Two hairiness meters read two moments

There are two ways to measure how hairy a yarn is and they have never agreed. One passes the yarn through a beam and reports the total length of protruding fibre per unit length of yarn, as a single index. The other counts individual hairs and reports how many exceed one, two and three millimetres. Both are careful instruments, both are repeatable within themselves, and the correlation between them across a set of yarns is poor enough that mills specify one or the other and argue about which.

The cloth #

The usual explanation is optical. One instrument measures obscuration and the other counts pulses; one is sensitive to fine fibres near the yarn and the other is not; the beam geometries differ, the calibrations differ, the thresholds differ. All of that is true, and none of it is the reason.

The reason is that the two numbers are different statistics of the same object, and the object is an exponential population with two parameters. A single yarn has a hair count N₀ and a decay length λ. An integrating meter reports N₀λ. A counter at three millimetres reports N₀e^(−3/λ). Those are two independent functions of two independent parameters, so one of them fixes nothing about the other unless λ is known as well — and neither instrument reports λ.

The claim #

An integrating hairiness index is the first moment of a hair population and a long-hair count is a tail probability. They are algebraically independent, and across the range of decay lengths real yarns occupy, two yarns with identical index differ by more than a factor of thirty in long-hair count. No improvement to either instrument can produce a correlation, because there is no correlation to find.

And the half that decides what anybody should do about it: the tail is the statistic that predicts. Pilling, prickle, a printed edge and a shot effect all need reach, and reach lives entirely in the tail. The index is an excellent measure of total fibre in the way and a poor measure of anything that happens at the cloth’s outside.

Two functionals, written out #

The population is N(h) = N₀e^(−h/λ): the number of hairs per unit yarn length standing at least h off it.

The index integrates. Total protruding length per unit yarn length is ∫N(h)dh = N₀λ. It is the population’s first moment, and it is dominated by whichever part of the distribution carries the most length.

The counter thresholds. Hairs over three millimetres are N₀e^(−3/λ), a tail probability times a count, and it is dominated by whichever part of the distribution reaches furthest.

Fix the index at H and vary λ. Then the tail is (H/λ)e^(−3/λ), which falls off a cliff as λ shrinks: halving the decay length doubles the count of hairs and annihilates the fraction of them that are long. Over the range of λ that cotton and wool yarns actually occupy — from under half a millimetre to over a millimetre — the tail moves by a factor of thirty at constant index.

The relation runs the other way too, and the file asserts it: fixing the tail leaves the index free by a factor of four. Neither number constrains the other in either direction.

Why λ is not a constant, which is what would rescue the correlation #

If every yarn had the same decay length the two instruments would agree perfectly, because both would be proportional to N₀. Somebody comparing two yarns of one fibre at one twist on one machine will find exactly that, and will conclude the instruments correlate.

They will stop correlating the moment the comparison crosses a fibre. The decay length is fifty-six fibre diameters — a property of the fibre with the yarn’s own diameter divided out of it — so it moves whenever the cotton grade moves, whenever a blend proportion moves, whenever wool is compared with cotton at all.

A hairiness correlation study that stays inside one fibre finds a correlation and one that crosses fibres does not, which is precisely the pattern in the literature and is usually reported as the instruments being unreliable outside a narrow range.

Where the eight-fold discrepancy comes from #

There is a second and larger reason the two disagree, and it is the one that sets the scale.

Setting this site’s modelled population against an integrating reading on the same yarn leaves a factor of about eight unaccounted for. The model puts N₀λ at about 0.6 and the instrument reports about 5. The model’s counts at one, two and three millimetres agree with the counter’s to within the spread of published values — so the model is right about the tail and short by a factor of eight on the integral.

The resolution is that there are two populations and only one of them is modelled. A dense cloud of very short protrusions — loops rather than ends, and fibre lying slack on the surface — carries most of the protruding length and reaches almost nowhere. The integrating instrument adds it up. The counter cannot see it, because it is below every threshold the counter has.

That is not a defect in either instrument. It is two instruments correctly reporting two physically distinct populations that happen to share a word.

What follows for an intervention #

The split settles a thirty-year argument about spinning systems in one line of arithmetic.

Compacting a spinning triangle removes ends that were unbound over a long stretch, which is the long population and nothing else. So it takes a large fraction off the tail and a small fraction off the index, and the ratio between the two drops is the reciprocal of the long population’s share.

