The answer is a set

Every rung of this anchor has ended with an answer of the same shape: a projection, ranked first, with a residual and a margin over the next candidate. The fit finds the projection when the projection is in the library; it does not when it is not; a careless copy hides itself; a datum hides inside the parameters; a rotation does not hide the way a shift does.

Not one of those answers came with a spread.

That is a strange omission in a collection whose founding rule is that a property is not printed until it is measured, and whose very first rung about coordinates is a coordinate is a number with a width. A best fit without a spread is a point estimate, and a point estimate on its own is not a measurement.

Where the spread comes from #

It costs nothing to get, because it is already in the problem.

A control point read off a map has a width: a digitising error, the thickness of the graticule line, the paper’s response to humidity, the scanner’s registration. Call it $\sigma$, as a fraction of the map’s own size, and it is between a tenth of a per cent for a modern scan of a modern sheet and several per cent for a point pricked off a folded historical chart.

A candidate whose residual after fitting is below $\sigma$ has not been rejected by anything. The data cannot tell it from the truth, and calling the best-ranked candidate the answer is choosing between things the measurement did not separate.

So the honest output is the set of candidates whose residual is inside the noise.

The test used here is the loosest sensible one — residual no greater than noise. A likelihood-ratio test with the right degrees of freedom is tighter by a factor of order one and needs the errors to be independent, which a digitised graticule’s are not: a slipped registration correlates every point on a sheet. The loose form is the honest one and the sets below are if anything too small.

What is in the set #

Over a sixteen-degree region at 45° north, at three parts in a thousand of noise, a map drawn in the conformal conic admits:

• the conformal conic,
• the equal-area conic,
• the polyconic.

Those three have nothing in common except a family resemblance. One is conformal, one is equal-area, and the third is neither and is true to scale along every parallel. A user told “this map is a conic” has been told something; a user told which conic has been told something the map does not support.

That is worth stating in the terms this collection uses everywhere else. Every projection minimises something, and the three admitted here minimise three different things. Identifying the projection is usually a step toward knowing what the map preserves, and over a small region that step does not deliver.

What buys the answer #

Read the table two ways.

Across a row, the set falls from ten to one — a factor of ten — over a twenty-fold increase in the region’s extent.

Down a column, it falls from ten to three over a thirty-fold improvement in the measurement.

Extent is worth more than precision, by a wide margin, and the practical consequence is uncomfortable: the extent of the map a practitioner has is fixed, and the precision is the only thing they can improve. A better scanner, a finer digitiser and more care buy a factor of three; the map stays the size it is.

The mechanism is rung 2’s: projections differ by quantities that vanish with the region’s size, so a small region is a place where every projection is nearly the same map. Improving the instrument chases a difference that is going to zero.

The set at two extremes #

Two more placements make the range concrete.

The pair is the whole argument. Nothing about the measurement changed between them; what changed is how much of the world the map covers, and that is not a property of the analysis at all.

What a practitioner should report #

Three things, and all three are already computed by the machinery that produces the current answer.

The set, not the winner. “A conic, and the data does not separate the conformal from the equal-area” is a complete answer and is shorter than the paragraph usually written round a single name.

The noise the set was computed at. A set is a function of $\sigma$ and the table above is what varying it does. Quoting a set without its noise level is the same defect as quoting a maximum without its sampler.

And the extent. Which is the quantity doing most of the work and is the one thing already in every published description of a map, so it costs nothing to relate the two.

What was computed, and how #

The graticule is drawn from a stated candidate at a stated centre, fitted by every member of the twenty-projection candidate library with a similarity transform absorbed — scale, rotation and offset removed, because a reproduction has all three — and the residual reported as a fraction of the map’s own size. That machinery is the anchor’s own from its first rung; nothing about the fitting changed.

What changed is one line: instead of returning the first row, return every row whose relative residual is below a stated noise.

Four assertions carry the rung and the first is the one that matters most.

The set must never exclude the truth. At every region size and every noise level in the table, the projection the map was actually drawn in must be in the set. A confidence set that excluded the truth would be a bug rather than a result, and it is the only check here that tests the construction rather than the maps.

The set must reach one somewhere — or identification never works, and the anchor’s eight rungs are about nothing. It must reach most of the library somewhere — or it always works, and this rung has nothing to say. And it must be monotone in both directions, which a set defined by a threshold on a residual has to be, and which would fail if the fitting were not finding its own optimum reliably.

Why the margin was not enough #

This anchor’s machinery has reported a margin since its first rung — the ratio of the second candidate’s residual to the first’s — and a reader might reasonably ask why that was not already the spread.

It is not, and the reason is a distinction worth keeping. A margin says how much better the winner is; it does not say whether the difference is measurable. A margin of five sounds decisive and is decisive only if the loser’s residual is above the noise. On a four-degree region the top five candidates have residuals differing by factors of two or three and every one of them is below a per cent of the map’s width, so the margins are large and the differences are invisible.

The two quantities answer different questions and only one of them involves the instrument. That is the same relation as the one between a difference and a standard error everywhere else in statistics, and it is why a coordinate is a number with a width is the essay this rung is really a descendant of: a number without its width supports no comparison, and a residual is a number.

What this does to the anchor’s earlier results #

Eight rungs’ findings survive, and it is worth saying which of them survive for which reason.

The ones about mechanisms are untouched. That a datum shift is absorbed into a conic’s parameters, that a rotation is not absorbed the way a shift is, that a careless copy leaves a shear — those are statements about what a fit does, demonstrated with exact data, and noise does not enter.

