A stepped hand passes for a tiring one on a short strip

A strip’s scatter points to the end drawn last found a second kind of evidence in a divided strip. The size of the bays cannot say which edge a painter began from, because a creep read backwards is a creep. The scatter of the bays can, because a hand that worsens as it works leaves its largest slips on the rows it reached last, and a scatter that grows is not the same read from the other end.

That essay ended on a rival it had not measured. A painter who marks each division by measuring from the one before — dividers opened to the last bay and walked on, a compass stepped along the strip — accumulates error. Every row inherits the slip of the row before it and adds its own, so the rows drawn later sit further, on average, from where they should, and nothing tires at all. The spread of such an error grows as the square root of the number of steps from wherever the stepping began, which is a scatter growing along the order of drawing. It ought to fool the fatigue reading, and the essay proposed a test for it: an accumulated error is correlated from row to row, so neighbouring residuals should lean the same way.

The test turns out to be weak for a reason that is more interesting than the test. Before any reading of the scatter begins, the habit and the creep are fitted to the rows, and a walk is exactly the kind of error those two numbers can take.

A walk, and what the fit takes from it #

The figures below paint strips of a floor splayed like the one in the earlier essays — splay 1.12, the even habit of 1, a creep of 2 per cent a bay — with one change to the hand, which the rows count hands, not cameras would call a change of procedure rather than of habit. It no longer sets each row fresh. It steps: the first row is placed from the ruled edge with a slip of 0.4 pixels, the second from the first with another 0.4, and so on. The two edges hold still, as before, because they are ruled before any dividing starts, so whatever the walk has accumulated by the last row drawn is taken up by the last bay.

The solid line is the error the dividers left, row by row, in the order they reached the rows. It wanders away from zero and does not come back: −0.45 pixels on the first row, −1.02 by the third, −1.84 on the last. That is what accumulation looks like. Each step’s slip is small and independent, but the errors on the page are sums of them, and a sum of independent slips drifts.

The dashed line is what the reading does first. It fits the two numbers every reading of these strips has fitted — the habit, which says how the painter spaces rows down a splayed surface, and the creep, which says how much each bay grows over the last. A tiring hand draws a different habit showed that on one strip those two are nearly the same thing, and together they describe a smooth bending of the spacing from one edge to the other. A walk that drifts steadily in one direction is, to a good approximation, a smooth bending of the spacing. The fit follows it.

What is left, the bars, is 17 per cent of the walk’s squared error on this painting. The slider shows that the share varies a great deal between paintings — 79 per cent on one, 17 on another — because a walk sometimes turns back on itself partway along, and a turn is not smooth. On average, over a hundred paintings of a twelve-bay strip, the fit takes about three-quarters of what the dividers did. That is the first finding, and it governs everything after it. The reading of the scatter never sees most of a stepped hand’s error, because by the time the reading starts, that error has been accounted for as a habit and a creep.

Why a walk is spent before it is read #

A fit of two numbers to a strip of eleven interior rows leaves nine directions free. If the hand’s slips were independent and the same size on every row, those nine directions would carry nine-elevenths of the error on average, since independent noise spreads evenly over every direction a fit can take or leave. The dotted line in the next figure is that share, $(n-3)/(n-1)$ for a strip of $n$ bays.

A tiring hand sits close to that line, and above it on long strips because its largest slips are on the rows furthest from where the fit’s slow shapes pivot. On six bays the fit leaves 74 per cent of a tiring hand’s error; on twenty-four, 94.

A stepping hand sits far below it, and does not climb. On four bays the fit leaves 16 per cent of the walk; on six, 16; on twelve, 24; on twenty-four, 23. More rows do not help, because the walk does not spread its error over more directions as the strip lengthens. A walk’s error lives in its slowest shapes whatever its length. Most of it is in one long drift from the edge begun at, a little in one bend, less in two bends, and so on — and the habit and creep are two slow shapes, so they take the drift and most of the bend at every length. The comparison is made fair by matching the two hands on the last row drawn, where the stepping hand has accumulated $0.4\sqrt{m}$ pixels over $m$ steps and the tiring one is set to reach the same.

This is the structure the whole sequence has run on, seen from a new side. A strip keeps its ratio, not the end it began found that the rows’ positions carry a ratio, and anything smooth that bends them is traded against that ratio. A walk is a random smooth bend. It is not fatigue in the painter’s hand, but it is fatigue in the rows: a stepped strip, fitted, reports a creep, and the creep it reports is the walk. On six bays of this floor the creep a stepped strip reports scatters by several per cent a bay from painting to painting, as much as the creep a tiring strip reports — so the size of the bays, which could not tell a tiring hand’s direction, cannot tell a stepping hand from a tiring one either.

The fatigue reading still names the edge #

What the fit leaves is a small, fast remainder. The question the earlier essay asked of it was whether it points to the edge the painter began from, and for a walk it does, a little less well than for fatigue.

