Phyllotaxis
the spirals are counted, not admired
Where the angle comes from

Six of six is not a measurement

A panel comparing two rules seed by seed reported the deeper one winning every one of six seeds at four correlation lengths out of six, and was read as a lattice with no corner in it. Six of six is the largest number the panel can print, so the flat middle was a reading of the ceiling — and raising the disturbance brings a corner out of it.
12 min read 24 figures State the claim, state the testCounted, not admired

Worth reading first: A disturbance with a memory · Where the noise gets in · A head is a set of points.

The depth thread’s central comparison is a paired one. Two rules — a deep neighbourhood and a shallow one — are given the same disturbance from the same seed, and what is counted is how many of six seeds agree about which rule let less of the drift through. Sweep the disturbance’s correlation length and the count traces a shape.

At the coarse and fine rises the shape is a crossing: the deeper rule loses on white noise and wins on long-correlated noise. At the middle rise it is not. There, the panel reported no crossing at all — the deeper rule ahead everywhere — and the result was written up as a lattice whose behaviour has no corner in it.

The same comparison at three sizes of disturbance. Each row is one panel: how many of six seeds the deeper rule beats the shallower one on, at six correlation lengths, on the lattice at a rise of 0.013. At the size of jostle the thread used, cells across the middle sit at six of six — the largest number the panel can print — so no feature could have appeared there whatever the lattice did, and the reading that this rise has no corner was a reading of that ceiling. Raised, the cells come down, and at the largest disturbance the panel shows a clean crossing: the deeper rule loses at white noise and takes every seed once the disturbance remembers itself for a few organs. The corner is there, and the flat middle was the instrument.
Fig. 1 The same comparison at three sizes of disturbance, with the cells that are pinned at every seed picked out.

The trouble is in the numbers themselves. Four of the six cells across the middle of that panel read six of six. Six of six is the largest thing the panel can print.

What a ceiling does to a reading

A cell at six of six cannot go up. So a change in the lattice, the rule, the correlation length or anything else that would have made the deeper rule more dominant produces no change in the number. The cell is not reporting how far ahead the deeper rule is; it is reporting that it is ahead, which the cell next door is also reporting, and the flatness between them is the flatness of a wall rather than of a landscape.

How many seeds the deeper rule takes, at three disturbances. The same six correlation lengths across the bottom, the share of six seeds the deeper rule wins up the side, one line per size of jostle. The top line is pinned against the ceiling across the middle of the range, which is the shape of a comparison that cannot report anything. As the disturbance grows the line comes down and tilts, and at the largest one it crosses the halfway mark between the third and fourth correlations. The ceiling was not evidence that this lattice has no corner; it was evidence that the panel had no room.
Fig. 2 The same panels as lines. The top one runs along the ceiling across the middle of the range.

That is enough on its own to withdraw the reading. “No corner here” was inferred from a flat middle, and a middle that cannot be anything but flat is not evidence of flatness in the thing being measured.

It is also enough to say what should have been done next, and what should not. The plan after that panel was a third exponent — a third rule, to see whether the corner appeared between other pairs. Three panels on a ceiling would have been three of the same non-measurement, at three times the cost. Taking the ceiling off first is cheaper and it has to come first.

A rule too long-ranged makes no pattern; every shorter one makes the same pattern. Each dot is 4 runs from a coarse start at one exponent, separated by 0.2° of placement noise. Below p ≈ 1.1 the divergence scatters by tens of degrees, which is what an arbitrary sequence gives. From p = 1.25 to p = 8 — a sixfold range — every run ends on 8/13 with the scatter between 0.75° and 1.06°.
Fig. 3 The exponents the third panel would have compared, none of which would have helped while the cells were pinned.

How the ceiling got past everyone

It printed a number, and the number varied.

Across the six correlation lengths the panel reads five, six, six, six, six, four. That is not a constant column; it has a shape, it rises and falls, and read quickly it looks like a measurement whose middle is flat. The two ends are away from the ceiling and do carry information, which makes the whole row look informative.

