Phyllotaxis
the spirals are counted, not admired
What a plant might be doing

A spread that grows with its window

A spread over a sample of a steady quantity does not depend on how big the sample is. These spreads triple when the window triples, on nearly every row of the census, which means the classes this thread calls steady are sliding — slowly, and invisibly at any single window.
14 min read 24 figures State the claim, state the testCounted, not admiredOne parameter, one family

Worth reading first: The damage has a period · Two windows on one stem.

The quantity this thread calls a spread is the worst standard deviation within a residue class of a wrecked stem’s displacement profile, measured over the last hundred and twenty organs of the run. Under ten degrees and the row is periodic; over ten and it is not.

Read the same rows at sixty organs and at a hundred and eighty and the spread is very nearly proportional to the window. That is the whole finding and it is not what a noise level does.

It came out of moving an instrument setting nobody had moved, which is the cheapest thing this collection does and the one that has produced the most corrections.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 1 Every cut’s class spread at three windows, on axes where proportionality is a line through the origin.

The arithmetic that makes it a finding

A standard deviation over a sample of a steady quantity with noise on it is flat in the sample size. Read sixty draws or a hundred and eighty from the same distribution and the estimate wobbles but does not trend.

A standard deviation over a sample of a quantity that is sliding grows with the sample, because a longer stretch of a slope has a wider spread of values in it. For a linear slope it grows in exact proportion.

So the ratio of the widest window’s spread to the narrowest is the diagnostic. Flat says noise; three says a linear drift; anything much above three says something else.

A period of 5, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 1.62° — so within a class the displacement is a constant. two classes sit at the common level. The three that do not sit at 67.7° and -149.5°, equal and opposite to within 75.3 per cent, and they are neighbouring residues. The stem's own divergence is 136.78°, so an exception is one organ's step.
Fig. 2 The classes a profile folds into, whose spread over the window is the quantity in question.

What the ratio is

Three, on nearly every row that keeps one verdict at all three windows. g020 cut at offset 4 reads 0.8, 1.6 and 2.4 degrees. g013 at offset 4 reads 0.3, 0.7 and 1.0. l013 at offset 4 reads 3.0, 6.1 and 9.1. g005 at offset 7 reads 0.5, 1.0 and 1.6.

The measured ratios run 2.0 to 3.3 against a prediction of exactly 3.0. That is close enough to proportional that the alternative — a flat noise level — is not a live reading of it.

Three rows are excluded from that count because their verdict changes across the windows, and their ratios are 21, 42 and 41. Those are a different effect entirely and mixing them in would have made the population look ragged.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 3 The same rows at the longer run length, where the proportionality holds unchanged.

Which means the classes are drifting

The classes are not steady with noise on them. They are moving, slowly, along the run, and the spread over any window is a measurement of how far they move across that window.

That is a small thing per organ. A spread of 1.6 degrees over a hundred and twenty organs is a drift of about a hundredth of a degree an organ, which no reading of a single profile would show and which a spread over one window returns as a small number with no direction attached.

How far every organ moved, 4 places back at a rise of 0.013. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 3 organs it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. three of those levels sit together and two do not.
Fig. 4 A wrecked stem’s displacement profile, in which a hundredth of a degree an organ is invisible.

Why a spread hides a drift

Because a standard deviation throws away the order. Take a hundred and twenty numbers climbing steadily from one value to another and shuffle them: the standard deviation is identical and the drift is gone.

So the statistic the thread has been using cannot tell a drifting class from a noisy one at any single window. Reading it at two windows can, and that is the whole method here.

This is the same weakness a summary statistic showed on a crowding question, where a number that discards structure was carrying an ordering that belonged to the instrument.

Both statistics, on the same stems, at a rise of 0.005. Five seeded stems at each disturbance, held at a fixed rise. Bars are how many returned the pair the position counter finds; open portions are refusals. The pair comes out from 0.1 to 0.25, and across that whole range the lag-one correlation of the same sequences is -0.33, -0.58, -0.59 — decisive, negative and flat. There is no trade between the two: one stem supplies both. Below the window the sequence has locked onto the sampling grid and is a cycle rather than a sample; above it there is no lattice left, at 116° of scatter.
Fig. 5 Two statistics over the same numbers, one of which keeps the order and one of which does not.

What a drift would look like directly

The obvious alternative measurement is a slope: fit a line to each class against the organ index and report the gradient. That would name the drift rather than infer it.

It is not what was done here, for a stated reason. Changing the statistic changes every earlier number in the thread, and the point of this reading is to say what the existing statistic has been measuring rather than to replace it. A slope is the right next measurement and it is a different piece of work.

The same argument was made when a second statistic was brought to a crowding question: the new statistic answered the question and the old numbers stopped being comparable, which is a cost worth paying once and not repeatedly.