The score is the one A strip’s scatter points to the end drawn last built: fit the habit and the creep, then ask whether the residuals are better explained by a slip that grows from the near edge or the same slip growing from the far. It knows nothing about walks. Given a stepped strip of six bays, it names the edge the dividers started from 64 times in a hundred, against 68 for a tiring hand; on twelve bays, 78 against 87.

That is the answer to the essay’s first question, and it is a mild yes. The residuals a walk leaves are still larger, on average, at the end the walk reached last. Not by much, because the fit took the large drift that made them so, but by enough for the fatigue score to read them. The four-bay strip gives nothing for either hand, for the reason the earlier essay found: a strip of four bays has three interior rows, the fit takes two, and one residual has a size but no position.

So the fatigue reading of a strip’s direction is robust to the mechanism in a way that matters for any real panel. A reader who does not know whether the painter stepped or tired can still use the scatter to name the edge, with a small loss when the painter stepped. What the reader cannot do from this score is say which of the two it was, and the earlier essay’s proposed test for that is the next thing to draw.

Neighbours lean together only on long strips #

A walk’s neighbouring errors share every step before them, so on the page, adjacent rows of a stepped strip err in the same direction: if row five is too low, row six, which was measured from row five, is likely too low as well. Independent slips have no such tie. The earlier essay proposed reading the correlation between neighbouring residuals, and predicted it would be positive for a stepped strip and near zero for a tiring one.

On long strips the prediction is right. At twenty-four bays a stepped strip’s neighbouring residuals correlate at 0.59 and a tiring strip’s at nearly nothing, which is a clear difference. On short strips it is wrong in a way that was worth measuring before trusting. On six bays a stepped strip’s neighbours correlate at −0.16 — they lean apart — and a tiring strip’s at −0.33.

Two things are happening, and both are the fit. First, residuals are not the errors. They are what is left after two slow shapes have been subtracted, and subtracting a best-fitting slow shape from anything leaves a remainder that must balance around it, which pushes neighbours apart: a remainder that is positive over one stretch has to be negative over another, and on a strip of three free residuals the stretches are single rows. That is why even independent slips lean apart on six bays. Second, the part of a walk that made its neighbours alike was precisely its slow drift, and the fit has taken it. The walk’s remainder is its fast part, and its fast part is nearly as uncorrelated as anything else.

The sign of the lean says “stepped” only from about eight bays, where the stepped curve crosses zero. Below that, the two hands differ only in how negative their lean is, and with three residuals to a strip the difference between −0.16 and −0.33 is far inside what one painting scatters by. A reader looking at one six-bay strip for neighbours that lean together will not find them, whichever hand drew it.

A proper test, and what it needs #

The correlation of neighbours is a rough statistic — a single number standing for a pattern, which is the complaint the residual has a shape makes about every scalar summary of a misfit. A sharper one writes down what each mechanism predicts for the residuals and asks which prediction fits. A tiring hand predicts independent slips whose spread grows along the order of drawing; a stepping hand predicts a walk, in which two rows share the slips of every step both inherited. Each prediction is a covariance for the rows. Carried into the directions the fit leaves free — the only directions a residual can occupy — it gives a likelihood for what is left, with the slip’s overall size fitted and each strip read at whichever edge suits each mechanism best, since a reader does not know the edge. This is restricted likelihood, and it uses everything the residuals carry about their mechanism.

On six bays it names the mechanism 63 times in a hundred, averaging over the two kinds of strip, where a coin names it fifty. On eight bays it does 64; on twelve, 79; on sixteen, 88; on twenty-four, 99. The mechanism of a single strip is readable, and it needs a long strip: a floor divided into sixteen or twenty bays — the receding surfaces that an inverse perspective is a leaning plane found carry the most divisions — a tiled pavement run to the back of a room, a colonnade of many intervals. The four- and six-bay strips of furniture and steps that make up most of a divergent interior carry almost nothing.

The reason is the count the earlier essays kept running into. A six-bay strip hands the reading three residuals. Three numbers are too few to show whether they came from the fast remainder of a walk or from a slip that grows as the hand works, because both are three numbers, somewhat larger towards the end drawn last, with no dependable pattern between neighbours once the fit has passed over them. Twenty-four bays hand the reading twenty-one residuals, and in twenty-one numbers the two kinds of remainder are different textures: a walk’s remainder still meanders over several rows at a time, and a growing slip’s does not.

One detail of the reading turned out to matter, and it is worth recording because it is the kind of choice that looks like a formality. A reader who does not know a strip’s edge can take the edge that suits each mechanism best, or can average over the two. The figure takes the better edge. Averaging looks more principled, and it gave a verdict that leaned to stepping on short strips whatever drew them, because a growing slip read from its wrong edge is a very poor fit while a walk’s two readings are much alike once its drift is gone — so averaging charged the tiring reading, on every strip, for being sharp. On the same paintings it never named a tiring workshop more than seven times in ten, however many strips were added. A choice about what to do with an unknown is a choice about which answer to favour when the evidence is thin, and on a six-bay strip the evidence is always thin.

