LEDs varied in their phosphors fall off every line

A camera turns its raw readings into colour with matrices blended by the lamp’s colour temperature, and under many lamps the result is off by a colour difference or two. A small ambient-light sensor beside the lens can estimate how far off. The design this thread arrived at reads three of the sensor’s narrow channels as departures — at 450, 500 and 640 nm, from what a smooth lamp of the same colour would give — places the lamp in a family by its nearest neighbour in those three numbers, and applies that family’s straight line in two of them: the ratio of the 500 and 450 nm departures, and the 640 nm departure.

Radiators and LEDs are one family to the sensor merged the census’s six radiators and daylights with its forty-four white LEDs into one smooth family and brought the estimate to ±0.127 ΔE₀₀ over fifty-four lamps, each left out in turn. New LEDs miss the line their cousins drew then held out fourteen lamps the census had never seen. Radiators, daylights and tinted lamps landed on the line. The new white LEDs missed it by ±0.196, half again the census’s error, and their misses followed the 640 nm departure at a slope of −0.72 where the line weighted it at −0.13. The diagnosis was that the census’s LEDs vary their blue pump, and the new ones vary their phosphors — the converter’s share of the pump’s light, and a red phosphor added or not — so the line had never seen the 640 nm channel move for a reason it could learn.

The prediction was that a census whose LEDs vary both would teach the line the red channel: its 640 nm weight would rise towards the −0.85 the new LEDs implied, the new LEDs’ miss would fall below ±0.13, and the radiators and daylights would stay within ±0.13, because their 640 nm departures are near nothing and a steeper weight acts on nothing there. The alternative named was that the radiators’ estimates move instead, which would mean one line cannot serve both kinds of lamp once LEDs differ in more than one way.

The red weight is learned and the LEDs still miss #

With the census’s forty-four LEDs replaced by forty-eight varied in pump, converter share and red phosphor, the smooth family’s line weights the 640 nm departure at −0.74, from −0.13 — 85 per cent of the way to the −0.85 the new LEDs implied. The new radiators and daylights stay within ±0.13, the worst at 0.113. But the new white LEDs miss by ±0.229, more than before, and the census’s own lamps, each left out in turn, are estimated to ±0.294, more than double its ±0.127. No other pool of LEDs tried brings either number back: the census’s error stays above ±0.23 and the new LEDs’ above ±0.16. A line fitted to the grid’s own LEDs alone misses them by ±0.29 whichever of the sensor’s departures it reads.

• The first part of the prediction holds. The line learns what the red channel means, nearly as much as the new LEDs asked.
• The second fails, and badly. Learning it moves the new LEDs to the other side of the line rather than onto it.
• The alternative was too narrow. It is not the radiators the rebuilt line cannot serve; it is the LEDs that taught it.

The three bars are the prediction’s first test. The census’s line barely uses the 640 nm channel: a weight of −0.13 means that moving an LED’s red departure across its whole range, about half a unit, moves the estimate by six hundredths of a colour difference. The new LEDs’ misses said the weight should be −0.85 — the census’s −0.13 plus the −0.72 slope their misses followed. The rebuilt line’s −0.74 is most of the way there. On the question the prediction asked first, whether varying the phosphors would teach the line the red channel, the answer is yes.

Forty-eight LEDs that vary three ways #

The grid’s LEDs are the collection’s own white LED — a blue pump line twenty-two nanometres wide, a broad yellow converter band at 565 nm, and, where present, a red phosphor band at 632 nm — on three settings at once. The pump sits at 440, 452 or 465 nm. The converter takes 0.5, 0.6, 0.7 or 0.8 of the pump’s light. The red phosphor is absent or at 0.3, 0.6 or full strength. That is three by four by four, forty-eight lamps, against the census’s forty-four. None of them shares a new LED’s settings: the heavier-converter LED’s 0.68, the deep-pump LED’s 445 nm, and the two red-phosphor LEDs’ 0.25 and 0.9 all fall between the grid’s steps, so the new LEDs are estimated from neighbours rather than from twins.

