Averaging tracings reads the easing, until the hand repeats itself

An easing is a seam with a width, and a drifting hand hides the width found that a compiled coast with one copied stretch, eased into its neighbours over eighteen points either side of each seam, is counted as one copy by a criterion that gives each seam a width and pays for it. It also found the limit of that result. On a tracing made by a hand that drifts — each point’s error mostly the error at the point before, over a correlation length of eighteen points — the count still comes out right once the drift is whitened, but the width of the easing cannot be read. Whitening discounts a departure by how much it looks like drift, and an easing eighteen points long looks exactly like drift eighteen points long.

That essay ended on a difference between the two departures that one tracing cannot use. The easing belongs to the sheet, so every tracing of the sheet carries it identically. The drift belongs to the hand, so every tracing carries its own. A second tracing tells a drifting hand from a copied stretch had already used the difference of two tracings, which cancels the sheet and leaves the two errors, to learn how long the drift is. The mean of several tracings does the opposite: it keeps the sheet whole and divides the drift’s variance by the number of tracings.

The proposal works, and the number of tracings it needs is different at the two seams of one copy, for the reason the earlier essay gave. What stops it working is not the number of tracings but a property of the hand that the difference of two tracings cannot see.

A dip four ten-thousandths of the map deep #

The coast is the one every essay on this question has used. Japan’s outline, 360 points, drawn on a conformal conic sheet, with the stretch from 55 to 85 per cent of the way round copied from a sinusoidal sheet and pinned to the host at both ends. The copy is eased over eighteen points either side of each seam with a smoothstep, and the walk round the coast starts opposite the copy so that no easing runs off its end.

What the easing changes is small. Pinned at its ends, the copy leaves the host at the seam with no step, and the easing only decides how fast the drawn coast moves from one sheet’s line to the other’s. Measured against a rigid joint — host up to the seam, copy from it — the eased coast differs by a dip that reaches 3.8 ten-thousandths of the map’s size and is gone eighteen points either side. That dip is the whole of the evidence an easing leaves.

The hand drifts by three ten-thousandths of the map, with a correlation length of eighteen points. So a single tracing carries an error of the same size as the dip and of the same length, and the thin line in the figure, wandering above and below the eased coast, is what one tracing gives the count to read. The thick line is sixteen tracings averaged: still not the eased coast, but closer to it by half over this stretch, and following the dip rather than inventing its own.

How the stack is counted #

Each tracing is the eased coast plus its own drift, drawn afresh from the same process the earlier essays used: an autoregression along the outline with a correlation of $e^{-1/18}$ from one point to the next, three ten-thousandths of the map in size, in both directions on the page. The stack’s mean is then counted exactly as a single tracing was. The count offers the coast up to six rigid seams and up to four eased ones, each eased seam paying one more number for its width, and the residual is whitened at the drift’s true correlation length, which is what the second-tracing essay showed how to learn.

Only one thing changes with the number of tracings, and that is the charge. The mean of N independent drifts with the same correlation length is itself a drift with that correlation length and with its variance divided by N. So the whitening is the same at every count of tracings, and the error the criterion is charged with is the single hand’s divided by $\sqrt{N}$. A stack of one is the earlier essay’s single tracing, point for point.

Four stacks were drawn at each count — 1, 2, 4, 8 and 16 tracings — each with its own hand, and each stack of sixteen contains its stack of eight, which contains its stack of four. Raising N adds tracings to the same pile rather than drawing a new one.

Two tracings read the strong seam, sixteen the weak one #

From one tracing the result is the earlier essay’s: the width reported at the first seam is 0, 0, 1 and 0 points against eighteen drawn, because the whitened criterion has discounted the easing as drift and found no reason to pay for a width.

Two tracings change that at the first seam. The widths come back as 14, 18, 14 and 16 — no longer a step, and within four points of the truth in every stack. Four tracings bring them to 16, 18, 18 and 18; eight and sixteen hold them within two points every time. The strong seam’s easing, invisible in one tracing, is read by averaging two.

