A condition imposed at points is not a condition

Nine rungs of this anchor state a condition and solve it. A projection written as a condition; three conditions one too many; solving for the map instead of choosing it; the nearest map to an impossible request.

Every one of those solves imposes the condition at a finite set of points, because a linear system has finitely many equations. Nothing else is possible, and nothing in the language of the ladder has said so. Equal-area means at every point of the sphere and is asked for at a few hundred; conformal means at every point and is checked at a few hundred more; and the two sets are different, which is the gap this rung measures.

The construction being tested #

Chebyshev’s criterion says the best conformal map of a region is the one whose scale factor is constant on the boundary. The site solves that by writing the map as a series with 2k + 1 real coefficients and requiring the boundary scale to be constant at a stated list of boundary points — a least-squares problem, exactly as every construction in this ladder is.

The question this rung asks is what the word constant means when it has been imposed at a list.

Two numbers answer it. The on-sample residual is the largest departure of the boundary scale from constant, measured at the points the fit used — which is what the solver returns and what every report of such a fit quotes. The between-sample residual is the same quantity measured on a boundary sampled twenty-four times more finely, which contains those points and the ones between them.

The measurement #

A square spherical cap of angular radius 25°, seventeen free coefficients, and the number of collocation points swept.

With twenty equations against seventeen unknowns, the solver reports a residual eight times better than the map’s. It is not lying: the departure really is 3.9 × 10⁻⁵ at the points it was asked about.

The finding is in the third column #

Read the two columns together and the obvious conclusion inverts.

The actual residual barely changes. It runs from 3.27 × 10⁻⁴ down to 1.98 × 10⁻⁴ and back up to 3.04 × 10⁻⁴ across a twentyfold refinement of the collocation — a factor of 1.65 either way, with no trend.

The reported residual rises by a factor of 7.8 to meet it.

So refining the collocation does not make the map better. It makes the report honest. The map a fit produces is about as good at twenty samples as at four hundred, because seventeen coefficients can only do so much; what the extra samples buy is knowing that.

That is the reverse of how such a refinement is usually described. Adding constraints sounds like tightening a condition; here it is auditing one.

And more freedom opens the gap #

The mechanism is the ordinary one and it has a name outside cartography. With few coefficients against many samples the fit is genuinely over-determined: the residual is spread across the samples and there is nothing left over to hide. As the coefficients approach the samples in number the fit gains enough freedom to drive the residual to zero at the samples specifically, and it does, because that is what least squares minimises.

The last row is past the crossing — twenty-five unknowns against twenty-four equations — and the system is under-determined. Everything falls apart at once, which is the right behaviour and is worth including because it is the failure a reader would notice.

The control #

A rung claiming that a fit conceals its own residual has to show the case where it does not, or the measurement is a fact about the comparison rather than about the fitting.

With four hundred samples against seventeen coefficients the two residuals agree to four significant figures and the hidden factor is exactly 1.000. The machinery can measure a case with nothing to hide, and it reports nothing.

What the profile shows that the numbers do not #

The two profile figures are worth putting side by side, because they make the mechanism visible in a way a ratio cannot.

At twenty samples the curve oscillates between the marks. Each mark sits on a near-zero crossing and the curve swings away between consecutive ones — which is exactly the shape a polynomial interpolant makes between its nodes, and it is why the phenomenon has the same name in numerical analysis as it does here.

At forty samples the oscillation has not gone away and its amplitude has barely changed. What has changed is that the marks are now dense enough to land on the excursions, so the largest departure the solver sees is the largest departure there is.

The map is oscillating either way. That is what seventeen coefficients do to this boundary, and no amount of collocation changes it — which is why the honest description of the refinement is that it stops the fit from being able to hide.

What this does to the ladder below #

Every construction in this anchor is a collocated fit, so every one of them has a residual it does not report, and the size of it depends on the ratio between samples and unknowns.

Two of the nine are worth naming.

