Least-squares — where it appears
Named by 47 essays across 8 fields — each of them below, with the objects they name alongside it.
Luther said when it would work
There is an exact condition under which a fixed three-by-three matrix converts camera raw to XYZ correctly for every spectrum in existence. It was stated in 1927, it is a theorem rather than a guideline, and no camera ever built satisfies it.
A camera profile is a fit
Since no matrix is exact, the one a manufacturer ships is a least-squares compromise over a set of surfaces somebody chose. Fitted on desaturated patches it is excellent on desaturated patches — and the sample set is a hidden argument to every camera profile in existence.
No matrix is right everywhere
The best three-by-three this sensor admits, fitted and tested on the same twenty-four surfaces, leaves a worst case of ΔE00 2.85. A control sensor built to satisfy Luther's condition reaches ten to the minus seven under the identical computation, which is what makes the first number a measurement.
Correcting colour costs noise
The matrix that turns a sensor's raw into XYZ has large off-diagonal terms of both signs, because that is what correcting a sensor which fails Luther's condition requires. Differences of large numbers are where noise grows, and the full correction amplifies photon noise by 1.66 times.
A fourth primary is a design
A display's fourth emitter is sold as more colour. Optimised instead against how far apart two hundred eyes are about its white — with the gamut held fixed so it cannot cheat — it buys agreement, and two and a half times closer together than three primaries reaching the same area, and the wavelengths it chooses are not the ones anybody would pick.
Fitted to an eye nobody has
A camera profile's residual against the observer it was fitted to is under a unit, and that is the number everybody quotes. Against a person drawn from a population it is 3.35 at the ninety-fifth percentile — and a camera with no spectral error whatever, reporting the standard observer's tristimulus values exactly, would carry 3.47. The camera is not the problem.
A gain needs a basis
Adaptation scales three signals, and which three is a choice. The basis in which a change from D65 to D50 is exactly diagonal can be computed in closed form from the two spectra, it beats every published transform on daylight by a factor of five, and it loses to all of them on a fluorescent tube.
A tolerance cannot cross a condition
If two measurement conditions disagree by seven units, the obvious repair is a correction matrix fitted between them. The best least-squares 3×3 over seventeen printed patches leaves 23 per cent of the disagreement and makes five patches worse than doing nothing — because the term it is trying to remove is proportional to how much paper is showing, and no linear map on three numbers can express that.
The chart decides the profile
A camera's colour matrix is nine numbers fitted to a set of patches somebody chose, and the number that comes with it is the error on those patches. On a chart with no chromatic range that number is 0.19 ΔE00 and the matrix delivers 1.65 — and a second matrix, indistinguishable on the chart, delivers 2.03.
A fit can be exact and empty
Every fitted object here reports one number, the residual on the data it was fitted to, and every one of them has two more that nobody publishes — how many of its parameters the data actually determine, and how large the family of equally good answers is. The third column is where the failures live.
A template cannot place a point
This collection's model of an eye is good to a few tenths of a per cent at predicting what a cone catches, which is far more than enough to place a spectrum. Asked where that eye's confusion points are, it puts the protanope's at (0.99, 0.20) against a measured (0.75, 0.25) and the deuteranope's anywhere from (1.1, −0.5) to (−18, 11) depending on which stimuli the fit was made over.
A chart decides what a camera scores
A camera profile's reported error changes by a factor of five when the test chart's saturation changes, with the camera and its matrix untouched. The elasticity is 0.67 — the same figure, to two per cent, that an entirely unrelated measurement over an entirely unrelated set of surfaces gives.
The objective nobody chose
Every camera profile here, and as far as can be told everywhere, is a linear least-squares solve in tristimulus space. That is an objective and it is on nobody's menu — it weights an error by how bright the patch is. Refitting the same matrix to minimise a real colour-difference formula improves the fit in all six, and moves the matrix, which means different pixels rather than a different report.
