A level for each temperature lets the curve be the error's

A camera’s colour error under a white LED can be read before the photograph is taken from an ambient sensor’s narrow channels, in two stages: the LED’s red phosphor from the sensor’s 600 nm departure, then the camera’s error from the estimated recipe by a line each pump’s LEDs draw. The converter gives the red phosphor away built that reading on thirty LEDs exactly on the Planckian locus and read their error within ±0.074 ΔE₀₀. Real LEDs sit in a bin a few thousandths of Duv either side of the locus, and across a bin, the camera’s error bends with the temperature found the line needing a slope and a curve in Duv for each of the three temperatures — ten numbers a pump — to read LEDs at ±0.003 within about ±0.05.

The bin’s curve holds between its points and breaks beyond them built 119 more LEDs at Duv values that line had never seen. Inside the bin, at ±0.0015, it read them within ±0.045 and ±0.043; at ±0.006, twice the bin’s edge, within only ±0.153 and ±0.161 — though the camera’s mean error at every temperature stays on the parabola through its three fitted values out to ±0.006 within 0.02. The line’s curve was not the error’s. Every pump’s line shares a constant, the estimated red, the mireds and their product across all three temperatures, those four terms cannot place each temperature’s level exactly, and the line had used the square of the Duv to put the bin’s edges right while leaving the locus LEDs +0.068 out at 4,000 K and −0.105 at 6,500 K. A parabola laid through that gap quadruples it at twice the distance.

The repair the mechanism asks for is small. A level for each temperature can be given by one more term, the square of the mireds, which with the mireds already in the line lets each pump’s error take any value at each of the three temperatures. The prediction was that with that term added the locus LEDs come within ±0.05, below the ±0.062 of a line with no Duv in it, and the LEDs at ±0.006 within ±0.08 on both sides, better than a slope per temperature’s ±0.093 and ±0.110 — because once the level is placed, the curve terms are left to describe the error’s own course. If the outer LEDs had stayed beyond ±0.10, the curve terms would have been absorbing something other than a level, and the bin’s line would be better learned with no curvature at all.

One number a pump, and the valley flattens #

With the square of the mireds in every pump’s line, the slope and curve per temperature — eleven terms — reads the thirty locus LEDs within ±0.025, each left out, the LEDs at the bin’s edges within ±0.052 and ±0.043, those at ±0.0015 within ±0.023 and ±0.018, and those at ±0.006 within ±0.049 and ±0.058. The same level added to the slope per temperature, eight terms, gives ±0.026 on the locus, ±0.024 and ±0.021 inside the bin, and ±0.084 and ±0.107 at ±0.006.

• Both predictions hold, by a wide margin. The locus is read at half the ±0.05 asked and the outer LEDs at well under the ±0.08.
• The locus is read three times better than the locus alone ever read it. The thirty locus LEDs that started this run were read at ±0.074 by a line learned on them; learned across the bin with a level and a curve, the same LEDs come within ±0.025.
• The curve now earns its terms where the earlier line lost them. At ±0.006 it reads under two thirds of the level-and-slope line’s residual on both sides; from −0.003 to +0.003 the two lines lie within 0.005 of each other.

The figure’s two dashed lines are the earlier pair, and the solid ones are the same lines with the level. Nothing about the first stage changes between them: the red phosphor is estimated by the same line in all four, within ±0.10 to ±0.12 at every Duv, as two stages catch the tube and bill the lamps they misplace required of a first stage that the second is to trust, so every difference in the figure is the second stage’s. The slope per temperature drops by about half everywhere inside the bin and barely moves outside it, which is what a line too plain for the error’s bend should do once its level is right: its remaining error is the curvature it does not have, and that grows with the square of the Duv. The curve, which was a valley climbing steeply at both ends, becomes nearly flat. Its two points at the bin’s edges now sit a little above its neighbours, which is the trace of leaving each LED out: at ±0.003 a left-out LED removes one of the three points its own temperature’s curve was fitted through, so its reading is a small extrapolation and the LEDs at ±0.0015 are read better than those at ±0.003.

The curve the line now draws is the error’s #

The test of the mechanism is not the residual but the curvature itself. Each pump’s line, once fitted, implies a curvature in Duv at each temperature — the coefficient of the Duv’s square, evaluated at that temperature’s mireds and doubled — and the camera error’s mean course implies one too, through its three values at nought and ±0.003.

The error’s mean course bends by +7,916 ΔE₀₀ per unit Duv squared at 2,700 K, +2,196 at 4,000 K and +363 at 6,500 K. The ten-term line, averaged over its pumps, implied +2,803, +14,591 and −13,349 — a third of the bend where the error bends most, six times it at 4,000 K, and the wrong sign at the temperature where the error is nearly straight. With the level, the line implies +8,198, +2,195 and +445.