A deeper rule passes more of a drift, not less. The wander left in a stem's divergences, against how many organs its disturbance stays correlated over, for rules whose neighbourhoods run from 3 organs to 182. The prediction under test said a rule should pass a drift once the drift outlasts its neighbourhood, so the shallow rules should be the leaky ones and each line should turn where its own depth is crossed. Every line rises smoothly and the deepest rule is the highest of them at every correlation length — 82 against 22 at the longest drift. There is no crossover anywhere in the sweep.
Fig. 6 A drift measured directly against a drift inferred from a spread, which are two different readings.

What it does to the threshold

The line is at ten degrees and the quantity is proportional to the window, so the line means different things at different windows. At sixty organs it admits a drift about three times steeper than at a hundred and eighty.

That is not a defect in the definition. This class moves less than ten degrees over the stretch that is read is a perfectly good property and it is what the threshold tests. It is not the window-free property the word periodic suggests, and the collection has been using the word without the qualification.

Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked.
Fig. 7 The classification the threshold produces at each window, which shifts by three rows.

The gap survives, which is the reassuring part

The threshold was chosen because the measured spreads separate into two populations with an empty gap between them: 0.12 to 6.09 degrees on the periodic rows and 10.3 and upwards on the rest.

At sixty organs the same rows read 0.1 to 3.0 and 4.0 upwards; at a hundred and eighty, 0.2 to 9.1 and 12.2 upwards. The gap is there at all three windows, so the line is sitting in a gap rather than cutting a continuum, whatever the window.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 8 The spreads the classification is drawn on, and the gap the threshold sits in.

And the two populations scale differently

That is the second half of it. The rows that are periodic scale by about three. The rows that are not periodic do not scale at all: l020 at offset 4 reads 12.8, 156.9 and 143.7 degrees, and at offset 5 it reads 149.6, 152.7 and 128.6.

Those are not proportional to anything. A class that is not periodic is wandering rather than sliding, and a wandering class’s spread saturates once the window is long enough to contain the wandering.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 9 Every row at every window, where the periodic rows climb in proportion and the rest do not.

So the two populations differ in kind

Which is a stronger statement than the threshold makes. The threshold says one group’s spread is small and the other’s is large. The three windows say one group’s spread is proportional to the window and the other’s is independent of it.

Two quantities that respond differently to an instrument setting are two quantities. That distinction is available for nothing once the readings are taken and it does not appear in the classification at all.

How steady each class is, over 300 organs and over 600. Each mark is one wrecked cut, placed across at the widest spread found inside any one of its residue classes over the shorter run and up at the same reading over the longer one. A mark on the diagonal is a row the two lengths agree about. The rules are the 10 degrees that separates a profile called periodic from one that is not: three rows fall in different quadrants at the two lengths, two of them becoming periodic and one ceasing to be. The gap between the two groups narrows from 1.69 times to 1.27.
Fig. 10 The same quantity read at two settings, where a difference in kind shows as a difference in slope.

A better test than the threshold

It follows that the ratio is a better classifier than the value. A row whose spread triples with the window is drifting; a row whose spread is flat in the window is wandering; and the two need no threshold at all to be told apart.

That is not adopted here for the same reason the slope is not. It is written down because it is available and because a classification that needs no line drawn in it is worth more than one that needs a line in a gap.

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. 11 Candidate rules scored against each other, which is how this collection settles which reading to keep.

What drifts, physically

A class is the set of organs at one residue modulo the surviving lag, and its level is how far those organs sit from where the control put them. A drifting class means that displacement is slowly growing or shrinking as the run goes on.

That is not a surprise once said. The cut stem and its control are two runs of the same rule from slightly different states, and nothing guarantees their difference is constant — only that it settles into a pattern at one lag, which is what the census measures.

What a cut moves, organ by organA stem counted at 5 and 8 spirals with the organ five places back from the tip removed, compared against a control that shares its history to the last digit. Each mark is one organ placed after the cut and how far its azimuth ended up from where the control put the same organ. The quantity folds at half a turn, so 152 degrees is near the largest displacement there is; and it does not decay with height, which is what a stem that never repairs means. There is therefore no organ that the cut disturbed most in any useful sense, and a reading that needs one has nowhere to stand.-200-100010020010203040organs above the removed oneazimuth moved (°)the removed organ was 5 places back5/8 at 0.013 · offset 5 · survivor 5generated from a stated rule, not drawn to look right
Fig. 12 How far each organ above a hole sits from where the control put it, which is what a class averages.

Whether the drift ever stops

A drift of a hundredth of a degree an organ over three hundred organs is three degrees. Over six hundred it would be six, if it kept going at the same rate.