The chart was measured, not photographed
A camera profile is fitted so that the camera's response maps onto the chart's tristimulus values, and those values came out of a spectrophotometer. So the profile's target carries the instrument's departures and the camera does not — a mildly translucent chart read through a four-millimetre aperture supplies targets 2.55 ΔE₀₀ from the truth, which is twice the profile's own fitting error.
The chart was measured by an observer too
A camera profile is fitted so that the camera's numbers reproduce the chart's measured tristimulus values. Those values were computed through the 1931 observer, so the fit inherits every departure in this round — and the fit's own residual, at 0.19 ΔE₀₀ on the chart, is fifteen times smaller than the term it cannot see.
A corner is corrected by one row
A grey-card gain map makes the corner of a frame exactly right on grey and leaves coloured patches 1.86 colour differences wrong at twenty-five degrees. A three-by-three matrix fitted at that position halves it — and six of its nine numbers do nothing, because the moved filter edge is in one channel. The three that matter rebuild the lost red from green and blue, they carry to another lamp better than a gain map in eleven cases of twelve, and in the twelfth, a row fitted under tungsten and used in daylight, they leave the grey 9.2 off.
A proof cannot be tuned for readers who disagree
A soft proof matched exactly for the standard observer is five colour differences wrong for one reader in twenty. Giving up that exactness to tune the display's three drives for a population instead moves the ninety-fifth percentile reader by 4 to 14 per cent even when the tuning is done on the very readers it is scored against — because what readers see is mostly each other's disagreement, and three drives act on every reader at once.
A profile interpolates light
A profile is exact at its patches and wrong between them, and how wrong has been measured twice. Which way it is wrong has not. At a nine-step lattice the mean signed lightness error between the nodes is +0.136 against a mean colour difference of 0.133 — the error is not a scatter but a bias, and it lightens. The repair costs nothing measured and is forbidden by how a profile is checked: let the table be wrong at its own patches.
A strategy keeps the refit only where black prints
Refitting an inverse table's nodes against their own error halves it, and in a four-ink table the refit has to leave the separation strategy's choice of black alone. Held to the strategy, it keeps the whole of its gain wherever the strategy prints black and almost none of it where the strategy does not — and the share it keeps equals the share of the grey ramp printed with black, to within three hundredths. The start of black generation, predicted to be where the refit would fail, is where the table needs it least.
One row for every lamp costs the lamps that lose least
A corner correction fitted under one lamp is right under that lamp and can be badly wrong under another, so a converter unsure of its lamp was offered a single row fitted under several at once. Pooled over four lamps, one row holds every lamp's corner between 1.33 and 1.56 colour differences — no lamp worse than a grey-card map, and none near the 4.91 a wrong row can leave. The price falls on the white LED and the fluorescent tube, which it leaves four times worse than their own rows, because they lose the least red at the corner and the pooled row is built to repair the lamps that lose the most. Two rows, one per class of lamp, keep almost all of both.
Two channels estimate what one can only sort
A narrow sensor channel was fitted to call lamps smooth or structured, and the label turned out to be a proxy for the thing a camera cares about: its own colour error under the lamp. Scored against that error directly, the best channel is still the one at 450 nm — the band where camera and observer differ most comes close and no closer. Read as an estimate rather than a verdict, a second channel at 500 nm, useless as a vote, earns its place: together they predict the camera's error to within 0.14 ΔE00 over fifty-four lamps, and deciding when to repair from that estimate raises a quarter of the false alarms the label raises.
A little black ends the cast
A four-ink inverse table errs by a warm cast along the grey axis wherever the separation strategy prints only three inks, and almost only in lightness wherever it prints black. Real strategies start black gently, and the gentle start was expected to keep the cast until black carried much of the grey. It does not. The cast falls to a tenth of its three-ink size once black supplies a tenth to a sixth of the grey's darkening, at every toe from nothing to thirty units wide — because inside the toe the table's chord runs along the trade of black against the other three inks, which errs in lightness. What the toe does cost is the segment where black begins, which goes from the table's best to its worst.