How many seeds the deeper rule takes, at three disturbances. The same six correlation lengths across the bottom, the share of six seeds the deeper rule wins up the side, one line per size of jostle. The top line is pinned against the ceiling across the middle of the range, which is the shape of a comparison that cannot report anything. As the disturbance grows the line comes down and tilts, and at the largest one it crosses the halfway mark between the third and fourth correlations. The ceiling was not evidence that this lattice has no corner; it was evidence that the panel had no room.
Fig. 4 The panel as it was read, against the next one down: a line with structure at both ends and a flat between them, and a line that has room to move.

What was needed was to ask what the maximum possible value of a cell is, and that question is not one anybody asks of a number that is already varying. It would have been asked immediately of a count of six out of six seeds if the count had been printed as a share — a cell reading 1.00 announces its own ceiling in a way that a cell reading 6 does not.

Agreement between two windows happens only on a slow enough shoot. Five stems at each of four rates and five disturbances, each read through two overlapping windows of 250 internodes. A filled mark is agreement — both windows reported the same pair; a half mark is a disagreement; a small mark is one window reporting and one refusing; an open mark is silence. Agreement appears 0 times in 25, 1 times in 25, 14 times in 25, 14 times in 25 at 130, 250, 400, 700 nodes per rung, and the two rates it is almost absent from are the two at which a rung is no longer than the window.
Fig. 5 A grid elsewhere in this collection printed as shares, where a cell at the top of its range is visible as one.

That is a small lesson with a specific fix: print a bounded count against its bound. It costs nothing and it is the difference between a reader seeing a plateau and a reader seeing a wall.

Taking it off

The knob is the size of the disturbance. Each stem is jostled by a small random displacement at every organ, and the thread’s own figure is a quarter of a degree. Raise it and the two rules’ outcomes move closer together, so fewer seeds agree, so the cells come down.

At a quarter of a degree, across the six correlation lengths: five, six, six, six, six, four of six. Four middle cells pinned.

At half a degree: four, five, five, five, five, three. Not one cell pinned, the deeper rule ahead everywhere, and still no crossing.

At one degree: two, four, four, six, six, six. A crossing — the deeper rule takes two seeds of six on white noise, four at the short correlations, and every one of them from a correlation of 0.8 upward.

The same comparison at three sizes of disturbance. Each row is one panel: how many of six seeds the deeper rule beats the shallower one on, at six correlation lengths, on the lattice at a rise of 0.013. At the size of jostle the thread used, cells across the middle sit at six of six — the largest number the panel can print — so no feature could have appeared there whatever the lattice did, and the reading that this rise has no corner was a reading of that ceiling. Raised, the cells come down, and at the largest disturbance the panel shows a clean crossing: the deeper rule loses at white noise and takes every seed once the disturbance remembers itself for a few organs. The corner is there, and the flat middle was the instrument.
Fig. 6 The three panels again, read as the sequence they are: pinned, unpinned, and crossing.

So the corner is there at the middle rise, and it was hidden by a comparison with no room above it.

The check the lift needed

A disturbance large enough to bring the cells down is a disturbance large enough to destroy what the panel is about. If a stem jostled at a degree is no longer counted at five and eight spirals, then a crossing found there is a crossing on some other arrangement and says nothing about this one.

So every stem in every cell is counted — from the point positions, by machinery that is never shown a divergence angle. At every amplitude up to a degree, at every correlation length, at every seed, both rules’ stems return the pair the rise carries.

Two readings of what a stem is, and one of them is wrong. The same runs read twice. On the left is the pair counted from the point positions by machinery that is never shown a divergence angle; on the right is the pair read from the divergence sequence, which is the column the depth thread has been quoting. Every exponent's stems are counted at 5/8 spirals from the points. Read from the angles, the deepest rule's stems come back as something else at every seed, and a pair with a 8 replaced in it is not a near miss but a family whose step is several times as long. The settled divergence moves by 0.118 degrees across all five exponents, so the rules are on one lattice and the disagreement is a defect in one of the two instruments.
Fig. 7 The check, and a second result that came out of it: two ways of reading what a stem is, which disagree on one rule.

