How many parameters a curve is worth

The model is what is fitted fitted the two-spin expression to an exactly computed chain of eight and got back a coupling of −66.7 where the sample had −50, with a residual of 0.998. The lesson was that a residual measures how well a two-parameter curve follows a smooth monotone function of temperature, which almost any two-parameter curve does, and that the mismatch goes into the $g$ factor quietly.

It ended by asking the question that would make all of that operational. Every fit there reported two numbers, as did the Curie–Weiss fits that invent a moment. Real fits report more — a monomeric impurity and a temperature-independent term are both standard, and four-parameter fits are published routinely — and whether a susceptibility curve contains four parameters’ worth of information had never been asked as an arithmetic question.

It is one. The answer does not depend on the data being noisy, on the fitting software, or on anybody’s judgement: it depends on the shape of the map from parameters to curve, and it can be computed before any data exist.

The four numbers a paper reports #

The sample is the same one used there: a Heisenberg chain of eight spin-½ centres, solved exactly by diagonalising the whole $2^8$-dimensional configuration space and computing the susceptibility from the complete spectrum. That is the sample, and it is exact in a way a real one never is.

What is fitted to it is not the same thing. Four parameters, and every one of them is something a paper reports:

• $J$, the coupling, which is what the measurement is for.
• $g$, which multiplies the whole curve, and which is where a model mismatch hides — the same place a force field puts a mismatch it has no coordinate for.
• $\rho$, the fraction of spins that are uncoupled monomer. A real sample has some — a broken bridge, a paramagnetic edge — and it shows as a rise in $\chi T$ as the temperature falls.
• $\chi_{\text{TIP}}$, a temperature-independent susceptibility from mixing with excited states. It adds $\chi_{\text{TIP}} T$ to $\chi T$ and so shows at the top of the range.

The last two are deliberately at opposite ends of the curve. That is not a device; it is why both are in every fitting program, because each is the standard repair for a discrepancy at one end.

Differentiating the curve rather than fitting it #

The quantity that decides everything is the matrix of $\partial \ln \chi T(T_i) / \partial \ln p_j$ — one row per measured temperature, one column per parameter. It says how the curve moves when a parameter moves, which is exactly and only what a fit has to work with.

Its singular values are the independent directions of that matrix, largest first, and each one is what a measurement of stated precision can resolve along its own direction. It is the same instrument a pair of band measurements is read with, asked of forty numbers instead of two. For the four-parameter fit over 20–300 K at forty points they are

$$12.411, \quad 2.026, \quad 0.149, \quad 0.025.$$

A span of five hundred, from a curve with no noise in it and no fitting anywhere. The first direction is what the whole curve’s height reports; the last is a combination the curve barely responds to at all.

Propagating one per cent measurement errors through that matrix gives what the fit can return:

A curve is worth about two parameters. Two are fixed to a fraction of a per cent; the third costs an order of magnitude in conditioning; the fourth costs another and comes back at thirty-seven per cent, which is not a measurement of anything. A parameter that never finds a value is the same finding reached from the other end, where a third parameter had no interior optimum at all.

The cost is not confined to the parameter that caused it #

The row that is easy to miss is the first column. Fitting $J$ and $g$ alone fixes $J$ to 0.47 per cent. Adding the two extra parameters — both of which are genuinely present in a real sample, both of which come back small — takes $J$’s own uncertainty to 2.95 per cent, a factor of six.

That is worth stating plainly because the intuition runs the other way, and because a moment counts electrons rather than orbitals — the quantity being reported is a sum over the whole sample, so a small population with a large moment is not a small term. An impurity fraction of two per cent is a small correction, and a small correction ought to perturb the answer by a small amount. It does. What it also does is open a direction in the four-dimensional parameter space along which the curve barely changes — and the coupling has a component along that direction. Everything is worse, not just the parameter that was added.