A straightened channel repeats the one beside it
A phone's narrow sensor channel at 570 nm ranks a camera's colour error under a lamp almost as well as the one at 450, and estimates it worse than guessing the mean, because one lamp reads ten times higher than any other. Read through a logarithm, the channel is rescued: its estimate improves from ±0.57 to ±0.23. It still does not reach the 450 nm channel's ±0.16, and put beside that channel it adds nothing at all, where a 500 nm channel cuts the error to ±0.14. Straightened, 570 nm sorts the lamps as 450 nm already does; and the two lamps it reads highest have the most different errors, which no transform of one reading can tell apart.
One ratio reads a peak and a trough
Two narrow sensor channels, at 450 and 500 nm, estimate a camera's colour error under a lamp to ±0.14 ΔE₀₀; the proposal was to read their ratio as one number instead. Taken as one channel's departure over the other's, the ratio is worthless — no better than the mean, because for smooth lamps both departures are nearly nothing. Taken as the departure of the two readings' ratio from what the camera predicts for it, one number estimates the error to ±0.141, as well as the two channels as two. The reason is a sign: under every structured lamp the channels depart in opposite directions, a peak at 450 and a trough at 500, and the ratio adds them; where a cyan fill lifts 500, it subtracts them, as the error falls. And no five per cent calibration error moves any of these estimates by a hundredth.
The tube a third channel misses has no family
Two narrow sensor channels read as one ratio estimate a camera's colour error under fifty-four lamps to ±0.141 ΔE₀₀, and their worst miss is the halophosphate tube, whose trouble lies in its long-wave phosphor. A third narrow channel in the long-wave gap was predicted to halve that miss near 610 nm. No channel does, at any centre or width: the best buys under a tenth of the whole error and leaves the tube's miss above three quarters of what it was. At 610 nm the camera already predicts the tube's share almost exactly. The tube's error follows where its main phosphor band sits, and a 620 or 640 nm channel does read that — inside the tube's own family. The census has one halophosphate tube, and a lamp left out of a census that holds nothing like it is a lamp no channel can be fitted to.
A dimmer session hides a lens band in the rods
A band of extra absorption in an older lens is read by a colour-matching session as lens age or as macular pigment, and it pulls the fitted rod weight with it. The rods see the lens and not the macula, so a dimmer session, where they carry a third of the signal instead of a tenth, was predicted to tell the band from the macula. It does the opposite. With the band at 450 nm left out of the fit, the rod weight's shift grows from 1.7 of its standard errors to 2.5, the misfit a session could notice shrinks by a third, and the macula and the band trade more tightly. The rods see the band — and the rod weight, which the fit is free to move, is exactly what absorbs it. What separates the band from the macula is fitting its amplitude as a fourth number, which works in daylight already.
A second lamp pair finds a lens band's centre
A band of extra absorption in an older lens is found by a colour-matching fit only if the fit has a parameter for it, and a band parameter at a fixed wavelength finds only a band near that wavelength. Freeing the band's centre as a fifth number asks whether the matches can place it. From one daylight-against-LED session they place a band above 430 nm to within six nanometres and one below it only to eight to twelve, where moving the band and ageing the lens change the matches together — and there they barely see the band at all. A second session, the LED against a three-emitter lamp, places every centre within five and a half nanometres. The fluorescent tube's line at 436 nm helps where it sits and nowhere else.
Two stages catch the tube and bill the lamps they misplace
A narrow-channel estimate of a camera's colour error missed the halophosphate tube because the census held nothing like it, and a line that learned the tube's family from four siblings cost every other lamp. The proposed cure was two stages: recognise a lamp's family, then apply that family's own line. Given each lamp's family, it works as predicted and better — the tube missed by 0.011 ΔE₀₀, the fifty-four lamps estimated to ±0.133 against the ratio's ±0.141. Found by the nearest other lamp in the sensor's departures, the family is right for the tube and wrong for five lamps, and those five, missed by ±0.41, take the whole census back to ±0.162. The gain is real; the classifier is where it is lost.
