What a quiet plant is worth

The specification this collection recorded a phase ago listed what each open question would cost in specimens, and it was built on an assumption so ordinary that nobody wrote it down: that a measurement is easier when the thing being measured is larger.

The divergence-sequence test breaks that assumption, and this essay is about what the break does to a survey.

The ceiling #

The statistic separates noise arriving before the placement rule’s choice from noise arriving after, by five sampling bands, on a stem whose divergence angles scatter by three quarters of a degree.

At nine tenths of a degree the separation is still clear. At 1.2° it has fallen by a factor of three, to about the width of the combined sampling band — which is not a clean crossing so much as a slope, and the slope is the point: the statistic is losing what it reads.

The reason is structural. What the informative kinds of noise preserve is the correlation of a lattice — the rule’s own self-correction, showing up as a relationship between consecutive angles. A lattice at one and a half degrees of scatter is at its tolerance boundary; its self-correction is losing to the disturbance, and what is being preserved is going away at the same time as the disturbance being characterised is getting large.

So the signal and the noise are not independent. The statistic reads a property of a working pattern, and a pattern with a lot of noise in it is not working.

Why that inverts a survey’s instincts #

Every sampling instinct a person has says: find the clearest example of the phenomenon. If the phenomenon is noise in phyllotaxis, find the stem with the most irregular angles, because that is where the noise is.

That is exactly backwards here. The stem with irregular angles is the one this test cannot read. The stem worth measuring is the one whose angles are tightest — a specimen that, by every other criterion this collection has, is the least interesting: a clean lattice, a stable pair, an unremarkable divergence sitting where the model says.

There is a name for a survey that selects on the wrong end of a distribution and this collection has already met it. Published spiral counts are drawn overwhelmingly from heads clear enough to count, which is a selection on exactly the property being reported and is why the Fibonacci share in the literature is not the Fibonacci share in the field. The same trap is available here in reverse: a survey that selected the noisy specimens because they looked like the ones with something to say would produce nulls, and the nulls would mean nothing.

The eligibility criterion is therefore part of the measurement rather than a precaution around it: measure the scatter first, and if it is above about nine tenths of a degree, stop. Reporting the fraction of specimens that failed that test is itself a datum, and it costs nothing.

What the ceiling does to the sample size #

The count in the previous essay — fifty-six internodes — assumed a stem at three quarters of a degree. The count goes as the inverse square of the separation, so it is worth writing out what the ceiling does to it.

At 0.75° the separation is 0.76 and the count is 56. At 0.9° it is about 0.30 and the count is roughly 350 — a long stem, and possible. At 1.2° the separation is about the width of the band and the count runs to several thousand internodes: no stem is that long, so the test is off in practice long before it is off in principle.

That last is the point worth carrying. The relationship between effect size and sample size in an ordinary design is a trade: a smaller effect costs more observations, and enough observations recover any effect. Here the trade has a wall in it. Past a certain noisiness, more data does not help, because what is being looked for has stopped existing.

A survey design that treated the scatter as a nuisance parameter to be averaged over would therefore be badly wrong. It is a gate, and specimens either pass it or are excluded.

What a gate does to a reported result #

A survey with an eligibility criterion reports two numbers rather than one, and the second is usually thrown away. It should not be here.

Suppose forty stems are examined, twelve pass the scatter gate, and those twelve come back with a correlation near zero. The result is not “the noise arrives after the choice”. It is “in the twelve stems quiet enough to read, the noise arrives after the choice”, and the twenty-eight excluded ones are a fact about the population that the analysis has deliberately set aside.

That distinction matters because the exclusion is not random. A stem fails the gate by being noisy, and whatever makes a stem noisy might be the same thing that decides which kind of noise it has. It is entirely coherent for quiet stems to be dominated by observation error — placement-like — while noisy ones are dominated by tissue irregularity, which is a jostle. A survey reporting only the eligible stems would find the first and never see the second, and would have found something true about a subpopulation while sounding like it had found something about plants.

The defence is arithmetic rather than clever: report the pass rate, and report the scatter distribution of the whole sample, not only of the eligible part. A reader can then see how much of the population the conclusion covers. This is the same discipline the collection’s counting essays ask for — a count without the radius it was taken at is a sample from an unstated mixture — applied to a different field of the same record.

The half of it that is genuinely bad news #

The uncomfortable reading is worth putting plainly, because the phase’s summary would otherwise sound triumphant.

This test works on plants whose noise is small. Its answer is about what kind of disturbance a quiet apex is subject to. If the interesting biology is in the noisy specimens — a plant under stress, a mutant with disrupted transport, a shoot growing irregularly — the test is silent about all of them, and those are the specimens where a developmental question is most likely to have a visible answer.

That is a real limitation on the scientific value of the instrument, as distinct from its statistical properties. An instrument that can only be pointed at the undisturbed case is not useless — the undisturbed case is what a model of the normal process should be judged against — but it is not the instrument anybody would have designed if they had a choice.

The half of it that is good #

Two things sit on the other side.

The eligible population is large. Three quarters of a degree is not an unusual scatter for a well-grown stem; the model’s own noiseless runs sit at half a degree, and a pattern is comfortable up to about a degree and a half. The gate excludes the worst specimens rather than most of them, and the fraction excluded is measurable rather than guessed.

The failure is honest. A stem that fails the gate fails visibly and in advance, before any analysis. That is a much better property than a test which silently returns weak answers on unsuitable data — which is what the site’s other survey questions do, and why they need exposures written down. Here the exposure is a threshold and the specimen either clears it or does not.

