Quadrature is exact in one room

A tolerance with an observer in it added six observer departures and got about three colour differences on an ordinary sample. The adding was done in quadrature — the square root of the sum of the squares — which is what anybody does with contributions taken to be independent, and which assumes the six point in mutually perpendicular directions.

Adaptation turns more pairs off than on measured the angles. Over the fifteen pairs they run from 18 degrees to 179, five pairs are obtuse at complete adaptation and nearly none is near a right angle. So the assumption underneath the allowance is plainly false, and the question is what it costs.

Wrong in both directions, and the room decides which #

Quadrature is exactly right at one degree of adaptation, too small below it and too large above it — so an allowance built that way is neither conservative nor anti-conservative but one or the other depending on the room, which is the property a safety margin is least able to absorb.

• The crossing is at a degree of 0.8618, an adapting luminance of 22 candelas a square metre, which is a lit living room.
• In a cinema the measured combination is 1.079 times the quadrature sum and under an overcast sky it is 0.864 — a swing of a quarter, with the same six eyes and the same surfaces.
• The allowance itself is a room. The six cost 2.30 colour differences under the sky, 3.24 in an office, 5.27 in a living room, 9.24 in a dim room and 12.45 in a cinema — a factor of 5.4.
• On individual surfaces quadrature is out by much more than the aggregate says. In a cinema the middle nine tenths of the surfaces run from 0.68 to 1.49 times the quadrature sum, and the weighting is the disagreement is why the spread is a property of the surfaces rather than noise.
• On the audit’s banded family the crossing moves to a degree of 0.9884 — 250 candelas a square metre, brighter than any room a sample is judged in, so quadrature understates the allowance everywhere on those surfaces.

What quadrature assumes, in the form it is assumed #

Six deviations added head to tail give a total, and the total’s length is what the six actually cost together. Three rules of thumb bracket it.

Summing the lengths assumes every departure points the same way, and is an upper bound. Taking the largest alone assumes only one matters, and is a lower bound whenever the others do not cancel it. Quadrature sits between them, and it is exact when every pair of deviations is perpendicular: the square of the total is then the sum of the squares with no cross terms left over.

The cross terms are the whole of the difference. The total’s squared length is the sum of the squared lengths plus twice the sum, over pairs, of each pair’s inner product — which is each pair’s two lengths times the cosine of the angle between them. So quadrature is right exactly when those cosines cancel, which needs either that every angle is a right angle or that the acute ones are balanced by the obtuse ones, weighted by how long the departures are.

The second is what happens here, and it happens at one place on the dial.

Below a degree of 0.8618 the cosines are on balance positive and quadrature is too small; above it they are on balance negative and quadrature is too large. The crossing inverts to an adapting luminance of 22 candelas a square metre in an average surround, which is a lit living room and is dimmer than every room a colour decision is supposed to be made in.

That is the first practical reading. In a viewing booth, an office or a design studio, quadrature overstates the observer allowance, by between one and fourteen per cent depending on how bright the room is. In a warehouse aisle or a shop floor it understates it.

The allowance is a room before it is a number #

The sign of quadrature’s error is the smaller of the two effects here, and the larger one is worth stating first because it swamps it.

The same six eyes and the same 120 surfaces cost 2.30 colour differences under an overcast sky and 12.45 in a cinema. An allowance of three, which is what gets quoted, is this calculation in an office and in no other room; a living room is 5.27 and a dim room 9.24.

So a specification quoting an observer allowance without a room has left out a factor of five, and the fourteen per cent that quadrature is wrong by is a correction to a number whose first digit is already wrong. The order to fix these in is obvious and is not the order they are usually discussed in. Two filters cancel only in a bright enough room made the same point about one pair and put the factor at twelve; six departures give five, which is smaller because the shape-like departures barely move.

The comparison with the largest departure alone is the other useful column. In every room the largest departure alone is between two thirds and three quarters of the measured combination — 2.15 against 2.30 under the sky, 8.82 against 12.45 in a cinema. An allowance built from the worst single departure and nothing else would be wrong by a quarter to a third, which is less wrong than leaving out the room and more wrong than quadrature.

