A stepped hand's walk is in the creeps it was given

A stepped hand passes for a tiring one on a short strip followed a painter who marks each division of a strip by measuring from the one before, and found that the error such a hand leaves is a walk: every row inherits the slips of the rows drawn before it, and the rows wander away from where they belong without anything tiring. It also found why that walk is so hard to see. Every reading of these strips begins by fitting two numbers, the habit that spaces the rows down a splayed surface and the creep that makes each bay a little larger than the last, and those two numbers bend the spacing smoothly from one edge to the other. A walk is slow. The fit takes about four-fifths of it before any reading of the residuals starts, and what is left names the mechanism only on long strips, or on sixteen short strips pooled.

The essay ended on the place the walk had gone. Each strip’s fit had absorbed its walk as a creep of that strip’s own, so a picture painted by a stepping hand is a picture whose strips report creeps that do not agree with one another. A tiring hand tires at one rate, and its strips report one creep, displaced on each strip only by the noise that strip’s slips put into the fit. The proposed test was to measure how far a picture’s creeps disagree and ask whether that is more than their slips can explain.

It is, by a wide margin, and the test needs half the strips the residuals needed.

Two ways to fit a picture, and the gap between them #

The measurement paints a divergent interior of the kind the whole sequence has used: strips at the splays of a floor (1.12), a footstool (1.20), a table (1.32) and a book (1.38), cycled for as many strips as the picture holds, every strip divided into six bays at a habit of 1 with a creep of 2 per cent a bay, its starting edge alternating from strip to strip. One hand paints it by stepping its dividers, slipping 0.4 pixels a step. Another paints it by setting each row fresh and tiring, its slip tripling from the first row drawn to the last. The two are matched where a strip’s scatter points to the end drawn last matched them, on the row drawn last, which the walk reaches at 0.89 pixels after five steps and the tiring hand is set to reach too.

The picture is then fitted two ways. First, each strip alone: its own habit and its own creep, found by the nested search the earlier essays built for the habit–creep valley and polished. Second, the whole picture at once: one habit and one creep for the hand, with each strip read at whichever edge suits it, because a reader does not know which edge a painter began from and a strip keeps its ratio, not the end it began showed the rows cannot say.

The bars measure the difference between the two fits, strip by strip and in pixels on the page: how far the rows one hand’s single habit and creep would draw sit from the rows that strip would have chosen for itself. That is the part of each strip that no shared creep can take — its disagreement with the rest of the picture. The ticks are each strip’s own residuals, what is left after the strip has had its own two numbers, and they are the only direct evidence the picture holds of how badly its painter’s hand slipped.

For the tiring hand in this painting, the bars sit below the ticks on every strip. Its strips disagree with one creep by 0.07 to 0.53 pixels, and their own residuals run from 0.32 to 0.78: the disagreement is about what the slips alone would put into eight separate fits. For the stepping hand the picture is inverted. Its bars run to 1.93 pixels and its ticks mostly sit under half a pixel, because each of its strips fitted its walk as a creep and left almost nothing. Seven of its eight strips disagree with the picture by more than they disagree with their own marks.

The slider shows the other paintings, and the pattern holds on most of them. A walk is a slow random bend, a creep is a slow bend, and a strip fitted alone cannot tell the one from the other. Two strips fitted together can, because the walk on one has nothing to do with the walk on the next.

The ratio a picture is read by #

The comparison in the figure has a name. It is the ratio an analysis of variance takes: the extra squared residual the shared fit leaves over the separate fits, per degree of freedom it gave up, divided by the separate fits’ own squared residual per degree of freedom they kept. A picture of eight six-bay strips has forty interior rows. Fitted strip by strip it spends sixteen numbers and keeps twenty-four residual directions; fitted once it spends two, so the difference between the fits lives in fourteen directions. The ratio, F, is the disagreement per direction over the slip per direction.

If every slip were independent and of one size, F would sit near one: a strip’s own fit would take its share of noise as a slightly different habit and creep, and those small differences are exactly what the residuals predict. A tiring hand is not quite that. Its slips grow along each strip, and the fit of two slow numbers takes less of a slip that is largest at one end than of one spread evenly — the stepped-hand essay measured the fit leaving 74 per cent of a tiring hand’s error in the residuals on six bays, more than the 60 per cent independent slips of one size would leave. So a tiring picture’s F sits below one: on eight strips its median is 0.54.

