A bad year does not average out

A branching plant whose growing points die can still be read by two numbers. What a scar is worth found that under one death chance $q$ for every point in every season, the living count grows at $(1-q)$ times the deathless rate and the scars per living point settle at $q/(x-1)$, so a rate and a scar share name the death chance, then the deathless rate, then how long a bud waits. Two ways to die, three things to count gave buds and apices different chances and found the two numbers still separable, badly conditioned, and owed a third.

Both essays drew every death independently: a coin for each growing point, fresh every season. Both named the weakness that assumption hides. A bad season — a late frost, a drought — does not kill points one at a time. It raises the chance for every point on the plant at once, and the next season it may be gone. This essay gives each season one death chance shared by the whole plant and asks what the two numbers are then worth.

One chance per season, shared #

The grammar is the one the counts were first read by: a mature apex makes a bud and stays, a bud waits two seasons and matures, and each season every living point dies with some chance before the survivors rewrite. What changes is where the chance comes from. Each season is bad with probability one in ten, and then every point dies with chance $q_{bad}$; otherwise every point dies with a smaller $q_{good}$, set so that the chance averaged over seasons is 0.1, the value the earlier readings used. At $q_{bad} = 0.1$ the good and bad seasons are the same and the model is the independent one exactly. At $q_{bad} = 0.5$ a bad year kills half the plant and the good years kill 5.6 per cent.

A plant is taken large enough that its own point-by-point draws average out within a season. Given its sequence of seasons, its counts are then the recurrence run with that season’s chance, and everything measured below is the spread the seasons alone produce. That spread is the part a larger plant cannot remove: a thousand points on one plant all see the same frost.

The expectation does not move #

The first result is the one the earlier essay predicted, and it holds exactly. Averaged over all 4,096 sequences of twelve seasons, each weighted by its probability, the expected living count and the expected scars at every season are the independent model’s at 0.1, to a part in a billion — the difference is rounding.

The reason is that seasons are independent of each other even though points within a season are not. The expected count after a run of seasons is an expected product of survivals, one per season, and the expectation of a product of independent factors is the product of their expectations. A bad year with a matching good year has the same average survival as two average years, and the average is all an expectation sees.

So everything the scar essay derived about expectations stands. What it does not describe is any particular plant.

The median plant falls behind #

Divide each plant’s living count by the expected count and the picture is plain. With bad years at 0.2 the median plant holds 97 per cent of the expected count after forty seasons and the middle ninety per cent of plants lie between 66 and 143 per cent. With bad years at 0.5 the median plant holds 54 per cent, and the band runs from 8 to 364 per cent. At 0.7 the median plant holds 16 per cent of the expected count, and a twentieth of plants hold under half a per cent.

The mean of the four hundred stays near one, as the expectation says it must, and it stays there by being carried by a few lucky plants. A plant that escaped its bad years grows far ahead of the expectation, and the plants that did not fall far behind it, and the average of a lopsided distribution sits well above its middle.

A plant settles below its expectation #

Run long enough, every plant’s rate approaches one number, and it is not the expected rate. With bad years at 0.2 it is 1.3183 against the expected 1.3190; at 0.3 it is 1.3154; at 0.5, 1.3001; at 0.7, 1.2624. The expected rate is the same 1.3190 at every point, because the averaged death chance is the same at every point.

The gap is the difference between the logarithm of an average and the average of a logarithm. A plant’s count after many seasons is a product of its seasons’ growth, and the rate of a long product is set by the average of the logarithms of its factors. The expected count is set by the average of the factors themselves. A factor of one half and a factor of one and a half average to one, but their logarithms average to less than nought, and a plant that meets each once has shrunk. The worse the bad years, the wider the spread of factors and the larger the gap: 0.05 per cent at 0.2, 1.4 per cent at 0.5, 4.3 per cent at 0.7. Compounded over forty seasons, 1.4 per cent a season leaves 56 per cent of the expected count, close to the 54 per cent the median plant was found holding; the rest is the early seasons, before the plant’s rate has settled.