A classifier that abstains gives up the tube
Found by the nearest other lamp, a lamp's family cost a two-stage estimate of a camera's colour error more than it bought, because five lamps were placed confidently in the wrong family. The proposed repair was a classifier that abstains — falls back on the census-wide line — unless a lamp's three nearest neighbours agree. It abstains on the wrong lamps. The misplaced ones have three agreeing neighbours, all white LEDs; the lamp it gives up is the halophosphate tube the two stages were built to catch, whose third neighbour is a cyan LED, and its miss returns from 0.011 to 0.445. Two agreeing neighbours look better than knowing every family. Chosen without looking at the lamp being estimated, the rule is worth exactly what the ratio alone is worth.
A lost lens band goes to age or the macula
Two colour-matching sessions determine the centre of a band of extra absorption in an older lens to about five nanometres, if a fit can find it. Fitted in full from eight starts across 400 to 470 nm on twenty simulated experiments, two sessions find a band at 415, 430 or 445 nm from every start in fifteen to eighteen experiments of twenty, against eighteen for a fit started at the truth. The prediction was that a low band would be lost from high starts to the macula's flank. The start barely matters with two sessions, and a lost band does go somewhere — but a band at 415 or 430 goes to lens age, three to five years of it, and only a band at 445 goes to the macula. Every loss is a worse fit than the session's best, so eight starts find the band as often as the truth does. With one session the start decides.
A family's line runs through its neighbours
The halophosphate tubes differ from one another by one number — where their main phosphor band sits — and so lie along a line in the three departures an ambient-light sensor reads. The proposal was to classify a lamp by the family whose line passes nearest, which would keep the tube a vote of neighbours gives up. It keeps the tube, under every version that does not rescale the distances. It also places eighteen white LEDs on the radiators' line, because that line — fitted to six radiators close together — runs along the LEDs' own and, extended, straight through the violet-pumped ones; and the radiators' estimate, extrapolated to them, misses the triphosphor tube by 7.6. Clamped to the stretch each family spans, the lines still cost ±0.285; given to the halophosphate family alone, its line claims a cyan LED and costs ±0.179. The one line classifier that comes in under the ratio line's ±0.141 does so by two thousandths, having placed the tube with the LEDs.
Radiators and LEDs are one family to the sensor
An ambient-light sensor's three narrow departures see the collection's six smooth radiators and forty-four white LEDs along one line: the radiators' own short line lies on the LEDs' long one. Estimating a camera's colour error family by family, the census had treated them as two, and every classifier that tried to tell them apart paid for it. Merged into one family of smooth broadband sources, found by the nearest other lamp as before, the two-stage estimate comes to ±0.127 over fifty-four lamps held out — below the ratio line's ±0.141, and below the ±0.133 the separate families managed when every lamp's family was simply given. The halophosphate tube is still caught. The radiators, which the prediction said the merge would cost a little, are estimated better inside it than inside their own family, because a three-coefficient line fitted over five radiators was worse for them than the census-wide ratio line.
New LEDs miss the line their cousins drew
An ambient-light sensor's three departures can estimate a camera's colour error under a lamp, and merging radiators and white LEDs into one smooth family brought that estimate to ±0.127 over fifty-four lamps, each left out in turn. Every one of those lamps had relatives in the pool. Fourteen lamps the census never saw test what the relatives were worth. The prediction was that new smooth lamps would land on the line and tinted incandescents fall off it by four tenths or more. The radiators and daylights land on it, within ±0.13 each. The tinted lamps do too — the pink-glass incandescent, placed among the LEDs by its nearest neighbour, misses by 0.12. The lamps that fall off are the new white LEDs: ±0.20, half again the census's error, two of them by a quarter. Their misses follow the sensor's 640 nm channel at a slope of −0.72, explaining four fifths of their scatter, because the line learned what that channel means from LEDs varied only in their blue pump. Adding the fourteen to the pool does not help.