A workshop’s strips pool the verdict #

The earlier essay found that one hand’s slip, fitted once across several strips, recovered a workshop’s common starting edge nineteen times in twenty from six strips where one strip could not. The mechanism is also a property of the hand, and a hand that stepped its dividers on one strip stepped them on the next, so the same pooling applies.

It works, and it works slowly. A stepping workshop is named 73 times in a hundred from four six-bay strips, 85 from eight and 95 from sixteen; a tiring workshop 80, 88 and 98. Each strip adds its three residuals’ worth of evidence about the hand, and the evidence is weak enough per strip that it takes a dozen or more strips to accumulate into a verdict a reader could rely on. Sixteen strips is a large panel, but it is not an unusual count for a picture of a whole interior, whose floor, walls, ceiling, furniture and steps are each divided by the same hand.

The pattern is the one that has run through this whole sequence. A tiring panel keeps its order, not its direction and its successors found that what the strips of a picture share can be read from many short strips, and what each strip decides alone needs a long one. The mechanism is shared across a hand’s strips, so it is the first kind; the rule one camera means one horizon found for geometry — fit what is shared once, across all the parts — lets sixteen strips that cannot say it one at a time say it together.

What the pooling cannot do is recover what the fit has already spent. Every strip’s slow drift was handed to its own creep before the pooling began, and the reading pools only the fast remainders. That is why it needs so many strips: it is reading the stepped hand from its weakest trace.

Where the stepped hand’s evidence actually is #

The measurements say where the better evidence sits, and it is not in the residuals. A stepped hand’s error is mostly drift, and the drift was given away to the creep. A reader who wants it back has to stop treating each strip’s creep as the painter’s intention.

On a single strip, that is impossible: the drift and a genuine creep are the same shape, and the rows under a splay measure the bays has already priced what each unknown costs in rows. Across a picture it is not impossible. A tiring hand tires at one rate, so its creep is one number shared by every strip, and a strip keeps its ratio, not the end it began used exactly that to recover the strips’ directions. A stepping hand’s apparent creep is a different random number on every strip, with no sign in common and no relation to the splay. So a picture whose strips report creeps that disagree beyond what their slips allow, and do not line up with the number of bays over the log of the splay as fatigue predicts, is a picture whose rows were probably stepped. That test reads the drift instead of discarding it, and it belongs to the whole picture rather than to any strip.

The measurement above did not make that test. What it established is the price of not making it: on the short strips that fill most divergent interiors, a stepped hand passes for a tiring one, and only a long strip or a large panel tells them apart. The fatigue reading of the edge survives the confusion, which is the useful half. The mechanism does not survive it on any one short strip, which is the cautionary half — a reading of fatigue from a six-bay strip is not evidence that the painter tired, only that the painter’s errors were larger at one end.

What was assumed #

The dividers are stepped from one edge to the other and never reset. A painter who steps from the near edge to the middle and from the far edge to the middle leaves two walks meeting in the centre, whose remainder is largest in the middle rather than at either end. The fatigue score would read such a strip as neither, and the reading of the direction would be meaningless for it.

Each step’s slip is independent and the same size. Dividers whose opening creeps as they are walked — a loose joint — add a growing systematic step to the walk, which is a creep the painter did not intend and the fit would take entirely. A painter who is both stepping and tiring leaves a walk whose steps grow, and the two mechanisms are then present together; the reading here assigns each strip to one or the other.

The edges are exact. The last bay absorbs the walk only because the far edge was ruled first and held. A painter who stepped the whole strip including its far edge leaves a walk with a free end, whose largest error is on an edge the reading treats as exact; the fit would then be wrong in a way no residual shows.

The rows are measured far better than the slip. As in the earlier essays, a real reading of a panel adds its own measuring error, which is independent from row to row and the same size everywhere. It dilutes a walk’s remainder with exactly the kind of noise the tiring model expects, and moves every verdict here further towards fatigue.

Still open: whether a picture’s disagreeing creeps convict a stepping hand #

The section before last proposes the test this essay did not make, and it is the one that reads the stepped hand’s evidence where the evidence is. A tiring hand leaves one creep shared across every strip of a picture, shifted on each by an amount fixed by the strip’s bays and splay. A stepping hand leaves a creep on every strip too, but it is the fitted image of that strip’s walk: its size scales with the slip, its sign is random, and it has no relation to the strip’s geometry.

The measurement that settles it paints a whole divergent interior both ways, fits each strip’s own habit and creep, and asks two things. First, whether the scatter of the per-strip creeps about the single shared creep fatigue predicts is larger for a stepping hand than for a tiring one with the same slip on its last row — larger, that is, than the slip alone can explain. Second, how many strips a picture needs before that excess scatter names the mechanism nine times in ten, and whether it is fewer than the sixteen six-bay strips that pooling the residuals needed to reach nineteen in twenty. If it is, the slow part of a walk, which every reading so far has handed to the creep, is the part that convicts it.