A known coast charges an office's habit, and a bending one takes it out

The spread between hands charges the mean, and an office’s habit hides in the sheet traced one compiled coast many times, by several hands, and found the right way to charge the pooled mean: the scatter of the hands’ means about each other, divided by the number of hands. That charge is the standard error of a mean taken over groups, it never needs to know how much of a hand’s drift is habit, and from three hands on it read the pooled mean’s true drift to within a fifth on every sheet. It failed in one way, and the failure was exact. A habit shared by every hand in the office is in every tracing, cancels from every difference between tracings, and is indistinguishable from the coast itself: tracings of a sheet with an office habit are tracings of a different sheet without one.

The essay ended on the repair the arithmetic allows. Something that is not a tracing of this sheet has to supply the office’s share. A control sheet — a coast known independently, traced by the same office in its ordinary way — shows the office’s habit as the difference between the office’s pooled tracing and the truth. What was left open was whether a few such sheets say enough, and whether what they say transfers: a habit learned on one coast’s shapes, carried to a coast of other shapes.

The second question turns out to decide the first, and it has two answers, because there are two things an office’s habit can be.

Two offices, and four coasts #

The model is the earlier one. Four hands each trace the sheet four times. A tracing’s error is the sum of three parts with one tracing’s variance between them: a habit the whole office shares, carrying a quarter; each hand’s own habit, carrying another quarter; and a wander drawn fresh for every tracing, carrying the half that remains. The counted sheet is the compiled coast with a copied stretch eased into it, whose seams the count is trying to find.

What changes is the office’s quarter, which is now modelled two ways.

In the first office it is what it was before: a drift along the coast, with the same correlation length as the hands’ own wander, drawn afresh for every sheet the office traces. It belongs to the sheet as much as to the office. Nothing about its shape on one coast says anything about its shape on another; only its size is the office’s.

In the second office the habit answers to the coast. Every hand in it draws a headland a little further out to sea than it is and a bay a little further in, by a fixed amount for each unit of curvature above or below the coast’s average. That is a habit with a known history. A draughtsman generalising a coast for a smaller scale is taught to keep its character, and exaggerating the features that carry it is one of the standard operators of generalisation, beside the simplification a tolerance is a promise about the picture priced. An office trained to do it does it everywhere, and does it by the coast’s shape. Its constant — how far out per unit of curvature — belongs to the office and is the same on every sheet. Its size on any one sheet does not: it is the constant times how much the coast’s curvature varies.

The figure shows why that matters. The counted coast is long and thin, and nearly all of its variation in curvature is at its two tips; the habit throws them outward and leaves the long flanks almost alone. Of the three controls, the thin ellipse varies in curvature about as much (15.6 against 16.9, in units of each coast’s size), the nearly round one hardly at all (0.1), and the coast of five headlands nearly twice as much (29.8). The same office, drawing all four, puts a habit of very different size on each.

One control restores a drifting habit’s charge #

A control sheet is read with the two quantities the counted sheet already supplies and one it cannot. The office traces the control four times by each of four hands and pools the sixteen. The pooled tracing’s mean-square departure from the known coast holds everything the pooled mean carries: the office’s share whole, each hand’s habit divided by four, the wander divided by sixteen. The spread between the four hands’ means on the control, divided by four, holds the last two and not the first, exactly as it did on the counted sheet. Subtract one from the other and what is left estimates the office’s share.

For the drifting office that is the whole repair. One control reads the share as 0.255 in the median, against the 0.25 drawn, and the middle eight offices in ten between 0.17 and 0.37. Added to the counted sheet’s own between-hand charge, it raises the charge from 0.53 of the pooled mean’s true drift to 1.01, and within a fifth of the truth for 94 offices in a hundred. The earlier essay found the between-hand charge alone falling to 0.52 of the truth at four hands of four, and no arrangement of hands or tracings on one sheet could raise it.

More controls narrow the share — 0.21 to 0.31 at four, 0.21 to 0.29 at eight — and leave the charge where it was, because the charge’s remaining scatter is the counted sheet’s own: four hands’ means give the between-hand spread three degrees of freedom at each independent stretch of the coast, and that scatter is in every office’s charge however well its share is known. One control does what eight do.

