Estimator — where it appears
Named by 51 essays across 7 fields — each of them below, with the objects they name alongside it.
How big a triangle it takes
Gauss proved that a surface-dweller can read the curvature off a triangle's angles. Doing it is another matter: a fifty-kilometre triangle has 5.49 seconds of excess, a one-second theodolite gives that excess a standard deviation of 1.73, and reading K to one per cent needs a side of 281 kilometres. Not one of the great surveys built a triangle within a factor of three of that.
The score is not stable at any scale
One boundary, read at eight resolutions from sixteen points to two thousand and forty-eight: the compactness score falls from 0.980 to 0.834 and is still falling. Changing the projection instead moves it by 0.69 per cent. The two decisions are made by the same person on the same afternoon and only one of them is ever reported.
Every reach set ever drawn is too small
An isochrone is drawn by walking out along a finite number of bearings and joining the points, so its vertices are on the true boundary and its edges are chords — which puts the drawn set inside the true one, always, at every count, for any convex reach set. The deficit falls as the square of the count, and a spherical cap loses less than a circle by exactly cos t (1 + cos t)/2.
How wrong a flat picture has to be
A determinant proves no flat picture of four places is exact and leaves the size of the failure where nobody can read it. Measured directly, the least error falls as the square of how much sphere the places span — fitted exponent 2.0088 — and the same exponent comes back from five different arrangements while the constant in front of it moves by a factor of seventeen.
A crossing is a chain of decisions
A route is handed back as a curve, and nothing anybody flies is decided once. Re-planning is worth exactly nothing when the forecast turns out right — a sub-path of an optimal path is optimal, and the chain is the plan. When it turns out wrong it recovers 19 per cent of the cost at two decisions and 83 at eighteen, and never all of it.
The best flat picture is not a map
A set of dots whose separations are as nearly right as separations can be made beats every projection in this collection — by a factor of thirteen on four world cities, and by eight per cent on sixteen. The collapse between those two numbers is not about cartography. It is that a picture of n places has 2n − 3 free numbers and n(n − 1)/2 distances to spend them on.
A drawn reach set stops at the river
A polygon drawn through a fan of bearings round a reach set is short by a chord deficit that falls as the square of the count — on ground with nothing in the way. Put a river three kilometres off, crossable at bridges, and the polygon misses 30 per cent of the reachable ground at every count from forty-eight bearings to a thousand, and a convergence check passes on it.
The exact map says the seam is smooth
The conformal seam's corner could only be bounded at about two degrees, because the series it was fitted with holds its boundary condition to three parts in a thousand. An exact map exists — the stereographic projection composed with ∫dt/√(1 − t⁴) — and it settles it: the corner falls in exact proportion to the arc the tangent is read over, fitted exponent 0.99, while the gnomonic's 21.4572° is the same to four decimals at every span.
The error belongs to a few of the places
The least error of any flat picture of sixteen cities is 53.3 per cent, and it is not spread over the sixteen. Six pairs hold it there and Sydney is in three of them: leave Sydney out and it falls to 25.5, while leaving out any of five other cities changes it not at all. Choosing which places to draw is worth more than any choice of how to draw them.
The span ladder, run on all five
A recorded shortfall said the exact map's span ladder was one loop away from settling the seam on the five Platonic solids. The loop was run and the answer is the other one: the series conformal face map has a corner that does not shrink with the measurement span on any of them — 4.10° on the tetrahedron, 0.77 on the cube, 0.004 on the icosahedron — and it belongs to the truncation rather than to conformality.
A place with a size can be drawn to scale
Points on a sphere have distances no flat picture holds. Places are not points: give every place a radius and the distance between two of them becomes a range, and some flat picture is right about every range once the radius passes a threshold — 38 kilometres for London, New York, Tokyo and Sydney, six metres for five towns in Britain, growing as the cube of the set.