LEDs varied in their phosphors fall off every line
The merged smooth family estimates a camera's colour error from an ambient-light sensor's departures to ±0.127, and five new white LEDs, varied in their phosphors where the census's forty-four vary their pump, missed by ±0.196 because the family's line had learned almost nothing about the sensor's 640 nm channel. Rebuilt on forty-eight LEDs varied in pump, converter and red phosphor, the line learns it: its 640 nm weight goes from −0.13 to −0.74, most of the way to the −0.85 the new LEDs implied. The prediction was that the new LEDs would then land within ±0.13. They miss by ±0.229, worse than before, crossing to the other side of the line, and the census's own held-out error more than doubles, to ±0.294. No pool of LEDs restores it, and no line in the sensor's departures reads the grid's own LEDs better than ±0.29. Across the grid the camera's error is a V about the Planckian locus, rising away from it on the green side and on the pink — and every one of the census's LEDs sat on the green side.
Two nanometres hide the yellow's notch from any fit
A plain Gaussian fit admits the reversed tube's 578 nm yellow line at every threshold on a two-nanometre table, and the ceiling built on its width fails from a column depth of two and a half. A fit with the notch in it — an emitted Gaussian with an absorbed one taken out of its centre — might see what the plain fit cannot. The prediction was that two nanometres leave too few entries to determine the notch, and one nanometre finds it from a depth of two. Two nanometres hide it completely: the best notch fit carries a negative notch, a spike, at every depth, and the readings a two-standard-error rule calls reversed are the same noise draws at every depth, each fitting a notch that erases the whole line. One nanometre finds it sooner than predicted — in every reading from a depth of one and half of them at a half — and its fitted depth climbs with the column. But its standard error cannot be trusted with seven entries and four numbers: the rule calls a third or more of unreversed readings reversed. The notch's fitted depth itself separates them cleanly.
The notch fit declares the true width, and the true width runs out
A lamp's table declares the narrowest line width its peaks admit, and builds on it a ceiling for how high any line can rise between the entries. A reversed tube breaks both: its lines are notched by cooler gas, a plain fit reads each notched line as a wider one, and on a one-nanometre table the ceiling falls below the true peak from a column depth of a half, until at two the table refuses every line and declares nothing. A fit with the notch in it reads the yellow's notch on that table, and the proposal was to declare the line it finds underneath — notch taken back out — with the prediction that the ceiling then holds to a depth of three. It holds to two. The notch fit returns the tube's own 1.2 nm core within two hundredths of a nanometre at every depth, clean and in noisy readings, and the ceiling built on it is safe to two and fails from two and a half, in the window beside the reversed blue line. So does the ceiling built on the true width itself, at the same depths to within a hundredth. The repair is perfect, and what fails past two is the ceiling's rule, which assumes the lines it bounds are not notched.
A ceiling that knows the notch holds to the deepest column
A lamp's table promises a ceiling for how high any line can rise between its entries, and for a reversed tube — every line notched by a cooler column — that ceiling fails on a one-nanometre table from a column depth of two and a half, because its rule assumes the line it bounds has no notch. The notch fit already reads the yellow line's notch, and the proposal was to build the ceiling for a notched line instead: in each window, the factor for a slit-blurred line carrying that window's notch. The prediction was that it holds to a depth of four. It holds to five, the deepest column the census runs, at every depth and in every one of 440 noisy readings. The fit's depth must be read as an optical depth, −ln(1 − d); taken as the first-order depth it fits as, the ceiling fails from four and a half. Part of the safety is the fit's own error: read that way, a deep notch comes out too deep — 4.25 for a true 2.29 at depth five — and an overstated notch only raises the ceiling.
A coarse table can borrow a notch, but not a phantom
A one-nanometre table can bound a reversed lamp by building its ceiling for the notch its own fit reads. A two-nanometre table cannot read the notch, and its ceiling, built for an unnotched line, fails from a column depth of three and a half. The proposal was to borrow the notch from the finer table, and the prediction that the coarse ceiling then holds to five with a wider margin. It does: 8 to 42 per cent above the reversed lamp's true peak at every depth, against the fine table's 1 to 29. But borrowed from the fine table's noisy readings, one notch in 440 sinks the coarse ceiling below the true peak — a notch of 0.13 the fit read in noise where there was no column at all. Under a wide slit a shallow notch flattens a line's top, two flanking entries see more of it, and a ceiling that assumes the notch is set lower; the phantom set it too low. Borrowing only notches the depth rule calls — deeper than 0.2 — keeps all 440 safe; borrowing those more than twice their error does not.
