What the ratio was hiding

The wander of a stem’s divergences is a ratio. Cut the divergences into consecutive blocks, take the variance of the block means, multiply by the square of the block size, and divide by the variance of the divergences themselves. The division is what makes it dimensionless, comparable between disturbances of different sizes, and equal to one for independent errors.

It is also what makes it a poor instrument for the question it was being asked.

Across a sweep of the rule’s neighbourhood — from three organs to a hundred and eighty-two, a factor of sixty-one — the wander rises from 22.4 to 82.3 at a disturbance correlated over thirty-three organs. That was read as the deep rule passing 3.7 times as much drift as the shallow one.

The quantity without the division #

The numerator of the wander, before it is normalised, is how far the block means move. Multiply by the block size and take a square root and it becomes a quantity in degrees: how much slow drift reaches the divergences, per organ, in the units a protractor would report.

Across the same sweep it runs 1.82°, 1.74°, 1.65°, 1.61°, 1.52° from the deepest rule to the shallowest.

That is a change of a fifth. A neighbourhood sixty times deeper passes twenty per cent more drift.

Put on a log scale the weakness is easier to hold. A factor of sixty-one in depth buying a factor of 1.20 in drift means the drift through grows as the depth to the power of 0.044 — about a twenty-third root. Extrapolating a fitted exponent is usually a bad idea, and here it is worth doing once for scale rather than for a prediction: at that rate, doubling the drift that reaches the divergences would take a neighbourhood of some twenty million organs.

That number is absurd and is meant to be. It is the honest way to say that the dependence is not weak in the sense of small but tuned; it is weak in the sense that the depth is very nearly not in the answer at all. A filter with a corner in it would show a region of steep dependence somewhere, and a power of a twenty-third leaves no room for one anywhere inside a sweep that spans sixty.

Meanwhile the denominator — the scatter between neighbouring divergences, which is the quantity a rule is supposed to be good at suppressing — runs 0.205°, 0.212°, 0.246°, 0.269°, 0.321°. A change of 1.57 in the other direction.

Multiply the two effects and the ratio moves by 3.7, which is what the wander does. Nearly all of it is the denominator.

Getting from the ratio to degrees #

The step from a dimensionless number to a quantity in degrees is one line of algebra and it is worth writing out, because everything here depends on it being the right line.

The wander at block size B is the variance of the block means, times B squared, divided by the variance of the divergences. So the variance of the block means is the wander times the divergences’ variance, divided by B squared. Take a square root: the standard deviation of the block means is the square root of the wander, times the standard deviation of the divergences, divided by B.

Multiply by B and what is left is the square root of the wander times the scatter — a quantity in degrees, and the natural one: it is how far a block of B organs is displaced from its neighbours’ blocks, scaled up by the block size, which is the size of the slow excursion the disturbance is producing.

Nothing in that derivation is specific to this thread, and its consequence is general: the square root of a normalised second moment, times the scale it was normalised by, is the unnormalised quantity. The reason it is worth spelling out is that it is available at no cost from every measurement this collection has already made with the statistic, and it was never taken.

Which is a result about the rule, just not the one it looked like #

Untangling the two does not leave nothing. It leaves two statements, both worth having and neither the one that was claimed.

A deeper rule corrects relative errors better. The scatter between neighbours falls by a factor of 1.57 as the neighbourhood grows from three organs to a hundred and eighty-two. That is exactly what a rule placing against neighbours should do: more neighbours in the comparison means a better estimate of where the next organ belongs, and less of the disturbance survives as scatter.

And a drift passes almost regardless. The degrees of slow drift reaching the divergences change by a fifth across the same range. The rule is not filtering the drift at any depth; it is passing nearly all of it at every depth, and what changes with depth is how clean the rest of the signal is.

Put together: the rule is a good corrector of the relative part and an indifferent filter of the shared part, at every depth tried. The wander rises with depth because it is measuring the corrector, not the filter.

How much of the denominator is the drift itself #

There is one honest complication and it limits how cleanly the two parts separate.

The denominator is the variance of the divergences, and the divergences contain everything: the relative scatter the rule failed to correct, and whatever neighbour-to-neighbour difference the drift itself produces. A drift correlated over thirty-three organs is not perfectly smooth from one organ to the next, so it contributes to the denominator as well as to the numerator.

That means the decomposition is not an identity. It is three measured numbers, and the statement “the ratio moved because the denominator fell” is a statement about their observed sizes rather than an algebraic fact.

How much does it matter here? The drift is correlated over thirty-three organs and the scatter is measured between adjacent ones, so the drift’s contribution to the denominator is small — it changes by about three per cent per organ over the correlation length, against a scatter of a fifth of a degree. That is why the three numbers multiply out as closely as they do: 1.20 times 1.57 is 1.87 against the square root of the wander’s own ratio, which is 1.92.

Agreement to within three per cent is the best available evidence that the decomposition is close to exact at this correlation length. At a much shorter one it would not be, and no claim is made there.

Why the normalisation was there in the first place #

It would be wrong to conclude that the division was a mistake. It was put in for a reason and the reason still holds.

Without it, the statistic has units of degrees and depends on the size of the disturbance driving the stem. Comparing a stem jostled at a quarter of a degree with one jostled at a half would then report the second as having twice the wander, whatever the structure of either disturbance — and the whole point of the statistic is to separate structure from amplitude.

The normalisation does that job correctly. What it cannot do is survive a sweep of a parameter that changes the denominator, and that is precisely what the depth sweep is. Every earlier use of the statistic in this collection compared disturbances at a fixed rule; this was the first comparison of rules, and it is the first place the denominator was free to move.