Each grid LED’s camera error and sensor departures are computed exactly as the census’s own, and the rebuilt pool keeps the census’s radiators, daylights, tubes and emitter unchanged. The grid’s camera errors run from 1.31 to 2.65 ΔE₀₀, higher than the census’s LEDs’ 1.09 to 2.17 and a quarter wider, and its whites from a Duv of −0.020, pinkish, below the Planckian locus, to +0.032, greenish, above it. A Duv of a few thousandths is visible as a tint to a careful eye; the grid spans tints a maker would never ship as white alongside ones a maker would, which matters later.

The new LEDs move and land on the other side #

The radiators and daylights barely move: each estimate shifts by about a hundredth, and the daylights come slightly closer. The LEDs move by as much as a third of a colour difference, and the deep-pump LED is the only one the move helps: from a miss of +0.29 to −0.06. The heavier-converter LED crosses from +0.18 to −0.17. The light-red LED, nearly on the line before at −0.05, falls to −0.24. The strong-red LED, already −0.24, goes to −0.33. The LED built from a green and a red band goes from −0.12 to −0.25; the census placed it beside a violet-pumped LED, and the rebuilt pool, with no violet LEDs left, places it beside a tube and estimates it from the census-wide line.

The pattern of the moves is the steeper weight doing exactly what it was asked to do and finding that it was asked the wrong thing. A weight on the 640 nm departure that goes from −0.13 to −0.74 raises every estimate in proportion to how deep that departure is: by about three tenths for the two LEDs with no red phosphor, whose departures sit near −0.49; by fifteen hundredths for the light-red LED at −0.24; by five hundredths for the strong-red one at −0.07. The deep-pump LED needed raising by three tenths and got it. The heavier-converter LED needed less than two tenths and got three. The two red-phosphor LEDs needed lowering, and were raised. A slope fitted through six misses was dominated by the two that were furthest apart, and the line learned that slope faithfully; what it could not learn from the slope was that the misses at the shallow end wanted the opposite correction. The misses the old line made along the red channel were real, and they were not the only thing wrong.

The three-band LED, a blue pump with narrow green and red phosphors, is the one non-white-LED lamp the grid helps: its miss goes from −0.18 to −0.12, inside the census’s error, because its nearest neighbour is now a grid LED with a strong red phosphor rather than one of the census’s violet-pumped LEDs.

The census loses its own accuracy #

The more serious result is not about the new lamps. Every lamp of the rebuilt census, left out in turn and estimated from the rest, is missed by ±0.294 — the grid’s own forty-eight LEDs by ±0.292, the census’s six radiators and daylights by ±0.207 against ±0.177 before, the two tubes by ±0.30. The census as it was estimated its own lamps to ±0.127. The rebuilt one does worse on every family it shares with the old.

This is the outcome the earlier essay’s alternative pointed at and drew too narrowly. It said that if the radiators moved, one line could not serve both kinds of lamp. The radiators do move, a little. But the large failure is inside the smooth family’s LEDs themselves: the line cannot serve LEDs that vary in two ways, whether or not radiators share it. A family’s line runs through its neighbours found that a lamp’s family line is worth only what its neighbours make it worth, and the grid gives every LED close neighbours. Close neighbours, it turns out, are not enough when the thing being estimated does not lie on a line through them.

No pool of LEDs restores it #

A single rebuilt pool could be unlucky in its grid. Three others test that.

Keeping only the thirteen grid LEDs whose white lies within 0.006 of the locus — the ones a maker would bin as white — gives the best rebuilt result for the new LEDs, ±0.168, and still misses the census’s lamps by ±0.236. Its line weights the 640 nm departure at −1.54, steeper than anything implied, and lets one new radiator slip to 0.163, outside the census’s error. Keeping the census’s forty-four and adding the grid’s forty-eight, ninety-two LEDs in all, puts the 640 nm weight back at −0.23, the new LEDs at ±0.198 — no better than the census alone — and the census’s lamps at ±0.242.

So the outcome does not hang on which LEDs are in the pool. Every pool that contains LEDs varied in their phosphors estimates the census’s own lamps to between ±0.236 and ±0.294 and the new LEDs to no better than ±0.168. The census’s ±0.127 was a property of a pool whose LEDs all varied the same way.