A stepping hand’s F sits far above one, and the same measurement says why. The fit leaves 16 per cent of a walk in a six-bay strip’s residuals, so 84 per cent went into the strip’s own habit and creep. Those are two directions per strip carrying most of the walk, against three directions carrying the small remainder. Once the picture is fitted with one creep, the two directions per strip are no longer free; everything the strips’ own creeps took comes back as disagreement, and F measures it against the small residuals. On eight strips the stepping median is 4.56.

The two clouds on eight strips barely touch. The line drawn through the gap at F = 1.5 was not chosen from these fifty pictures of each. It was set on a hundred other paintings of eight strips by each hand, where it is the value at which the two kinds of mistake are equal — 94 tiring pictures in a hundred fall below it and 95 stepping pictures above — and it is then held fixed for every picture and every count in this essay. On these fifty it leaves 46 tiring pictures below and 49 stepping pictures above.

The slider moves the count of strips, and the count is the whole story of the test’s power. On two strips, a picture’s disagreement has two degrees of freedom to show itself in, and F is a ratio of two numbers each estimated from almost nothing: 47 of 50 tiring pictures fall below the line but only 19 of 50 stepping pictures rise above it. On four strips the stepping pictures are 43 of 50 above. On sixteen the clouds part entirely, fifty and fifty.

Half the strips the residuals needed #

The earlier essay’s best reading used the residuals: for each strip, the remainder of its own fit, scored for how well it looks like a walk’s fast part or a tiring hand’s growing slip, one slip for the whole hand. It reached nineteen in twenty on sixteen six-bay strips. The fair comparison runs both readings on the same paintings.

The creeps reach 89 per cent at four strips and 95 at eight. The residuals reach 86 per cent at eight strips and 94 at sixteen, which is where the earlier essay found them. The creeps do in eight what the residuals needed sixteen for, on the same paintings, with the same slips and the same edges.

Split by hand, the two readings fail differently. On eight strips the creeps name 92 per cent of tiring pictures and 98 per cent of stepping ones; the residuals name 90 per cent of tiring pictures and 82 per cent of stepping ones. The residuals’ weakness is the stepping hand, whose evidence they read from its weakest trace. The creeps’ weakness, on two strips, is also the stepping hand: 38 per cent, because two strips give the disagreement too few directions to rise above a line placed to protect tiring pictures. That trade is set by where the line sits, and a reader who cared more about catching a stepping hand than about wrongly accusing a tiring one would set it lower.

The reason the creeps do better is the arithmetic of where the walk went. The residuals see about a fifth of a stepped strip’s error. The creeps see the other four-fifths, the slow part that made the walk a walk. A test that reads the four-fifths can afford fewer strips than one that reads the fifth, and the gain is roughly the ratio of what each can see, softened by the fact that a creep is two numbers where a residual is three. This is not a better statistic in general; it is a statistic pointed at the part of the error that carries the mechanism.

What the strips’ length buys #

A picture’s F is a ratio of two estimates, and both come from the strips. The disagreement between strips sets the top of the ratio. The strips’ own residuals set the bottom, and a strip with few bays leaves few residuals to set it by.

On strips of four bays a picture of four strips names the hand 68 times in a hundred; on strips of twelve, 89. The four-bay strip is the one the stepped-hand essay found could say nothing about its own mechanism, because two numbers fitted to three interior rows leave a single residual. Here it is not useless, because the disagreement between strips does not need the residuals to have a shape; it needs only their size. But one residual per strip sets that size badly, and the ratio is judged against a slip that is itself barely measured, so the stepping pictures fall back towards a coin.

Between six and twelve bays the gain is modest: four strips of six bays already name the hand 86 times in a hundred, and four of twelve, 89. The test is limited by the count of strips more than by their length, which is the opposite of the residual reading, whose power on a single strip came entirely from its length. For the divergent interiors this sequence has studied — many short strips, a floor, a table top, a step, a book, a footstool — that is the right way round.

The assumption the test cannot relax #

The ratio asks whether the strips of a picture can be drawn by one habit and one creep. A stepping hand fails that because each strip has its own walk. It is not the only way to fail it.

Two painters who both tire, sharing a picture strip by strip, with habits one apart — one spacing rows nearly as a camera would, the other spacing them evenly by eye — are called a single stepping hand 70 times in a hundred. With habits two apart, every time. The rows count hands, not cameras found exactly this: a divergent picture’s strips reporting different habits are the signature of more than one hand, and one habit shared across strips is a claim the rows can test. The same disagreement that convicts a walk convicts a second habit, because both put into each strip a slow bend that the others do not share.

