What a second model changed

This site is built out of libraries that each model one thing carefully. The spatial filter has no state. The appearance model has no clock. The difference formula has no population. The printing model has no observer position, and the camera has no people in it.

Every one of those absences is defensible on its own and every one of them is a claim the model cannot make. So a collection assembled from them will state single-model answers with no way of knowing which of them a second model would have moved — and no way of knowing which of them were never answers at all.

There is already an instrument for this shape of question. An earlier audit took the seven claims here that read a filtered signal and asked each of them twice, at a point and as components, because the two readings can differ by a factor of a hundred. Its value was not the seven answers; it was that the question could be asked of the whole collection at once and that the answer sorted the claims into two groups nobody had noticed were different.

The claim #

The kind of thing a second model supplies predicts what happens to the answer, and the subject does not.

• Ten of thirteen claims move by half again or more. The largest is a factor of twenty-one and the smallest is a third.
• Three of them are not a matter of degree. The first model’s answer is exactly zero because it has no variable for the quantity, so the second model did not correct an answer — it supplied one.
• Every one of those three is a join that supplied a state or a device, and not one of them is a join that supplied a population or a second channel. There are five of the latter.
• And one row barely moves, which matters as much as the rest: an audit in which every claim moved would be an audit of the auditing.

What a row is #

A claim one of these essays makes, the number one model gives for it, the number two models give, and what kind of thing the second model supplied. Every number is computed here from the same libraries the figures are drawn from, so a row whose arithmetic stopped agreeing with its essay stops the build rather than printing something stale.

The four kinds are not subject areas. They are shapes of absence — the four ways a model can be missing something rather than merely approximating it.

A state is a variable that carries the past: a clock, a position on the retina, an adaptation pool with an extent. A model without one is not a model of a slightly simpler system; it is a model of a system that has no history.

A device is a thing in the world the model treats as a condition: a lamp with a temperature, a booth with a geometry, a camera with a shutter.

A population is a spread where the model had a point. Two hundred eyes instead of one.

A second channel is the same computation run again on a different axis: the red–green and blue–yellow filters beside the luminance one, or an eye’s reading beside an instrument’s.

The three with no answer #

Whether a metameric match survives being looked at differently. One observer says a match is a match — the residual is 2 × 10⁻¹⁴, which is floating-point zero. An observer with a position on the retina says the same pair is ΔE00 5.9 apart ten degrees off axis. The first model is not wrong about the residual; it has no variable in which a position could be written.

What a stated degree of adaptation is. The appearance model as written predicts no further change once it has finished, because it has no time in it. The same model with a clock predicts 1.67 more units still to come. The first answer is zero and it is zero by construction.

How much colour a viewing booth adds to a judgement. A booth read as a stated illuminant contributes exactly nothing across its sample plane. A booth read as the phosphor-converted luminaire it is contributes 2.4 units.

In all three the first model’s zero is not a measurement. It is the shape of a variable that is not there, and it reads as a zero because a quantity nobody computed and a quantity that came out zero are written the same way.

Why populations and channels never do that #

A population is a spread around an answer the first model already gave. Ask two hundred eyes about a colour and the median is near what one eye said, because the population was constructed to have that property and because an average of people is what the standard observer is. So a population join can widen a number, shift it, or reverse a ranking — and it cannot produce a number where there was none, because the thing it varies is already an argument.

A second channel is the same computation run on a different axis. The chromatic filters have the same form as the luminance one with different constants; an eye’s reading of a halftone and an instrument’s are both integrals. So a channel join can change an answer by a large factor — the second largest non-silent row here is one — and cannot produce an answer from nothing either.

A state or a device is different in kind. A clock, a retinal position, a lamp’s temperature: none of these is an argument the first model takes, so there is no value of anything that would make the first model produce the second model’s answer. That is the division, and assertTheSilentClaimsAreStatesAndDevices holds both halves of it — every silent row is a state or a device, and no population or channel row is silent.

The one that survives #

Naming, at a factor of 1.31.

Changing the distance function renames 20.5 per cent of the displayable gamut, and changing the room renames 26.8. Those are close, and the closeness is the result: the partition was already known to depend on an unrecorded choice, and it depends on a second one about as strongly.

An audit in which every row moved would be evidence about the audit rather than about the claims — it would mean the second models were chosen to be the ones that made a difference. A row that confirms a single-model answer is what makes the moving rows readable, and assertSomethingSurvives requires at least one.

Reading the table the other way #

Sorting by factor rather than by kind puts the rows in an order with no subject structure at all.

The largest is fading, at twenty-one: a temporal filter says a stabilised pattern keeps all of its contrast and a pool with an extent says a coarse one keeps five per cent of it. Second is a halftone against an eye, at eighteen. Third is a display’s fourth primary, at seven.

Those three are a vision result, a printing result and a display result, and they have nothing in common but that in each case the second model held something the first had no slot for or ran the same computation on a different detector. The two printing rows in the table — the halftone at eighteen and the booth, which is silent — are further apart than either is from rows in other fields.

So what a model is missing predicts more about what happens to its answers than what the model is about does. That is the finding, and it is the kind of thing only an audit across a whole collection can see, because within one subject there is nothing to compare against.

The thirteen rows #

The units differ from row to row and deliberately so: each is the unit its own essay argues in, and forcing them into a common one would be a fifth model nobody asked for.

Reading down the second column is the whole point. There is no clustering by field — but there is a relation between the kind and where the row sits, and it is not only at the top where the three with no answer are two states and a device. The section after next is that relation, which had to be written after the table rather than before it.

