New LEDs miss the line their cousins drew

A camera turns raw sensor readings into colour with a matrix chosen by the lamp’s colour temperature, and under some lamps that matrix is wrong by a colour difference or two, because the camera’s sensitivities are not the eye’s and a lamp’s spectrum decides how much that matters. A small ambient-light sensor beside the lens, reading a few narrow channels, can estimate how large the error will be. The thread that built that estimate found it working through three of the sensor’s readings — departures, at 450, 500 and 640 nm, from what a smooth lamp of the same colour would give — and through a two-stage design: place the lamp in a family by its nearest neighbour in those three departures, then apply that family’s straight line from the ratio of the 500 and 450 nm departures and the 640 nm departure.

Radiators and LEDs are one family to the sensor found that the radiators and the white LEDs are better treated as one family of smooth broadband sources, and that the merged design estimates the camera’s error to ±0.127 ΔE₀₀ over the census’s fifty-four lamps, each left out in turn — below the ratio line’s ±0.141 and below the separate families given their true labels. It ended on the obvious hole. Every lamp it estimated had relatives in the pool it was fitted on: forty-four white LEDs, most of them variants of one design with its blue pump moved, and six radiators and daylights. A lamp unlike any of them could be placed confidently in a family by its nearest neighbour and estimated from a line it does not follow.

The prediction was that new smooth lamps would be estimated within the census’s own error, ±0.13, because they share the smooth family’s structure, and that tinted incandescents would fall off the line by four tenths or more, as the census’s pink-tinted radiator had sat uneasily between families. If so, the smooth family would be a real family for any lamp whose spectrum is smooth, and a tint would be the one thing the three departures could not read.

The smooth lamps hold; the new LEDs miss #

Fitted once on the whole census and applied to fourteen lamps it never saw, the merged design estimates every new radiator and daylight within ±0.13 of the camera’s true error, ±0.086 over the five. The tinted and filtered lamps do not fall off, against the prediction: the incandescent behind pink glass, placed among the violet LEDs by its nearest neighbour, misses by 0.117, and the three together by ±0.070. The lamps the design misses are the new white LEDs — ±0.196 over five, half again the census’s error, with the deep-pumped LED missed by 0.29 and the strong red-phosphor LED by 0.24 — and the three-band LED, by 0.18. Across the six LED-type lamps the miss falls with the sensor’s 640 nm departure at a slope of −0.72, explaining 82 per cent of their scatter, where the merged line weights that departure at −0.13. Adding the fourteen to the pool and leaving each out in turn changes nothing: the new LEDs’ miss goes from ±0.196 to ±0.203, and the census’s own error stays at ±0.127.

• The smooth family is real for smooth lamps, as predicted — including temperatures outside the census’s and daylights it never had.
• A tint is not what breaks it. A smooth lamp tinted by a glass stays smooth to the camera, and its error is small whichever family it is put in.
• What breaks it is a white LED built differently from the census’s. The census’s LEDs differ in their pump; the new ones differ in their phosphors.
• One new lamp of each kind is not a family. The line cannot learn a dependence from lamps that each vary it once.

Each bar is one lamp’s miss, coloured by what the lamp is. The radiators and daylights are the short bars at the top, the largest the two daylights at 6,500 and 7,500 K, estimated too high by 0.12 — the census’s daylights stopped at 5,000 K, and the line extended past them a little too far. The tinted lamps sit among them. Every bar outside the dotted band is an LED: the heavier-converter LED at +0.18, the deep-pumped one at +0.29, the strong red-phosphor one at −0.24, the three-band LED at −0.18. The LED with a light red phosphor and the green-and-red LED fall inside, at −0.05 and −0.12.

Fourteen lamps built from the collection’s own spectra #

The new lamps are made from the same pieces the census’s are, put together in ways the census did not. Five are smooth: radiators at 2,700, 2,856 and 3,400 K, and daylights at 6,500 and 7,500 K, where the census had radiators at 3,000 and 4,000 K and daylights at 4,000 and 5,000. Three are smooth lamps seen through glass: an incandescent behind a blue daylight-correcting glass that lifts it to about 3,500 K, the same behind a pink glass that takes a broad bite out of the green and pushes it 0.016 below the Planckian locus, and a daylight behind a warm filter. Five are white LEDs varied in their phosphors rather than their pump — a heavier yellow converter, a deeper 445 nm pump with a lighter one, a red phosphor at a quarter and at nine tenths of the census’s strength, and a green-and-red phosphor pair with no yellow — and one is a three-band LED of a narrow blue pump and narrow green and red phosphors.

Each lamp’s camera error and sensor departures are computed exactly as the census computes its own: put through the same functions, four of the census’s own lamps return their census rows to the last digit. The camera’s error ranges over the new set from 1.04 ΔE₀₀ under the 6,500 K daylight to 2.17 under the deep-pumped LED, inside the range of the census and its four tube siblings, 1.09 to 2.55.