The agreement at 4,000 K is to a thousandth, at 2,700 K within four per cent, and at 6,500 K within a quarter of a curvature so small that the parabola there barely differs from a straight line. Nothing in the fit asked the line to match the mean course: it was fitted to individual LEDs, recipe by recipe, with the red phosphor estimated and every pump separate. That it now reproduces the mean course’s bend at all three temperatures is the evidence that the earlier line’s curvature was a level in disguise.

It also explains why the earlier failure looked so strange. A curvature of −13,349 at 6,500 K, against a true +363, would be invisible at ±0.003 if the line’s level there were set to absorb it, and it was: the line read the bin’s edges well precisely because its wrong curvature and its wrong level cancelled at those two Duv values. At ±0.0015 they cancel three quarters of the way; at ±0.006 they add.

No bend left in the misses #

With the level, the line’s mean miss lies within ±0.018 at every temperature and every one of the seven Duv values. The largest are at 2,700 K at ±0.006, −0.017 below the locus and +0.018 above; at 4,000 and 6,500 K no mean miss exceeds 0.004. Where the ten-term line’s means formed three parabolas of opposite sign, the eleven-term line’s form three flat lines along nought.

The two misses at 2,700 K are the trace of the one discrepancy the mean course itself showed. The parabola through 2,700 K’s three fitted values misses the mean error at ±0.006 by 0.015 and 0.020, the only temperature at which the error departs measurably from its parabola, and the eleven-term line, whose curve now follows the parabola, misses there by the same amounts with the same signs. What remains in the line’s mean is what remains in the error’s departure from a quadratic, and nothing of the line’s own making.

What is left is the warm whites’ scatter #

At 4,000 and 6,500 K the eleven-term line reads every Duv within ±0.040, and inside the bin within ±0.03. At 2,700 K it reads the locus within ±0.026 and the LEDs at ±0.0015 within ±0.026 and ±0.024, but those at the bin’s edges only within ±0.068 and ±0.062 and those at ±0.006 within ±0.069 and ±0.087. The overall residual at the bin’s edges and beyond is almost entirely the warm whites’.

The mean miss at 2,700 K is small, so this residual is scatter, recipe against recipe, not a shared offset. LEDs varied in their phosphors fall off every line found that kind of scatter first, before the reading had a red phosphor stage at all: no line in the sensor’s departures read a grid of LEDs varied in pump, converter and red phosphor better than ±0.29, because the camera’s error is a V about the locus whose depth the departures alone do not see. LEDs built on the locus still fall off the line found the camera’s error a V about the locus across LEDs varied freely in their recipes, with its depth depending on the phosphors, and at 2,700 K, where most of an LED’s light lies in the long wavelengths on which the camera and the eye disagree most, the depth of that V would differ most from recipe to recipe. A curve per temperature carries one depth for each; a warm white whose red phosphor is strong bends more than one whose red is weak, and the line cannot tell them apart by the Duv.

That is a statement the census can check. Of the 208 LEDs, eight are missed by more than 0.1 and all eight are at 2,700 K, seven of them pumped at 440 or 452 nm; the largest miss, 0.180, is an LED at +0.006. The locus LEDs are read pump by pump within ±0.030, ±0.021 and ±0.024, with no pump left behind.

Four lines side by side #

Set side by side, the four lines say what each term is for. The level costs one term a pump and more than halves the locus’s residual in both lines, from ±0.058 to ±0.026 with a slope and from ±0.073 to ±0.025 with a curve. The curve costs three terms a pump and, with a level, cuts the residual at −0.006 from ±0.084 to ±0.049; without a level it raised it from ±0.093 to ±0.153.

So a term’s value depends on the terms beside it. Read alone, the earlier comparison said the curve was a liability outside the bin; it was a liability only while it had a level to carry. Radiators and LEDs are one family to the sensor found the converse among lamps: two families estimated apart paid for every classifier that tried to separate them, and merged into one family they were read better. Here the same eighty-nine LEDs read worst as one group with shared levels and best as three groups given a level each — what a line should treat as one population is decided by where its terms let the populations differ, not by the lamps’ names.

Every LED’s miss #

Plotted LED by LED, the LEDs the line was fitted on and the LEDs it never saw are indistinguishable at 4,000 and 6,500 K: both lie in a band a few hundredths wide about nought, and the outer LEDs at either end of each temperature’s spread are no further out than the inner ones. At 2,700 K the spread widens towards both ends, held-out and never-seen alike, which is the warm whites’ scatter rather than an extrapolation failing.