So the conclusion the test supports is narrower than “this painter stepped”. It is: this picture’s strips were not drawn by one hand following one steady rule. A walk is one explanation, a second hand another, and a painter who changed habit between the floor and the ceiling a third. What separates them has to come from outside the ratio — from the strips’ residuals, which a second habit leaves untouched and a walk shapes, or from which strips disagree with which. Two hands split a picture into two groups that agree internally; a walk makes every strip disagree with every other.

That split is itself measurable. A stepping hand’s per-strip creeps are independent draws, so the disagreement is spread evenly across the picture, with no grouping of strips that would reduce it. Two hands’ disagreement collapses to almost nothing once the picture is fitted with two habits instead of one, and the reduction in F from that single extra number would be large for two hands and small for a walk. The measurement was not made here; it is the obvious one.

What the creeps can and cannot convict #

Put together, the measurements answer both halves of the question the stepped-hand essay left. The creeps of a stepping hand’s strips scatter far more than a tiring hand’s with the same slip on its last row: a median F of 4.6 against 0.5 on eight strips, where F is the scatter per direction over what the slips put into each strip’s own residuals. And the excess names the mechanism 89 times in a hundred from four strips and 95 from eight — half the sixteen the residuals needed for the same verdict on the same paintings.

The larger point is about what a fit throws away. Every reading of these strips, from a tiring hand draws a different habit onwards, has fitted each strip’s habit and creep first and asked questions of what was left. That is the right order when the question is about a strip. It is the wrong order when the question is about a hand, because a hand’s habit and creep are its own, shared by every strip it painted, and fitting them strip by strip hands each strip the freedom to absorb whatever its own error happens to look like. One camera means one horizon, not one point made the same argument about cameras: fit what is shared once, across all the parts, and the parts’ disagreements are evidence rather than noise.

What the creeps cannot do is see a walk on a single strip. A strip alone has no other strip to disagree with, and its creep is exactly as consistent with a walk as with a steady hand. The single-strip reading still belongs to the residuals, and to long strips. The two readings are complementary rather than competing: the creeps say that a picture’s strips were not drawn by one rule, and the residuals, on the strips long enough to have a texture, say what kind of rule-breaking it was.

What was assumed #

One hand, one habit, across the whole picture. The test’s reading is a disagreement with a single habit and creep, and the measurement above shows how a second hand looks the same. A picture known to be a workshop product, with several painters, has to be read in groups, which needs the groups.

Every strip’s creep is the hand’s. A tiring hand’s creep is modelled as one rate shared by every strip. A painter who rests between strips, and so starts each strip fresh and tires along it, shares a rate but not a starting state, which the model allows; a painter who tires across the whole panel, so that later strips creep faster, does not share a rate, and a tiring panel keeps its order, not its direction measured what that looks like. It would add disagreement between strips and push F up.

The line is set on eight strips of this interior. The value 1.5 balances the two errors on pictures of these four splays, six bays a strip. On a picture with a different make-up the balance point moves, and the line would have to be set on paintings of that picture’s own strips. That is a small simulation, but it is not optional.

The rows are measured far better than the slip. A real reading of a panel adds its own measuring error, independent from row to row and the same size everywhere. It inflates every strip’s residuals and so pulls every F towards one, which helps a tiring hand’s pictures and hurts a stepping hand’s. How far it moves the balance was not measured here; a measuring error as large as the painter’s slip would put as much into every strip’s residuals as the slip does, and the count of strips the test needs would rise with it.

Still open: whether one more habit separates two hands from a walk #

The test here convicts a picture of not having been drawn by one steady hand. It leaves open which of the alternatives drew it, and the measurement that separates the two commonest is short to state.

Refit each picture with two habits instead of one, the strips split into the two groups that best reduce the disagreement, and record how much F falls. A picture painted by two hands should drop close to one — everything its strips disagreed by was the gap between two habits, and one extra number takes it. A picture painted by one stepping hand should barely move, because its disagreement is spread across every strip independently and no single split into two groups absorbs more than a small share of it. The question with a number in it is how large a drop separates the two, how many strips a picture needs before the split is found correctly rather than chosen from noise, and whether the split a two-hand picture yields matches the strips each hand actually painted — which would turn a verdict of “not one hand” into a map of who drew what.