The ones about extents are strengthened. Two projections that cannot be told apart measures the region size at which a pair’s residual falls below a stated tolerance, which is this rung’s construction applied to one pair. This generalises it from a pair to a library and reports the count rather than a threshold.

And the one worked identification is the one that needs restating. Wherever this anchor has said the map is in projection X, the honest form is X, and n others the data does not separate, with n from the table. On the regions this anchor has used — mostly fifteen to twenty-five degrees — n is between one and four.

Where the model stops #

Twenty candidates. The set’s size is a count within a stated library, and enlarging the library enlarges the set. When the answer is not in the library is the complementary failure and it is not cured by this rung: a set of size one is not a proof, it is a statement that nineteen alternatives were rejected.

Noise as a single number. A real digitising error has structure — larger along a fold, correlated within a sheet, worse where the graticule is faint — and a scalar $\sigma$ is a summary of it. The consequence is that the sets here are the sets for the best case, and a correlated error of the same size admits more.

One centre and one true projection. The table is a conformal conic at 45° north. A cylindrical truth at the same size gives a different table, because its rivals are different; the qualitative shape — extent beating precision by a wide margin — holds in every case tried and only three were tried.

And the loose test is a choice. A tighter test gives smaller sets and needs an error model this problem does not have. Stating which test produced a set is part of stating the set, exactly as the tolerance that decides the verdict requires for a ranking.

The set at the sizes this anchor works at #

Most of this anchor’s worked examples are at fifteen to twenty-five degrees of half-extent, which is thirty to fifty degrees across, and at that size the table says the set is between one and three at any realistic noise.

That is a comfortable place to be and it is worth saying why it is comfortable: at thirty degrees across the projections in the library differ by quantities of the order of a per cent of the map’s width, which is the same order as a careless digitisation and an order above a careful one. The anchor’s examples sit exactly at the boundary where identification starts working, which is not an accident — it is where the question is interesting.

The generalisation #

The rule is one this collection applies everywhere else and had not applied here: an answer to an inverse problem is a set, and reporting its best member is reporting a summary of the set without saying which summary.

That is exactly the mistake a residual has more than one explanation identified from the other direction. There, one residual admitted several causes and the rung’s finding was that the fit cannot choose between them. Here, one dataset admits several projections, and the difference is only that the alternatives are enumerable — so the set can be printed rather than merely acknowledged.

The habit this suggests is small and general. Wherever a procedure returns a best member of a library, ask what the second-best is and whether the data separated them. The margin over the next candidate is already computed by this anchor’s machinery and has been reported since its first rung; what was missing was the comparison of that margin against the noise, which is one division.

What it costs to report #

Nothing, and that is the argument for doing it.

The residuals of every candidate are already computed — the ranking requires them — so the set is a filter on a list the machinery is holding anyway. The noise level is the one new input, and it is a number the person who digitised the map is better placed to state than anybody who later reads the result.

The output is shorter than what it replaces. “A conic; the data does not separate the conformal from the equal-area at 0.3 per cent digitising noise” is one sentence, and it says what the current output’s single name plus a residual plus a margin does not: whether the answer is one projection or three.

Who found it, and when #

Confidence sets are Neyman’s, from 1937, and the practice of reporting a set rather than a point is standard in every field that fits models to noisy data. Model selection — choosing which of several models the data supports — has its own large literature, and the modern form of it, from Akaike onwards, is explicit that a selection without a measure of separation is not a result.

None of that reaches the cartographic identification literature, which is small and practical. Identifying the projection of a historical map is done by fitting candidates and reporting the best, and the good papers report the residual; the question how much better than the second is that is asked informally when the answer is obviously yes and not at all when it is not.

The reason is probably that identification is usually attempted on maps large enough for the answer to be unique — a world map, or a continent — where the set really is a single projection and there is nothing to report. The failure lives on small maps, which is where the historical questions are, and where the number of surviving sheets is largest.

What a catalogue entry should say #

The finding has a practical form, and it is about how an identification gets written down rather than how it is computed.

A catalogue entry naming a projection is a point estimate presented as a fact. Sinusoidal in a record has no residual attached, no runner-up, and no indication of whether the second candidate fitted almost as well or not remotely as well — so the reader of the catalogue cannot tell a determination from a preference, and neither can the next researcher who builds on it.

Three fields would fix it and none of them is expensive. The best candidate and its residual; the runner-up and its residual; and the size of the set at whatever separation the cataloguer is willing to defend. That is a line of output from a fit that has already been run.

When the set has more than one member, the honest entry names what its members share. Very often that is the family rather than the projection — a cylindrical equal-area, standard parallel undetermined — and a family is a real finding: it says the map-maker chose to preserve area, which is the historically interesting decision, while declining to assert a parameter the sheet cannot support.

And a set of size one is worth recording as such, because it is the case where the current practice happens to be right and there is currently no way to tell it apart from the cases where it is not. A catalogue in which every entry is a bare name has thrown away the distinction between its strongest and weakest determinations, and the strongest ones are the ones somebody should build on.

Where the ladder goes next #

Nine rungs have taken identification from a fit to a set. Every one of them has assumed the map was drawn in some projection from a library, correctly, once. A great many historical maps were not: they were compiled from other maps of different projections, joined, and adjusted by eye, so no single projection fits any of them and the residual is a mixture of construction and compilation. Telling those two apart is the next thing this anchor cannot currently do.