Modelling unknown errors as random variables

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Modelling unknown errors as random variables
Thomas Svensson, SP Technical Research Institute
of Sweden, a statistician working with Chalmers
and FCC in industrial applications with companies
like Volvo Aero, SKF, Volvo CE, Atlas Copco,
Scania, Daimler, DAF, MAN, Iveco. SPs customers
with fatigue testing and modelling, and the
evaluation of measurement uncertainty.
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Problems

• Introduce statistics as an engineering tool for
• measurement uncertainty
• reliability with respect to fatigue failures

Identify all sources of variation and uncertainty
and put them in a statistical framework For both
applications we use the Gauss approximation
formula for the final uncertainty,
where the covariances usually are neglected.
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Sources of variation and uncertainty

• We have random variables such as
• Electrical noise in instruments
• A population of operators
• Material strength scatter
• Variability within geometrical tolerances
• A population of users, drivers, missions, roads

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Sources of variation and uncertainty

• We have also non-random variables which are
  sources of uncertainty
• Calibration error for instruments
• Non-linearity in gage transfer functions
• Sampling bias with respect to
• suppliers
• users
• Statistical uncertainty in estimated parameters
• Model errors .

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sources of model errors
Fatigue life assessment by calculations
corrections
material properties
material properties
residual stresses
Equivalent stress process, one or two dimensions.
External force vector process
Multiaxial stress process
Cycle count
Damage number
transfer functions, static, dynamic
reduction by principal stress, von Mises, Dang
Van
rain flow count, level crossings, narrow band
approximation,
Empirical relationships, Wöhler curve, Crack
growth laws
Model errors are introduced in all steps,
reduction is necessary in order to compare with
material strength, only simple empirical models
are available because of the lack of detailed
information, defects as microcracks, inclusions
and pores are not at the drawing.
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Modelling unknown errors as random variables
A total measurement uncertainty depends on
several sources, both random and
non-random. Fatigue life prediction depends on
several random variables, uncertain judgements
and possible model errors. Can these different
type of sources be put in a common statistical
framework? How can we estimate the statistical
properties of the sources?
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Example 1, Calibration
An instrument is constructed in such a way that
the output is proportional to the value of the
measurand. Calibration and linear regression
gives the proportionality constant, the
sensitivity b.
The mathematical model
The statistical model
Statistical theory gives prediction intervals for
future usage of the instrument
based on n observations from the calibration
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Example 1, Calibration
A more true model
There is a systematic model error which violates
the assumptions behind the prediction
interval. Solution hypothetical randomization.
The measurement uncertainty by means of the
prediction interval should be regarded as a
measure of future usage where the level x is
random. Restriction Reduction of uncertainty by
taking means of replicates will not be in control
unless the variance for the random part is
known. Regression procedure If replicates are
made on different levels, the regression should
be made on mean values to get a proper estimate
of the standard deviation. But, for the
prediction interval this estimate must be
adjusted since we actually estimate
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Example 2, model error, plasticity
In a specific numerical fatigue assessment at
Volvo Aero they can, by experience, tell that the
true plasticity correction is expected to be
between the values given by the linear rule and
the Neuber rule. We calculate the fatigue life
assuming the linear rule, keeping all other
variables and procedures at their nominal values.
The fatigue life prediction is
We calculate the fatigue life assuming the
Neuber rule, keeping all other variables and
procedures at their nominal values. The fatigue
life prediction is
We now regard the unknown systematic error as a
uniform random variable with variance
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Example 2, model error, plasticity
How can this procedure be justified? What are the
implications? Hypothetical randomisation by
regarding the air engine chief engineers as a
population? Hardly! In fact, the choice will
introduce a systematic error for all VAC engines
and the statistical measure will not comply with
observed failure rates.
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Example 3, Instrument bias
A testing laboratory buys an instrument that is
specified to have the accuracy, say 0.2. What
does this mean? Usually it means that the
systematic error is less than 0.2 of the maximum
output. How can this systematic error be handled
in a statistical sense? For global comparisons
one can regard it as a random bias, which
hopefully has mean zero. It can then be included
as a random contribution for global uncertainty
statements. In a single laboratory there may be
several similar instruments and by assuming that
the operator always makes a random choice, also
the local bias may be regarded as random and be
included in the laboratory uncertainty. For
comparisons the systematic error can be
eliminated by using the same instrument and be
excluded from uncertainty statements for
comparative measurements.
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Discussion
In some engineering problems often uncertainties
are far more important than random variation.
This has resulted in the rejection of statistical
tools and worst case estimates, conservative
modelling and vague safety factors are kept in
use. By putting also uncertainties in the
statistical framework it is possible to take
advantage of the statistical tools, compare all
sources of scatter and uncertainty, and be more
rational in updating and refinement of models.

• Reduction of variance by mean values of
  replicates is out of control.
• The resulting uncertainty measure in reliability
  cannot be interpreted as a failure rate.
• Large systematic errors should be eliminated by
  classification, better modelling, or more
  experiments.
• Are there more problems?