The second seam does not move until much later. From two and four tracings it reads 0 to 3 in six stacks of eight, with an occasional value far off — 74 points from one stack of two, 23 from one of four — which is what a width the residual barely constrains looks like: the criterion can put it nearly anywhere. At eight tracings two stacks read 16 and 20 and two still read 0 and 2. Only at sixteen does it come back as 12, 18, 18 and 18, three exactly right and the fourth six points short.

The difference between the two seams is the one the easing essay measured on a single tracing with independent error. Over the easing at the first seam the copy departs from the host by 27.6 times the digitising error that essay used; at the second, by 6.9. The width of an easing is only visible against the departure it shapes — the same rule where the control points are found for every parameter of a fit — and the second seam’s copy stays close to its host over the whole easing. On one quiet tracing that made the second width an estimate with a spread of about five points either side — a value and a width, as the answer is a set asks of every number this method produces; here it makes the second seam need eight times as many tracings as the first.

A rough rule follows from the figure. A stack reads an easing’s width once the drift left in its mean is small beside the departure the easing shapes. Two tracings leave 71 per cent of one tracing’s drift, which is small beside the first seam’s departure and not beside the second’s; sixteen leave a quarter of it, which is small beside both. The count of tracings needed scales with the square of the ratio between the drift and the departure, so a seam whose copy departs a quarter as strongly needs about sixteen times as many tracings — and here it needed eight.

The rigid count wakes up as the drift falls #

The count itself behaves in a way that is easy to misread as a failure of averaging, and it is the opposite.

The eased count calls two seams in every stack at every count of tracings — twenty stacks, twenty right. The rigid count, which cannot model an easing, calls two seams for every stack of one or two tracings, then three in most stacks of four and eight, then four, six, four and five from sixteen.

That is the drift losing its cover rather than the mean going wrong. With a drifting hand at full strength, the whitened rigid count was right on every tracing in the earlier essay because the easing looked like drift and the criterion had discounted it as drift. As the drift shrinks, the charge shrinks with it, and the easing is no longer small beside the error the criterion expects. The rigid count then does exactly what a copy eased into place loses its seams before its source found it doing on a quiet tracing: it buys pieces to explain a smooth departure no projection explains, the failure a residual cannot count the pieces of a map first named. At sixteen tracings the mean’s drift is three-quarters of a ten-thousandth of the map, close to the independent error at which the rigid count was first measured, and the rigid count is back to calling four to six seams for one copy.

So the averaging has not created the problem; it has uncovered one the drift was hiding. And the only model that survives it is the one that contains the easing. That is a better argument for the eased model than the single-tracing essay could make. There, on a drifting tracing, the rigid count was right and the eased count added nothing but a width it could not read. Here, as soon as the evidence is good enough to read the width at all, it is good enough to break the rigid count — the width and the count arrive together.

A hand that repeats itself keeps part of the drift in every tracing #

Everything above assumed that each tracing drifts afresh, that the hand’s error in one tracing tells nothing about its error in the next. A draughtsman does not work that way. The same hand on the same coast leans the same way on a long stroke, overshoots the same convex headland, starts each tracing from the same corner with the same pressure. Some of the drift is a habit, and a habit is shared.

The model here splits the drift into two parts with the same correlation length: a habit, drawn once for the hand and present in every tracing, and a wander, drawn afresh for each tracing. A share s of the drift’s variance is habit and 1 − s is wander, so that one tracing still drifts exactly as before. The mean of N tracings keeps all of the habit and divides the wander by N, and the variance left is

$$\sigma^2\Bigl(s + \frac{1 - s}{N}\Bigr).$$

It falls with N towards $s\,\sigma^2$ and never below it.

The figure puts the two measurements above against that floor. An independent hand read the strong seam’s easing once its mean held 71 per cent of one tracing’s drift, which took two tracings, and the weak seam’s once it held 25 per cent, which took sixteen. A hand that shares a quarter of its variance between tracings can bring its mean to 50 per cent and no lower; one that shares half, to 71. On that arithmetic a quarter shared still lets the strong seam be read, at about three tracings, and never the weak one; half shared puts the strong seam on its threshold however many tracings are averaged, and the weak one out of reach.