That is a general shape and this collection has met it before. A ratio is safe across a sweep of anything its denominator does not depend on, and misleading across a sweep of anything its denominator does depend on — and which of those a new sweep is cannot be read off the statistic’s definition without thinking about the mechanism.

The repair #

The repair is not to abandon the ratio. It is to report the two parts beside it, which costs nothing since both are computed on the way to it.

So the measurement is now three numbers per cell: the wander, the scatter, and the drift in degrees. The first is comparable with everything this collection has already measured; the second says what the rule did to the relative part; the third says what reached the sequence. Any claim of the form the rule passes more or less of a drift has to be made in the third.

The same repair has been made twice before here in different clothes. A period was reported without the resolution of the grid it was computed on, and it turned out to be the grid; the fix was to return the span of the motif beside the period. A dip width was reported without the window it was integrated over, and it turned out to be the window; the fix was to report the width at several windows. The rule of thumb they add up to is short: a summary statistic returned without the scale of what it summarises can be read as anything.

The third number, and why it is reported per cell #

Adding a number to a table is cheap and adding the wrong one is not, so it is worth saying why the drift in degrees is reported at every cell rather than once.

A single headline — the drift through changes by a fifth across the sweep — would hide the thing that makes it convincing, which is that it changes monotonically and by small steps: 1.52°, 1.61°, 1.65°, 1.74°, 1.82°. Five values in order, none of them out of sequence, over three seeds each. A quantity that wandered between those values would be reporting seed noise; one that stepped cleanly is reporting something small and real.

It also matters that the wander and the scatter are reported at the same cells rather than summarised. The claim is a relation between three quantities and relations between summaries are how a decomposition goes wrong: two effects that each look monotone in a mean can be uncorrelated cell by cell.

The cost of all this is three numbers where there was one, and the benefit is that the next sweep of anything in this thread will show immediately whether it is moving the numerator, the denominator or both. That is the whole of the repair, and it is smaller than the mistake it is fixing.

What this changes about the earlier claim #

The claim this thread started from is that a placement rule sharpens a drift rather than correcting it, and it was made from a single pair of numbers: a rule’s stems giving a wander of 46.5 against a kinematic lattice’s 30.0 at the same disturbance, while the scatter fell from 0.553° to 0.246°.

Both halves of that measurement stand. What changes is which of them carries the claim.

The scatter falling by more than half is the rule correcting the relative part, and it is the larger of the two effects. The wander rising is what a ratio does when its denominator halves. So sharpens is the right word for what the rule does to the visibility of a drift, and the wrong word for what it does to the drift: it does not amplify anything, it cleans up everything else.

That is a smaller claim and a more useful one, because it is a claim about an instrument a botanist would use. A plant whose placement rule corrects well is a plant in which a slow disturbance is easier to see, not one in which a slow disturbance is larger.

What a botanist would do with this #

The practical form of the correction is worth stating, because it changes the advice this thread gives about a real measurement.

A slow drift in a plant’s divergences — a wander over tens of organs rather than a jitter between neighbours — is a distinct kind of disturbance and the collection has argued that it is worth looking for, because it is the one signature of a disturbance with a memory in time rather than in space. The question a botanist faces is whether a placement rule would hide it.

The answer, corrected, is no and for a slightly surprising reason. The drift comes through nearly intact at any depth, so what reaches the sequence of divergences is essentially what the disturbance produced. What the rule changes is the background the drift has to be seen against: a rule that corrects relative errors well leaves a quieter sequence, and the same drift is then a larger fraction of what is left.

So the practical claim is that a well-corrected plant is a better subject for this measurement than a poorly corrected one, and the earlier wording — that the rule sharpens the drift — was right about the observation and wrong about the mechanism. Nothing about how many organs a botanist has to count changes; what changes is that the number now follows from a quantity in degrees rather than from a ratio whose parts had not been separated.

What this does not say #

It does not say the wander should not be used. It is the right statistic for comparing disturbances at a fixed rule, which is what it was built for and what every earlier measurement here does with it — including the ones taken once the neighbourhood became a stated hypothesis rather than a loop bound.

It does not say the twenty per cent is nothing. The drift through does rise with depth, monotonically, across the whole sweep. It is a real and small effect, this essay does not have the seeds to say whether its shape is interesting, and the extrapolation above is a way of stating its size rather than a claim about neighbourhoods nobody will ever grow.

It does not say the scatter is the whole denominator story. The variance of the divergences includes whatever the drift itself contributes to neighbour-to-neighbour differences, so the two parts are not perfectly separable. The decomposition here is a good approximation rather than an identity, and it is reported as three measured numbers rather than as an algebraic split.

And it does not settle what a real inhibition’s range is. The exponent is a handle on depth in a model, and it is the only handle: the loop bound the sweep looked like it was testing does not bind at all. What it corresponds to in tissue is the open question this collection has repeatedly named as its most testable one.

The check that would refuse it #

Two assertions, and their relation is the content.

The first is that the drift reaching the divergences changes by less than a factor of 1.5 across the whole depth sweep. That is the claim that the numerator is nearly flat, and it is the one that would fail if the filter reading were right after all — a rule with a corner in it would show a large change in the numerator somewhere in a sweep spanning a factor of sixty in depth, and the corner that is there sits at two or three organs whatever the depth.

The second is that the wander changes by more than twice as much as the drift through does. That is what makes the essay’s title a measurement rather than an opinion: it says the ratio moves for a reason its numerator does not supply, and it fixes how much of the movement has to be unaccounted for before the claim is allowed. Measured, the wander moves by 3.7 and the drift through by 1.20, so the margin is comfortable and the assertion has room to fail.

Both are computed from the same three seeds per cell as the figures, and the spread between those seeds is carried rather than hidden — because a decomposition of an effect into two smaller effects is exactly the kind of claim that a noisy measurement can be talked into.