A turn is seen one trace late

A drift put back at the table’s price estimated the traces of a sequence of operators whose spectra drift, carrying each trace’s measured spread forward to set the next trace’s probe count and multiplying it by the ratio of the last two spreads to account for the drift. Over 400 sequences that trend carry restored the coverage the plain carry had lost: 67.8 to 68.9 per cent of estimates inside a 3% target at one standard error, against the normal table’s 68.3.

Every sequence it ran drifted one way. It asked what happens when the drift turns. “A sequence whose decay falls for eight traces and then rises would make the ratio of the last two spreads the wrong extrapolation for one trace at the turn. The prediction with a sign is that the trend carry’s coverage dips at the trace after the turn by about twice the plain carry’s lag — four or five points — and recovers within one trace, so that its average over the sequence stays within a point of the table; and that a guard which caps the factor at the ratio of the last three spreads removes the dip for no measurable cost.”

Three of those four clauses can be checked, and two of them hold. The dip is the clause that fails, and it fails in a way that depends on a detail the prediction did not specify: which way the spread was moving when the drift turned.

Two sequences that turn #

The operators are the earlier essays’: symmetric, 60 × 60, with fixed random eigenvectors and eigenvalues $d^k$ for $k = 0, \ldots, 59$, so the trace is a geometric sum and the decay $d$ is the only thing that changes along the sequence. The spread of a Rademacher probe — Hutchinson’s estimator’s standard deviation over the trace — grows as the decay falls, because a faster-decaying spectrum concentrates the trace in fewer eigenvalues and a random probe’s estimate of a few dominant terms is noisier, relative to the total, than of many. A rate that belongs to the matrix found Hutchinson’s own convergence exponent near a half on every spectrum it measured, which is why a single number — the spread — is all a probe count needs: the rate is fixed and only the constant moves, and along these sequences it moves with the decay and nothing else.

The sequence the prediction described has the decay falling from 0.9 by a fixed step for eight traces and then rising by the same step, so the spread rises to a peak at the ninth trace and falls back. Its mirror has the decay rising to 0.9 at the ninth trace and falling again, so the spread falls to a trough and rises. At a step of 0.02 the peak sequence’s spread runs from 0.264 to 0.505 and back; the trough sequence’s runs from 0.505 down to 0.264 and back up. The two sequences visit the same operators in opposite orders.

The four rules are the earlier essay’s. A fresh pilot of eight probes every trace; the plain carry, which uses the previous trace’s measured spread; the trend carry, which multiplies that spread by the ratio of the last two; and a guard, which takes whichever of the last two ratios is nearer one and carries the spread flat when the two lie on opposite sides of one — a turn read off the history. Each trace’s probe count is then set from the spread so that one standard error is 3% of the trace, and coverage is how often the estimate lands inside 3% of the truth, over 400 seeded sequences.

The extrapolation’s error has the turn’s sign #

The figure at the top of the page is the trough. Before the turn the trend carry’s spread is within one per cent of the truth, which is what the earlier essay found for a steady drift; the plain carry runs about ten per cent high, a step behind a spread that is falling. On the trace after the turn — the tenth — the trend carry uses 0.749 of the true spread. It had seen the spread fall by twelve per cent a trace for eight traces, extrapolated one more fall of twelve per cent, and met a rise of fourteen. One trace later it uses 1.001.

The dial shows the peak, and the picture is the same reflected. The trend carry had seen the spread rise by five per cent a trace, extrapolated another rise, and met a fall: it uses 1.110 of the true spread on the tenth trace, and 0.995 on the eleventh. At 0.01 a trace the same two shapes give 1.060 and 0.863.

The size of the error follows from the steps. The trend factor at the turn is the last ratio, $r$, applied to a spread that is about to move by a ratio near $1/r$, so the error is about $r^2$ in the direction the history pointed. At the peak the spread is large and the decay moves it by five per cent a trace, so $r^2 \approx 1.05^2 \approx 1.10$. At the trough the spread is small and the same step in the decay moves it by twelve to fourteen per cent, so $r^{-2} \approx 0.88^2 \approx 0.77$. The two turns are not symmetric because the spread does not depend on the decay linearly, and the trough is where it moves fastest.