The nearest equal-area map to an impossible request minimises a departure over a sample of the sphere, and its residual is quoted on that sample. Its sample is far larger than its coefficient count, so by the measurement above it is in the safe regime — but nothing in that rung says so, and nothing in it would have noticed if it were not.

Not every distortion can be asked for measures how far the nearest achievable map falls short of a stated request. That shortfall is a difference of two residuals, so a hidden component in either one propagates into it directly.

The repair is cheap and is now standard practice on this site: measure the condition on a set the fit did not use. It costs one extra evaluation pass and it converts a residual that cannot fail into one that can.

What would actually improve the map #

Since refining the collocation does not, it is worth saying what does — three things, in order of how much they buy.

More coefficients, up to the conditioning limit. Going from nine to seventeen takes the actual residual from 7.3 × 10⁻⁴ to 2.9 × 10⁻⁴, which is a real improvement in the map rather than in the report. Going past twenty-four takes it to 0.32, because the system is then under-determined.

Better-placed collocation points. The oscillation is largest near the boundary’s corners, and an even spacing puts no more equations there than anywhere else. Clustering them where the map is working hardest is the standard remedy and this rung does not make it.

A different basis. The series here is a global polynomial in a conformal chart, and a global basis is what makes a fit able to buy accuracy at one sample by spending it at a distant one. A local basis cannot, which is why the hidden factor would be much smaller with one — and why a spline or a finite-element construction fails differently.

The first is the only one the site’s own machinery currently does, which is worth recording rather than glossing: the constructions in this ladder are all global fits with evenly spaced collocation, which is the configuration this rung shows to be the worst case for the failure it measures.

Where the model stops #

One condition and one construction. The boundary-scale condition on a series is measured because it is the one the site solves most often. A condition imposed over an area rather than a curve, or on a basis that is local rather than global, would have different numbers — and in the local case a much smaller hidden factor, because a local basis cannot buy accuracy at one sample by spending it at another far away.

The boundary is a square cap. On a circular cap the condition is exactly satisfiable — the stereographic does it — so every residual is 10⁻¹⁶ and there is nothing to measure. That is itself worth knowing: a fit to a satisfiable condition hides nothing, because there is nothing to hide. The phenomenon needs a condition the basis cannot meet.

The site’s own conformality gate is in the same position, and is safe. assertConformal samples several hundred points and requires the angular deformation at every one to be zero to arithmetic noise. It has no free coefficients to spend, because it is testing a closed formula rather than fitting one — so there is nothing that could buy accuracy at the samples, and the hidden factor is structurally one. The failure this rung measures needs a fit, and a test is not one.

And the refinement is a refinement of the same boundary. Nothing here samples the region’s interior, which is where Chebyshev’s criterion claims the boundary condition delivers its benefit. Whether the interior spread inherits the hidden factor is a separate measurement.

The rule of thumb, stated #

The measurement supports a rule with a number in it, which is worth extracting because it costs nothing to apply.

Samples fewer than about 1.5 times the unknowns: the report is untrustworthy. At twenty against seventeen the hidden factor is 8.3; at twenty-four against twenty-one it is 3.05.

Samples about twice the unknowns: the report is within a fifth. At forty against seventeen it is 1.20.

Samples three times the unknowns or more: the report is the truth. At sixty against seventeen and above it is 1.00 to three decimals.

That is a cheaper test than the out-of-sample measurement it replaces, and it is available before anything is computed — a ratio of two integers, both of which the person writing the solve already knows.

It also says the out-of-sample measurement is only needed in the first band. What makes it worth doing anyway is that nothing in a solver’s output distinguishes the bands, so a construction written once and reused at a different resolution can cross from the third band into the first without anything announcing it.

One more consequence, for reading anybody else’s construction. A published solved projection quotes a residual, and the residual is on the samples: it is the number the solver returned, because there is no other number to have. So the question to ask of such a figure is not whether it is small but how many equations produced it against how many unknowns — and that ratio is usually stated, because a paper describing a fit says how the boundary was sampled and how many terms were kept. Two integers, both present, and a factor of eight between what they imply and what is quoted.