The answer is a set
Fitting a projection to control points produces a best fit — one projection, ranked first, with a margin. A best fit without a spread is not a measurement, and the spread is free: a control point has a width, and every candidate whose residual is inside that width has not been ruled out. At one per cent noise a four-degree region admits ten of twenty candidates and a forty-degree one admits exactly one.
The first break is mostly its denominator
Three earlier measurements have fitted the near-optimal set's fracture threshold against region size and read the answer as a statement about the landscape. It is a ratio, and separating it takes one multiplication: the best score the region admits at all rises with exponent 0.92 and carries 65 to 70 per cent of the threshold's movement on three projections, while the pass's own depth falls slowly and steadily, with exponent −0.42 to −0.49.
A jet moves the floor that a uniform wind cannot
A uniform wind leaves the least radius at which the North Atlantic can be crossed almost where it was, because it blows the same way on both halves of every gap between airfields. A jet does not. A southerly of 200 km/h past the southern tip of Greenland lowers that floor by 36.9 kilometres, the same jet 400 kilometres further west raises it by 23.3, and one integral of the wind along each gap predicts every such move to within a few kilometres.
The average was a choice of norm
Ten projections, one region, one measured quantity, and the only free decision left is how to turn a field into a number. Over the world's scale departure the ordering at the mean and the ordering at the worst case have a rank correlation of −0.04, all ten maps change position, and the exponent that produced each answer is stated nowhere.
The sheet moved before it was measured
Every method of reading a projection off a map takes its control points from the sheet and assumes the sheet is the one the cartographer drew. Paper shrinks across its grain three times as fast as along it, and on a map whose grain runs along its own axis that shrinkage is EXACTLY a change of standard parallel — one per cent moves the recovered parallel by 0.57 degrees with the residual sitting at the solver's floor.
A cocked hat holds the ship one time in four
Three lines of position with any error in them bound a small triangle, and the instinct is that the ship is inside. With independent errors it is inside one time in four, whatever the triangle's shape. A common error in every sight makes that almost always when the stars surround the ship and almost never when they do not. And on a sheet that forgets the latitude, the triangle keeps its size and moves off the ship altogether.
A map with no graticule
Identifying a projection starts from control points, and a great many maps have none. Handed an outline with no labels on it at all, the method still works — and works better: the correspondence between ink and ground is recoverable exactly, because a similarity preserves ratios of arc length, and the margin on clean observations is 1.6 × 10¹⁰ against a graticule's 9.9 × 10⁶. What breaks it is noise, at three parts in a thousand.
A river boundary goes where the river goes, or stays where it was
A boundary that follows a river moves when the river moves, and the law of every such boundary turns on one word: whether the river moved gradually or suddenly. Over one meander's life, gradual movement changes the country of about twenty-eight square kilometres of land a loop. A single flood that cuts the loop off strands twenty-six — nearly the same land, arriving as one parcel on the wrong side of the river.
The residual reports the error the fix was immune to
A fourth sight gives a four-line fit something three lines never had: a leftover. The leftover is a real measurement of the sights, and it is useless exactly where it is needed. One number computed from the four azimuths alone — before any star is observed — says how much of a common error reaches the residual, and it is 1 when the bodies stand at the quarters of the compass, where such an error moves the fix by nothing at all, and 0.004 when they are bunched within sixty degrees, where it moves the fix by two kilometres.
What the extra unknown costs where nothing can see it
Four lines of position can estimate a common error in every sight as well as the position. The estimate is honest — three kilometres put in comes back as 2.8 — and where the residual is blind, its scatter is plus or minus thirty-one, and the fix that carries it is scattered fourteen kilometres from the ship against the biased fix's two and a half. Letting the residual choose between the two is worse than either, because a test with no power is a one-in-twenty lottery on the worse answer.
An ellipsoid computed to a nanometre is known to a decimetre
The meridian series is exact to seventy-six nanometres on WGS84, and WGS84 is exact by definition: its axis and flattening are conventions with no uncertainty at all. The ellipsoid that actually fits the Earth is a measurement, known to sixteen centimetres from equator to pole, and WGS84 puts the pole sixty-seven centimetres away from it. From its second term, the series is more accurate than anything it is used to compute.