One session has a second valley
Every essay on older observers since the second session has read its fitted lens bands and macula shifts from one census: nine simulated observers, three designs, twenty noisy experiments each, every fit a damped descent from two starts. Two of forty experiments refitted in the last essay found a better fit than the census had. The prediction was that the census misses one fit in twenty, almost all where the band is small, and that the misses change no count it reported by more than a few hundredths. Refitted from three more starts, 66 of its 540 experiments have a better fit — 45 of 180 under one session, 9 and 12 under two and three. Sixty of the 66 better fits move the macula's peak more than two nanometres further short, most shrinking the band, and 31 lie lower by under one unit of χ²: a second valley, nearly level with the first. Two sessions' counts stand to within one or two experiments. One session's do not: with the best fits an unmoved macula is reported moved in eight experiments of twenty, where the census said one.
Seven channels end the trade and lose the tube
A three-channel ambient sensor fitted to a room's forty-seven lights trades its error on LEDs against its error on lamps, and a fourth channel moved that trade without closing it. The prediction for more channels was that six, spread from 445 to 650 nm, bring every lamp, mixture and LED under a ΔEITP unit with the lamps within 0.0025 in the CIE 1960 plane, and that nothing short of eight reads the halophosphate tube. Six channels leave the 2,200 K lamp beyond a unit and the lamps at 0.0040; seven bring every lamp, mixture and LED in, with the LEDs and lamps both under 0.001, and the trade is gone — both families improve together. The halophosphate tube is read from six. But the triphosphor tube, held out of its own fit, is read only by three and four broad channels and lost by every finer sensor, its tint growing to thirty-three units at twelve channels. A sensor learns the lights it was fitted on, and a finer one extrapolates further from a light with no relative in the census.
Relatives, not twins, teach a sensor a tube
An ambient sensor of seven evenly spread channels, fitted to a room's forty-seven lights, reads every lamp, mixture and LED within a ΔEITP unit with each held out of its own fit, and loses the triphosphor tube, which nothing else in the census resembles — thirty-three units out at twelve channels. The prediction was that four triphosphor siblings, the same phosphors mixed for 3,000 to 5,000 K, would bring the tube within a unit at seven and eight channels and stop its rise past eight. They do both: the tube reads at 0.04 with seven channels and 0.06 with twelve, and seven or eight channels then read all fifty-one lights. A twin is not what does it. Three siblings, every one warmer than the tube and the nearest sixty mireds from it, read it within a unit at every count from five. What the siblings cost is paid at the finest sensors: at twelve channels the Planckian lamps' worst error rises fivefold, while from five to eight it moves by under a quarter.
A bound on the macula trades false moves for hidden ones
Fitted thoroughly, one session of an older observer's colour matches finds a second valley in a quarter of its experiments — the macula's peak moved ten or more nanometres from the model's, where no real macula sits — and reports a moved macula where there is none in up to eight experiments of twenty. The prediction was that holding the macula's shift within six nanometres either way, the range real pigment mixtures span, restores every observer's one-session counts to within two of the census's and leaves about one fit in four pressed against the bound. The false moves vanish: an unmoved macula is reported moved in one, two and one experiments of twenty for the three bands, against eight, six and two unbounded. Seven of nine observers keep their count of moved maculas within two of the census's. But only one fit in six ends on the bound, and beside the band at 445 nm the bound hides real moves: a macula truly three nanometres short is found in three experiments of twenty, where the census found seven. Two sessions barely notice the bound.