It does not. At the longer run length the same rows read the same spreads at each window — 0.8, 1.6 and 2.4 on g020 at both lengths — so the drift is a property of the window rather than something accumulating without bound. Whatever slides, slides locally.

One cut's profile over 600 organs, with both onsets marked. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control sharing its history. The two vertical rules are where the pattern is said to start: the left one is what a 300-organ run reports and the right one is what this 600-organ run reports. They are not both present. The onset is measured against the levels the classes hold at the end of the run, so a profile that drifts slowly is compared against different levels at each length and the reading moves with the length.
Fig. 13 One row’s profile at two run lengths, which is where a drift that accumulated would show.

Which narrows what the drift can be

That is worth pausing on because it rules something out. If the classes were separating steadily — the cut stem’s arrangement slowly walking away from the control’s — the spread at a fixed window would grow as the run got longer.

It does not. So the classes are not separating; the profile is locally sloped in a way that is the same everywhere along it, which is closer to a systematic offset between neighbouring organs than to a divergence between two runs.

How steady each class is, over 300 organs and over 600. Each mark is one wrecked cut, placed across at the widest spread found inside any one of its residue classes over the shorter run and up at the same reading over the longer one. A mark on the diagonal is a row the two lengths agree about. The rules are the 10 degrees that separates a profile called periodic from one that is not: three rows fall in different quadrants at the two lengths, two of them becoming periodic and one ceasing to be. The gap between the two groups narrows from 1.69 times to 1.27.
Fig. 14 The same spread at two run lengths, which is flat and is what rules out an accumulating separation.

The rows where the ratio is off

Not every row gives three. A handful come back at 2.0 to 2.5, which is under proportional, and those are rows whose spreads are smallest in absolute terms — 0.2, 0.3 and 0.4 degrees.

At that size the reading is close to whatever floor the azimuth grid imposes: the grid is 1,536 samples of the circle, a quarter of a degree a step, so a spread of a fifth of a degree is under one grid step and is being quantised rather than measured.

What the finer grid does to the rises already published. The two rises this site has argued from and the one it published as having no answer, each read at both azimuth grids, five stems apiece. A filled mark agrees with the position counter, a half mark contradicts it, an open mark is a refusal. At 0.013 and 0.005 the readings are identical at both grids, so nothing already written depends on the sample count. At 0.008 they are not: 384 azimuths gives 5/8, 5/8 and 1152 gives 8/13, 8/13, against a counter that says 5/8. That rise was chosen in the earlier work because the three shortest lattice offsets there are within a fifth of each other, and a stem with no answer answering differently on a different grid is the object behaving as it was said to.
Fig. 15 The grid the azimuths are placed on, which is the floor any small spread is measured against.

So the smallest spreads are not measurements

That is worth stating rather than leaving in a footnote. A spread of 0.2 degrees over a hundred and twenty organs on a grid whose step is 0.234 degrees is a statement about the grid.

It does not affect the classification, because every one of those rows is periodic at every window by a wide margin. It does mean the proportionality should be read off the larger spreads, and it is: the rows at 3.0, 6.1 and 9.1 degrees are several grid steps at every window.

A block that was the grid rounding a constant is the standing example here of a quantity read below the grid’s own step, and it is why every small number in this collection is checked against the step before it is believed.

The 13/21 rung, at two azimuth grids. Five stems at each of three disturbances, all at a rise of 0.0019, read through a lag window of 45 from a seed of 40 nodes. A filled mark is a stem that returned 13/21, which is what the position counter finds in it; an open mark is a refusal. The only difference between the two rows is how many azimuths the placement rule samples when it takes its minimum: 384, which every run on this site has used since the earlier work, against 1152. At the coarse grid the rule locks onto its own sample points and the readout refuses 14 of 15 stems; at the fine one it reads all 15. The ceiling was a parameter of the program.
Fig. 16 What a finer grid would do to a quantity sitting near the grid’s own step.

What this does not touch

The classification’s content is unchanged. Twenty-five of thirty rows are periodic at the working window, the same twenty-five as before, and the five that are not are the same five.

What changes is the sentence that goes with the number. It was these classes are steady and it is now these classes move less than ten degrees across the stretch that is read, and they are moving.

Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked.
Fig. 17 The classification at the longer run length, whose content this reading leaves alone.

Where the same shape has appeared before

This is the third quantity in the collection to turn out to be partly about its own instrument. A residual that looked ordered belonged to the window it was measured over; a fragility belonged to a reading rather than to a lattice.

Both of those were found the same way: read the same object through a different setting and compare. Neither needed a new measurement, only a second value of an old one.

Two windows on a shoot at 400 nodes per rung. A stem grown at 400 nodes to the rung with a disturbance of 0.01, its rise falling from 0.4 to 0.004 over 1914 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads 8/10; the lower reads 8/10. The verdict is agree, and the window holds 0.63 of a rung.
Fig. 18 The same quantity read through two settings of one instrument, which is how each of these was caught.