An undetermined direction is a property of the parameter space, not of a parameter. A fit with one badly determined parameter does not have three good ones and a bad one; it has a region, and every parameter’s error bar is a shadow of that region on its own axis.

Where in the curve each parameter lives #

The window is where the argument becomes concrete, because a monomer fraction is a rise as $T \to 0$ and there is nothing else it can be.

Holding everything else fixed and moving only where the measurement starts:

A window starting at eighty kelvin returns the monomer fraction with an uncertainty larger than the parameter itself. That is the arithmetic form of this measurement contains no information about that quantity, and it is a much sharper statement than saying the fit is poorly constrained: the curve is consistent with there being no monomer at all and with there being three times as much as assumed.

And the coupling — the quantity the whole experiment is for — degrades from half a per cent to twelve, because it shares the space with a direction that has gone.

What the cold end is worth, in points #

The obvious response to a badly determined parameter is more data. The right question is how much more, and the answer is computable rather than rhetorical.

Uncertainties fall as the square root of the number of points — that is what independent measurements of equal precision do, and checking it rather than assuming it is what converts a window into a price. Over a factor of sixteen in the point count, every parameter’s uncertainty tracks the square-root law to within sixteen per cent, and the departure is systematic and in one direction: extra points crowd into a stretch of curve already measured and are worth slightly less than the same number of independent ones. So the law is an upper bound on what data buys.

Now price the cold end. Forty points reaching down to 2 K fix the monomer fraction to 0.82 per cent. Starting instead at 20 K:

Sixteen thousand points starting at twenty kelvin still do not match forty points that reach two. Four hundred times the data, and the parameter that lives at the cold end is fixed slightly less well than by the small measurement that goes there.

The contrast is what makes it a result rather than a complaint. The temperature-independent term lives at the hot end, which both windows have, and four thousand warm points beat the cold forty comfortably — 3.84 per cent against 10.62. The cold end is worth four hundred times the data for the parameter that lives there and nothing at all for the one that does not, which is a statement about where information is and not about how much of it there is.

Why this is not the residual’s problem #

Nothing above involves a residual, and that is the whole difference from the two-spin fit.

The two-spin fit’s failure was a wrong model fitting well: the residual was 0.998 and the answer was out by a third, because a residual asks whether the curve goes through the points and a smooth two-parameter family goes through almost any smooth points. The failure here happens with the right model and a residual that will be excellent, because the fit really can reproduce the data — along a whole valley of parameter values that reproduce it equally well.

The two failures are independent and they compound. A published four-parameter fit to a warm-started curve can be wrong in the coupling because the model is not the sample’s, and wrong again because the coupling is entangled with a direction the data never touched, and both times the residual will be beautiful.

What is quoted, and what is computed #

Nothing here is quoted. The sample is an exact spectrum, the parameter point is stated — a chain of eight at $J = -50$ K, two per cent monomer, $\chi_{\text{TIP}} = 2 \times 10^{-4}$ — and the one per cent precision is a hypothesis about an experiment rather than a report of one.

That is the honest shape for the question. An uncertainty computed from a design matrix is knowable before any measurement is made, which is exactly when it is useful: it is a statement about which experiment to do, and it stops being available the moment somebody has done one.

Every derivative is a central difference taken at two step sizes a factor of two apart, and a derivative whose two estimates disagree is refused rather than returned. The exact spectra come from the same diagonalisation used for the two-spin fit, and each coupling’s spectrum is computed once and reused across the whole curve — a finite difference in $J$ needs one new spectrum, not one per temperature.

The condition numbers are ratios of singular values of the design, and the singular values come from the eigenvalues of its Gram matrix, computed by the Jacobi solver that produces every spectrum drawn here.

What this cannot say #

A well-conditioned fit is not a correct one. Everything here assumes the model is the sample’s. The two-spin fit shows what happens when it is not, and the two questions are independent: this arithmetic is equally happy to report that a wrong model’s parameters are beautifully determined.