That check is not a formality and it did not come back clean everywhere. Pushed to two degrees the stems stop being readable at all, which is why the lift stops at one. And the check turned up a disagreement between two ways of naming a pair that is a separate result and a more uncomfortable one.

A lattice or a wreck, with nothing in between. Every run in the sweep, at both kinds of noise. The line at 6° is where a run stops being counted as a lattice, and nothing lands near it: the intact runs reach 1.97° and the destroyed ones start at 8.06°, a factor of 4.1 away.
Fig. 8 What a larger jostle does to a stem’s scatter, which is the quantity the lift is trading against the lattice.

What the new panel says, and what it does not

The crossing sits between no memory at all and a memory of about one and a half organs. That is short, and it is roughly where the crossings at the other two rises sat — which is the point of the exercise, since the middle rise was the one said to be different.

The deeper rule against the disturbance's memory, at three contact scales. How many of six seeds agree that the deeper rule passed more drift, swept across the correlation length of the disturbance, at three rises. The rule, the amplitude and the run length are identical in every panel; only the rise differs, and with it the contact numbers — 3 and 5, then 5 and 8, then 8 and 13. The shapes are not the same: 3/5 is crossing, 5/8 is no corner, 8/13 is crossing. A corner that sat at a fixed number of organs would look the same in all three, and it does not.
Fig. 9 The corners at the other rises, which the middle rise now joins rather than contradicting.

But the new panel has a ceiling of its own, at the other end. At one degree the three long-correlation cells read six, six, six. So the lift has not removed the ceiling; it has moved it, from the middle of the range to the top.

How many seeds the deeper rule takes, at three disturbances. The same six correlation lengths across the bottom, the share of six seeds the deeper rule wins up the side, one line per size of jostle. The top line is pinned against the ceiling across the middle of the range, which is the shape of a comparison that cannot report anything. As the disturbance grows the line comes down and tilts, and at the largest one it crosses the halfway mark between the third and fourth correlations. The ceiling was not evidence that this lattice has no corner; it was evidence that the panel had no room.
Fig. 10 The old panel and the new one, with the ceiling in a different place on each.

That is what a crossing looks like when one side of it is a rout, and it means the same caution applies to the new reading. Nothing about the shape past the crossing can be read: whether the deeper rule’s advantage keeps growing with the correlation length, whether it plateaus, whether it eventually falls back — none of that is visible in three cells all reading six of six.

A memory manufactures nothing. The largest comb mean found in a kinematic lattice whose azimuth errors are an AR(1) process, against the coefficient of that process, over eight seeds at each point. The dashed line is where the rule's own stems sit, at 0.64; the shaded strip is three sampling bands. Every point is inside the strip — 0.028, 0.026, 0.022, 0.014, 0.015 at ρ = 0.3, 0.5, 0.7, 0.9, 0.97 — and the readout returns nothing on 40 runs out of 40. A correlated error is not a periodic one.
Fig. 11 The correlation lengths the panel sweeps, whose upper end is where the new ceiling is.

Why the panel is built this way at all

It is worth defending the design before criticising it further, because the paired count is not a careless choice.

The obvious alternative is to compare the two rules’ means — how much drift each let through, averaged over seeds. That was tried and it does not work here, because the seed-to-seed spread inside one cell is a factor of one and a half to two and a half, which is larger than the difference between the two rules. A comparison of two means with that much spread is a comparison of two numbers that overlap.

Three kinds of noise, matched at 0.75° of divergence scatter. The amplitudes differ — field 0.0056 (fraction of the barrier), jostle 0.15 (degrees of azimuth), placement 0.18 (degrees of azimuth) — and are in different units, so they cannot be compared directly. What can be compared is what they produce, and matched here they are within 27% of one another. Everything a finished pattern records about its noise is shared between the three.
Fig. 12 The spread the paired design exists to get around: two settings whose distributions overlap while their pairs do not.

Pairing by seed removes the spread, because the same disturbance drives both rules and what is counted is which of the two handled it better. That is a real improvement and it is why the design was adopted.

What pairing costs is resolution at the extremes. A paired count over six seeds has seven possible values, and two of them are walls. The fix is not to abandon pairing but to keep the cells away from the walls — which is what the amplitude knob does, and which nobody thought to check because the numbers looked like a result.

