What the protractor has to be

Every open question in this collection has now been priced, and the prices are in two different currencies. Most are in specimens: tens of plants, sometimes hundreds, to establish a share or a rate across a population. Two are in internodes: fifty-six to separate two kinds of noise, sixty to read a parastichy number out of the angles.

An internode is much cheaper than a specimen, and both of the internode-priced tests look like an afternoon’s work. This essay is about the third currency, which neither of them was quoted in and which turns out to be the binding one.

The measurement a botanist actually makes #

A person reading divergence angles off a stem does not measure angles. They mark where each organ is, around the stem, and subtract.

That distinction is the whole essay. An error in one mark enters two consecutive divergences — once positively and once negatively — because the mark appears as the end of one interval and the start of the next. It is not an independent error on each angle; it is a correlated error with a specific structure, and the structure is the one that does the damage.

What it does to the peak #

The readout works by finding a peak in the autocorrelation of that list, at the lag corresponding to a parastichy number. Adding an independent error to every mark adds variance to the sequence — at every lag, flat, with no structure — while the pattern’s own signal stays exactly where it was.

So the correlation at the peak is divided by a larger total variance, and it falls by a factor that can be written down without measuring anything:

$$\text{peak}(\varepsilon) = \text{peak}(0)\cdot\frac{\sigma^2}{\sigma^2 + 2\varepsilon^2}$$

where $\sigma$ is the pattern’s own divergence scatter and $\varepsilon$ is the error on a single mark. The factor of two is the two consecutive divergences each mark enters.

Nothing in that expression is fitted. It has one measured input, the pattern’s scatter, which is a number the botanist has anyway.

Measured against simulated readings, it holds: at a quarter of a degree the predicted peak is 0.61 and the measured one 0.61; at half a degree, 0.40 against 0.41. The agreement runs out where the arithmetic says it must, at the point where what is left of the signal is smaller than the floor an argmax produces from noise alone.

Where the expression comes from #

It is three lines and worth doing, because the shape of it is what makes the conclusion unavoidable.

Let $m_i$ be where the rule actually put node $i$ and $e_i$ an independent reading error on the mark, with variance $\varepsilon^2$. The recorded divergence is

$$d_i = (m_i - m_{i-1}) + (e_i - e_{i-1}).$$

The two terms are independent, so the variance of the recorded sequence is the pattern’s own $\sigma^2$ plus $2\varepsilon^2$ — the two coming from the two marks each divergence uses.

An autocorrelation is a covariance divided by a variance. At a lag of $m$ — the parastichy number, several nodes away — the error terms contribute nothing to the covariance, because $e_i$ and $e_{i+m}$ are independent and $m > 1$. The numerator is therefore untouched. The denominator has grown by $2\varepsilon^2$.

Hence the ratio, and hence the two features that matter: it does not contain the number of internodes, and it does not contain anything about the pattern except its scatter.

The same three lines at a lag of one give a different answer, and that difference is the subject of a later section. There $e_i$ appears in both $d_i$ and $d_{i+1}$, with opposite signs, so the error contributes $-\varepsilon^2$ to the covariance as well as $2\varepsilon^2$ to the variance. It moves the numerator, and it moves it downward.

The number #

Read the other way round, the expression says what the instrument must be.

To keep half the peak, the error on a mark must be under $\sigma/\sqrt{2}$. For a pattern scattering by three quarters of a degree — which is a quiet, healthy stem — that is about half a degree, and half a degree is where the readout starts failing: right on two runs of five instead of five.

To keep the readout reliable, the requirement is about a quarter of a degree, at which it is right on five of five.

A quarter of a degree, on the azimuthal position of an organ around a stem. On a stem ten millimetres across that is a positional precision of about twenty micrometres.

That is not a protractor. It is photogrammetry, or a rotating stage with an encoder, or a serial-section reconstruction — and it is a different kind of undertaking from the afternoon with a specimen that the internode price implies.

Why more internodes do not help #

The instinct on being told a measurement is too noisy is to take more of it, and here that instinct fails in a way worth being precise about.

Averaging reduces the sampling band — how much an autocorrelation computed from a finite sequence wanders on its own. The band is $2/\sqrt{n}$, so quadrupling the internodes halves it, and with enough internodes the band can be made as small as anyone likes.

It does nothing to the peak height. The dilution factor above contains no $n$. A reading error of a degree takes the peak from 0.74 to about 0.17 whether the sequence is sixty internodes long or six hundred, because it is a ratio of variances and both variances are properties of the process rather than of the sample.

So the two costs are independent and they must both be paid. Internodes buy resolution of the peak against the band; precision buys the peak itself. A thousand internodes measured with a protractor gives a very precise measurement of a number that is no longer there.

The failure mode is a confident wrong answer #

This is the part that would cause the damage, and it is why the essay exists rather than being a paragraph of caveats.

The readout is an argmax over thirty lags. Handed a sequence in which the signal has been destroyed, it does not fail, hesitate or widen. It returns a number, promptly. At a reading error of two degrees the five runs return five numbers, all of them confidently, none of them right.

What changes is the peak height, and the peak height is the thing a user would not think to report. The clearance test — three sampling bands, with the threshold set from white sequences rather than from these ones — catches every one of them. Without it, a study with an inadequate instrument would report parastichy numbers that are simply the tallest bar in a noise spectrum, and they would look exactly like the right kind of answer: small integers, varying between specimens, sometimes Fibonacci.

