Errors In Models Dr. Jim Holten. CS 351/ IT 351 Modeling and Simulation Technologies. Errors in Models. Sources of Errors Characterizing Errors Using Error Bounds Interpreting Error Implications. Sources of Errors. Input Values (measurements) Machine Inaccuracies Algorithm Inaccuracies - PowerPoint PPT Presentation
Text of CS 351/ IT 351 Modeling and Simulation Technologies
CS 351/ IT 351 Modeling and SimulationTechnologiesErrors In Models
Dr. Jim Holten
CS 351/ IT 351Errors in ModelsSources of ErrorsCharacterizing ErrorsUsing Error BoundsInterpreting Error Implications
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CS 351/ IT 351Sources of ErrorsInput Values (measurements)Machine InaccuraciesAlgorithm InaccuraciesBad models
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CS 351/ IT 351Measurement ErrorsMeasurement granularityGranularity accuracy ==> Error intervalsTypes of measurements
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CS 351/ IT 351Machine Errors: RepresentationFloat: 7 decimal places, E+/-38, or subnormal E-45 (fewer digits of precision)
Double 16 decimal places, E +/-308, or subnormal E-324 (fewer digits of precision)
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CS 351/ IT 351Machine Errors: RepresentationEquality comparisons (does 0.0F == 0.0D?)Overflow (too big an exponent)Underflow (too small an exponent)Mismatch (1.000E19D + 47.3D = ?)
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CS 351/ IT 351Machine ErrorsDivide by zero (+/- Inf), or divide zero by zero (NaN)Propagate bad valuesWorst-case scenarios, not seen as errorsNear zero results of add or subtractNear zero denominator
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CS 351/ IT 351Algorithm Sources of ErrorsInaccurate representation of real worldInaccurate representation of ideal worldComputational errors
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CS 351/ IT 351Real World to Ideal ModelMath Models are IdealisticReal world has many perturbationsStatistical estimates are only best fit to observed measurementsResults in an inaccurate ideal model
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CS 351/ IT 351Ideal Model to ImplementationMachine errors in number representationsMachine errors in arithmetic calculationsResults in even worse implementation model values
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CS 351/ IT 351Computational ErrorsNumerical calculation to approximate math functionsNumerical IntegrationNumerical differentiationTechniques used determine the error behaviors
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CS 351/ IT 351Controllable ErrorsUnderstanding sources and behavior of errors empowers you to control them and predict their effects on the results.
Identifying sources and effects of errors allows you to better judge the quality of models.
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CS 351/ IT 351What Gives Bad Models?Wrong equationsWrong numerical methodsDetails gone awryAll irrationally affect results.
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CS 351/ IT 351Characterizing ErrorsError Forms (Probability Distributions?)Error propagation effects on error formsLimitations versus needs
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CS 351/ IT 351Error CharacterizationssError probability distributionsThe normal distributionZoo of common other distributionsArbitrary distributionsError boundsGeneralized error estimation functionsEnumerated values and false negatives
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CS 351/ IT 351Error Probability DistributionsMeasurement error characteristicsCalculation error characteristicsIntroduced algorithmic error terms
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CS 351/ IT 351Measurement ErrorCharacteristicsDiscrete sample on a number lineSpacing determines range for each measurement pointActual value may be anywhere in that range
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CS 351/ IT 351Calculation ErrorCharacteristicsRound-offDivide by near-zeroDivide by zeroAlgorithm inaccuracies
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CS 351/ IT 351Algorithmic ErrorCharacteristicsDepends on the algorithms/solvers usedDepends on the problem sizeDepends on inter-submodel data sharing patterns and volume
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CS 351/ IT 351Errors: Normal DistributionsEasy to characterizePropagates nicely through linear stagesUseless for nonlinearities, special conditionsNot always a good fit
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CS 351/ IT 351Errors:Generalized DistributionsNot commonly usedEasy to represent (histograms into PDFs)Propagate through nonlinear calculations?Awkward: histograms, PDFs, CDFs for each variable
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CS 351/ IT 351Errors: BoundedNot commonly usedEasy to represent (+/-error magnitude)Can be propagated through nonlinear calculationsStill awkward for some calculations
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CS 351/ IT 351Errors: Propagating a DistributionHighly dependent on the distribution and the calculations being performed.Generally only linear operations give easily predictable algebraic results.Others require piecewise approximations
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CS 351/ IT 351Error BoundsExpected value, +/-error magnitude, or min/max
Propagates through calculations?
More complex forms may be needed after propagation bounded piecewise linear distributions
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CS 351/ IT 351Errors: Unhandled ImplicationsMisinterpretation of results
Misplaced confidences
Chicken Little, The Boy Who Cried 'Wolf', and ignored real consequences
