\varepsilon_{rel}),$$ which is independent of $S$ — exactly the same at $M{=}1$ and $M{=}100$ ($\Delta(\log S)=\log(1.02)=0.0198$ throughout, for $\varepsilon_{rel}=2\%$). A constant relative error is therefore exactly homoscedastic in log space, which is precisely the assumption ordinary least squares needs — making $\log S=\log a+b\log M$ the statistically correct linear model to fit whenever the underlying error is a constant percentage of the signal.

Question 5(c)/(d) — log-space error at low vs. high signal
Error model$\Delta(\log S)$ at $S=2$$\Delta(\log S)$ at $S=2000$Homoscedastic in log space?
Constant absolute error (0.05)0.02470.000025No — do not log-transform
Constant relative error (2%)0.01980.0198Yes — log transform is valid

e) Plotting the raw measurement error (or residual scatter) against the signal value directly reveals which error model applies: a scatter band of roughly constant width across the whole signal range indicates constant absolute error, whereas a scatter band whose width grows in proportion to the signal (a widening "funnel" or "megaphone" shape) indicates constant relative (percentage) error. This diagnostic plot tells the experimenter directly whether to fit $S$ or $\log S$, rather than assuming one or the other without evidence.