The gap also grows faster than the bad years do. A bad year worse than the average by 0.1, 0.2, 0.4 and 0.6 costs the settled rate 0.05, 0.27, 1.43 and 4.29 per cent: four times the excess costs twenty-seven times the gap, and six times the excess eighty-one times. The average of logarithms is most sensitive to the seasons that take the most, since the logarithm of a survival falls steeply as the survival approaches nought, so a rare season that kills seven points in ten does more to a plant’s long-run rate than its share of the average suggests.

This is the first thing the earlier readings could not see. The rate they read a plant by is the expected rate, $(1-\bar q)$ times the deathless root. Under shared deaths no plant has that rate, however long it is watched, and the reading takes the plant’s slower rate as evidence of a higher death chance or a longer wait.

The scar share jumps at every bad year #

The second number behaves worse. Under independent deaths the scars per living point settle at 0.313 and stay there. Under shared deaths a bad year at 0.5 kills half the plant at once: the scars jump by half the living count and the living count halves, so the share rises by about one in a single season. Between bad years the plant outgrows its scars, and the share decays by roughly the plant’s growth each season.

So the share is a sawtooth, and it never settles. Three plants read at season sixty stand at 0.184, 1.807 and 0.145 — one of them a season after a bad year, the other two in long good stretches. The independent model’s 0.313 is not a value any of them sits at; it is roughly where their sawtooth averages.

It is set by the last bad year #

Sorting plants by how long ago their last bad year was turns the scatter into a curve. With bad years at 0.5, a plant read the season after one averages a share of 1.35 — four times the independent model’s 0.313. A season later it is 0.99, then 0.77, 0.63 and 0.48, and it passes below 0.313 about six seasons on. Twelve seasons after the last bad year the share is 0.17, half the independent value, because the good years kill only 5.6 per cent and the plant has had twelve seasons to outgrow its old scars.

With bad years at 0.3 the same curve runs from 0.63 down to 0.23, a factor of under three rather than eight. The share a plant is read at therefore carries one thing above all: the date of its last bad year. The seasons before that are there, but diluted by every season of growth since, so the share remembers the recent past and forgets the rest.

The floor the share decays to #

The decay between bad years has a floor, and it can be named. A plant long past its last bad year has been living under good years only, with a death chance of 0.056 at $q_{bad} = 0.5$. Under that chance alone it would grow at 1.3842 a season and settle at a share of $0.056/0.384$, which is 0.145. That is exactly the fifth percentile of every population of plants read here, at every window from ten seasons to eighty: the plants with the lowest shares are the ones whose last bad year is furthest behind them, and they sit on the good years’ own value. With bad years at 0.3 the same arithmetic gives 0.221, and again it is the fifth percentile.

Each season on the way down the share falls by a factor near the plant’s growth — 0.74, 0.77, 0.81 and 0.76 over the first four seasons after a bad year at 0.5 — because the scars stop being added at the bad year’s rate and the living count keeps multiplying. So the sawtooth has a shape that can be written down: a jump of about one at each bad year, a decay by the growth rate, and a floor at the good years’ share. None of the three is the independent model’s 0.313.

Which reading steadies with time #

The two-chance essay asked whether the scar share, accumulating over every season, would be the steadier of the two numbers. The measurement says the opposite, and says why.

Across plants, the spread of the rate narrows as the window lengthens. With bad years at 0.5 the logarithm of the ninety-fifth percentile over the fifth is 0.159 over ten seasons and 0.079 over eighty; at 0.3, 0.069 and 0.034. It narrows the way an average does, as more seasons’ growth goes into it.

The spread of the scar share does not move. At 0.5 it is 2.19 over ten seasons and 2.28 over eighty; at 0.3, 1.04 and 1.06; at 0.7, 3.9 and 4.1. A share accumulates over every season, but it does not average over them, because each scar is weighed against the living count at the moment of reading, and the living count has grown by a different factor since each season. Old scars are divided by a large number and new ones by a small one. The share is a record of recent seasons however many are behind it, and no length of observation makes it converge.