A confidence ellipse is honest only where the sheet keeps angles
The cocked hat covers seven square kilometres and holds the ship a quarter of the time; the confidence ellipse from the same three sights covers fifty and holds it ninety-five per cent, which is what it was built to do. On a plotting sheet whose longitude is not corrected for latitude that ellipse fails in two unrelated ways — its centre is displaced, which costs coverage and depends on the navigator's own assumed position, and its shape is misread, which costs nothing but meaning and depends only on the latitude.
A bend in the barrier puts the drawn polygon over water
A straight river leaves the near side of a reach set convex, and that is what makes a fan of bearings fail in one direction only: every chord cuts a corner, nothing is claimed that is not there. A meander breaks that: the near side is no longer convex, and a chord between two bearings can cross water. So the drawn polygon now over-claims as well as omitting — by a seventh of a per cent, at every bend measured, and more bearings do not reduce it. The question was whether the two errors partly cancel. They do not, and the reason is not their signs: the over-claim never reaches a hundredth of the omission, because one scales with the whole set and the other only with the river.
A low sight is worth keeping only if it is weighted
A body five degrees above the horizon is seen through so much air that its refraction correction is ten times a high body's, and a tenth of that correction uncertain doubles the sight's error. Given the same weight as the others, a sight below seven degrees makes the fix worse than throwing it away; weighted by its variance it never does. The ellipse drawn without weights is turned twenty degrees from the right one, and a navigator who must guess the weights loses far less by guessing high.
Six projections have two curves, and the refusal has company
Three projections at four latitudes gave three pass-depth exponents, and the question was what sets them. Six at nine latitudes say that most of them have none to set: the Hammer's pass depth rises with what the region sees at R-squared 0.95 and Mollweide's at 0.81, Eckert IV's half-follows, and Robinson's, the Winkel tripel's and the sinusoidal's do not follow it at all. Neither candidate explanation predicts a left-out projection's exponent better than guessing the average — and the sinusoidal, once the lone refusal, is one of three.
The area weighting was a readership all along
Twelve measurements have taken a projection comparison apart and found the comparer's own freedom in every part of it. All twelve rested on one thing none of them examined: that a square kilometre of empty ocean counts exactly as much as a square kilometre of city. That is not the absence of an assumption about the reader — it is the assumption that everybody who will ever look at the map is spread evenly over the globe. Named as one readership among five, it produces a different winner from three of the other four, and against a North Atlantic readership its choice is fifth of twenty and 33 per cent worse.
Weighted by what it disagrees with, the fix trusts the sight it leans on
With six sights and two unknowns, the residuals seem to offer a way to find the bad sight without knowing the weather: weight each line by how little it disagrees, and solve again. One step of that gives a low body almost exactly the weight its altitude would, 4.5 per cent against 4.0. Every further step walks it away. Iterated until the weights settle, the fit is worse than weighting nothing in all four arrangements tried. Where the low body stands alone in its direction it ends with six times the weight it earns, because the fix leans on its line and its residual shows only 46 per cent of its error.
A copy eased into place loses its seams before its source
A stretch of coast lifted from another sheet and dropped in rigidly leaves two sharp corners where it joins, and both of them are findable. A compiler does not work that way: they pin the ends and ease the middle until it looks continuous. Easing halves the corner within five per cent of the outline and the seams survive it — until forty per cent, where the located seam jumps from four per cent of the coast out to twenty. The source projection survives twice as far, to seventy-five. The piece goes on naming where it came from after the method has lost track of which piece it is.