Where the new lamps sit #

The merged design places a lamp by its nearest neighbour in all three departures and estimates it from two of them, so the second view is the one that decides the estimate. The smooth lamps, new and old, sit near the origin, where a lamp’s departures from a smooth lamp of its own colour are small by construction. The census’s white LEDs sit far out along the ratio’s departure, in two clumps along the 640 nm axis: the blue-pumped LEDs and their violet variants from −0.49 to −0.40, the cyan-pumped variants near +0.05. Between them the census has one lamp, its single red-phosphor LED, at −0.15.

Two of the new LEDs — the ones given red phosphors — land in that gap, and the three-band LED lands beyond the cyan clump. Every new lamp is placed in the merged smooth family, because the nearest pool lamp to every one of them is a radiator or an LED; the pink-glass incandescent’s nearest is a violet LED 0.098 away, and the three-band LED’s a violet LED 0.28 away, five times the census’s median spacing. Nearness does not predict the miss, though: the two new LEDs nearest the pool — the heavier-converter LED, 0.034 from the census’s warm LED, and the deep-pumped one, 0.043 from its neutral LED — are two of the four the design misses most.

Why a tint does not break it #

The prediction singled out the tinted lamps because the census’s own pink-tinted radiator had been awkward: a family’s line runs through its neighbours found it sitting nearer the LEDs’ line than its own, and a classifier that abstains gives up the tube found the family structure struggling with exactly such lamps. An awkward placement is a problem only if the family it lands in estimates it badly, and for a smooth lamp the families barely differ near the origin. The pink-glass incandescent lands among the violet LEDs and is estimated from the merged line, whose value near the origin is fitted by every smooth lamp in the pool, and it misses by 0.117. Its camera error, 1.45, is among the largest of the smooth lamps — the pink glass takes a broad bite out of the green, where the camera’s sensitivities differ most from the eye’s — and its ratio departure, 0.36, larger than any radiator’s, carries most of that excess into the estimate.

The lamps two channels miss are smooth to the camera found that the violet-pumped LEDs escaping both narrow channels were the ones the camera itself sees as smooth, so that their escape cost nothing. A glass over a radiator is smooth to everything: its absorption is broad, the camera and the eye see much the same spectrum, and the error the camera makes follows the lamp’s colour almost as it does for an untinted radiator. A tint moves a lamp off the locus; it does not give it structure, and structure is what the camera’s error is about.

What the sensor’s channels see of a phosphor #

The new LEDs differ from the census’s in where their phosphors sit, and the sensor’s three channels see those differences unequally. The 450 and 500 nm channels sit on the pump and on the shoulder between pump and converter, which is why one ratio reads a peak and a trough found their ratio reading the pump’s position so well. The 640 nm channel sits on the red end of the converter band, where a red phosphor adds its light; it was added to the design — after a straightened channel repeats the one beside it had found a 570 nm channel adding nothing beside the 450 — because the tube a third channel misses has no family found no single long-wave channel rescuing the halophosphate tube, and the two-stage design then used it inside families instead.

So the channel that should read a red phosphor is in the design, and the line gives it almost no weight, for the reason the next figure shows.

The misses follow the red channel #

The six LED-type misses lie close to a straight line in the 640 nm departure: a slope of −0.72, explaining 82 per cent of their scatter. An LED whose 640 nm departure is deeply negative — little red, as in the heavier-converter and deep-pumped LEDs — is estimated too low; one whose red phosphor fills the 640 nm channel is estimated too high. The merged line has a term for that channel, and weights it at −0.13. The new LEDs say the channel matters nearly seven times as much.

The census’s own LEDs explain why the line learned so little. Its three phosphor-converted LEDs without red — cool, neutral and warm — share one 640 nm departure, −0.491, because they differ only in how much of the pump’s light the yellow converter takes, which the 640 nm channel sees almost nothing of. Its forty variants move the blue pump, which moves the 450 and 500 nm departures and sends the 640 nm departure to one of two places. A single LED has a red phosphor. The line’s weight on the red channel was fitted on lamps that hardly vary it, or that vary it together with everything else, and a lamp varied in the red channel alone is estimated as though that channel were nearly irrelevant.

A family’s line runs through its neighbours found the radiators’ line lying along the LEDs’, which is what made merging them sound; this is the same geometry seen from the other side. A line through a family is a line through the directions the family happens to vary in, and a new member varied in a direction the family never explored is extrapolated along a slope nobody measured.

Every kind, against the census #

Read by kind, the new set divides cleanly. The new radiators and daylights are estimated to ±0.086 and the tinted lamps to ±0.070 — better than the census’s own ±0.127, because their errors are small and the line near the origin is well fitted by forty-odd lamps. The new white LEDs are estimated to ±0.196 and the three-band LED misses by 0.18. The census-wide ratio line, with no families and no 640 nm channel, does nearly as well on every kind: ±0.096 on the smooth lamps, ±0.058 on the tinted ones, ±0.221 on the LEDs. Over all fourteen the merged design misses by ±0.140 and the ratio line by ±0.158, the same margin as on the census.