Singeing does the opposite and is worth more. A flame does not scale the population, it truncates it: everything standing clear of the cloth is burnt back, and what is left is the short cloud. So a singeing takes almost all of the tail and — because the tail carries so little length — an eighth or less of the index.

An intervention judged by the index therefore looks marginal whatever it does, and an intervention judged by the tail looks dramatic. Both readings are correct and the second is the one that predicts the cloth.

What each instrument is genuinely good for #

Neither reading is useless and it is worth saying which question each answers well, because the trade’s habit of picking one and defending it has obscured that they are complementary.

The index predicts what happens in the shed. Lint shed onto a loom, clinging between neighbouring ends, the amount of size a warp needs to survive weaving, the fly in a weaving room — all of those are about total fibre standing in the way, integrated over the whole population, and the short cloud contributes to every one of them because it is most of the fibre. A weaving manager specifying on the index is specifying correctly.

The tail predicts what happens to the cloth. Reach is what lets a hair bridge from one thread to the next and become a pill, from a printed area to an unprinted one and become a feather, from the cloth to the skin and become a prickle. None of those can be done by a protrusion of a few micrometres however many of them there are.

And nothing predicts warmth, because that needs both — a canopy needs density and length, and it needs their product in a combination neither instrument reports. This site’s own criterion for it, n_A λ², is a count times the square of a length and is not proportional to either reading.

What was counted, and how #

Two constructions and two assertions, and both are about independence rather than about a value.

The first fixes the index and sweeps the decay length over the range real yarns occupy, requiring the tail to span at least eightfold. It spans thirty.

The second fixes the tail and requires the index to span at least fourfold. It does.

The pair together is the claim: neither number determines the other. A single assertion in one direction would have been consistent with a one-way relation, which is a thing that exists — a bound rather than a correlation is still useful — and there is not one here.

The counts themselves are checked against the instruments and were not fitted to them. The model puts a twenty tex ring-spun cotton at 17,800 hairs per hundred metres over a millimetre, 3,550 over two and 709 over three. Those are the numbers such instruments report on such yarns; the escape fraction that produced them was set once, from the mass comparison, and the three counts came out together.

Where the model stops #

The short population has one number attached to it and no model. Its share is a measurement, taken by difference. Everything in this essay that involves the index therefore has that number’s uncertainty running through it, and the file quotes it as a range rather than a value.

The exponential is fitted at three points. Three counts on a log axis determine a straight line and cannot test whether the line is straight. The model predicts the exponential from migration being irregular, which is a stronger statement, but the evidence for it in these instruments’ output is thin.

And a real distribution is a mixture. Two exponentials with different decay lengths do not add to an exponential, so a counter reading at one, two and three millimetres on a yarn with a large short population is reading a curve with a knee in it. Where the knee sits is not known, and a counter whose lowest threshold sat below it would report a decay length that is neither population’s.

Nothing here is a calibration. No instrument’s actual optical threshold is used, and the model cannot say what a particular machine will report on a particular yarn.

The one measurement that would close it #

The distribution is the thing to measure, and both instruments nearly measure it.

A counter reading at four or five thresholds instead of three would show whether the line is straight and where the knee is, and the ratios between adjacent thresholds are a measurement of the decay length that carries no calibration at all. That is the number neither instrument reports and both could.

With λ in hand, the index and the tail become two readings of one population rather than two rival hairiness numbers, and each becomes predictive of what it should be predictive of. It would also settle this site’s own claim that λ does not move with the yarn count, which is a ratio measurement inside one instrument on one day.

What the ratio measurement would actually cost #

The proposal above — read the decay length off the ratios between adjacent thresholds — is worth putting numbers on, because its whole appeal is that it needs no new instrument and the obvious objection is that counting noise will swallow it.

The arithmetic is one line. If the population is exponential, two counts at heights h₁ and h₂ give

λ = (h₂ − h₁) ÷ ln(N₁ / N₂),

with no calibration constant anywhere in it, because every multiplicative factor an instrument might carry — its optical efficiency, its threshold offset, the length of yarn it looked at — cancels in the ratio.