The coast a control is drawn on does not matter to this office at all. The thin ellipse, the round one and the coast of headlands read the same share to the third decimal, sheet for sheet, because a drift’s size does not depend on what it drifts along. That is the one thing about this office that transfers, and it transfers whole.

The slope tells the two offices apart #

The two offices would leave identical tracings of any one sheet, as the earlier essay found of any office and any sheet. A control does more than restore a charge: it shows the habit itself, and the habit’s shape on a known coast distinguishes them.

Read the control’s pooled departure from the truth at every point, take its outward part — the component across the coast, along which a headland is pushed out — and set it against how much more or less curved than its average the coast is there. Fit a line through the origin. For the office whose habit answers to curvature, the slope is the office’s constant, and one thin control reads it to within twelve per cent either side for eight offices in ten, four to within six. For the drifting office the slope is nothing: a drift along the coast has no reason to point outward at the tips more than at the flanks, and the slope reads zero, within about a quarter of the overshooting office’s constant from one thin control and a tenth from four. The coast of five headlands, which varies most in curvature, reads both more tightly still.

So a few control sheets answer a question the counted sheet could not even pose. An office whose departures follow its coasts’ curvature, control after control, has a habit of shape; an office whose departures do not has a habit that is particular to each sheet. From a single thin control the two offices’ ranges do not overlap, and a digitiser holding one can tell which office traced the counted sheet.

The round control cannot. Its curvature barely varies, so the outward departure has nothing to be set against: the slope it reads is the hands’ and the wander’s own scatter divided by a very small spread, ranging from minus three and a half to three and a half times the true constant from one control, and still from minus one to nearly two from four. A circle is the coast on which every point bends alike, and on it an office that overshoots headlands has no headlands to overshoot.

A size transfers as the coasts’ curvature does #

For the overshooting office the first repair — read the habit’s size on a control and add it — still works when the control bends as the counted coast does, and stops working when it does not.

The habit a control carries is the office’s constant times the control’s spread of curvature. Read on four thin controls, whose spread is 92 per cent of the counted coast’s, the share comes out as 0.235 for the 0.25 the counted sheet carries, and the charge as 0.98 of the pooled mean’s true drift — as good as the drifting office’s. Read on four round controls, whose curvature varies by one per cent of the counted coast’s, the share is 0.002: the habit is barely there to be read, and the charge of 0.54 is no better than charging from the hands alone. Read on four coasts of five headlands, with 176 per cent of the counted coast’s variation, the share is 0.439 and the charge 1.26 — too high by a quarter, which is safer than too low and spends the count’s power on errors that are not there.

The three points lie on one curve, and the arithmetic of the earlier essay draws it. Between hands the counted sheet’s mean is charged at 0.094 of one tracing’s variance; it truly carries 0.344, the office’s 0.25 included. A share read on a coast whose curvature varies r times as much as the counted coast’s is 0.25 r, and the charge it gives is the square root of (0.094 + 0.25 r) over 0.344. A size transfers exactly where the shapes agree — the earlier essay’s guess, now with a number on it — and the number to agree on is not the coasts’ outline but how unevenly they bend.

A habit that answers to curvature can be taken out #

The slope does more than diagnose. A habit of shape, once its constant is known, is known everywhere: on the counted sheet it is the constant times the counted coast’s own departures from its average curvature, along its own outward normal, and it can be subtracted from the pooled mean before anything is counted. That is a different kind of repair from the drifting office’s. A charge admits the habit is in the mean and widens the allowance for it; a subtraction takes it out, and the mean then draws the coast where the coast is rather than where the office would put it.

The counted coast’s curvature is not known — it is the coast being measured — so it is read from the office’s own pooled tracing, smoothed over thirteen points to keep the wander’s small jerks out of a second difference. One thin control’s slope, applied that way, leaves 1.4 per cent of the habit in the mean; four leave 1.1. The floor of about one per cent is the smoothing, which rounds the counted coast’s two tips slightly and so reads their curvature slightly low; it is not the controls, since a coast of five headlands, whose slope is the best-read of all, reaches the same floor. Every repair here applied after that one starts from a mean that carries a hundredth of the office’s quarter.