Every open question here needs under 28 specimens. The sample size at which each comparison reaches 80 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.
Fig. 13 The general arithmetic of how many runs a comparison needs, which says nothing about where its ceiling is.

More seeds, which is the other fix

Six seeds is the reason the ceiling is at six. Twelve seeds would put it at twelve and would resolve a cell at six of six into anything from seven to twelve of twelve, and it would cost twice as much.

That is worth stating because it is the fix a reader will think of first, and it is a worse fix than the amplitude for two reasons.

It does not remove a ceiling; it raises one. A rule that beats its rival on every seed will beat it on twelve as easily as on six, and the cells that are pinned here would very likely be pinned there.

A lattice survives about 1.8° of scatter, whichever way the noise arrives. The largest divergence scatter at which a run is still a lattice, from each kind of noise at the largest amplitude that leaves one. The two routes share no code below the placement rule: one displaces the node after the choice, the other perturbs the energy the choice is made over. They agree to 0.33°.
Fig. 14 A comparison at ten runs rather than three, which sharpens a distribution and does not move a wall.

And it is the more expensive of the two. Doubling the seeds doubles every cell of every panel; changing the amplitude costs the same number of stems as before. Where a knob and a sample size would both work, the knob is the one to reach for first — and where the knob is the thing the panel is about, as it is here, it is also the more informative.

What a lattice survives depends on the colour of the disturbance, sixfold. five stems for each of seven disturbances at each of six displacements, every stream normalised by its own measured spread so that a displacement of half a degree is half a degree in every row. A filled mark is a stem that still has a lattice — a divergence scatter under 2° — and an open one is a stem that does not. Independent errors survive to 0.5°; errors that remember the last one to 1.5°; errors inherited from the contact neighbours only to 0.25°. The number beside each row is the scatter a protractor would record where the lattice is standing, and it is the quantity that explains the table: what destroys a lattice is not how far an organ moves, but how far it moves relative to the organs it is placed against.
Fig. 15 The amplitude as a subject rather than as a dial, which is how the rest of this thread has used it.

What this does to the thread’s earlier readings

Two of them need re-reading and one does not.

The corner at the coarse and fine rises was read off panels whose cells were not all pinned — the crossing is visible because cells on both sides of it are away from the walls — so those readings stand.

The reading that the middle rise has no corner does not stand, and it is withdrawn here.

The deeper rule against the disturbance's memory, at three contact scales. How many of six seeds agree that the deeper rule passed more drift, swept across the correlation length of the disturbance, at three rises. The rule, the amplitude and the run length are identical in every panel; only the rise differs, and with it the contact numbers — 3 and 5, then 5 and 8, then 8 and 13. The shapes are not the same: 3/5 is crossing, 5/8 is no corner, 8/13 is crossing. A corner that sat at a fixed number of organs would look the same in all three, and it does not.
Fig. 16 The three rises as the thread reported them, one of which was reading a ceiling.

And the conclusion drawn from all three together — that the corner is not a fixed number of organs, because it moves and at one rise does not exist — needs its second clause removed. It still moves. It exists at all three rises measured, once the middle one is measured with room to report.

The deeper rule against the disturbance's memory, at three contact scales. How many of six seeds agree that the deeper rule passed more drift, swept across the correlation length of the disturbance, at three rises. The rule, the amplitude and the run length are identical in every panel; only the rise differs, and with it the contact numbers — 3 and 5, then 5 and 8, then 8 and 13. The shapes are not the same: 3/5 is crossing, 5/8 is no corner, 8/13 is crossing. A corner that sat at a fixed number of organs would look the same in all three, and it does not.
Fig. 17 The same comparison with the middle rise corrected, where the corner is present at all three.

That is a partial rescue of the contact-scale reading rather than a confirmation of it. Three crossings at three rises, all short, is consistent with a corner that tracks nothing much — and the thread’s original question, whether the corner is the contact scale or simply any memory at all, is where it was. What has changed is that the evidence against a fixed corner is now one observation weaker: the sweep that refuted it rested partly on a rise where no corner appeared, and that rise now has one.