Two stages catch the tube and bill the lamps they misplace found the two-stage design’s gain coming from one lamp it placed correctly and its cost from the lamps it placed wrongly. On new lamps there is nothing for the families to catch: every new lamp is placed in the same family, and the merged design’s advantage over the ratio line is the 640 nm term, which helps the LEDs a little and does not help enough.

What relatives would buy #

The natural repair for lamps unlike the pool is to add them to it. With the fourteen added and every lamp left out in turn, the new LEDs’ miss goes from ±0.196 to ±0.203, and the census’s own held-out error from ±0.127 to ±0.127.

The bars barely move because the new LEDs are each a family of one. The strong red-phosphor LED, left out, is estimated from a line fitted on the census and the light red-phosphor LED — one more lamp varied in the red channel, among fifty-odd that are not — and the line’s weight on that channel hardly moves. A dependence is learned from lamps that vary it, in numbers, and fourteen lamps each varied a different way teach a line almost nothing. A classifier that abstains gives up the tube met the same limit from the other end, a lamp that no family’s members resembled; here the lamps are resembled well enough and estimated badly anyway.

How the new lamps were built and read #

The fourteen spectra are built on the collection’s five-nanometre grid. The radiators are Planck’s law at their temperatures and the daylights the CIE daylight series. The blue glass transmits linearly from all at 380 nm to a twentieth at 900; the pink glass removes up to 45 per cent in a Gaussian band 40 nm wide at 545 nm; the warm filter transmits from 35 per cent at 380 nm rising linearly to all at 780. The white LEDs are the collection’s phosphor-converted LED model — a pump line and a broad yellow converter band, with an optional red band at 632 nm — at converter shares of 0.55 to 0.74, a pump at 445 or 452 nm, and red phosphors at a quarter and nine tenths; the green-and-red LED is a 452 nm pump with Gaussian phosphor bands at 528 and 630 nm, 40 nm wide; the three-band LED a 450 nm pump with bands at 530 and 625 nm, 32 and 28 nm wide. Each lamp’s camera error is the census’s: the camera’s matrix blended between its tungsten and daylight profiles by the lamp’s correlated colour temperature, held neutral on the lamp’s white, and the mean ΔE₀₀ over the test colours. Its departures are the census’s three narrow channels’ readings against a smooth lamp of its own camera white. The merged design is fitted once on the census’s fifty-eight lamps — fifty-four and the halophosphate tube’s four siblings — with each family’s line from the ratio’s departure and the 640 nm departure, and each new lamp assigned the family of its nearest pool lamp in the three departures.

What the census does not reach #

Model lamps. The new spectra are built from the collection’s own parts, which is what makes their computation exact, and real lamps have shapes no Gaussian band reproduces. A real two-phosphor LED would be tested the same way, and the prediction from this census is that it would miss in proportion to its red channel.

One camera. The camera’s error under a lamp depends on its sensitivities; a camera whose red channel is nearer the eye’s would care less about a lamp’s red phosphor, and the 640 nm channel’s true weight would be smaller.

Fourteen lamps. Enough to separate the kinds and to fit a slope through six, not enough to say how widely the slope varies across real LEDs.

Still open: whether a design built on phosphor-varied LEDs keeps the radiators #

The census’s LEDs vary their pump; the new ones vary their phosphors; the line knows only the first. A census whose LEDs varied both — converter share and red-phosphor strength crossed with the pump positions already there — would let the line learn the 640 nm channel’s weight from lamps that vary it.

The calculation is the census rebuilt with its forty-four LEDs replaced by a grid of the same size in converter share, red-phosphor strength and pump wavelength, the merged design refitted, every lamp left out, and the fourteen new lamps estimated again. The prediction is that the 640 nm weight rises towards the −0.85 the new LEDs imply, that the new LEDs’ miss falls below ±0.13, and that the radiators and daylights stay within ±0.13, because their 640 nm departures are near nothing and the steeper weight acts on nothing there. If the radiators’ estimates move instead, one line cannot serve both kinds of lamp once the LEDs are allowed to differ in more than one way, and the merged family was a convenience of a census that varied its LEDs in only one.

A held-out test is only as new as its lamps #

The habit here is about what leaving one out tests.

Leaving each lamp out of its own fit is an honest test of a design on lamps like the ones it was fitted on, and the census’s ±0.127 was honest in exactly that sense. It could not say what happens to a lamp unlike all of them, because there was always a near relative left in — forty of the forty-four LEDs were one design with its pump moved, and leaving one out left thirty-nine siblings to stand in for it.

The failure mode is to read a leave-one-out error as an error on new cases when the cases come in clusters. The test is to build cases outside the clusters on purpose, varied along directions the clusters do not cover, and here the first such direction tried — the red channel — was enough to find the line’s blind side.