The gate is the whole of the difference between an instrument and a random number generator with good manners, and it costs nothing to apply.

What the same error does to the other test #

The two internode-priced tests are damaged by reading error in different ways, and the contrast is sharp enough to be useful.

The noise-discrimination test reads the correlation at lag one and asks whether it is positive or near zero. Reading error contributes a term at lag one with autocorrelation $-\tfrac{1}{2}$ — which is the same signature as placement noise, the hypothesis being tested. So an inadequate instrument does not merely obscure that test’s answer; it manufactures one of the two answers. A perfectly quiet stem with a jostle in it, measured with half a degree of error, reports placement noise.

The parastichy readout reads a peak further along the same curve, where reading error has no structure at all. It contributes variance and nothing else. So an inadequate instrument makes the readout weaker and eventually silent, and it never makes it say something false — provided the clearance test is applied.

That is a real difference in kind. One test is confounded by measurement error and the other is merely defeated by it. A defeated test that knows it is defeated is much the better position, and it is the reason the clearance threshold is worth more than the number it guards.

The requirement scales with the specimen, and the wrong way #

The threshold is not a fixed number of degrees. It is a fraction of the pattern’s own scatter, which means it moves with the specimen — and it moves in the direction that makes the good specimens hardest.

Written out: the reading error must be under about a third of the divergence scatter for the readout to stay reliable. So

• a stem scattering by 1.0° needs marks good to about 0.35°
• a stem scattering by 0.75° needs 0.25°
• a stem scattering by 0.5° needs about 0.17°
• a stem scattering by 0.3° needs 0.10°

A quiet stem is a harder measurement than a noisy one, on this test, by exactly the factor its angles are tighter.

That is the second time this thread has produced a requirement that runs backwards from the usual one, and the two are not the same. The noise-discrimination test needs a quiet plant and gets harder as the disturbance grows. This test needs a disturbed plant — a sequence with no variation in it has no spectrum — and gets harder on the instrument side as the disturbance shrinks.

Between them they define a window in the specimen’s own scatter rather than an inequality: too quiet and the instrument requirement becomes unreachable, too noisy and the lattice is failing. The middle of that window is somewhere near three quarters of a degree, which is where every measurement in this thread has been made, and that was a choice made for other reasons before this consequence was visible.

Why this was not obvious from the internode price #

It is worth asking why the previous phase, which priced its own test carefully and in the right currency, did not arrive here.

Because the calculation it did was the right one for the question it asked. Two correlations are distinguishable when they differ by more than their combined sampling bands; the band is $2/\sqrt{n}$; therefore $n$ internodes. That arithmetic is correct, it is the arithmetic this collection uses for every sample-size question, and it produced fifty-six.

What it assumes, silently, is that the quantity being measured is the quantity of interest. The sampling band reports how well it has measured the correlation of the recorded sequence. Whether the recorded sequence’s correlation is the pattern’s correlation is a separate question, and it is the question measurement error answers.

That is a general shape and it is worth naming: a sample-size calculation prices precision and says nothing about bias. Every “how many would it take” figure in this collection has the same silent clause, and most of them are safe because the quantities they estimate are not systematically displaced by the ordinary errors of measuring them. This one is not safe, because the ordinary error of measuring an angle sequence has a specific structure that lands on the statistic.

The check that would have caught it is cheap and is now part of the site’s habit: after pricing a measurement in samples, ask what the instrument’s own error does to the estimator, and do it with the same three lines of variance arithmetic used to price the sample.

What a survey would have to say about its instrument #

The site’s survey specification already lists what a study of this subject would have to report, and it is mostly about the sample: how many specimens, at what radius, counted how. This adds a line to it, and the line is unusual in being about the apparatus rather than the design.

Report the angular precision of the position measurement, per organ, and report it as a number. Not “measured carefully”; a figure in degrees, obtained by remeasuring the same organs and reporting the spread.

The reason is that the precision and one of the hypotheses are the same operation on the same numbers, so a study that does not report its precision has not reported enough to distinguish its result from its instrument. This is not a general methodological homily — it is specific to the fact that measurement error on a position is a moving-average term at lag one, which is exactly what one of the two candidate mechanisms produces.

And the remeasurement is cheap: the same stem, twice, by the same person, with the difference reported. That single number decides which of the tests in this thread the specimen can be used for, and it is obtainable before any analysis is done.

The honest price #

Two numbers, and the second was missing until now.

Sixty internodes, on one stem, in one rung of the ladder, which is an afternoon.

A quarter of a degree per organ, which is an instrument most fieldwork does not carry and which cannot be reached by measuring more organs.

The first made this look like the cheapest open question in the collection. With the second attached it is not cheap — it is merely small, which is a different virtue: it needs one specimen rather than a population, and one good instrument rather than a long campaign. Whether that is easier than tens of plants depends on what is to hand, and this collection is not in a position to say which.

What it is in a position to say is that the price was quoted incompletely, and that the missing term is the one that decides.

There is a consolation and it is not a small one. The precision requirement is the same for both of the internode-priced tests, because both read the same sequence off the same marks. So an instrument good enough for one is good enough for the other, and the specimen that qualifies for the noise test — quiet, long, visibly climbing — can be run through the parastichy readout at no additional cost in fieldwork. Two answers from one campaign, and the second of them checkable against a spiral count taken with a camera.

That is the argument for building the instrument rather than for abandoning the question.