[remislp]

What this adds

ekdist.mixtures — two functions for the case where a set of fitted estimates is not one population.

Fit a mechanism to many simulated records and the results do not always form one cloud. Some records take the optimiser to a second maximum — typically one where most events are missed — and the estimates come back as two groups an order of magnitude apart. Any summary that ignores this is meaningless: the mean of such a set describes neither group.

	for
mixture2	two groups that overlap or nearly do. Fits two Gaussian components in log space; reports the proportion in each and their geometric means
gap_split	a handful of runaways cleanly separated from the body. Finds the largest multiplicative step between neighbouring sorted values
proportion	a proportion and its binomial SE, both per cent

Why not just pick a threshold

Choosing a cut by eye and counting either side makes the answer a property of the cut. Both functions determine the split from the data, so an exclusion can be stated — "16 values above a 55-fold gap" — rather than assumed, and someone else can check it.

This is not hypothetical. In the work this came from I first reported a cluster mean from an ad-hoc alpha2 < 2500 threshold and got 2025; the mixture fit gave 2038.8, because the threshold had stranded five points on the wrong side.

Why log space

Rate constants are positive and their errors are multiplicative, so a group of them is roughly log-normal, not normal. Fitting in logs keeps every fitted mean positive and makes each group closer to the Gaussian a mixture assumes. It also means Mixture.means are geometric means, which is said explicitly rather than left to be discovered.

Tests

23, and the interesting thing about them is what they defend against.

A mixture fit cannot look wrong. It returns two components whatever it is handed, with plausible means and a plausible proportion, and raises nothing. So every recovery test runs on data generated from a known mixture, where the proportion and both means are known before the fit runs — never against values the function produced itself. Proportions are recovered at 10 %, 27 % and 50 % so the tolerances are not tuned to one case.

Two tests check nothing about the arithmetic and exist to keep the suite honest:

• test_a_unimodal_set_is_split_without_a_gap — records that one population comes back as two, without complaint, because that is the mistake a user will make. Nothing here tests whether two components is the right number; the separation of the means is the only thing that tells the caller, and the test pins what "no real split" looks like (within a factor of two, against the eightfold separation of a genuine one).
• test_the_recovery_tests_could_fail — shows the tolerances would reject the most likely way for the fit to be broken while still looking reasonable (returning the overall mean for both components).

Plus determinism, ordering (means[0] is always the lower group, so it can be indexed blind), responsibilities summing to one, and that gap_split's mask follows input order rather than sorted order.

Suite goes from 75 to 98.

Notes

• numpy only. No new dependency, and the clean-install job needs no change — it enumerates modules since Measure the correlation between an opening and its neighbouring shut times #8 and picks this up as the eighth. That fix paid for itself within a day.
• Named mixtures, deliberately not clusters: "cluster" already means a run of openings between long shuttings, and reusing it here would be actively confusing.
• Written for the reproduction of Colquhoun, Hatton & Hawkes (2003), whose Fig. 2 is this situation exactly. Their examples appear in the docstrings because a concrete case reads better than an abstract one, but nothing in the module is specific to that paper or to ion channels.

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