The wander climbs because its denominator falls. Three quantities across the same sweep of the rule's depth, each divided by its own value for the shallowest rule so that they share an axis. The wander rises by a factor of 3.7 as the neighbourhood goes from 3 organs to 182. The scatter between neighbouring divergences falls by 1.57, because that is the part of a disturbance a placement rule corrects and a deeper rule corrects it better. What is left — how many degrees of slow drift actually reach the divergences — changes by 1.20, from 1.52 to 1.82 degrees. The rule barely filters a drift at any depth.
Fig. 18 The quantity the corner is read out of, whose denominator moves with the exponent and hides the crossing in a ratio.

The cost of finding this out

Three panels rather than one, which is three times six correlation lengths times six seeds times two rules — two hundred and sixteen stems of nine hundred organs each. On one core that is an hour or so, and every one of the stems is cached against the code that produced it, so re-reading the panels costs nothing.

Against that, the third exponent the plan called for would have been another panel of the same size, and it would have been pinned in the same places. The lift is not merely the correct order of operations; it is also the cheaper one, because that is what says whether the next panel can report anything.

The same comparison at three sizes of disturbance. Each row is one panel: how many of six seeds the deeper rule beats the shallower one on, at six correlation lengths, on the lattice at a rise of 0.013. At the size of jostle the thread used, cells across the middle sit at six of six — the largest number the panel can print — so no feature could have appeared there whatever the lattice did, and the reading that this rise has no corner was a reading of that ceiling. Raised, the cells come down, and at the largest disturbance the panel shows a clean crossing: the deeper rule loses at white noise and takes every seed once the disturbance remembers itself for a few organs. The corner is there, and the flat middle was the instrument.
Fig. 19 The panel as the thread had it, whose middle four cells were the whole of the reading being drawn.

The general version is worth stating because it applies to any bounded statistic: before spending on more cells, check that the cells already paid for are able to move. A cell that cannot move is a cell that will not answer a new question either.

Where each kind's lattice gives way. The largest amplitude at which every run still has a lattice, and the scatter it produces there. The amplitudes are incomparable — field 0.015 (fraction of the barrier), jostle 1 (degrees of azimuth), placement 0.8 (degrees of azimuth) — and the scatters agree to 19%. The boundary belongs to the pattern rather than to the disturbance: a lattice fails at about a degree and a half of scatter, and which of three mechanisms produced it does not move where.
Fig. 20 Another bounded comparison in this collection, whose cells were checked against their range when it was built.

The reading this replaces, in the ledger

It is worth being explicit about what is withdrawn, because this collection records refutations of its own claims in the same place as everything else and a vague withdrawal is worse than none.

The claim was: at a rise of 0.013 the deeper rule wins at every correlation length, so the comparison has no corner there. The first half is what the panel printed at a jostle of a quarter of a degree; the second half is the inference, and it is the inference that fails. At a jostle of one degree the same comparison at the same rise on the same lattice has a crossing between the first and second correlation lengths.

The same comparison at three sizes of disturbance. Each row is one panel: how many of six seeds the deeper rule beats the shallower one on, at six correlation lengths, on the lattice at a rise of 0.013. At the size of jostle the thread used, cells across the middle sit at six of six — the largest number the panel can print — so no feature could have appeared there whatever the lattice did, and the reading that this rise has no corner was a reading of that ceiling. Raised, the cells come down, and at the largest disturbance the panel shows a clean crossing: the deeper rule loses at white noise and takes every seed once the disturbance remembers itself for a few organs. The corner is there, and the flat middle was the instrument.
Fig. 21 Both panels, which are the same comparison at two settings of a knob nobody was varying.

What is not withdrawn is anything about the coarse or fine rises, or about the drift-through quantity, or about the neighbourhood depths the exponents correspond to. Those were measured on cells that could move.