A spread too large is never a dip #

A spread that is too large asks for too many probes. The estimate is then more accurate than the target requires and lands inside it more often than the table says: on the peak sequences the trend carry covers 70.5 and 66.5 per cent on the trace after the turn, against a table of 68.3 and a draw-noise standard error of 2.3 points — both inside one error of the table, the first above it. The dip the prediction expected on this sequence is not there, and cannot be, because the extrapolation’s mistake is on the side that costs probes rather than coverage.

On the trough sequences the mistake is on the other side. A spread a quarter too small asks for about $0.75^2 \approx 0.56$ of the probes the target needs, the standard error is a third larger than the table assumes, and coverage falls to 59.5 per cent at 0.02 a trace and 62.3 at 0.01 — 8.8 and 6.0 points below the table, three and a half and two and a half draw-noise errors. The prediction’s four or five points was the right order and the wrong sequence.

Trace by trace, the trough’s dip is one point on a noisy line, and the trend carry does not have it to itself. A single trace’s coverage over 400 sequences scatters by 2.3 points. The plain carry, which had been running ten to thirteen per cent high while the spread fell, is caught by the turn as well: on the tenth trace it uses 0.877 of the true spread and covers 60.0 per cent, against the trend carry’s 59.5. The spread error is in the ratio the prediction named — the trend carry’s 25 per cent is twice the plain carry’s 12 — but the coverage the two lose is the same to within the draw noise, because both are on the steep side of the curve that turns a spread error into missed targets. Both are back by the eleventh trace. A drift put back at the table’s price found that a single trace was noisier than the effect it was looking for; here the effect is large enough to see, but only just, and only by reading the spread ratio, which is the quantity that moves first and moves cleanly.

Why the ratio is the right thing to watch #

Coverage is the quantity the caller cares about and the worst one to measure a turn with. Each trace’s coverage is a proportion of 400 and moves by 2.3 points from draw noise alone; the effect of a turn on one trace is two to four of those. The spread ratio — the median over 400 sequences of the spread a rule used over the operator’s true spread — has no such noise, because it is a median of continuous quantities and because the true spread is computed exactly from the spectrum. It moves before coverage does, and it moves by the amount the arithmetic of the extrapolation says it should.

That is also why the earlier essays could measure the carried rules at all. A rule that reads only its own probes found the sequential stopping rule calibrated in the middle and not at the edge, and the edge is where single-trace coverage lives. A rule’s spread error is the cause and its coverage is a noisy consequence; a turn is visible in the first at one trace and in the second only as one low point among sixteen. The translation from one to the other is steep on the low side and flat on the high side: a spread a quarter too small costs nine points, a spread eleven per cent too large gains two. That asymmetry, not the shape of any particular turn, is why the peak and the trough look so different in coverage and so alike in the ratio.

It is also the reason the trace estimator in counting what cannot be looked at needed a spread at all. Hutchinson’s estimator promises an unbiased trace and says nothing about how many probes make it accurate; every carry rule measured so far is a way of reading that number off the probes, and every one of them reads it a trace late on a sequence. The turn is simply the place where being a trace late is most expensive.

The average does not notice #

Averaged over the fourteen traces that carry a spread, one bad trace is a fourteenth of the effect. The trend carry covers 69.4 and 68.3 per cent on the peak sequences and 67.4 and 68.6 on the trough sequences — all within a point of the table’s 68.3, which is the third clause of the prediction exactly. The plain carry’s averages, 67.3 to 69.3, are inside the same band, because a sequence that turns spends half its length drifting each way and the plain carry’s lag changes sign at the turn: running high while the spread falls and low while it rises, it averages out. A drifting sequence that never turns was where the plain carry lost coverage, and a turn is what gives it back.