The kinds do predict how far, and the table says so #

Two sentences above claim the ordering has no structure in it — that there is no run of one kind and that the kinds predict whether an answer exists rather than how far it moves. Reading the sorted table by kind rather than by row withdraws both.

Number the thirteen rows from the largest factor to the smallest. The state rows occupy positions one, two, four, twelve and thirteen — that is, four of the five most extreme places in the table, at both ends of it. The population rows occupy positions six, nine and ten, which is the middle and nothing else. The two channel rows are at five and seven; the three devices at three, eight and eleven. That is not a table with no structure in it. It is a table in which one kind holds the extremes and another holds the centre, and the two kinds in between are in between.

The spreads say it more sharply than the ranks do. Setting the silent rows aside, the three surviving state rows run from a factor of 1.3 to a factor of 20.8 — a range of sixteen. The devices run from 2.3 to 4.7, a range of two. The populations run from 3.0 to 6.8, a range of 2.3. The channels run from 5.3 to 18.2, a range of 3.4. A state is between five and eight times more variable in its effect than any other kind of second model, on this table, and that is the pattern the essay’s own thesis predicts without saying so.

It predicts it for the reason the essay gives about silence, extended one step. A population is a spread around an answer that already exists, so it widens that answer by roughly the population’s own width — a moderate factor, every time, because the width of a population of eyes is not a free parameter. A second channel is the same computation with different constants, so it moves an answer by roughly the ratio of the constants — again a moderate factor, and again for a structural reason rather than by luck. Both of those are bounded operations on an existing answer, and bounded operations produce a narrow range of factors.

A state is not an operation on an existing answer at all. Adding history to a model either engages the phenomenon or does not: if what is being computed depends on what happened before, the answer changes utterly or arrives from nowhere; if it does not, the state sits there and the number barely moves. There is no mechanism by which a state produces a middling change, which is why the state rows are at 20.8, at 2.1, at 1.3 and at two silences and at nothing in between.

So the finding is one step stronger than the essay makes it. It is not that the kinds sort the claims into silent and not silent while the factors are noise. It is that the kinds sort the claims by variance: two of the four move an answer by a predictable moderate amount, and one of them is a coin toss between everything and nothing.

That changes what the table is good for at the end. The essay closes by using it to decide what to build next, and predicts weakly that a state join will produce an answer where there was none. The variance reading sharpens the advice and makes it less comfortable. A state join is the highest-value thing to build and the least predictable — three of the five here paid enormously and two paid almost nothing, and there is nothing in the table that says in advance which a given one will be. A population or channel join is the safe purchase: a factor of three to seven, reliably, with no chance of a silence and no chance of a twenty.

What was computed, and how #

Every row’s two numbers are computed live from the libraries the essays draw their figures from — none is transcribed. That is the property that makes an index like this worth more than a list of opinions, and it is the same discipline the refutation index runs on.

A row is marked silent when the first model’s answer is under a thousandth of the second’s. That threshold is a judgement and it is not a close call anywhere in the table: the silent rows are at 10⁻¹⁴, at exactly zero and at exactly zero, and the smallest non-silent first answer is 0.61.

The factor is taken whichever way round is larger, so a second model that reduces an answer counts the same as one that raises it. Two rows do reduce — the fading residual and the screen’s ceiling — and treating a reduction as a smaller change than an increase of the same ratio would be an artefact of which way the subtraction went.

Three rows, drawn #

The table is a list of ratios and the ratios are not the argument. Three of the rows are worth seeing as pictures, because each is a different one of the four kinds and the difference between them is visible.

Where it stops #

The thirteen rows are the thirteen joins one phase made, which is not a sample of anything. A claim enters this table because somebody built the second model for it, and somebody built the second model because the join looked interesting — so the table is biased toward joins that pay, and the ten-of-thirteen moving is not an estimate of how often a second model matters in general.

What is not biased is the division. Whether a silent row is a state or a population is decided by the arithmetic rather than by the choice to compute it, and the fact that every silent row falls on one side is a property of the four kinds rather than of the selection.

The four kinds are themselves a construction and there could be a fifth. A second observer — not a population but a different standard — is arguably its own kind, and so is a second geometry. Both are represented here inside the existing four and both could be argued into their own.

And a factor is a crude summary. Two answers a factor of two apart in a quantity where two is a tolerance are further apart in any sense that matters than two answers a factor of twenty apart in a quantity nobody acts on. The table’s ordering is by ratio because ratio is the only thing comparable across thirteen different units, which is a real limitation and is the same one every audit of this shape has.

Who found it, and when #

Nobody, because it is a question about a particular collection of models rather than about colour.

What it borrows is a habit, and the habit has a literature in other fields. A sensitivity analysis asks how an answer moves when an input moves; a structural sensitivity analysis asks how it moves when the model changes. The second is much rarer, because it needs two models rather than one, and building the second model is the expensive part.

What makes it cheap here is that the second models exist for their own reasons. Every row in this table is a computation some essay needed, and the audit is what happens when they are all asked one question afterwards. The cost was one file and the answer was a division nobody had.

Where the ladder goes next #

The table has a use beyond describing itself: it is a way of deciding what to build next. A model with no state, no device, no population and no second channel is a model with four things that might be missing, and the audit says which kind of absence has historically produced the largest surprises here.

The obvious application is to the models this site has that have not been joined to anything. The spectral rendering machinery has no observer position. The gamut solids have no population. The instrument model has no viewing geometry. Each of those is a row that could be added to this table, and the audit predicts — weakly, from thirteen points — that the ones supplying a state will be the ones that produce an answer where there was none.

That is a prediction with a stated basis and a stated weakness, which is the most this kind of audit can offer. The alternative is to keep computing single-model answers and to keep not knowing which of them are zeroes.