This is also why the floor is invisible from the tracings themselves. Every tracing in the stack drifts by the same amount whatever s is — that was the construction — and the stack’s members agree with each other more closely than an independent hand’s would, which looks like a steadier hand rather than a repeating one. Only the mean’s error against the sheet shows the habit, and the sheet is what is unknown.

Sixteen tracings misread the way four did #

The trials agree with the arithmetic, and add the thing the arithmetic leaves out.

Charged at the mean’s true drift — the habit included — the count is still right: two seams in every stack, with a quarter shared or half, from four tracings or sixteen. The strong seam’s width is read less well than an independent hand allowed. With a quarter shared, four tracings give 10, 18, 16 and 4 against eighteen drawn, where an independent hand’s four gave 16, 18, 18 and 18; sixteen tracings improve it to 10, 14, 18 and 18. With half shared, sixteen tracings are no better than four: 8, 8, 16 and 26 against 8, 18, 14 and 4. The weak seam is not read at all.

What the trials add is which stacks go wrong. The first stack, with a quarter of the drift shared, reads the strong seam as 10 points from four tracings and as 10 from sixteen; with half shared, as 8 from four and 8 from sixteen. Its twelve extra tracings changed nothing, because its error was its habit, and every one of the twelve repeated it. An independent hand’s errors are different in every stack and average out across stacks; a habit’s error is the same in every tracing of a stack and does not average out within it. Read across many draughtsmen, the habit is noise. Read within one draughtsman’s stack, it is a bias.

The spread between tracings cannot see the habit #

The charge in those trials was the mean’s true drift, which is a number nobody tracing a coast is given. In practice it has to be learned from the tracings, and the natural way is the one a second tracing tells a drifting hand from a copied stretch used for the drift’s length: compare tracings with each other. The difference between two tracings cancels the sheet, and the scatter of the tracings about their mean measures the hand.

It measures only part of the hand. The habit is in every tracing identically, so it cancels from every difference just as the sheet does. A digitiser who learns the drift’s size from the spread between tracings learns $(1 - s)\,\sigma^2$, divides it by N for the mean as the arithmetic says to, and charges the mean with $\sigma^2(1 - s)/N$. The mean actually carries $\sigma^2(s + (1-s)/N)$. With a quarter shared and sixteen tracings that is a charge of 22 per cent of one tracing’s drift against a truth of 54 — too small by a factor of 2.5, so every unit of misfit is counted as more than six.

The count collapses. Charged that way, the rigid count calls six seams in fourteen stacks of sixteen and the eased count four in twelve — in each case the most it was offered — where two were made. The only stacks that escape at all are the least averaged: with a quarter shared and four tracings the wander still dominates, the charge is too small by only a third, and the eased count is right in two stacks of four. Every other case is a count that has spent every piece it was allowed.

That inverts the reason for taking more tracings. With an independent hand, each tracing added makes the count better and the width more readable. With a hand that repeats a part of itself, each tracing added shrinks the part of the drift the spread can see while leaving the part it cannot, so the charge learned from the spread grows more wrong as the stack grows. A digitiser who traced the coast sixteen times and learned the hand from the tracings is in a worse position than one who traced it once and learned the drift from a separate test.

The habit is a property of the hand, and the sheet is not #

None of this is peculiar to coasts. The split between an error repeated in every measurement and an error drawn afresh each time is as old as the adjustment of observations, and every trade that averages repeated measurements meets it. A surveyor reading an angle on both faces of a theodolite averages the readings to cancel a collimation error that changes sign between faces, and knows that no number of rounds on one face will. A week of twilights learns what a sight is worth met the same distinction in a navigator’s sights, where the refraction an evening shares across every star does not average away with the number of stars, and tonight’s residuals measure tonight’s air badly and weight tonight’s fix well found that one evening’s sights estimate that evening’s shared part only badly, however carefully each is reduced. The weights are a guess the solve believes met it in the weights themselves: a variance estimated from the scatter of one group of observations is the variance of what varies within that group, and nothing more.