Run it on the counts this essay already quotes. Seventeen thousand eight hundred hairs over a millimetre and three thousand five hundred and fifty over two give a ratio of 5.01 and a decay length of 0.62 mm. Three thousand five hundred and fifty over two and seven hundred and nine over three give a ratio of 5.01 and 0.62 mm again. The two agree because the model is exponential by construction, so this is a check on the arithmetic rather than evidence about a yarn; on a real instrument the two would differ, and how much they differ is the test of whether the distribution is a straight line at all.

Now the noise. A count is a count, so its uncertainty is the square root of itself, and the uncertainty in the log of a ratio is √(1/N₁ + 1/N₂). On the one-and-two millimetre pair that is 1.8 per cent, and since the fractional error in λ is that figure times λ divided by the gap between the thresholds, it comes to about one per cent. Even on the sparsest pair available — two and three millimetres, where the higher count is only seven hundred — it is two and a half per cent.

So the parameter the trade has been missing for thirty years is recoverable to one per cent from output a standard instrument already prints, and nobody takes the ratio.

And with λ in hand the two instruments stop disagreeing. The tail count at a known threshold, multiplied by e^(h/λ), gives the long population’s own N₀; multiplied again by λ it gives that population’s contribution to the index; and subtracting it from the integrating instrument’s reading gives the short population’s share, measured on that yarn rather than assumed from one global constant. Every quantity in this essay that is currently carried as a range becomes a per-yarn measurement.

That is the strongest form of the argument this essay makes, and it inverts the complaint the trade has been making. The two instruments do not correlate and never will. But taken together they over-determine a two-parameter population — three counts and one integral against two unknowns — so a mill that owns both machines already owns a complete description of the hair layer and is throwing it away by reporting the two numbers side by side and choosing between them.

The one thing the pair still cannot do is separate a genuinely non-exponential distribution from a badly calibrated threshold, because both show up as the two ratios disagreeing. Distinguishing those needs the fourth and fifth thresholds argued for above, and that is the only change to an instrument anything here asks for.

The comparison this site can make and a mill cannot #

There is one advantage a computed population has over both instruments, and it is worth being explicit about because it is the reason this essay exists at all.

The model reports N₀ and λ separately, because it built them separately: the density from a count of fibre ends through a shell, the length from a migration residual. Every functional of the population is then available — the index, the tail at any threshold, the areal density on a cloth, the canopy criterion, the coverage at any height — and all of them are consistent by construction.

An instrument reports one functional and cannot recover the parameters from it. That asymmetry is why a model with one fitted constant can settle an argument between two careful machines: not because it is more accurate than either, but because it is parameterised and they are not.

The corresponding caution is that the model’s separation is only as good as its one fitted constant, and the constant multiplies N₀ alone. So every claim here about λ is independent of the fit and every claim about the index’s level is not. The claims that matter — that the two statistics are independent, that the tail predicts and the index does not — are all in the first group.

The generalisation #

Two statistics of one distribution are not two attempts at one number, and treating them as such produces a decade of correlation studies that cannot succeed.

The transferable form is that a mean and a tail are related only through the shape, and the shape is usually the thing nobody measures. Wherever two instruments disagree persistently on what looks like one quantity, the first question is which functional each computes; if they are different functionals, no amount of careful comparison will help and the right response is to report the shape.

This site has been here before with a different pair. A cloth is a population and not a thread showed that a quantity linear in a diameter is unbiased at any spread while a fourth power is fourteen per cent high at fifteen per cent variation — the same lesson, that what a distribution does to a number depends entirely on which function of it is being taken.

Who found it, and when #

The two instruments are Uster’s, from the 1980s in its modern form, and Zweigle’s counting method, which is older. The poor correlation between them is documented in dozens of papers and is standard knowledge in any spinning mill.

The explanations offered have been optical and instrumental. The arithmetic explanation — a first moment against a tail probability, algebraically independent — appears not to have been written down, which is surprising, because it takes two lines and needs nothing but the observation that the distribution is exponential, and that observation has been in print since the instruments existed.

Where the ladder goes next #

Out of the yarn and into the cloth. The population is now built and measured; what it does begins with a light touch never reaching the crowns, which asks what a plate meets at a stated pressure and finds a crossover an order of magnitude below the pressure a thickness gauge presses at.

The tail — the statistic this essay argues is the predictive one — is what a pill is anchored by, what a printed edge feathers to and what a prickling fibre needs. Every one of those needs reach, and every one of them is invisible to the instrument the trade specifies on.