Scatter breaks the later ties and never the first

A painter’s rate of tiring signs every strip but the first found that two painters who tire on their own work at different rates leave the difference in a panel’s rows, and that a rate read on one panel and carried to another gives a strip to its painter, with exceptions. Two strips with the same creep cannot be told apart by creep, however well it is read. A painter’s first strip carries no rate at all, so two first strips are always exchangeable. And wherever one rate times one count equals the other rate times another, as with the first painter’s third strip and the second painter’s second at rates of one and two, the same applies.

That essay ended by pointing out that a tiring hand does more than creep. A strip’s scatter points to the end drawn last found that a hand’s slips grow as it tires within a strip, and if they also grow from strip to strip, then a strip painted late in a painter’s sequence is more ragged than one painted early, whatever its creep. Two strips with the same creep need not have the same scatter. The first painter’s third strip has had two earlier strips of fatigue, the second painter’s second has had one. They ought to differ.

The measurement is in three parts: whether scatter breaks a later tie, and how often; whether anything breaks the tie between two first strips; and how much the scatter knows by itself about the order of a painter’s strips. That last one matters, because the rate reading had to find an order from the creep alone, by a search through every candidate, and this is a second reading of the same thing.

A scatter is a measurement with few rows behind it #

Let a painter’s slips be Gaussian, with a standard deviation $s(j) = s_0(1 + gj)$ on their $j$-th strip, counting from zero, so that $g$ is how much the slip grows per strip, in units of its fresh size. A strip with $n$ rows has its rows fitted to a law with two numbers in it, a habit and a creep. What is left is the residual, and its sample standard deviation is the reading of that strip’s scatter.

A sample standard deviation from few points is a very noisy number. With $n$ rows it has $n - 2$ degrees of freedom left, and its relative scatter is about $1/\sqrt{2(n - 2)}$. With five rows that is about 40 per cent, which is comparable with a difference of one strip’s fatigue at any reasonable growth. So the tie is being broken by comparing two noisy readings of two nearly equal numbers. How often the larger reading is the later strip depends on how much larger it should be and how much noise is on each.

With no growth at all the rule is a coin, to within the sampling of the measurement: 50.7 per cent over 6,000 trials. With slips growing by a half per strip, five rows settle 68 per cent of such ties, ten settle 79, twenty settle 89 and forty settle 96. With a stronger growth of one per strip, five rows settle about 74 per cent and twenty 96.

The strips on these panels have four to six bays, so five rows is the right number to read. On the panels the earlier essay used, then, this tie is broken about two times in three. That is real evidence, and a long way from certainty, and it costs almost nothing, since the residuals are already there from fitting the creep. It should be taken as a weight on an assignment and not as a verdict, and it can be combined with whatever else a panel has to offer.

Which ties it can reach #

The size of the gap matters, and several kinds of tie arise.

Third against second, a gap of one strip, comes to 68 per cent. Fourth against third, also a gap of one but with less proportional difference between them, comes to 64 per cent. Fourth against second, a gap of two strips, comes to 79 per cent. The last is the kind of tie that rates of one and three produce, so such a pair of painters is not only separated by larger creep differences elsewhere, but also has an easier time with its one tie.

Rates of one and two leave strips three and two tied, settled 68 per cent of the time. One and one and a half, four and three, 64 per cent. One and three, four and two, 80 per cent. Two and three, four and three, 64 per cent. Rates such as one and 1.7 have no later tie at all, as the earlier essay found, so scatter has nothing to add to them. That table gives an unexpected reason to prefer one pair of rates over another if a person were designing such an experiment, though nobody is designing painters.

The first strips stay tied #

Two first strips are a different thing altogether, and the figure above shows them as a coin. It is 50.7 per cent over 6,000 trials, with growth in the slips, which is the same as no information.

This is not an accident of the model. A first strip is a first strip. Both come from a hand that has not yet tired. Their creep is the starting creep, and that is common to both, since the rates have not begun to act. Their scatter is the fresh scatter. Nothing about tiring distinguishes them, because neither has done any tiring. If the two painters happen to have different fresh slips, then something else separates them: a difference in skill, or a brush, as measured elsewhere. In that case it is not a tie at all. But with the two painters’ fresh hands equal, there is nothing in those two strips that could say which is whose, and any further instrument in this family would be reading noise.

So there is a clean division. The ties that come from different histories can be broken, in principle, by anything that records history. The tie between two first strips comes from identical histories, and nothing that records history can break it. That gives a precise statement of what the signature in the title of the earlier essay can and cannot be: it can sign every strip but each painter’s first, and it stays that way, not for want of precision but because a signature is what a hand acquires. Two painters on their own clocks drew a related boundary around the interleaving of two sequences. This is the other edge of the same limit.

It also says what would be needed. Some fresh difference between the painters must be brought in from outside the tiring model. A habit is one such thing, when two painters have different ones. On the panels of the earlier essay they deliberately shared one, to make the rates carry the weight, and a real workshop might have the same reason to train everybody to one habit. Then the first strips are truly anonymous, and a person hoping to recover two hands in such a workshop should stop expecting to name them. Groups in a row has already found how little a panel says about who painted a whole group, even with all of its strips read. A claim about one hand’s first strip is a claim about something a panel can only record by accident, such as a mark or a mistake that is not part of the method.

A run of panels #

A single panel gives one comparison. Two painters who work together repeatedly give several, and each panel’s tied pair is an independent draw. If the same two people always come in at the same places in their own sequences, then pooling the log of the scatter across panels gives a reading with far more behind it.