The drifting office has no slope, so the same subtraction removes nothing from it, and that is harmless where the control bends: thin and five-headland controls read a slope of nearly zero and subtract nearly nothing. On round controls it is not harmless. A slope read there is mostly the scatter of the hands divided by a curvature that barely varies, and the counted coast’s tips, which do vary, carry it at full size. From one round control the drifting office’s mean, so corrected, carries five and a half times the variance its habit did; from eight, it still carries the habit and more than a quarter again. The overshooting office’s habit, corrected from round controls, keeps 89 per cent of itself from one and 11 from eight, and the fall is the slope’s scatter shrinking, not the habit being learned.

Subtract, then charge what is left #

The two repairs combine into one that does not need to know which office it is serving. Read the slope on the controls and subtract it from them as from the counted sheet. Whatever departure the controls still show, beyond their own between-hand charge, is the part of the habit the slope did not explain — the whole habit if it is a drift, almost none of it if it answers to curvature. Add that to the counted sheet’s between-hand charge.

From one thin control the result charges the drifting office’s corrected mean at 1.00 of its true drift and the overshooting office’s at 1.04, within a fifth of the truth for 93 and 88 offices in a hundred. A control of five headlands does as well. The overshooting office’s charge sits a few points high in the median, and further high than the drifting office’s at the top of its range; a few points of overcharge is the safe direction to err, and the counts below show it costing nothing.

Round controls fail the combined repair both ways, at 0.49 and 0.56 from one, still under nine tenths from four, and within a fifth of the truth for between a fifth and a little over half of offices. That is the cost of a control on which nothing bends: it cannot distinguish the offices, its slope is noise, and noise subtracted at the tips is added to the mean.

Four sheets counted #

The arithmetic says which repairs restore the charge; the count is where that has to show. Each of four sheets was traced four times by each of four hands, pooled, and counted exactly as the earlier essays counted their stacks: whitened at the drift’s own length, charged as stated, and offered up to six rigid seams and four eased ones, with the criterion a residual cannot count the pieces of a map introduced, which charges each piece for the numbers it spends.

The drifting office, charged between hands plus the share one thin control reads, counts two eased seams on every sheet — what was made. Its charges came out at 0.97, 1.27, 1.08 and 0.98 of the truth, and the strong seam’s half-width reads 18, 16, 16 and 16 against eighteen drawn. That is as well as the earlier essay’s four hands did when they shared no habit at all; the office’s quarter, charged rather than removed, costs nothing the count can see. The weak seam reads 8, 0, 34 and 38, never near its eighteen: the mean still holds the office’s whole quarter, which charging allows for and does not remove, and an easing is a seam with a width, and a drifting hand hides the width found a width to be the first thing a drift takes.

The overshooting office, charged from four round controls, collapses on every sheet: four eased seams, the most offered, and six rigid. Its charges were 0.52 to 0.57 of the truth, no better than the hands alone, and the collapse has a place. Two of the four eased seams sit near the copy’s seams, and the other two at points 151 to 153 and 331 to 333 — the counted coast’s two tips, where its curvature is greatest and where the office pushed the coast out. Undercharged, the office’s habit is not merely noise the count spends pieces on; it is drawn where the coast bends, and the count reads it as two more places where the map changes.

The same office, its habit subtracted on one thin control and charged for what the control still showed, counts two seams on every sheet again, at 18, 18, 16 and 18. Its charges, 1.14, 1.14, 1.01 and 1.11, sit a little high, as the combined charge does across offices, and a slightly generous charge costs nothing here. The rigid count, which can only place steps, spends one seam more than was made on each of these four sheets, against one sheet in four for the drifting office; the eased count, which can give a seam a width, does not.

A standard traced to find the hand #

Learning an observer’s habit by having the observer measure something already known is older than the tracing table. In 1796 the Astronomer Royal, Nevil Maskelyne, dismissed his assistant David Kinnebrook for recording the transits of stars across the Greenwich wires most of a second later than Maskelyne did; a quarter of a century later Friedrich Bessel, reading the episode, compared his own transit times with other astronomers’ and found that every observer carried a steady difference of his own — the personal equation, measured by setting observers against each other and against a standard and applied as a correction rather than a scatter. The two offices here are the two ways that correction can be meant: a constant to be charged, or a rule to be subtracted.