Which pairs carry the error #

The cross terms are not an abstraction to be bounded. They are fifteen numbers, they can be attributed to pairs, and almost all of them are carried by three.

In an office the fifteen sum to −1.43 against a summed square of 10.51 — fourteen per cent — and the largest single term is −2.21, which is larger than the total. Unadapted the same fifteen sum to 92.7 against a summed square of 744.7, with individual terms of +670 and −264. The cross terms are small in every room not because they are individually small but because they cancel, and the cancellation is between named pairs rather than a statistical accident.

The three largest are the same three at both ends of the dial and they are the three that involve two filter-like departures. Unadapted, the lens against the macular pigment contributes +670 while the field size against the lens contributes −258 and the field size against the pigment −264. Adapted, the lens against the pigment has swung to −4.18 and is the largest negative term, while the lens against the rods at +2.08 and the field size against the lens at +1.41 are the largest positive ones.

That is the fifteen-pair picture read as arithmetic rather than as angles, and it says something the angles alone do not. Quadrature’s error is dominated by three pairs whose cross terms are each an order of magnitude larger than the error they leave, so a small discrepancy is being produced by a near-cancellation of large quantities — which is the condition under which a small discrepancy is least stable. Change one departure’s size by a tenth and the total moves by far more than a tenth.

So the crossing at a living room is not a coincidence of fifteen small effects finding a balance. It is the degree at which one large positive term — the lens against the macular pigment, still mostly shared yellowing — has shrunk enough to stop outweighing the two large negative ones. Two departures that partly cancel is where that pair’s behaviour was first priced, and it is doing the same work here one level up.

The median sits where the surfaces do not #

Everything above is a ratio of two medians over 120 surfaces, and a ratio of medians is not what happens to a surface. The two statistics disagree, and the disagreement is larger than either effect measured so far.

Taken surface by surface, the middle nine tenths of the ratios run from 0.71 to 1.05 under the sky and from 0.68 to 1.49 in a cinema. The median of those per-surface ratios is 0.927 under the sky and 0.986 in a cinema — below one in every room, where the ratio of medians crosses one at a living room.

Those are two different questions and both get asked. How large should a blanket allowance be for this whole product range? is a question about aggregates and the ratio of medians answers it. Is the allowance right for this sample? is a question about a surface, and the answer is that quadrature is out by up to a half in either direction and which direction depends on the surface’s own shape.

This is the shape a mean is not a difference is about, arriving in a new place. An aggregate statistic that sits near one is not evidence that the thing it summarises is near one; it is evidence that the errors on either side are balanced, and a specification applied to one sample at a time never sees the balance.

The banded family moves the crossing out of every room #

The 120 smooth natural reflectances are the set the pairwise work was done on. The audit’s own family — surfaces with a single absorption band, at four reflectance levels — is closer to what a specification covers, and it puts the crossing somewhere else entirely.

On banded surfaces quadrature is too small in every room a sample is actually judged in. The ratio is 1.173 in a cinema, 1.168 in a dim room, 1.107 in a living room and 1.036 in an office; it only falls below one above about 250 candelas a square metre, which is brighter than a graphic-arts viewing booth.

The reason is the same reason the pairs behaved differently on this set. A banded surface makes the two filter-like departures disagree less about shape, so their residuals point more nearly the same way, so more of the cosines stay positive at high degrees of adaptation. Two filters cancel only in a bright enough room found the extreme case: on banded surfaces the lens and the macular pigment never cancel at any degree.

The costs themselves barely move between the two sets — 2.41 against 2.30 under the sky, 13.13 against 12.45 in a cinema — while the relation to quadrature inverts. That is worth noticing on its own: the allowance is robust to which surfaces it is computed over, and whether the usual way of computing it is conservative is not.

What an allowance would have to be #

Three things follow, and none of them is a correction factor.