The drift is not the same at every lag

One reading the three windows make available and nobody had asked for. Sorting the rows by the lag they kept, the rows at a lag of 8 have the smallest spreads — 0.4 to 1.6 degrees at the working window — and the rows at a lag of 5 the largest among the periodic ones.

That is the direction a sampling argument would predict rather than a physical one: a class at a lag of 8 has fifteen samples in a hundred and twenty organs and a class at a lag of 4 has thirty. More samples of the same slope give the same spread, so the ordering is not about the count, and no account of it is offered.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 19 The census’s spreads sorted by the lag each cut kept, which is a reading the classification does not use.

What a proportional spread means for comparing rows

It means two rows read at the same window are comparable and two rows read at different windows are not. Every row in the thread is read at the same window, so nothing published is affected.

It also means a spread should not be quoted without its window, in the way an angle should not be quoted without saying whether it is folded. Both are conventions that are invisible while only one setting is ever used, and both become mistakes the moment a second appears.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 20 The same rows at the longer run length, where the same convention applies and the numbers differ.

The cheapest version of this check

Two windows, not three. The third adds confidence about linearity and the first two are enough to tell a drift from a noise level: if the spread roughly doubles when the window doubles, it is a drift.

That costs one extra read of runs already grown. It is the same shape as the run-length check — one alternative value, run once, compared — and it is the habit this collection keeps having to relearn.

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. 21 The same object read twice through one setting, which is the whole method behind every correction of this kind.

What a second window costs and buys

Four minutes, over the whole census, because the stems are already grown and a window is a read rather than a run.

What it bought was a change in what the word periodic means in this collection, a distinction between two populations that the threshold does not make, and three rows whose verdict turned out to be the window’s. That is a good return on a setting that had a justification attached to it and had never been varied.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 22 The reading a second window buys, which is a slope rather than a number.

What a reader should carry

That a spread proportional to its window is measuring a drift, and that these are. The classes a wrecked stem settles into are sliding at about a hundredth of a degree an organ, which no single reading would show.

And that the two populations the classification separates differ in more than size: one scales with the window and the other does not, which is a cleaner distinction than the threshold and costs nothing once the readings exist.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 23 The rows that keep one verdict, climbing in proportion to the window they are read over.

What the picture at the top shows

One line per wrecked cut, from its spread at sixty organs through a hundred and twenty to a hundred and eighty. The dashed guide is exact proportionality, drawn from the mean of the narrow-window readings.

Nearly every line runs parallel to it. A flat set of lines would have said the classes are steady with noise on them; a set climbing faster than the guide would have said something was accumulating. They climb with it.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 24 Every row, including the three whose lines climb far faster than the guide and are a different effect.

The one line

Class spreads are proportional to the reading window — 0.8, 1.6 and 2.4 degrees at sixty, a hundred and twenty and a hundred and eighty organs on one row, and the same triple on nearly every row that keeps a verdict.

A spread that scales with its sample is measuring a drift, so the classes this thread calls steady are sliding at about a hundredth of a degree an organ, and the ten-degree threshold is a line on a quantity that depends on how much is read.

Where this goes next

Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked. The window nobody moved Three instrument settings sit between the ablation census and every statement it makes. Two have been varied and both decided answers. The third is a hundred and twenty organs at the top of a run, it has never been moved, and moving it changes the verdict on three rows. Where each window begins, against where the pattern begins. The run is drawn left to right, one bar per cut. The pale stretch is the disturbance still healing, up to the organ from which the profile holds its levels for the rest of the run; the dark stretch is the pattern repeating. The marks are the organ each of the three windows starts at, since a window is the last N organs of a run. On these rows the widest window starts inside the pale stretch, so it is reading the healing and calling it the pattern — which is why widening the window takes them off the periodic list. Three rows a window moves Three of the census's thirty cuts are periodic when a hundred and twenty organs are read and not when a hundred and eighty are. Their spreads do not grow in proportion to the window, they grow by twenty and forty times, and the reason is that a window is the tail of a run. How far every organ moved, 6 places back at a rise of 0.01. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 18 organs it settles into a repeating pattern of eight levels, one per residue class modulo 8, which is the lag whose hop this stem kept. six of those levels sit together and two do not. The damage has a period Every wrecked stem in the census has had two numbers read out of its displacement profile and the profile itself read out of none of them. Folded on the lag the stem kept, twenty-five of the thirty are constant inside each residue class to between 0.12° and 6.09°.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

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

Named objects

A flat tag is an object no other essay names yet.

ArtefactClaim testingDamage profileDriftHonest limitsInstrument settingMeasurement errorNoise floorPeriodicityReading windowResidue classSummary statistic