The cartographic version is the control point. Comparing an old map with the coast as now surveyed, through points identifiable on both, is how its systematic errors are read — Waldo Tobler set the statistics out as bidimensional regression, and Bernhard Jenny’s MapAnalyst, from the 2000s, draws the distortion such points imply over a whole sheet. Those comparisons read a map’s error at the points where the truth is known. A control sheet reads an office’s error on a sheet where all of it is known — the sheet moved before it was measured met the paper’s own version of such an error, which every tracing inherits — and carries it to a sheet where none of it is, and what can be carried is set by what the error depends on. A thousand features are wrong in the same direction found a simplification’s area loss systematic over a population and unpredictable on one feature; an office’s overshoot is the same kind of error with a rule in it, and the rule is what the controls learn.

The pattern recurs wherever a shared error is invisible from inside. One survey cannot tell an optimistic report from a moved mark found a local survey unable to learn its control’s error from its own residuals; a week of twilights learns what a sight is worth found a navigator’s evening of sights unable to measure the evening’s own refraction; a season learns the stretch from where the streak was learned a forecast ensemble’s shared bias from crossings whose outcome was known; the weights are a guess the solve believes met the estimator that splits an error among groups of observations. In each the repair is a measurement of a thing already known, and its worth is set by how much that thing resembles the thing that is not.

What the controls were held to #

No office habit, no share. Controls traced by an office with nothing shared must read an office share near zero; over forty offices the median is within five hundredths of it.

A drifting habit’s share is read without bias. Eight thin controls must read the drifting office’s quarter to within a tenth of itself in the median, and the restored charge must read the counted mean’s true drift to within a tenth. They read 0.254 and 1.01.

The finding has to be there to fail. For the overshooting office, a size read on four round controls must charge the counted sheet more than a fifth short, and the slope read on one thin control must leave less than a twentieth of the habit in the mean. They charge 0.54 and leave 1.4 per cent.

The counted coast’s curvature is read from the tracing. Every subtraction applies the slope to curvature computed from the office’s own pooled tracing, smoothed, never from the coast as drawn, which a digitiser does not have; the controls’ curvature is the known coast’s, because a control is a coast whose truth is known.

Independent tracings are independent. Every drift is seeded from its office, hand, tracing and sheet with the seed mixed before use, as in the earlier essay.

Where the controls stop #

Two kinds of habit. A real office’s shared habit is likely both: some exaggeration that answers to shape and some drift that answers to nothing. The combined repair is built for that case, since it subtracts what the slope explains and charges what it does not, but the mixture was not drawn and the shares of each were not varied.

One rule of shape. The overshoot here is linear in curvature and acts along the normal. An office that exaggerates small headlands and smooths large ones, or that answers to a feature’s length rather than its bend, would need its own regressor; the slope test would show its absence by reading zero on a coast that bends, which is the useful failure.

Controls traced as the counted sheet is. Each control was traced by the same four hands, four times each, in the same way. An office that traces its controls more carefully than its working sheets, because it knows they are controls, teaches the controls a smaller habit than it has.

Coasts at one size. Every coast was drawn at the counted sheet’s map size. The habit’s constant is in map units per unit of curvature, so a control at another scale carries it at another size, and the regression would have to be done in the units of the paper, not of the ground.

Still open: whether a control can be found in the sheet #

A control sheet is a coast the office drew and somebody else surveyed. Most archives that hold an office’s compiled sheets also hold some of its sources — the surveys, charts and earlier sheets the compilation was traced from. Where a stretch of the counted sheet was copied from a source that survives, that stretch is a control inside the sheet: the office’s tracing of a known coast, drawn the same day with the same pens, with no doubt about whether it was traced as carefully as the rest.

Whether the copied stretches of a compilation, read against their own sources, are enough to read the office’s slope; whether a stretch that short, between two seams, bends enough to separate the two offices; and whether the seams themselves — where the office joined what it copied to what it drew — carry the office’s habit at its strongest, are questions controls traced on separate sheets cannot ask.