That is the property the ten-term line was missing. A line whose LEDs at a new Duv look like its LEDs at a fitted one is describing the quantity, not the sample. A family’s line runs through its neighbours found the opposite case among lamp families, where a line fitted on six radiators close together, carried past them, missed the triphosphor tube by 7.6 — a line describing its sample, read where the sample did not reach.

What a calibration engineer should take from this #

Before adding curvature to a line, give each group its own level. A curve in one variable can absorb an offset between groups that share the line’s other terms, and the residual at the training points will not show it. One term a group is cheap and removes the confusion.

Check a fitted curvature against the quantity’s own course. The ten-term line and the eleven-term line read the bin’s fitted Duv values within a few thousandths of each other and implied curvatures that differed by an order of magnitude and in sign. The second agreed with the error’s mean course; the first did not.

A bin’s calibration LEDs can serve the locus better than the locus’s own. Beside the locus the converter still gives the red away treated the LEDs off the locus as a harder case to be brought up to the locus’s standard; with a level per temperature they are the reverse, the data that pins down what the locus LEDs alone could not. Learned across the bin with a level and a curve, the line reads LEDs on the locus three times better than a line learned on them alone, because the bin’s LEDs fix the level and the slope that the locus LEDs alone leave loose.

How the lines were fitted #

Each line has two stages. The first reads the red phosphor’s strength by least squares from the sensor’s 600 nm departure, the mireds, their product and the Duv, over all LEDs in its training set; the second reads the camera’s error within each pump by least squares from the estimated red, the mireds and their product, the square of the mireds over 250 when the line has a level, and the Duv’s slope per temperature — the Duv, its product with the mireds over 250 and with their square — with the same three terms in the square of the Duv when the line has a curve. At nought and ±0.003 each LED is left out of both stages and read by the line fitted on the other eighty-eight; at ±0.0015 and ±0.006 every LED is read by the line fitted on all eighty-nine. A line’s curvature at a temperature is twice the sum of its three curvature coefficients weighted by one, the mireds over 250 and their square at that temperature, averaged over the three pumps; the mean course’s is its second difference at nought and ±0.003.

What this does not reach #

Three temperatures only. A level per temperature, given as the square of the mireds, places three levels exactly and says nothing about an LED at 3,000 or 5,000 K, where the quadratic in mireds interpolates. Real bins span the whole range between.

The warm whites’ scatter is not resolved. Its pattern — 440 and 452 nm pumps, the bin’s edges and beyond — suggests a curvature that depends on the recipe, which the line does not carry.

One camera, one sensor. The error’s shape across the bin is this camera’s; the level’s importance is general, its size is not.

Still open: whether the red phosphor sets the warm whites’ bend #

The eleven-term line reads 4,000 and 6,500 K within ±0.04 everywhere, and 2,700 K within ±0.03 inside the bin but only within ±0.087 outside it, eight of its LEDs beyond 0.1, every one of them warm. The scatter is recipe against recipe, and the recipe the line already estimates is the red phosphor.

The calculation is the eleven-term line with the square of the Duv multiplied by the estimated red phosphor added — one term more a pump — read on the eighty-nine LEDs with each left out and on the 119 new LEDs. The prediction is that the 2,700 K LEDs at ±0.006 come within ±0.05 and every LED within 0.1, while the cooler temperatures stay within ±0.04, because a warm white’s bend across the locus is set by how much of its light the red phosphor puts where the camera and the eye disagree. If the warm whites’ scatter stays near ±0.08, their bend depends on something the first stage does not estimate — the converter’s peak, or the pump — and the sensor has to read it directly.

The red phosphor tilts the warm whites and bends nothing made that calculation: the bend term moves the warm whites by a thousandth, the red enters the slope instead, and what is left is the first stage reading the warm whites’ red squeezed toward the middle.

A curve that was carrying a level #

The habit here is about reading what a fitted term is doing.

The ten-term line’s curve terms fitted well where they were fitted and implied curvatures nobody had checked. When they were checked against the quantity’s own course, one of them had the wrong sign, and the reason was that they were doing a different term’s job: placing each temperature’s level, which the shared terms could not. Given that term, the same curve terms reproduced the error’s bend to a few per cent without being asked to.

The failure mode is to read a fitted coefficient as the effect it is named for, when the fit is free to use it for any difference its form can absorb. The test is to compute what the coefficient implies about the quantity directly and compare it with the quantity’s own course — and where they disagree, to ask which missing term the coefficient is standing in for.