A week of twilights learns what a sight is worth
One fix cannot tell how good its low sight was, but every residual of every fix is a point on one straight line whose slope is the square of the refraction share, once each is corrected exactly for its own fix's geometry. After one twilight the estimate is zero four times in ten; after a week it is within half again of the truth nearly four times in five. And the lesson is worth little: a guess within a factor of two already costs under 4 per cent of the fix, a week's estimate 1.3, and a share that changes from night to night by half again costs more than the learning buys.
A forecast's error in where the streak is becomes a margin
A jet streak drifting along a gap moves the diversion floor by up to forty kilometres during a crossing, and an operator knows where the streak is only from a forecast. Planned to the worst over the forecast's own crossing window, the floor needs a margin of about eight kilometres for every hundred of error in the streak's position, and stops paying for itself against simply ignoring the streak near an error of two hundred and fifty. Planned to the departure analysis alone, it is worse than ignoring the streak at any error at all.
A residual cannot count the pieces of a map
A compiled map does not say how many sheets went into it, and trying every count and keeping the best fit cannot tell. Seven pieces always fit better than two, because they contain them. A criterion that charges ln N for each number a piece spends recovers the true count in every trial, for one copied stretch and for two, until the digitising error reaches a thousandth of the map. AIC charges two and takes every piece it is offered. And an eased copy defeats the right criterion the other way: it is not a projection, so BIC buys pieces to explain it, three seams, then four, then six.
Tonight's residuals measure tonight's air badly and weight tonight's fix well
A week of twilights learns the typical refraction share and nothing about tonight's, and a share that changes from night to night costs more than the week buys. With the sextant's error taken from the week, one twilight's residuals estimate tonight's share alone — badly: within half again of the truth 43 times in a hundred with three bodies near the horizon, and zero 22 times. Used as a weight, that poor estimate costs the fix between two and four per cent whatever the night, so it beats the typical share once nights differ by more than half again, and a blend of the two, leaning on tonight as the nights grow more variable, beats both. A weather predictor has to explain most of the nightly change to do better.
An ensemble carries its own margin, if its spread is honest
A fixed margin of about eight kilometres for every hundred of forecast error covers the diversion floor in nineteen crossings of twenty, but it needs the error's size and it is spent on every crossing alike. Nineteen forecasts of the same crossing need neither. If the truth is one more draw from the spread they come from, the highest of them covers the floor met in nineteen crossings in twenty by rank alone — whatever the error, the streak or the drift — and it holds a fifth to two fifths less in hand than the fixed margin, because it spends nothing on the crossings the streak cannot reach. What it cannot do is notice that its own spread is too small: at half the true spread it covers 88 in a hundred.
A loop reads a root's depth from its own pattern, and a triangle reads none of it
A levelling loop and a geodetic triangle respond to a buried root in proportions that change with its depth, so their ratio looked like a gauge of how deep a range is compensated. It is not, because a triangle's response to a root at 30 km is a hundredth of its own noise. The depth is there in the loops alone: a root changes the pattern of misclosures from loop to loop as it deepens, and 1,600 loops of 50 km place a 10 km root within a factor of 1.5 three times in four and a 30 km root about half the time. A 60 km root is not placed even by 6,400, and when the depth is fitted rather than stated, such a root is read as a light shallow one carrying 25 per cent of the mass where it carries 60.
A second tracing tells a drifting hand from a copied stretch
A hand tracing a coast drifts rather than jitters, and a drifting error is a smooth departure no projection explains — so the criterion that counted copied stretches correctly under independent noise counts four or five seams where two were made, and three or four on a coast with no copy at all. Undo the drift before fitting and every count comes back, for no copy, one and two, in every tracing. That needs the drift's correlation length from outside the coast, known to within a factor of three on the short side and much less on the long, and a second tracing of the same coast supplies it: the difference between two tracings holds nothing but the two errors, and read that way the count is right in 22 tracings of 24, and never too high.