What the tracing case adds is a way to break the habit. Since it belongs to the hand, a second hand does not share it. A stack built from tracings by several draughtsmen, each contributing a few, has a habit for each hand and divides the habits among the hands as it divides the wander among the tracings. The mean of four tracings by four hands has the variance of four independent tracings, whatever any one hand repeats, and the spread among them then measures what the mean carries. That is the old instruction to vary the observer, recovered from the arithmetic.

The other way is to change what the hand does between tracings. A habit that belongs to a direction of stroke reverses when the coast is traced the other way round; a habit that belongs to the sheet’s orientation on the table changes when the sheet is turned. The sheet moved before it was measured found the paper’s own shrinkage to be a directional departure present in every tracing, which is the extreme case: a habit not of the hand but of the sheet, which no change of hand removes and which a stack will therefore read as part of the coast.

What the trials were held to #

A stack of one is the single tracing. With nothing shared, a stack of one tracing must reproduce the earlier essay’s drifting tracing to a billionth of the map, so that every count here continues that essay’s rather than restarting it. It reproduces it exactly, and the stack of one reads the first seam’s width as 0, 0, 1 and 0, as that essay found.

Averaging divides only the wander. Over twenty-four stacks, the mean of sixteen tracings with nothing shared must carry a quarter of one tracing’s drift, and with half shared $\sqrt{1/2 + 1/32}$ of it, each to within a tenth. They carry 0.254 and 0.720 of it, against 0.250 and 0.729.

The count was offered what it needed and more. Up to six rigid seams and four eased; two were made. A count that reached the most it was offered — six or four — has bought every piece it could, which is the failure the spread-learned charge produces and the reason it is reported as such rather than as a number.

The search is the earlier essay’s search. The eased fit starts from the rigid count’s exact optimum at each count and descends over seams, widths and sources from five starting widths, and is never worse than the rigid fit it starts from. That it finds the best eased fit is not guaranteed, and the widths read within two points of the truth from four tracings on are the evidence that it does here.

Where the stack stops #

One hand, one habit, one length. The habit and the wander are drawn with the same correlation length, so the mean is still a drift of that length and the whitening never changes. A habit at the scale of the whole coast — a hand that leans the same way for a hundred points — is a slower departure than the wander, and would be whitened wrongly at the wander’s length.

Shares, not habits, are stated. The trials fix the share of the drift a hand repeats at a quarter and a half. What share a real draughtsman repeats between tracings of one coast, and how it falls with time between tracings, is a measurement of hands rather than of geometry, and nothing here supplies it.

Four stacks a setting. The widths and counts are four stacks at each count of tracings and each share, so a statement like eight times as many tracings is read off counts of tracings that double at each step, with four stacks at each. The order of the two seams and the collapse of the spread-learned charge are far outside that resolution; the exact count of tracings a seam needs is not.

One copy, one easing. Every stack is of the same coast eased over eighteen points. An easing longer than the drift was already readable from one tracing; whether a stack shortens the easing it can read in proportion to the drift it removes is the natural next measurement, and not made here.

Still open: whether the habit can be measured from tracings at all #

The spread between one hand’s tracings cannot see what that hand repeats, and a stack of tracings by several hands breaks the habit only if the hands do not share one. Draughtsmen trained in one office, working from the same sheets with the same pens, may share habits with each other as well, and a habit common to every hand is indistinguishable from a property of the sheet.

There is a measurement that would separate them: the spread between hands’ means against the spread within each hand’s tracings. With independent hands the first is the second divided by the number of tracings each contributes; with habits, the first is larger by the habit’s share. That is a variance-component problem of the kind the weights are a guess the solve believes solved for groups of observations, applied to tracings. Whether a few hands and a few tracings each are enough to estimate the habit’s share well enough to charge the mean correctly — and so to make the count right again — and whether a share common to every hand can be told from the sheet at all, are questions a single hand’s stack cannot ask.