More seasons read the wait worse #

Put the two together and read the plant as the scar essay does: the death chance from the share times the rate less one, the deathless rate from the rate over the survival, the wait from the deathless rate. With bad years at 0.2 the reading names the two-season wait for 84 per cent of plants over ten seasons and 96 to 98 per cent over twenty or more; mild bad years cost little.

With bad years at 0.5 it names the right wait for 57 per cent of plants over ten seasons, 43 over twenty, 32 over forty and 22 over eighty. With bad years at 0.7 it falls from 49 to 11. Reading more of a plant’s history makes the answer worse.

The two findings above account for it. The longer window makes the rate precise, and the value it becomes precise about is the settled rate, below the expected one the reading assumes. The share stays as scattered as ever. A reading that combines a precisely wrong number with an unconverged one does not improve with patience; the short window is right more often only because its rate is still loose enough to land near the expected value by chance.

What the reading names instead #

The failures are not scattered evenly, and they are silent. Over eighty seasons with bad years at 0.5, the reading names a three-season wait for 37 per cent of plants — more than name the true two — and one season for 18 per cent, four for 11 and no wait at all for 12. It refuses one plant in a thousand. The death chance it reads averages 0.138 against the true 0.1 over eighty seasons, and 0.097 over ten, so the long reading is not only scattered but pushed one way: the slower settled rate and the share’s heavy upper tail both push towards more death and a longer wait.

That is the pattern the delay essay warned of when it made the wait the thing a Fibonacci count measures: a longer wait is less tolerant of death, and a reading that overstates the death chance compensates by overstating the wait. Under shared deaths the compensation is systematic. Every reading still returns an answer, and nothing in the answer says it came from a plant that had a bad year — the kind of failure a count that carries no error is written against.

What this means for a survey #

How many plants would it take sized its surveys on the assumption that specimens are independent draws. Plants on one site share their seasons. A frost that halves one plant halves its neighbours, so thirty plants from one meadow are thirty readings of one sequence of seasons, and their spread understates the spread a second meadow would show. For the rate this is a bias shared by every plant in the meadow; for the scar share it is worse, since a whole site read the season after a bad year reads high together.

The remedy the arithmetic suggests is to sample across sites, or across years, rather than across plants, because the seasons are the unit that varies. That is the same shape as the census’s lesson about counting: a sample’s size is not its number of specimens when something every specimen shares is the thing being measured.

What a shared season leaves out #

The model draws bad years independently, one season in ten. Real bad years come in spells — a drought runs two or three seasons — and a spell would widen every spread measured here, since several bad seasons in a row behave like one very bad one.

It gives buds and apices the same bad year. The two-chance essay found reason to think buds are the more fragile, and a frost may well take buds and spare apices, which would move the composition of the plant as well as its size.

And it treats the plant as large. A small plant adds its own point-by-point draws to the seasons’ spread, which is the independent model’s noise, and the two add; a small plant is never better read than the large plant measured here.

Findings that would undo it #

A sequence of independent seasons whose expected count differs from the independent model’s at the same averaged chance. A plant, watched long enough, whose rate settles at or above the expected rate while its bad years are worse than its good ones. A scar share whose spread across plants narrows as the window lengthens. Any of the three would mean the account of what a shared season does is wrong, not just incomplete.

Still open: scars dated by the season they were made #

Everything here reads the scars as one total. On many plants a scar can be dated — by its position along a shoot, by the growth ring it sits in — and a dated scar count is a series of deaths per season rather than a single share. From it the bad years can be picked out, each season’s death chance read as that season’s scars over the points then living, and the averaged chance recovered as an average over seasons rather than as a ratio the last bad year dominates. The measurement is how many dated seasons it takes to recover the averaged chance and the wait to the precision the independent model gave, and whether the bad years’ own chance, which the plant’s rate is most sensitive to, can be read from the same series.