How to check a series for path dependence#

Fit the series with direction="both" and look for SEQUENTIAL_PATH_DEPENDENT, and pass verify_discontinuities=True so that each step the series flags is refitted from cold on its two patterns alone.

Each fit in a series starts from its neighbour’s answer. An error in one pattern can therefore pass to every pattern after it and still draw a smooth curve. These two checks separate a trend in the data from one the chain made.

Run the chain both ways#

This fits the seven synthetic LaB6 patterns of How to fit a temperature series from 300 K up to 900 K and again from 900 K down:


from _lab6_ramp import TEMPERATURES, TRUE_A, ramp, simulate, starting_model

import rietx as rx

patterns = ramp()
series = rx.refine_sequential(patterns, *starting_model(), plan="mccusker_default",
                              x=TEMPERATURES, x_label="T (K)", direction="both")

print([d.code for d in series.diagnostics])
forward = series.trajectory("phases.0.cell.a")
backward = series.backward.trajectory("phases.0.cell.a")
gap = max(abs(f - b) / e for f, b, e in zip(forward.value, backward.value, forward.stderr))
print(f"largest forward-backward gap: {gap:.2f} esd")

['SEQUENTIAL_PERSISTENT_FINDING', 'SEQUENTIAL_PERSISTENT_FINDING']
largest forward-backward gap: 0.00 esd

series.backward is the downward chain’s own SeriesResult, and the entries of series are the upward one’s. A SEQUENTIAL_PATH_DEPENDENT diagnostic names each parameter whose two values differ by more than their esds allow, and its message quotes both values. Read the two values rather than the multiple of σ, because a parameter near zero with an esd near zero can report a gap of thousands of σ that has no physical size.

No such diagnostic fired here, and the largest gap in the cell edge is under 0.01 esd. SEQUENTIAL_PATH_CHECK_INCOMPLETE is absent too, so the comparison ran on every pattern. The two checks together mean the comparison was made and found nothing. Without the second, an empty list can also mean the downward chain was stopped or raised before it finished.

The downward chain repeats the work of the upward one, so direction="both" takes about twice as long.

Check a step against two independent fits#

SEQUENTIAL_DISCONTINUITY flags a step much larger than the series’ typical step. Such a step is either in the data or made by the chain, and the diagnostic cannot tell which. verify_discontinuities=True refits the step’s two patterns from your starting model, each on its own, and reports the step they give. This replaces the 700 K pattern with one whose cell is 1.5 % larger:

jumped = list(patterns)
jumped[4] = simulate(TRUE_A[4] * 1.015, seed=15)  # the 700 K cell 1.5 % larger
series = rx.refine_sequential(jumped, *starting_model(), plan="mccusker_default",
                              x=TEMPERATURES, x_label="T (K)", verify_discontinuities=True)

for d in series.diagnostics:
    if d.code == "SEQUENTIAL_DISCONTINUITY" and d.where == ["phases.0.cell.a"]:
        print(d.message)
for entry in series:
    print(f"{entry.label}  {entry.rung}")

phases.0.cell.a steps by 0.06454 between p003 and p004 (T (K) 600 → 700), 31× its median step over the series. Refitted cold and independently, the two patterns step by 0.06454, 1.00× the chain's.
p000  cold
p001  warm
p002  warm
p003  warm
p004  warm_staged
p005  warm
p006  warm

The diagnostic’s value is the cold step over the chain’s. Near 1 the step is in the data, as here. Near 0 the chain made it, because the two patterns agree when nothing passes an error between them. Near −1 the cold pair stepped the other way. series.discontinuities holds the flagged pairs, with SeriesStep.indices and SeriesStep.step.

The check costs up to two full fits per flagged step, and nothing else in the result changes. Here cell.b and cell.c are flagged at the same step as cell.a, because the cubic cell ties them to it, and the three flags share one pair of refits.

Read where each fit started#

The jump also broke a warm start, at the jumped pattern. The 700 K fit, started from the 600 K answer, reached an Rwp over 0.9, far above the rest of the series. The series fitted it again with the whole plan, still from the 600 K answer, and reached 0.04. The 800 K fit, started from that answer, needed no second attempt. entry.rwp_warm keeps the Rwp the first attempt reached.

A cold entry after the first pattern means no warm start could be rescued, and the pattern was fitted from your starting model. Its values do not continue the chain, so read them as a separate measurement.

The full script is examples/guides/path.py.

See also#