Exponent error — the series

A tree built at 3, measured to 2%, reads 2.957 on the informative band. The exponent recovered from 100 junctions of a tree built at exactly 3, against the relative error placed independently on the parent and on both daughters — half a per cent is a machined section under a microscope, one to two per cent is callipers on a clean branch, five is a branch that is not round, ten is a radius read off a photograph. Each line is a band of daughter ratio and the whiskers are the central 90% of 300 replicate samples. Every band is displaced downward at every error and never upward: at 2% the informative band read 2.957 and the informative band read 2.957. Below, the same rows with the displacement and the spread drawn as separate bars, because only one of them falls when more junctions are measured.

The exponent an error moves

Every real measurement of a branch radius carries error and no synthetic tree does, so the question is what a symmetric error does to a fitted exponent. It does two things — a bias and a spread — and the bias runs downward at every error level and in every band, by an amount derivable from the daughter ratios alone.

part 1 · branching

The fragile junctions are the informative ones

That is the obvious worry once the radii are uncertain, and it is false. Across the whole range of asymmetry a junction's contribution to the bias moves by a factor of 1.36 while its leverage moves by a factor of 308,352, so the junction that says nothing damages the answer as badly as the one that says everything — and a sample is spoiled by counting rather than by weight.

part 2 · branching

Between 3% and 5% of radius error, no sample size answers — 50 junctions among them. One row per error level. The pale bar is the sample sizes whose interval is narrow enough to state a claim from — half-width under ±0.25 and excluding 2 — and it starts where precision arrives. The second bar is the sample sizes whose interval still contains the 3 the tree was built at, and it ends where the displacement overtakes the width. Where the two overlap there is a usable window; at 5%, 7%, 10% they do not overlap at all, so below 50 junctions the answer is too wide to state and above 30 it no longer contains the truth.

The window that closes

The spread of a fitted branching exponent falls as the reciprocal root of the sample and its displacement does not fall at all, so there is a count past which every further junction buys confidence and no accuracy. Between three and five per cent of radius error the count arrives before the answer does, and no sample size both states a claim and contains the truth.

part 3 · branching

Where three and two become one

Two trees, one built to obey Murray's law and one to obey Da Vinci's, are measured through the same fifty junctions with the same instrument. At twelve per cent of error on each radius the two answers overlap, and above twenty and a half the tree built at three measures lower than the tree built at two.

part 4 · branching

What a swelling does to the exponent read from the informative band, by where the swelling is. 50 junctions of daughter ratio 0.6–1, built at exactly 3 and read with no random error, but with one or more radii measured fat. A parent read fat lowers the reading: 1% gives 2.864, 3% gives 2.631, 10% gives 2.075, and at 11.3% the tree reads Da Vinci's 2. Daughters read fat raise it: 1% gives 3.150, 3% gives 3.499; from 8% some junctions have a daughter measured wider than their parent, which no exponent balances, and the line stops. All three radii read fat by one factor return 3.000 at every swelling — the unswollen reading exactly.

A swelling at the fork

A branch thickens where it forks, so a parent measured just below a junction and daughters measured just above it carry three different amounts of the same swelling. A swelling that fattens all three alike moves a fitted exponent by exactly nothing. A parent read one per cent fat moves it by as much as 3.6 per cent of random error on every radius, in a sign known in advance, and a tenth of a radius turns a tree built at Murray's three into one that reads Da Vinci's two with no noise at all. Added to the noise, it does not bring the two rules together any sooner: the two errors do not add.

part 5 · branching

Adding error to carry a fitted exponent back to none, at 12%. At 12% of error on every radius each replicate is refitted with more error added at four levels, and the dots are the means over 300 replicates, the tree at 3 above and the tree at 2 below. Uncorrected they read 2.035 and 1.676. Carried back to no error through the mean points, the tree at 3 reads 2.695 by a quadratic curve, 2.380 by a linear curve, 2.914 by a rational curve; the tree at 2 reads 1.961 by a quadratic one, 1.877 by a linear one, 1.982 by a rational one. The curve carried back is a choice the method does not make.

A correction that keeps the overlap

The duel between a tree built at Murray's exponent and one built at Da Vinci's ended by saying the displacement is the geometry, and that no better estimator removes it. Correcting every replicate by simulation-extrapolation removes 92 per cent of the tree at three's displacement at five per cent of error and 68 per cent at twelve, and the error at which the two means cross leaves the measured range altogether. It pays in spread — the corrected readings are twice as wide at twelve per cent — so the error at which the two trees' intervals overlap does not move. Of the duel's two numbers, the inversion was the estimator's and the overlap is the question's.

part 6 · branching

Measured radii against the tips each branch carries, at 12% of error on every radius. One replicate of a 50-junction tree, 101 segments, every radius measured with 12% of relative error and drawn against the number of tips that segment carries. The count has no error in it, so the slope is not displaced: the tree built at 3 gives a slope of 0.3404, an exponent of 2.937, and the tree built at 2 a slope of 0.5071, an exponent of 1.972. On the same measured radii the junction-by-junction fit reads 2.218 and 1.736.

A count carries no error

Fitting r₀ᵏ = Σrᵢᵏ junction by junction puts a measured radius on both sides of every equation, and at twelve per cent of error a tree built at Murray's three and one built at Da Vinci's two stop being told apart, however the fit is corrected. Fit the same measured radii against the number of tips each branch carries instead — a count, which nobody measures with error — and the two trees read 2.996 and 1.997 at twelve per cent and 3.015 and 2.001 at thirty, never overlapping. The twelve per cent belonged to the junction fit, not to the tree. The count fails in its own way, and the way is stated.

part 7 · branching

The exponent read from a tree that has lost tips, tips lost one at a time. The 50-junction tree built at 3 and at 2, 12% of error on every radius, 300 replicates of loss and measurement at each level. Shaded: the central 90% of the exponent read against the tips still there. Dotted: the mean read against every tip ever grown. Solid: the junction fit's mean over the junctions that survive. The count's two intervals are still apart with 70% of the tips gone, reading 2.366 and 1.576, and overlap by 80%. The junction fit's intervals overlap by 20%. At the heaviest loss drawn, 90%, the scars still read 2.998 and 1.997.

A count that has lost tips

Radii read against the tips each branch carries keep Murray's three apart from Da Vinci's two where junction fits cannot, because a count has no measurement error in it. A count of the tips a tree has is not a count of the tips it grew. Losing them lowers both trees' readings by one factor that belongs to the losses and not to the rule, so the count stops being right long before it stops telling the trees apart: on fifty junctions at twelve per cent of error, to seventy per cent of the tips lost one at a time, and only to about a quarter lost in whole limbs. Counting scars repairs single losses exactly. Nothing countable repairs a shed limb.

part 8 · branching

A loss that picks its tips can read high

A tree that has lost tips at random reads its branching exponent low, by one factor shared by every rule. Let the loss choose which tips go and the factor is still shared — the ratio of three to two survives any choice — but it is no longer below one. On fifty junctions with half the tips gone, the choice of which half moves Murray's tree from 2.23 to 3.52, and past about half the tips a single tree's reading can no longer say which rule built it. The ways a crown plausibly chooses stay within a few hundredths of a fair loss, all of them low, and counting scars repairs every one.

part 9 · branching