An agency can map where its published covariance is optimistic
The local surveys tied to a regional network each estimate, badly, how far the network's published covariance understates the truth, and the agency that receives them can pool what they say. When the understatement is confined to one part of the region, pooling it into one number dilutes it: a patch covering 15 per cent of the region with errors four times their published size reads as 1.30 for the whole. Counted place by place, the same surveys find the patch — two agencies in three find it with 400 surveys and nearly every one with 800, flagging almost nothing outside it. Fed back as one number the scale removes little of the false alarms inside the patch and costs nearly a third of the test's power outside; fed back as a map it cuts the false alarms inside from 20 per cent to 3 and costs little outside.
An easing is a seam with a width, and a drifting hand hides the width
A copy eased into a compiled coast is not a projection, so a count that explains the coast in rigid pieces buys four, five or six seams for one copy. Give each seam a width — the copy blended into the host over that width, the same linear fit at every width, the width charged as one more number — and the count is two seams at every easing up to 17.5 per cent of the coast, and the width it reports is the one drawn, wherever the copy departs from the host by many times the digitising error. Where it departs by seven, the width is a guess. On a tracing made by a drifting hand the easing disappears: whitened for the drift, the rigid count already calls two seams in every tracing, because an easing as long as the drift is a stretch of drift, and its width can no longer be read until it is twice the drift's length.
A season learns the stretch from where the streak was, not from the floors
An ensemble whose members spread four fifths as widely as the forecast's real error covers the diversion floor in 94 crossings of a hundred instead of 95, and the repair is to stretch every member away from the centre by a factor learned from past crossings. Learned the way the rank histogram teaches it — from where the truth's floor fell among the members' floors — a season of a hundred crossings lands within a tenth of the right stretch one time in three, and a season of eight hundred two times in three. Learned from where the streak turned out to be, which the analysis after each crossing supplies whether the streak reached the gap or not, a hundred crossings land within a tenth 85 times in a hundred and two hundred almost always. The floor is the quantity the plan is about; it is the wrong quantity to learn the spread from.
Gravity is worth four loops a station, and the background still hides a deep root
Levelling loops alone place a range's compensating root at ten kilometres three times in four from 1,600 loops, and do not place one at sixty from 6,400. Gravity measured on the ground reads the root's attraction directly, and a station is worth about four loops for a shallow root and two for one at thirty kilometres. It does not place the deep root either: 1,600 stations put a sixty-kilometre root within a factor of 1.5 of its depth in fewer than one survey in five. With the background field taken away, a hundred stations place it every time. What hides the deep root is not the reading but the geoid's own mass that the range does not explain, and either half of its spectrum is enough to do it. Fitted for depth, gravity reads the deep root as too heavy, where the loops read it as too light.
Averaging tracings reads the easing, until the hand repeats itself
One tracing by a drifting hand cannot show how far a copy was eased, because an easing as long as the drift is a stretch of drift. Average several tracings and the easing stays while the drift shrinks: two tracings read the easing's width at the seam where the copy departs strongly, sixteen read it at the seam where it barely departs, and as the drift falls the rigid count starts buying pieces for the easing again while the eased count stays at two. All of that assumes each tracing drifts afresh. A hand that repeats part of its drift in every tracing leaves that part in the mean, where it sets a floor no number of tracings goes below; and a digitiser who learns the hand from the spread between tracings measures only the part that averages away, and charges the mean too little, so that both counts buy every piece they are offered.
A revised forecast is relearned in as many crossings as the learner remembers
An ensemble's stretch is a property of the forecast system, and forecast systems are revised. When a revision narrows the members from four fifths of the real spread to three fifths, a learner that weights every crossing since the start of the winter equally still holds 1.44 for a new stretch of 1.67 three hundred crossings later, and leaves the plan under the floor in 93.8 crossings of a hundred instead of 95. A learner that forgets regains ninety-five per cent coverage after about as many crossings as its memory — 31, 57, 113 and 206 for memories of 25, 50, 100 and 200 — and pays for forgetting only when nothing has changed. A memory of fifty costs nothing measurable over a winter. And a bias the whole ensemble shares, which the stretch alone reads as extra spread, is found to a kilometre by a learner that tracks the mean departure as well.