There is a general form worth extracting. A test whose signal is a property of the regime being disturbed has a ceiling, and the ceiling should be published as an eligibility criterion rather than discovered as a null. Anything measuring the disruption of a structure by reading a property of that structure has this shape: the reading fails where the structure does.

Where else this shape turns up in the collection #

It is worth checking whether the ceiling is a peculiarity of one statistic or an instance of something the site has met before, and it is the second.

Counting. A parastichy count is worth a factor of one over the product of the two numbers — a report of 34 and 55 pins the divergence to a tenth of a degree of arc, and a report of 2 and 3 leaves nearly forty. So high counts are worth enormously more. And high counts are taken at the rim, where the families are most numerous, most nearly parallel, and hardest to trace correctly. The precision rises with exactly the property that makes the measurement error-prone, and a wrong 34/55 excludes the truth confidently where a wrong 2/3 excludes almost nothing.

Jugacy. Whether a stem is genuinely multijugate is decided by the rotational symmetry of its point set, which is a measurement on positions. A noisy point set has no exact symmetry, so the measurement degrades precisely on the specimens whose counts are ambiguous — which are the specimens the question is about.

Varying exponents. An organ’s local rise exponent can only be recovered from several rings, and two rings cannot show a varying exponent at all. The measurement needs the thing to be extended, and an organ small enough for the variation to matter is often too small to carry the rings.

Four instances of one shape: the regime where the question is interesting is the regime where the instrument is worst. It is not a coincidence and it is not bad luck. All four instruments read a structure, and all four questions are about that structure being stressed — so in each case the question asks about the conditions under which the reading fails.

The transferable form is a habit rather than a technique. When specifying a measurement, ask what happens to it at the extreme of the effect being looked for, and expect the answer to be worse, not better. If it is worse, the extreme belongs in the specification as a gate, stated in advance.

What it changes in the specification #

The previous phase’s specification named three sample sizes and a set of fields to record. This one adds an entry and changes one of the fields.

The new entry is the sequence test: fifty-six usable internodes on a single quiet stem, with the scatter gate, the remeasurement control for observer precision, and the requirement that the counted pair agree at the two ends of the measured stretch.

The changed field is what a survey records per specimen. The old list had the counted pair, the counting radius, the rotational symmetry order and the mean divergence. It now needs the divergence angles in order, individually, rather than summarised.

That is a smaller change than it sounds and a larger one than it looks. Smaller because the angles are already being measured — a mean divergence is computed from them, so they exist in a notebook somewhere. Larger because the convention is to report the summary and discard the sequence, and a dataset published as means and standard deviations cannot be re-analysed this way however good it was.

What the phase leaves undone #

Three things this thread could have measured and did not, listed so that a later phase does not have to rediscover that they were available.

The ceiling as a function of the rise. Everything here is measured at one place on the ladder, with runs ending at a rise of 0.004. The tolerance a lattice has is known to tighten as the pattern gets finer, and the correlation the statistic reads is a property of the same self-correction, so the ceiling almost certainly moves along the ladder too. A stem at a coarse rise might be readable at scatters where a fine one is not, or the reverse. It is one sweep.

The negative lobe against the counted pair. The noiseless autocorrelation goes mildly negative around lag five, which looks like the correction being passed round a parastichy family. If the lag of that lobe tracked the smaller parastichy number as the rise falls, it would be a real result about how a lattice propagates a correction — and it would give the sequence a second, independent statistic, which is exactly what the mixture problem needs.

A correlated jostle. The model displaces each primordium independently, and real tissue deformation is correlated over some length. A correlated displacement field moves whole groups, which is much closer to what field noise does, and would test whether the phase’s prediction that a jostle “behaves like field noise” gets more true with realism or less.

None of the three is expensive. All three were visible from inside this phase and were left because the phase already had more than it could write up, which is a better reason than not having noticed and is still worth recording as a reason.

The general lesson about summaries #

This thread has produced one methodological result that has nothing to do with plants, and it belongs at the end of the field that records what the collection cannot do.

Four phases of this site measured a divergence sequence with statistics that are invariant to shuffling: a mean, a spread, a classification of the counted pair. Every one of them is a legitimate measurement and every one of them discards the order. The order turned out to contain the only quantity that separates two hypotheses the summaries cannot.

The general form: a process that acts sequentially cannot be characterised by a statistic that does not care about sequence. That sounds obvious written down and it was not obvious in practice — the summaries were the natural things to measure, they answered the questions being asked, and the questions they could not answer were attributed to the plant rather than to the statistic. Two phases concluded that two kinds of noise were indistinguishable, which was true of the statistics used and not true of the data.

The cost of finding that out was one function. The autocorrelation of the sequence is four lines of arithmetic over data every run had already produced, and it was available from the first phase.

What kept it from being written is the thing worth guarding against. The summaries were not chosen carelessly; they were chosen because they are what the subject reports. A phyllotaxis paper gives a divergence angle and a parastichy pair, because those are what the field means by a measurement, and a collection built to check that field’s claims naturally computes the same quantities. Adopting a subject’s vocabulary is how a piece of work becomes legible to it, and it is also how a piece of work inherits the subject’s blind spots.

The general form is the one to carry: the statistics a field reports are a selection, and the selection encodes what that field has historically been able to ask. Reproducing them is necessary and is not the same as looking. The one place this collection has consistently done better is in refusing to trust a count without its radius — and that is a case where the field’s own summary was found to be incomplete by the same route, which is somebody computing the thing the summary was a summary of.