No line in the sensor reads the grid #

If no pool works, the question is whether the problem is the merged design — its nearest-neighbour families and its two inputs — or the sensor. A line fitted on the grid’s LEDs alone, with no radiators, tubes or families to confuse it, isolates the second.

The merged design’s two departures read the grid’s own LEDs to ±0.291; all three of the sensor’s departures, used separately, to ±0.298 — no better. The departures carry no more about the grid’s camera errors than the two the design already uses. The 640 nm channel was itself a late addition: the tube a third channel misses has no family found a long-wave channel reading where a tube’s phosphor band sits, but only inside the tube’s own family — the same shape of limit the grid now shows for LEDs. Adding a fourth input, the white’s distance from the locus, which a sensor with a white-point reading could in principle supply, brings the miss to ±0.240: better, and still nearly twice the census’s error.

That localises the failure. It is not the merged family, nor the nearest-neighbour step, nor the choice of two departures over three. Over LEDs that vary their phosphors, the camera’s error is not a linear function of anything the sensor reads. Two channels estimate what one can only sort first turned the sensor from a sorter into an estimator, and every essay since has been refining which straight line to use. The grid says that for this kind of lamp no straight line is good enough.

More converter helps, then hurts #

The reason a line fails is visible in a single slice of the grid. With no red phosphor, the camera’s error is least at the smallest converter share and climbs with it, from 1.91 at 0.5 to 2.53 at 0.8. With any red phosphor it falls and then rises: at full red strength, from 2.11 at 0.5 to 1.51 at 0.7 and back to 1.64 at 0.8. The minimum sits at a different converter share for different red strengths.

The sensor’s departures do not turn. As the converter takes more of the pump’s light, the 450 nm departure rises steadily — from 0.72 to 0.94 for the LED with no red phosphor pumped at 440 — and the 500 nm departure rises steadily too, while the 640 nm departure stays almost fixed. A straight line in readings that move one way cannot follow an error that goes down and then up. The census’s LEDs never showed this because their converter share hardly varied; the turn was outside the range they sampled.

The pump adds a third, separate effect: the grid’s LEDs pumped at 440 nm average an error of 2.09, those at 465 nm 1.63. The census’s violet- and cyan-assisted LEDs explored the pump direction carefully, and in that direction a line was enough — and half of them, the lamps two channels miss are smooth to the camera found, left the camera’s colours as accurate as a smooth lamp does.

The error is a V about the locus #

What turns is the white. Raising the converter share moves an LED’s white from bluish towards yellowish and, with no red phosphor, up off the locus into the greens; adding red phosphor pulls it back down and, at a low converter share, below the locus into the pinks. Across the grid the camera’s error rises with the white’s distance from the locus on both sides — correlated at 0.68 with the distance on the green side and at −0.64 on the pink — and is lowest near the locus. It is a V, and the converter-share curves are slices through it.

The camera’s profile is built for this. Its two matrices are blended by the lamp’s colour temperature, which is defined along the locus, and a matrix is fitted under one light described what happens when the light is somewhere else. A white well off the locus is a light the blend was never fitted for, on either side.

Every one of the census’s forty-four LEDs sits on the green side, at Duvs from +0.007 to +0.034. That is not by itself why the census’s line worked: within the census’s LEDs the camera’s error is barely correlated with the distance, at 0.10, because what varies among them is the pump, and the departures read the pump well. But it means the census never contained an LED on the far arm of the V, or one near its point, and the new red-phosphor LEDs are both — the light-red LED at +0.011, the strong-red one at +0.004, close to the locus where the grid’s errors are lowest.

There is a subtlety against an earlier finding. Two matrices do not reach a white LED found that across kinds of lamp, distance from the locus does not decide the profile’s error — a triphosphor tube sits nearly on the locus and fares worst — and lines in the spectrum do. Both hold. Across kinds, structure dominates; within one kind of lamp, built from the same parts in different proportions, the distance from the locus is what varies, and it matters.