The spread between hands charges the mean, and an office's habit hides in the sheet
One hand's tracings cannot see the part of its drift it repeats, so a mean of sixteen by one hand is charged for a third of the error it carries. Sixteen tracings by four hands carry half the error of sixteen by one, and the spread between the hands' means, divided by the number of hands, charges that mean correctly to within a fifth on every sheet from three hands on — without ever estimating how much of a hand is habit. The same spread reads a hand's share of habit from two hands of two. What it cannot read is a habit every hand in the office shares: tracings of a sheet with one are tracings of a different sheet without it, and the charge they give falls further short the more hands and tracings are added.
A known coast charges an office's habit, and a bending one takes it out
An office whose hands all share a habit leaves it in every tracing of every sheet, where the spread between hands cannot see it. One control sheet — a coast known independently, traced the office's usual way — reads the habit's size and restores the charge, if the habit is a drift. If the habit answers to the coast's own curvature, a size read on a rounder coast charges a thinner one at little more than half, but the same control read differently does better than charge the habit: its departures, set against its curvature, give a slope that is the office's own on every coast, and subtracting it leaves about one per cent of the habit in the mean. The slope also tells the two kinds of office apart. A coast that hardly bends teaches neither, and corrects worse than nothing.
The worst point is not on the grid
Every maximum distortion this collection has printed is a maximum over a sample, and a maximum over a sample is a lower bound. On Mercator over a sixty-degree band a twelve-by-twelve grid reports 3.464 where the answer is exactly 4, and the shortfall does not go away with refinement so much as decay at a rate that says where the extreme is hiding.
A mean that does not exist can still be printed
Mercator's area-weighted mean areal factor is artanh(sin Φ)/sin Φ, and it has no limit. A sampler asked for it returns the logarithm of its own sample count plus 1.512 — measured slope 1.001 against ln n — so the number is a property of the person who computed it. The Kavrayskiy number for the same map over the same sphere settles at 0.52124 and is a number.
There is no equal-area lattice on a sphere
Every number in this collection is an average over a point set, and the three point sets available all fail to be equal-area in different ways. The equal-area ring sampler this site has used since its early essays is the one whose outermost ring sits half a step inside the rim — which is how a measured scale spread once came in 3.42 parts in a thousand below a proved bound.
A refinement that stops moving
Doubling the sample and watching the answer settle is how every quadrature in every field is checked. On the Robinson projection the doubling ladder — 4, 8, 16, 32, 64, 128 — converges beautifully, with its increments halving at every step, on a limit that is wrong by a factor of twenty-seven. Whether a grid finds the answer is decided by whether n is a multiple of four.
Which of these numbers are the sampler's
Four rungs have shown that a sampled maximum understates, a sampled mean can be a report on the sampler, no arrangement of points is neutral, and refining until the answer settles proves nothing. So the collection re-measured itself. The means move by at most 1.1 per cent, the worst points by up to 19, and the rankings — which is what the essays actually argue with — do not move at all.
The area is unbiased and the perimeter is not
A boundary measured from noisy vertices comes out long, always, by σ²/d on every leg. The area enclosed by the same vertices comes out exactly right, because a shoelace is bilinear and the cross terms vanish. So densifying a boundary makes its area five times more precise and its perimeter three thousand times more wrong, and every compactness score computed from it falls short.
The sample was drawn on the page
Every mean in this collection integrates over the sphere, because that is where the ground is. A raster, a pixel loop and any figure that walks its own canvas integrate over the page instead, and the difference is exactly the covariance between the quantity being measured and the map's own area distortion — 7.2° of mean angular deformation on Miller becoming 18.0°.
Named alongside it
The objects these essays reach for when they reach for this one.
VerificationToleranceLeast-squaresResidualClosed formPurposeNoiseConvergence rateSeamConvergenceNavigationRanking