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Growth Rates, and Why R² Lies

Source: PHY653B Ch. 4

Intuition

Measuring a growth or damping rate is the most common quantitative claim in this field, and it is done badly more often than not. The procedure looks trivial — take a log, fit a line — and every step of it hides a way to get a confident, precise, wrong number.

The factor of two, again

Linear theory gives the amplitude rate: \(\delta \sim e^{\gamma t}\). Energy diagnostics give \(\mathcal{E} \sim e^{2\gamma t}\). Decide which you are plotting and say so. Half or double the right answer is the most common error in the subject, and both are plausible-looking numbers.

Why R² is not a goodness-of-fit here

\(R^2\) measures how much variance the line explains. For data that spans several decades in \(\ln\mathcal{E}\), almost any line through it explains most of the variance, because the variance is dominated by the overall rise. You can fit a window that is half linear-phase and half saturation, get a slope that corresponds to no physical process at all, and still see \(R^2 = 0.998\).

The diagnostic that actually works is stability of the slope under changes to the fit window. Slide the window; if the fitted \(\gamma\) moves by more than its own error bar, you are not measuring a clean exponential and there is no single rate to report.

A defensible procedure

  1. Plot \(\ln\mathcal{E}\) against \(t\) and look at it. Identify by eye: initial transient, linear phase, saturation. If there is no visually straight segment, stop — there is no exponential to fit.
  2. Fit the envelope, not the raw signal. An oscillating mode makes \(\ln\mathcal{E}\) a sawtooth; regressing through it biases the slope. Fit through the local maxima.
  3. Scan the window. Fit over \([t_1, t_2]\) for a family of choices. Report the mean and the spread. That spread is your uncertainty — it is almost always larger than the formal least-squares error.
  4. Establish the noise floor. In PIC especially, show the mode rose above the noise before you started fitting. A "growth rate" fitted to noise is a measurement of your marker count.
  5. Check resolution independence. Re-run at finer grid, smaller \(\Delta t\), more markers. A rate that moves is a numerical parameter, not physics.

Dispersion relations from raw data

For a field \(\phi(x,t)\) on a periodic grid, a 2-D FFT in \(x\) and \(t\) gives power in \((k,\omega)\). The bright ridges are the dispersion relation — measured, not assumed. Overplot the analytic branch and the comparison is immediate and total: every mode at once, rather than one rate at a time.

Practical points: window in time (a rectangular window smears the peaks badly), remember the Nyquist limits in both variables, and note that the run must be long enough for \(\Delta\omega\) to resolve the structure you care about.

Uncertainty

Every number in a paper needs one, and "the least-squares error on the slope" is usually the wrong one, because it assumes independent Gaussian errors on points that are neither. Better sources:

  • Window-scan spread (above) — the honest workhorse.
  • Ensemble spread over different random seeds, for anything noise-driven.
  • Resolution scan — the difference between the two finest runs bounds the discretisation contribution.

Common mistakes

  • Reporting \(R^2\) as evidence. See above.
  • One run, one number, no error bar.
  • Fitting through saturation. It flattens the slope and always in the same direction.
  • Not stating the window. A rate without its fit window is not reproducible.

Knowledge graph position

Prerequisites: least squares, FFT, verification and validation. Leads to: every quantitative claim in the course.

Quiz

Q1 (conceptual). Why is \(R^2 = 0.998\) not evidence that a growth-rate fit is good?

Answer

Over several decades of \(\ln\mathcal{E}\) the total variance is dominated by the overall rise, so nearly any line through the data explains most of it. A window straddling the linear phase and saturation gives a physically meaningless slope with an excellent \(R^2\). Slope stability under window changes is the real test.

Q2 (conceptual). How would you obtain an uncertainty on a measured growth rate?

Answer

Scan the fit window over a family of reasonable choices and report the spread of fitted slopes; for noise-driven problems add an ensemble over random seeds, and check the result against a resolution scan. The formal least-squares error is an underestimate because the residuals are correlated.

Q3 (MCQ). A 2-D FFT of \(\phi(x,t)\) in both \(x\) and \(t\) produces:

  • (a) the growth rate directly
  • (b) power in the \((k,\omega)\) plane, whose ridges are the measured dispersion relation
  • (c) the distribution function
  • (d) the noise floor only
Answer

(b). It measures the dispersion relation from the data rather than assuming it — the most complete single comparison you can make against linear theory.