What a sensor would need #

The V explains a share of the grid’s scatter and not all of it: the best line with the distance from the locus added still misses by ±0.240. The remainder is the pump effect and the way red phosphor changes the profile’s error at a given distance, neither of which is linear in the departures either.

What would read it is a different shape of estimate. A family line per construction of LED — no red phosphor, some, a lot — would put the V’s arms on separate lines; two stages catch the tube and bill the lamps they misplace built exactly that kind of design for tubes, and its cost was the lamps the first stage misplaces. A term in the distance from the locus that is allowed to bend would follow the V itself. And a design that recognises when a lamp sits on a part of the V the pool never sampled could decline to estimate, as a classifier that abstains gives up the tube found worth doing for the one lamp with no relatives.

None of these is computed here. What is computed is that the census’s design, the best this thread has built, was fitted to LEDs that varied along the one direction a line could follow.

How the grid was built and read #

Each grid LED is the collection’s white LED model at the stated pump wavelength, converter share and red-phosphor strength, the pump line twenty-two nanometres wide and the converter band 118 wide at 565 nm, the red band at 632 nm, normalised to its peak. Its camera error is the mean ΔE₀₀ over the collection’s test chart when the camera’s two-matrix profile, blended by the lamp’s correlated colour temperature, renders the chart with neutrals held; its departures are the sensor’s 450, 500 and 640 nm channels read against the camera’s white. Four census lamps put through the same computation reproduce the census’s own numbers to twelve places.

The merged design is refitted on each pool exactly as before: nearest neighbour in the three departures to choose a family, that family’s least-squares line in the absolute ratio departure and the 640 nm departure, the census-wide ratio line when a family has fewer than four members. Held-out errors leave each census lamp out of the pool it is estimated from; the new lamps are never in the pool.

What the census does not reach #

A coarse grid. Four converter shares and four red strengths sample the V’s arms at a few points each, and the turn in each converter curve falls between two of them.

Whites a maker would not sell. Thirty-five of the forty-eight sit more than 0.006 from the locus. The thirteen-lamp pool tests the binned ones alone, but thirteen is a small family for a three-number line.

One camera and one sensor. The V belongs to this camera’s profile, and a profile blended differently, or a camera with sensitivities closer to the eye’s, would shape it differently.

Still open: whether LEDs built on the locus fall on a line #

The grid varied its settings independently and let the white fall where it fell, which put most of its LEDs on the arms of the V. A maker does the opposite: chooses the converter share for each pump and red strength so that the white sits on the locus at the colour temperature wanted. A census built that way removes the V by construction and keeps the variation in phosphors.

The calculation is a grid of LEDs at three pumps, four red-phosphor strengths and three colour temperatures — 2,700, 4,000 and 6,500 K — each with its converter share solved to put its white within 0.002 of the locus, the merged design refitted on the census with those thirty-six in place of its LEDs, every lamp left out, and the new lamps estimated again. The prediction is that on the locus the camera’s error is nearly linear in the departures again, that the census’s held-out error returns to within ±0.15, and that the new red-phosphor LEDs, which sit near the locus, land within ±0.13 while the heavier-converter LED, at a Duv of +0.023, keeps its miss. If the census’s error stays above ±0.2 on the locus as well, the red phosphor changes the profile’s error in a way no line in the departures reads, and the sensor needs a different channel rather than a different pool.

A weight that moves the right way is not a model that fits #

The habit here is about what a prediction that half comes true is worth.

The earlier essay diagnosed its misses well: they followed the 640 nm departure, the line weighted it at a sixth of their slope, and the census’s LEDs had never varied that channel for a reason. The repair it predicted was the natural one, and its first step came true — the rebuilt line weights the channel almost as the misses asked. A reader stopping there would call the diagnosis confirmed.

The failure mode is to test a diagnosis by whether the parameter it names moves, rather than by whether the estimate it was meant to fix improves. The weight moved; the estimate got worse; and the reason was that the diagnosis was correct and incomplete — the misses followed the red channel because the census sampled one arm of a V, and teaching the line the red channel only tilted it across the V. The test is always the held-out number, on the lamps that motivated the change and on the lamps that were fine before it.