The ratio follows the disturbance, not the rule. The ratio of the second comb to the main comb on stems grown by the placement rule and jostled by seven different disturbances, all at 0.25° of displacement per organ and all on the same rule. Independent errors and errors with a memory return 0.76–0.81, which is the value this site measured for the rule. A periodicity at the smaller parastichy number takes it down to 0.45; errors inherited from the contact neighbours take it up to 1.09, most of the way to the 1.24 a transported disturbance gives with no rule in it at all. So the quantity separates arrangements by how their errors are related, not by whether anything computed the positions.
Fig. 22 The quantity underneath all of it, whose own measurement is unaffected by where a paired count’s walls are.

What is left

The third exponent, now that it would mean something. With the cells off the ceiling at one degree, a comparison between other pairs of exponents can report where their crossings sit, and whether a crossing’s position depends on how far apart the two neighbourhoods are.

Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by three different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.
Fig. 23 The exponents and their neighbourhood depths, which is the quantity a third panel would sweep.

And the amplitude itself as a subject. Everything above treats the jostle as a dial for getting a readable panel, which is a use rather than a question. Whether the crossing’s position moves with the amplitude is a two-panel measurement that nobody has made, and it bears directly on whether the corner is a property of the lattice or of the size of the kick. This collection has measured what an amplitude does to a stem’s own tolerance and never what it does to a crossing.

The awkward possibility is worth naming rather than left to a reader. If the crossing moves with the amplitude, then the corner is a property of the disturbance rather than of the arrangement, and the whole thread’s central quantity is about the kick and not about the lattice. Two panels would settle it, and they have not been run because the question only became askable once the cells came off the ceiling.

Placements that went to a different minimum, per thousand. The rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.
Fig. 24 What a larger kick does elsewhere in this collection, where it is a subject rather than an instrument.

Where this goes next

Where the deeper rule changes hands, and where it never does. One row per rise, with the contact numbers on the left and what the comparison does on the right. At the coarsest the deeper rule wins at both ends and loses in the middle; at the middle rise it wins throughout; at the finest it crosses once, from losing to winning. The contact scale runs over a factor of 2.6 across these three rises and the behaviour is not a translation of one curve — it is three different curves. That refutes a corner fixed at a short correlation, and it does not by itself establish one that tracks the contacts. The panel with no corner Sweeping the rise gave three shapes where one was expected, and the middle one is the informative panel: at the 5/8 contact scale the deeper rule wins at every correlation and there is no crossing to locate. That is either a fact about the lattice or a fact about the pair of exponents, and one measurement separates them. The deeper rule against the disturbance's memory, at three contact scales. How many of six seeds agree that the deeper rule passed more drift, swept across the correlation length of the disturbance, at three rises. The rule, the amplitude and the run length are identical in every panel; only the rise differs, and with it the contact numbers — 3 and 5, then 5 and 8, then 8 and 13. The shapes are not the same: 3/5 is crossing, 5/8 is no corner, 8/13 is crossing. A corner that sat at a fixed number of organs would look the same in all three, and it does not. The corner moves with the rise The corner was either the contact scale or simply any memory at all, and nothing in the thread had ever varied the rise — the one knob that moves the contact numbers while leaving the rule, the amplitude and the run length alone. Swept over it, the comparison does not keep its shape. Two readings of what a stem is, and one of them is wrong. The same runs read twice. On the left is the pair counted from the point positions by machinery that is never shown a divergence angle; on the right is the pair read from the divergence sequence, which is the column the depth thread has been quoting. Every exponent's stems are counted at 5/8 spirals from the points. Read from the angles, the deepest rule's stems come back as something else at every seed, and a pair with a 8 replaced in it is not a near miss but a family whose step is several times as long. The settled divergence moves by 0.118 degrees across all five exponents, so the rules are on one lattice and the disagreement is a defect in one of the two instruments. The pair read from the angles Checking that a jostled stem is still the lattice its panel is about turned up a disagreement. Counted from the point positions every rule's stems return five and eight spirals; read from the divergence sequence, the deepest rule's stems come back as five and seven at every seed. The points are right, and this site's founding rule is why.

What links here

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Essays that name at least two of the same things, and that neither author linked.

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ArtefactAutocorrelationCeiling effectClaim testingControlEnsembleExponentHonest limitsMeasurementNegative resultNeighbourhood depthNoiseNull modelResolution