The classical correction, and why it is the wrong size here #

Statistics has a formula for exactly this deflation and it is worth putting beside the measured column, because the comparison says something about what kind of quantity is being reported.

Fitting p free parameters to n observations by least squares leaves a residual whose mean square is smaller than the true one by the factor (n − p)/n, since p of the n directions have been used up. The root of that is the deflation a report inherits, and it is available from the two integers alone. For the configurations in the tables above it predicts hidden factors of 2.58 at twenty samples against seventeen coefficients, 2.83 at twenty-four against twenty-one, 1.32 at forty against seventeen, 1.18 at sixty and 1.02 at four hundred.

Against the measured 8.34, 3.05, 1.20, 1.00 and 1.00 that is the right shape and not the right size, and it is wrong in both directions.

At the tight end it under-predicts by a factor of three. The reason is that the quantity reported here is a maximum over the samples and the formula is about a mean square. A least-squares fit with barely more equations than unknowns can drive the residual very close to zero at the samples specifically, and a maximum over a handful of near-zeros is far more flattering than the root mean square of the same set. The classical correction therefore describes the wrong statistic, and it is the optimistic one.

At the loose end it over-predicts. With sixty samples against seventeen coefficients the formula still expects an eighteen per cent deflation and the measurement finds none at all — because by then the finely sampled boundary’s maximum and the coarsely sampled one’s maximum are landing on the same excursion, and the two numbers are two readings of one peak rather than two statistics.

So the degrees-of-freedom formula cannot be used as the rule of thumb, which is a pity, because it would have been derivable rather than fitted. What survives from it is the variable: the whole of the deflation depends on n and p through their ratio and on nothing else about the problem, and that is why a table measured on one boundary with one basis transfers to another. The bands in the previous section are the empirical version of a quantity the theory says should exist, measured on the statistic that is actually quoted.

The practical reading is unchanged and slightly sharper. A report of a maximum residual from a barely-over-determined fit is optimistic by more than the textbook correction says, so a construction in the first band cannot be repaired by applying a factor — it has to be measured somewhere else.

The generalisation #

A residual measured on the data the fit used is not a residual.

That sentence is the whole of the rung and it is not cartographic. It is the same statement the rule scored out of sample makes about a rule of thumb, and the same one the in-sample number flatters makes about a datum transformation: a quantity fitted to a sample and then evaluated on it reports the fit rather than the thing.

What is distinctive here is that the sample is not data. There is no noise, no measurement, no population — the condition is exact, the boundary is exact, and the “sample” is a list of places the equation was written down at. The failure is purely a discretisation, and it produces exactly the shape of error that over-fitting produces on noisy data, which is worth noticing because the usual remedies for over-fitting are all about noise.

The transferable rule: whenever a condition is enforced by a solve, count the equations against the unknowns and measure somewhere else. The ratio predicts the size of what is hidden, and the second measurement is one pass.

Who found it, and when #

Collocation is one of the standard ways to solve a boundary-value problem and its failure mode is well known in numerical analysis: a collocated solution interpolates the condition at its nodes and can oscillate between them, which is the same phenomenon as Runge’s for polynomial interpolation and is why spectral methods choose their nodes rather than spacing them evenly.

The cartographic constructions in this ladder do not choose their nodes. They sample the boundary evenly, because that is what a boundary generator produces, and even spacing is the worst case for this failure. Choosing Chebyshev nodes instead would reduce the hidden factor substantially and is a repair this rung does not make.

What appears not to be written down anywhere is the size of it for a conformal projection fit, which is what the tables above supply — and the more useful half, that refining the collocation improves the report and not the map.

Where the ladder goes next #

Ten rungs state conditions, solve them, find them impossible, find them under-determined and now find them enforced only where they were written down. What none of them prices is a condition imposed on a quantity that is not a function of position at all: a bound on the worst distortion anywhere, or on the total, is a condition on the map as a whole rather than at any point, and it cannot be collocated because there is nowhere to write it.