A second break on a flat profile

A practitioner who has fitted one break has an obvious next question. The persistence changed once; did it change twice?

The search is the same construction one dimension up: fit three regimes at every ordered pair of admissible positions, take the pair whose profile likelihood is highest, and compare. What makes it possible to run at all is arithmetic rather than patience — the segment sums are prefix sums, so a segment’s contribution costs a constant number of operations and a search over about two thousand pairs costs what a search over seventy-three positions used to.

What it costs in the other sense is the subject of this essay.

A second search is as dear as the first #

Over two hundred draws on a law with no break at all, a search over one position reports 5.697 units of likelihood ratio. A search over pairs reports 4.278 more, at a standard error of 0.117, with a 95% point of 6.944.

Now the same measurement on the law that has exactly one break, where a second break is as absent as the first was in the row above. The single search reports 9.442 and the second adds 5.800.

Looking for something that is not there costs the same whether or not something else was there to find. The excess is slightly larger on the broken law, not smaller, because the first break’s presence widens the range over which the second search finds structure to chase.

So the honest charge for a two-break fit is not twice the one-break charge and it is not one and a half times: it is the one-break charge plus a second charge of about the same size, and a three-regime model selected against a one-regime model by a criterion counting coefficients is over-fitting by about ten units of ratio before the data has said anything.

Why the second search is not cheaper #

The result is worth an argument as well as a measurement, because the intuition runs the other way. A second break has fewer places to go — it must be at least twelve rows from the first and inside the trim — so a naive count of positions says the second search maximises over fewer things and should manufacture less.

Two effects cancel that and one of them is the interesting one.

The pair is searched jointly. The maximum is over pairs rather than over second positions given the first, so moving the first break to accommodate the second is allowed, and the surface being maximised has about two thousand points on it rather than seventy-three.

Each candidate pair carves a shorter middle segment, and a shorter segment has a noisier coefficient. A noisier coefficient is a coefficient that can, by chance, be further from its neighbours — which is exactly what the likelihood rewards. So the shorter the segments a search is allowed to carve, the more likelihood it can manufacture from a series with nothing in it, and the constraint that keeps the segments long is the thing keeping the excess down to four units rather than ten.

That second effect is the reason a minimum gap has to be imposed rather than left to the data, and it is the same reason the trim exists in the one-break search. Every constraint on a search is a constraint on how much the search can manufacture, and none of them is a modelling choice: they are all about the arithmetic of short segments.

In parameters, and in effective tries #

Converted at half a unit a parameter, the first search costs 2.85 parameters and the second 2.14 — five parameters for a rule that has fitted two break points and would be charged two by a count.

The second figure can be read one step further, into how much of a search a hundred and twenty rows actually supports. Treating the manufactured excess as the maximum of a chi-square on one degree of freedom over m effectively independent tries, 4.278 units corresponds to about twenty-six, against about ten for the single search’s 5.697.

So the pair search’s roughly two thousand configurations behave like twenty-six independent ones — a compression of about seventy-seven fold, against the single search’s seventy-three positions behaving like ten, a compression of seven.

The pairs are ten times more redundant with each other than the single positions are, which is what they should be: moving either member of a pair by one row leaves a three-segment fit almost unchanged, and there are two ways to do it.

That is also why the second search is not free and not nearly as dear as the count of configurations suggests. Two thousand tries at the price of twenty-six.

One more property is worth recording because it is convenient rather than deep. The excess has a standard error of 0.117 on two hundred draws, so its own standard deviation is 1.66, and its 95% point of 6.944 sits 1.61 standard deviations above its mean — very nearly the 1.645 a normal would give. The manufactured excess is close to normally distributed, which is not what a maximum over many correlated tries has to be, and it makes a charge based on its mean and spread more defensible than one based on the mean alone.

Where the pair lands #

Under the law with no break the two positions average 45.3 and 78.3. Under the law with one break at row 60 they average 66.1 and 90.3 — so the first of the pair lands near the truth, six rows late, and the second lands two thirds of the way through the remaining data.

That second number is what a practitioner would read as a finding. There is nothing at row 90. What is there is the same maximum-of-noise the single search finds when there is nothing at all, running now on the rows the first break left over.

The minimum gap between the two positions is twelve rows, imposed so that no segment is too short to have a coefficient. That constraint is doing work: without it the pair search would place two breaks a row apart and carve out a single-row segment whose lag-one coefficient can be anything.

The profile is shallow #

The reason a second break finds so much is visible in the first search’s own profile, and it is not subtlety — it is flatness.

Over two hundred and fifty draws on the law with a break at row 60, the positions within two log-likelihood units of the best one number 13.0 of the 73 searched, and the whole profile’s range from its highest point to its lowest is 6.78 units. A function whose entire range is under seven and whose top two units cover a sixth of its domain is a shallow function, and the maximum of a shallow function of a noisy series is not a well-determined thing.

On the stationary law the set is 22.9 positions — nearly a third of the range — over a profile range of 5.39. There is no break there for the set to contain, and the set is half again as wide.

The same flatness, read as a cost #

There is a second way to see the flatness, and it is the one the laws field already measured: sweeping where the break is placed and reading off what each placement costs. That sweep is nearly flat over twenty rows, which is the same statement as the two-unit set covering a sixth of the range — one measured in likelihood and the other in regret.

The two agree, and their agreeing is worth something. A profile could be shallow in likelihood and steep in consequence, in which case the search’s uncertainty would matter a great deal; or steep in likelihood and flat in consequence, in which case it would not matter at all. Here it is shallow in both, so the search’s inability to locate the break precisely is cheap — which is the reason a rule that splits at the wrong place still beats a rule that does not split.