Name the room. An observer allowance is a function of the adapting luminance before it is anything else, and quoting it without one leaves out more than every other effect here put together. A tolerance has no light level makes the same point about the tolerance the allowance is added to, and the two multiply. The rooms a specification is read in are computable from their illuminance — a perfectly diffusing grey of twenty per cent returns illuminance times reflectance over π — so a booth at 2,000 lux is 127 candelas a square metre, an office at 300 lux is 19 and an aisle at 100 lux is 6.

Do not rely on quadrature being conservative. It is not conservative; it is right in a living room and wrong in both directions on either side, and on the banded surfaces a specification is most likely to cover it is too small in every ordinary room. A specification that wants a margin has to put one in, rather than hoping the arithmetic supplied it.

And say which question the number answers. A blanket allowance over a product range and an allowance for one sample are different numbers with different spreads, and here they differ by more than the room does. An observer is a contract argued that the standard observer is an agreement rather than a description; an allowance is the part of that agreement that says how much disagreement is tolerated, and it cannot be stated without saying over what.

How the six were combined #

The six departures are the audit’s own, each a pair of observers differing in one respect, with the basis change left out because it is a change of curves rather than of an eye. Each deviation is read on each surface under the daylight source, divided by its own white’s luminance, and adapted towards D65 by CAT16 at a stated degree.

The measured combination is the price of the sum of the six deviations, taken as a single displacement from the surface’s own reading; quadrature, the sum and the largest are computed from the six individual prices on the same surface. Every price is ΔE₀₀ at the reading, so all four quantities are in one unit and are comparable. The aggregate figures are medians over the surface set and the per-surface figures are the distribution of the ratio; the two are reported separately because they disagree.

The crossing is found by bisection on the degree, and the room it implies is the model’s own formula for the degree inverted — D = F · (1 − exp((−La − 42) / 92) / 3.6) with F at 1 for an average surround.

What this does not settle #

The six departures are two-standard-deviation constructions rather than a population, so the combination is what a particular constructed observer differs from the standard one by, not what a distribution of readers does. A population would put a distribution on each departure and the cross terms would then be expectations rather than products; whether that changes the sign of the aggregate error is a calculation nobody here has run.

The degree comes from CIECAM16’s formula and has no reader’s eye in it, so both the crossing and the rooms it names inherit the standing limit on every result of this shape — the one a discount nobody measured records.

And the surfaces are two constructed families. The costs agree closely between them and the relation to quadrature does not, which is the honest summary: the conclusion about the allowance’s size is robust and the conclusion about quadrature’s direction is a property of the surfaces, and a real product range would need its own computation.

Still open: whether a real product range sits with the smooth set or the banded one #

The two surface families give opposite answers to the practical question. On smooth natural reflectances quadrature overstates the allowance in every room a colour decision is made in; on banded surfaces it understates it in all of them. A specification has to know which of the two its own products resemble, and that is a measurement rather than a choice.

It is a small measurement. A few hundred measured reflectances from the range a specification covers — a dyehouse’s shade library, a paint manufacturer’s fan deck, a press’s ink set on its own substrate — put through the same computation return the crossing directly, and the crossing is the whole answer: above it quadrature is safe and below it it is not.

The more interesting outcome would be a range that straddles. If a product set’s surfaces have crossings on both sides of the room it is judged in, then no single allowance is conservative for the range, and the honest specification is one that states the allowance per shade rather than per contract — which is a change in how a tolerance is written rather than in what it says.

An assumption inside an arithmetic step is still an assumption #

The habit is about where modelling decisions hide.

Nobody writes down that the observer’s departures are mutually perpendicular. What gets written down is a square root of a sum of squares, because that is what one does with independent contributions, and the perpendicularity is inside the operation rather than beside it. It is a claim about the world wearing the clothes of a calculation.

The move is to ask what each arithmetic step would have to be true for it to be exact, and then measure that thing directly. Quadrature is exact when the cross terms cancel; the cross terms are cosines; the cosines are computable from the same deviations the lengths came from, at no extra cost. The measurement was one loop away from a number already published here.

The failure mode is to audit the inputs and not the operator. Every one of the six departures here was measured, sourced and priced carefully, and then combined by a step nobody examined — which turned out to be the step whose error changes sign inside the range of rooms the whole exercise is about.