The fresh pilot covers 66.5 to 67.6 per cent, a little under the table on every sequence. The table’s 68.3 per cent is what one standard error buys when the misses are a normal distribution’s, which the miss a normal table already priced measured them to be; a spread measured on probes it does not average then separated the pilot from the probes it sizes, so that the spread is estimated from samples that are not averaged and the stopping rule selects nothing. What is left, a point or so under the table, is the eight-probe pilot’s own scatter, and it is the same on a sequence that turns as on one that does not.

The guard fires one trace late #

The guard was written to catch exactly the event the trough sequence produces, and on the trace that matters it does almost nothing. It uses 0.824 of the true spread at 0.02 a trace where the trend carry uses 0.749, and covers 57.3 per cent where the trend carry covers 59.5 — milder in the ratio, no better in the outcome, and the outcome’s difference is inside the draw noise.

The reason is when the turn becomes visible. The spread used on the tenth trace is extrapolated from spreads measured on the eighth and ninth, and on the trough sequence both of those were falling: the ninth trace is the trough itself, the lowest spread of the sequence. The two ratios the guard compares are both below one; nothing in the history says a turn has happened, because it has not yet appeared in any measurement. The guard sees the turn on the eleventh trace, when the tenth’s measured spread has risen — by which time the trend carry has seen it too and is back to 1.001 on its own.

That is not a defect of this guard. Any rule that sets a trace’s probe count from spreads measured on earlier traces extrapolates from a history that ends before the turn, and the first trace after a turn is the one trace whose spread no such history predicts. A spread carried from the trace before made the same point about a constant drift: the carried spread is always one trace old. A turn is the drift changing, and the change is one trace old too.

What the guard costs is as small as the prediction said. Over the sixteen traces it spends between 0.992 and 1.011 times the trend carry’s probes: 2,892 against 2,915 on the faster peak, 2,848 against 2,834 on the faster trough. The fresh pilot spends two to nine per cent more than the trend carry on every sequence, the eight pilot probes a trace that carrying saves. The guard is free, and buys nothing.

What one bad trace is worth #

The practical question is whether a trace estimator used along a sequence of operators should do anything about turns, and the measurements here say no. On the peak, the extrapolation’s error is a few dozen extra probes on one trace. On the trough it is one trace in sixteen with coverage nine points low, which the sequence’s average absorbs, and any rule that could prevent it would need to know the next operator’s spread before measuring it. A caller who cannot afford one low trace in a sequence can take a fresh pilot on every trace, which covers within two points of the table at two to nine per cent more probes; a caller who can afford it gains nothing from a guard.

The decay rate being the only thing that moves is a simplification that matters here. A real sequence — the operators of a Newton iteration, or of a parameter sweep — might move its spread in steps of every size, and the error at a turn is about the square of the last step. At a step of 0.02 in the decay that is a quarter at the trough. At a step three times larger, it would be a factor of two.

What 400 sequences of one family do not show #

Every sequence turns once, at the ninth trace, by a symmetric reversal of the same step; a turn that slowed before it reversed would give the trend factor a trace’s warning. The eigenvectors are fixed along the sequence, so only the spectrum drifts; a sequence whose eigenvectors rotated would change the spread through the off-diagonal entries, which no ratio of spreads tracks. Coverage on one trace is a proportion of 400 and carries 2.3 points of draw noise; the claims about single traces are made about the spread ratio, a median of 400, which does not.

Still open: a step that grows, and the contour #

A turn that is sharper than the steps before it. The trend carry’s error at a turn is about the square of the last ratio. A sequence whose decay moves by 0.02 a trace and then reverses by 0.06 would make the error at the trough about $0.88 \times 0.69 \approx 0.6$ of the true spread; the prediction with a sign is that coverage on the trace after that turn falls below 55 per cent, that a fresh pilot on that trace alone — which no history-reading rule can know to take — would restore it, and that the sequence’s average still stays within two points of the table.

The contour. Counting what is inside a circle estimates a trace whose exact value is an integer, and a spectrum drifting across the contour changes it between traces. That is a turn of a different kind — a jump in the trace itself rather than in its spread — and the prediction is that the ratio of the last two spreads stays near one across it, so that neither carry sees the jump and both miss its trace by the jump’s size.