That is a comfort about the decision and not about the reported position. A practitioner who splits at row 78 when the break is at row 60 gets a whitening nearly as good as the right one and a sentence about the data that is wrong by eighteen rows. Only one of those two is checked by anything.

The interval that is not an interval #

The two-unit set is not a curiosity. It is what a practitioner reads off a profile and calls a confidence interval, because that is what a two-unit set delivers for a parameter whose likelihood is quadratic near its maximum.

Here it contains the true break on 41.2% of draws. Turning the law over — the same break, the persistent regime second — takes that to 20.8%, because the estimate leans towards the persistent side and the set leans with it.

Neither number is 95% and neither is any fixed number at all. The convention comes from a quadratic expansion around a true parameter value, and a change point has neither: the likelihood is a step function of a discrete index, and under the null there is no true value to expand around. What the set delivers is whatever the profile’s flatness gives, and the flatness is a property of the draw.

Two different laws, two coverages a factor of two apart, from the same convention. A rule whose delivered coverage depends that strongly on which direction a break runs is not delivering a coverage.

What a criterion would have to charge #

Putting the pieces together gives a rule a practitioner could actually apply, and it is worth writing out because the arithmetic is not the obvious one.

A three-regime model has two more coefficients than a one-regime model and two searched break points. A criterion counting coefficients charges 4. Charging what the searches manufacture charges the one-break figure of about 5.2 plus the second search’s 4.3, so about 9.5 — and that is on top of the 4 for the coefficients, since the manufactured figures are ratios against the one-regime fit and already contain the coefficient’s own unit.

The comparison worth making is between a three-regime model and a two-regime one, which is the comparison a practitioner reaches after already deciding to split once. That charge is the second search’s excess alone: about 4.3 units on average and 6.9 at its 95% point, against the 2 a criterion charges for the extra coefficient. So the same factor of two and a half appears at the second step that appeared at the first, which is what says the charge is per search.

What this means for a third search #

The three results compose in a way worth stating, because the composition is what a practitioner actually does.

A first search costs about five units. A second costs about four more. And each of them is run on a profile so shallow that its own answer is not well determined — the first break’s position is uncertain over about a sixth of the sample, and the second search is conditioned on that position.

So a rule that searches for two breaks and then reports where they are is reporting two numbers, one of which is uncertain to twelve rows and the other of which is a maximum over the residue. What the essay on a window for every candidate says about a tuning parameter applies here one level up: a minimum over a list is not a criterion, and a maximum over a list of maxima is not a location.

The honest response is not to search for fewer breaks. It is to charge for each search and to report the flatness beside the position, which costs nothing and is the one thing a profile can say reliably.

Reporting the flatness is not a consolation prize. The two-unit set’s width is a measurement with no convention in it — it is a count of positions the likelihood cannot separate, which is exactly what it is — and it is the width rather than the coverage that tells a reader what the profile knows. Thirteen positions of seventy-three says the sample locates the break to about a sixth of its own length; twenty-three says it locates it to a third, which on a stationary sample is the correct answer, because there is nothing there. What the convention adds to the width is a coverage claim, and the coverage claim is the part that is not delivered.

What the field leaves standing #

Four essays in, the break point has been described, priced, spent and doubled, and it is worth saying which of the four numbers is the one to carry.

It is not the charge. Five units for one search and four for a second are properties of this trim at this sample length on this model, and the essay that measured them says so.

It is not the split rate either. Fifteen per cent against ninety-nine is a large improvement in what a rule says, and what it buys in the decision runs the other way on the laws that matter most.

The number to carry is the ratio: a search for a change point manufactures more than a real change point in this design is worth. Eight point eight against five point one, on a break from 0.95 to 0.65 in a hundred and twenty rows. Everything else in the field follows from that one comparison, including the parts that disagree with each other, and it is the comparison nothing before this could make — because the field that found the break had no way to run the same search on a series it knew had nothing in it.

What is claimed here, and what is not #

This essay takes what a search over a pair of break points costs. The claims are that a second search manufactures 4.278 units of ratio on a law with no break and 5.800 on a law with exactly one, so the second search is as dear as the first; that a pair search on a law with one break at row 60 lands at 66.1 and 90.3, the second of which is a maximum over the residue; that the one-break profile’s whole range is 6.78 units on the law with a break and 5.39 on the law without, so the surface being maximised is shallow; and that the two-unit set holds 13.0 of 73 positions and contains the truth on 41.2% of draws under one law and 20.8% under the same law turned over.

What stays out, and is named as a decision: a three-break search, and a search over the number of breaks. Both are the same construction again and neither is run. The reason is not cost — the prefix sums make a third dimension affordable — it is that a hundred and twenty rows split four ways leaves segments of thirty, and a coefficient estimated from thirty rows with a searched boundary is a quantity whose properties would have to be established before anything measured about it meant anything. The field stops where the segments stop being long enough to be about the data.

The boundary against the essay that measures the charge is that it prices one search and this one shows the price is per search rather than per model — which is what makes the charge a property of the procedure and not of the family of models it selects from.

The checks, and the refusals that make them mean something #

Three claims are gated. The excess a second search reports is required to be positive on both laws, which is what says a search cannot lose. The two-unit set is required to cover a substantial share of the positions searched, on every law, because a set that were narrow would mean the profile is peaked and the whole argument here would be about something else. And the fast profile the pair search is built on is required to agree with the direct loop at every candidate position to ten decimal places, because a search over two thousand pairs is only possible through the prefix sums and an error there would be invisible in every number this field reports.

The refusal is the interval convention. A two-unit set for a break point read as a 95% interval is refused, with the delivered coverage printed for both directions of the same break, and with the reason: the convention comes from a quadratic likelihood in an identified parameter and a change point is neither.