A soft prior on the macula frees the lens band
One session of an older observer's colour matches, fitted thoroughly, can drift into a valley where the macula's peak sits ten or more nanometres from any real eye's, and a hard bound at ±6 nm removes those false moves at the cost of hiding some real ones — beside a lens band at 445 nm, a macula truly three nanometres short is found in three experiments of twenty where the census found seven. The prediction was that a Gaussian prior four nanometres wide, which lets a fit go further when the matches insist, keeps the false moves away and restores those real ones. It keeps the false moves away — none, one and one for the three bands. It does not restore the real moves: it finds them as seldom as the bound or more seldom, two of twenty beside 445 nm, because it pulls every fitted shift towards nought, to a mean of 1.2 to 1.9 nanometres for moves of three. What it buys instead is the lens band. With the macula held near the standard, the band beside it is found in ten to twelve experiments of twenty, where the census found three to seven.
A wider prior gives the moves back by spending the band
One session of an older observer's colour matches, fitted under a Gaussian prior four nanometres wide on the macula's peak shift, finds the lens band beside the macula in ten to twelve experiments of twenty and real three-nanometre moves of the macula as seldom as a hard bound does. The prediction was that a prior six nanometres wide keeps the false moves at two or fewer, finds the real moves within two of the bound's counts and keeps two thirds of the band's gain, and that at eight the far valley reopens at 415 nm. Three parts hold. The fourth fails: six nanometres keeps 55 per cent of the band's gain. Read across the widths the prior traces one trade — 14 real moves and 44 bands at four nanometres, 22 and 34 at six, 25 and 25 at eight, against the unconstrained fit's 27 and 22 — and the hard bound, at 19 and 20, is worse than six nanometres on both.
A second session leaves the prior one job
Under one session of an older observer's colour matches, a prior on the macular pigment's peak shift trades the lens band against real moves of the macula, six nanometres wide at the knee of the trade. Two sessions already find the band in twelve to twenty experiments of twenty and real three-nanometre moves in twelve to eighteen, and still report an unmoved macula moved in two to three. The prediction was that the six-nanometre prior, applied to two sessions, removes those false moves and costs nothing else. It removes none of them — two, two and none remain, about the rate at which a test at twice the standard error calls a move by chance. It costs nothing at five observers of six. Beside 415 nm, with the macula moved short, it finds four fewer moves: three of the best fits there were in the far valley, eight to twelve nanometres short, and counted as moves because their sign was right. With two sessions the far valley is the prior's only remaining job.
A third session finds the edge band and counts bands misplaced
Two sessions of colour matches find an older observer's lens band in eighteen to twenty experiments of twenty at every place but one: a band at 415 nm, beside the lens's own short-wave edge, found in twelve. The prediction was that a third session — a phosphor LED set against a lamp with a cyan line — finds it in at least seventeen, and that the experiments losing it under two sessions fit the lens three or more years old. Three sessions find it in eighteen, and every loss under two did fit the lens older, but by under three years in five of eight. The count hid something else. It credits a band wherever a fit puts it, and up to twelve of 180 fits put it sixty to eighty nanometres from its place, on the macula's long flank. With three sessions the six-nanometre prior no longer closes that valley.
A band held near the lens closes the valley the prior could not
Three sessions of colour matches find an older observer's lens band beside the lens's own edge as predicted, but twelve of 180 fits place the band sixty to eighty nanometres away on the macula's long flank, each with the macula fitted far short — a valley the six-nanometre prior on the macula's shift no longer closes. The prediction was that holding the band's centre to 400–460 nm, where a lens band can be, empties that valley, finds the 415 nm band with the macula short in at least eighteen experiments, and raises the moves found beside 430 nm from nine to twelve. The first two hold: no fit is left in the valley, and the band is found in nineteen. The third does not: 430 nm stays at ten. And the valley turns out to have been nearly level with the truth — holding the band costs ten of the twelve fits under four units of the fit's chi-squared, and every one's macula comes home.
Named alongside it
The objects these essays reach for when they reach for this one.
Camera rawModelling assumptionHeld-out validationIdentifiabilityCalibrationFluorescentMacular pigmentWhite LEDColour matrixStandard errorLens yellowingLuther condition