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Deriving a specification from field data

A random or shock specification written from measurements starts as a set of spectra: one PSD or SRS per flight, per run, per event, or per segment of a mission. No two are alike, so the specification is written over a statistical limit of the set, a level at each frequency that the next member of the set is not expected to exceed, and the limit is then covered by a few straight lines on log-log axes. Those lines are the breakpoint table a laboratory receives. core.field_limit makes both steps reproducible and says on each result what it rests on.

The set

gather reads the records onto common lines: the records of one PSD or SRS, every record of several, or chosen (object, record) pairs. Records that cannot be read one way are refused, each with its cause named:

  • records of different quantities, or with units declared on one side and not the other;
  • a PSD mixed with an SRS, two SRSs at different Q or of different kinds, a cross spectrum, or a specification (a requirement, not a measurement);
  • frequency ranges with nothing in common, or a band reaching past the range every record covers;
  • a NaN, or a level at zero or below, at a line inside the band. A level of zero has no logarithm, and a dead channel is not a quiet flight.

Records on different lines are read onto the first record's lines over the range they share, log-log between their own lines, and the result notes it. The 0 Hz line is left out, since a level on log axes needs a positive frequency.

The limits

The levels are taken in decibels, 10 log10 of a density for a PSD and 20 log10 of a peak for an SRS. At each line, over the n records:

Statistic The limit What it rests on
envelope the largest record nothing about the distribution
distribution_free the largest record, as a limit at a stated coverage and confidence independent records, and at least ln(1 − confidence) / ln(coverage) of them
mean_k the mean in dB plus k standard deviations the standard deviation estimated from the records
tolerance the normal tolerance limit, the mean plus a factor times the standard deviation the levels log-normal across records (normal in dB)

Whatever the distribution, the largest of n independent records stands above a share p of the population with probability 1 − pⁿ: eight records give 34 percent at 95 percent coverage, and a distribution-free limit at 95 percent coverage and 50 percent confidence needs 14. A set smaller than that is refused with the count it needs.

The normal tolerance limit stands above a share coverage of the population with probability confidence. With the standard deviation s estimated from n records, the limit is the mean plus k s, where

\[ k = \frac{t'_{\gamma}\!\left(n - 1,\; z_p\sqrt{n}\right)}{\sqrt{n}} \]

with z_p the normal quantile at the coverage and t′ the quantile at the confidence γ of the noncentral t distribution. Ten records at 95 percent coverage and 95 percent confidence give k = 2.911; eight at 95 and 50 give 1.719. At 50 percent confidence the factor falls toward z_p = 1.645 as the set grows.

When the set is too small to estimate a standard deviation, one can be stated (sigma_db), and it is then taken as known: the factor is z_p + z_γ / √n. One record with 3 dB stated gives a limit 1.645 × 3 = 4.9 dB above it at 95 percent coverage and 50 percent confidence, and (2.326 + 1.282) × 3 = 10.8 dB at 99 percent and 90 percent. The result says the scatter was stated and not measured.

The envelope over the limit

fit_envelope covers the limit with straight lines on log-log axes by stated rules. The lines are at or above the limit at every line. The breakpoints are at lines of the limit with their levels free, set so the lines stand as little above the limit as they can on average over log frequency. Breakpoints are added, each where the span that over-tests most turns, until no line is more than max_overshoot_db under the envelope or max_breakpoints is reached; the result says when the allowance was not met, by how much and where. Breakpoints the allowance does not need are then removed. The area the envelope adds over the limit is reported in decibels.

A narrowband limit is ragged from line to line, and lines that stay within a few decibels of every valley need a breakpoint at almost every other line. The limit of spectra banded onto octaves first (Psd.to_octave) is the one a few lines can follow.

A cover always raises the rms. With rms='conserve' (a PSD only, since an SRS has no area) the covering lines are lowered until the envelope's area is the limit's, and the lines where it then falls below the limit are counted with the deepest named.

The margin a document asks for is added when the specification is written, in decibels on the same convention: EnvelopeFit.specification gives a Specification at the breakpoints for a PSD (with the starter bands of the specification sheet) or a ShockSpecification at the SRS's own Q, kind and spacing for an SRS.

A script

Eight flights on sixth-octave lines, with a resonance near 300 Hz and 1.5 dB of scatter from flight to flight:

from visualdynamics.core.field_limit import derive
from visualdynamics.plot import plot_field_limit

fit = derive(flights, statistic='tolerance', coverage=0.95,
             confidence=0.5)
print(fit.limit.describe())
# normal tolerance limit P95/50, k = 1.719 over 8 records
print(fit.limit.rests_on)
# 8 records; the levels taken as log-normal (normal in dB) across
# records, the standard deviation estimated from them
print(fit.describe())
# 6 breakpoints, at most 2.87 dB over the limit, +1.07 dB of area added
fit.frequencies          # 20, 178.3, 317, 447.7, 1588.7, 2000 Hz
spec = fit.specification(margin_db=3.0)

plot_field_limit(flights, statistic='tolerance', coverage=0.95,
                 confidence=0.5, margin_db=3.0, path='field-limit.png')

Eight flights' spectra drawn faint on log-log axes from 20 to 2000 Hz, their mean in dB dotted, the normal tolerance limit P95/50 heavy above them, and the straight-line envelope over the limit with six numbered square breakpoints, the gap between envelope and limit hatched and named in the legend as 1.07 dB of area added. A dashed copy of the envelope 3 dB higher is the envelope with its margin. The title states the limit, that it rests on eight records taken as log-normal, and the breakpoints and overshoot

The same set asked for a distribution-free limit is refused, since eight records are not enough for it:

derive(flights, statistic='distribution_free')
# ValueError: the largest of 8 records stands above 95% of the
# population with only 34% confidence; a distribution-free limit at 95%
# and 50% needs at least 14 records

Project.derive_specification writes the specification into a project, linked to the set and recorded so that it is worked out again when the set changes (Project.refresh); Project.field_limit gives the same reading without adding anything.

In the app

Select a plain PSD or SRS, or several of one kind, and press Field Limit on the bar. The set is drawn with the limit and the envelope as above, and the panel beside it holds the settings (the limit, its coverage and confidence or k, a standard deviation to state, the overshoot allowed, the most breakpoints, whether the rms is exceeded or kept, and the margin) and reads out the records, the factor, the scatter, the breakpoints, the overshoot, the rms of the limit and of the envelope, the area added, what the limit rests on and the notes. A refusal is said in their place. Make Specification adds the specification the reading shows, with its margin, beside the set.