The uncertainties are linearised. A one per cent region is an honest description of a small region; a 235 per cent one is a statement that the region is large, not a description of its shape. Where the linearisation fails it fails in the direction that strengthens the conclusion.

Real errors are not one per cent everywhere. A susceptibility measured at 2 K is a small number with a large relative error, and a real cold measurement is worse than the one assumed here. That moves the conclusion the right way for honesty and the wrong way for the recommendation: the cold end is worth what is claimed only if it can be measured, and what a thermometer can find is the same limit in another guise.

And the chain is eight spins. A real chain is long, and its susceptibility has a different low-temperature form; eight is what can be diagonalised exactly, which is the price of having a sample with no model error in it at all.

What was checked #

The four singular values span more than two orders of magnitude, at every window tried — checked as a ratio rather than by looking at whether the smallest is small.

Two parameters are well conditioned and four are not, by a factor of more than twenty in the condition number. The two-parameter case has to come back good, or the claim would be that fitting is hard rather than that these particular extra parameters are nearly free directions.

The monomer fraction’s uncertainty is monotone in the window’s cold end — a sequence rather than two points, because a single comparison would be consistent with a fluctuation.

The square-root law holds to within a fifth over a factor of sixteen in the point count, which is what licenses pricing a window in points at all.

And no warm run tried matches the cold reference for the parameter that lives cold, while the parameter that lives hot is matched. Both halves are checked; either alone would read as a statement about how much data is enough.

A version of the test that needs no singular values #

The arithmetic here requires decomposing a design matrix, which is not what a chemist fitting a susceptibility curve is going to do. There is a version of the same test that needs nothing but the fitting routine already in use, and it answers the same question.

Fix each parameter in turn at a deliberately wrong value and refit the rest. If the residual barely moves, that parameter was not determined by the data: the other three absorbed the change, which is exactly what an ill-conditioned direction means. If the residual rises sharply, the parameter is carrying real information.

The test costs four extra fits and it reports the same thing the singular values do, in units a reader already understands. It also reports it per parameter rather than per direction, which is less informative and is what a paper’s four quoted numbers implicitly claim.

There is a cruder rule of thumb behind both, and the numbers here support it. The information in a curve is where the curve is changing shape, and a susceptibility measurement of a simple antiferromagnetic dimer has one shape change in it — the maximum in χ, or equivalently the fall in χT. A curve with one feature can support the two parameters that place the feature and roughly one more that scales it. A fourth parameter is being fitted to a region of the curve that has nothing in it.

That gives a check to make before the fitting starts rather than after. Count the features, then count the parameters. A measurement that does not reach cold enough to show the maximum has no feature at all, and its curve is a smooth monotone one — which carries a height and a slope, and that is two numbers however many are requested.

None of that replaces the conditioning analysis, which says which combinations are determined and by how much. What it does is make the failure visible to somebody who was never going to run one, which is most of the people quoting four parameters.

Still open: an odd ring, and which combination is determined #

The obvious open question is the odd ring, and the reason to run it here is different from the reason it was first proposed. A ring of an odd number of spins has a frustrated ground state whose susceptibility diverges at low temperature, in the way a half-filled band’s own degeneracy makes itself felt — which is a feature, at the cold end, in exactly the place shown here to carry the information. Whether a feature there buys back the fourth parameter, and how many points’ worth it is, is the same arithmetic run on a different sample, and it would turn measure colder into measure a sample whose curve has something in it.

The nearer question is the one set up here and not asked: which combination the undetermined direction actually is. The singular vector belonging to the smallest singular value is a direction in the four-parameter space, and it is not the $\chi_{\text{TIP}}$ axis — it is some mixture, and knowing which mixture would say what a paper’s four quoted numbers are jointly constrained to rather than what each is separately. Reporting the combination that is determined instead of four numbers that are not is a different way of writing a result, and an underdetermined structure raises the same possibility about a set of bond lengths.