With five rows and a growth of a half, one panel settles 69 per cent of ties, four settle 81 per cent and sixteen settle 95 per cent. With a growth of a quarter, thirty-two panels are needed to reach 94 per cent. With a growth of one, four panels are already close to 90 per cent. So a body of work carries evidence no one panel has, which is a familiar kind of result, and it comes with a specific condition. Pooling strips across panels assumes those strips stand at the same place in each painter’s sequence, and that is only known once the order on each panel has been read. A pooled reading needs the same rate and the same order. On panels where the order is itself in question, the pool is a mixture, and the evidence falls.

The analogy with a stepped hand’s walk is close. There too a single panel’s reading was too weak to act on, while a model of the whole work could say something firm about it, and the condition on the pool was the same: an assumption about what stays the same from one strip to the next.

Scatter alone and the order of a painter’s strips #

The third question was how much scatter knows by itself about the order. The earlier reading searched through all the orderings of each painter’s strips, using creep alone. Could scatter have found them?

With five rows, and a growth of a half, all four are in the right order only 23 per cent of the time, against 4 per cent by chance. A pair of neighbouring strips is ranked correctly 69 per cent of the time. At a growth of one, 36 per cent and 76 per cent. So scatter alone is a weak way of finding an order. It does better than nothing, by about a factor of six on the full order, and it is much weaker than the creep at any reading better than a pixel. It cannot replace the creep.

It may be better thought of as a check. Creep reads an order only up to reversal, as a tiring panel keeps its order found: a steady rate of tiring read forward or backward looks the same. Scatter has a direction. It grows with the count and it does not shrink, so it has a chance of fixing which end of the panel was painted first, and a modest one is enough: picking one of two directions, with a sum over several strips behind it, is a much easier problem than finding one of twenty-four orders. This is a useful division of labour. The creep finds the order up to reversal, and the scatter chooses the direction. That has not been measured here, and it comes with its own question about how much of the panel’s evidence it uses, which the reading below takes up.

What the slip’s growth has to be #

All these figures depend on $g$, and nothing on the panel says what it is. The fatigue reading of a strip’s scatter used a slip that tripled across a strip. That is a growth within a strip. Whether a hand that has finished one strip starts the next at the larger slip, or rests, or recovers a part of it between strips, is a separate matter, and it has not been measured anywhere. If a hand recovers fully between strips, then $g$ here is zero, however strongly it tires within each one, and a tie stays a coin. In that case, scatter differs between strips only as much as the strips’ own lengths make it differ.

That sets a limit on what can be claimed. The arithmetic above is exact, given a growth. So whether scatter breaks a tie on real panels is an empirical question about how real hands behave from strip to strip, and the most this essay can do is say how large a growth would make it detectable, and with how much evidence. With five rows and one panel, a growth of a quarter gets a tie up to 61 per cent, hardly better than chance. So a real workshop’s hands would need to show quite strong carried fatigue for one panel to say anything, while thirty-two panels could detect a growth of a quarter comfortably.

A panel with a growth large enough to be useful would also show it directly. Fitting a growth in scatter against each painter’s count is a test in its own right, and if the tie-breaking were carried out, a sensible first step would be to confirm that the growth exists at all, on strips whose order is not in question. Those are the strips with no ties. A further check on this is that a panel with no growth would give a scatter difference between tied strips that averaged to zero, so the reading can detect that its own assumption has failed.

A less steady painter separates a tie by skill alone #

Slips are independent from row to row. A hand that drifts within a strip has correlated slips, and those are partly absorbed by the fit as a creep, which would lower the scatter reading, by different amounts on different strips. A stepped hand passes for a tiring one has a large example of how much a fit can absorb.

A strip’s rows are measured with no error beyond the hand’s. Reading error adds to each strip’s scatter, by the same amount in each, so at a fixed reading error it shifts the comparison toward a tie. On a photograph read to a tenth of a pixel, with hand slips of a pixel or more, this is unimportant. It would matter on a lower-resolution picture or for a very steady hand.

The two painters have the same fresh slip. That is the assumption that makes the first strips tie, and it also keeps the later comparison simple. If one painter is simply less steady than the other, then their scatters differ everywhere by a constant factor, and a tied pair is more easily separated, as a matter of skill rather than of history. A reading that uses it should say so.

The growth is linear. A hand might tire quickly at first and then level off. That would change the sizes of these numbers without changing the argument.

Still open: whether the scatter’s direction is worth its share #

The section above claimed that creep finds an order up to its reversal and scatter chooses the direction. The two readings are not independent uses of the same rows, though. The residuals that are read for scatter are what is left after a fit has taken its habit and creep out, and using the same fit to select an order means selecting the residuals too. The order with the best fit tends to have the smallest residuals, and that can hide part of the growth the reading depends on.

The measurement that settles what is left goes through full panels whose order is unknown, searches for the order by creep, reads the scatter from the residuals of that best fit, and then asks whether the direction chosen by scatter is right as often as the strip-by-strip analysis suggests; how much the selection of the best-fitting order biases the scatter it leaves behind; and whether a reading that fits the creep and the scatter together, weighting each strip by its own slip, finds the direction and the order better than the two readings one after the other.

A signature is what a hand acquires. Anything that records history can tell two strips apart if their histories differ, and nothing can tell two first strips apart, because their history has not begun.