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CS 351/ IT 351 Modeling and Simulation Technologies

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

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CS 351/ IT 351 Modeling and Simulation Technologies

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  1. Errors In Models Dr. Jim Holten CS 351/ IT 351 Modeling and SimulationTechnologies

  2. CS 351/ IT 351 Errors in Models • Sources of Errors • Characterizing Errors • Using Error Bounds • Interpreting Error Implications

  3. CS 351/ IT 351 Sources of Errors • Input Values (measurements) • Machine Inaccuracies • Algorithm Inaccuracies • Bad models

  4. CS 351/ IT 351 Measurement Errors • Measurement granularity • Granularity accuracy ==> Error intervals • Types of measurements

  5. CS 351/ IT 351 Machine Errors: Representation • Float: 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)

  6. CS 351/ IT 351 Machine Errors: Representation • Equality comparisons (does 0.0F == 0.0D?) • Overflow (too big an exponent) • Underflow (too small an exponent) • Mismatch (1.000E19D + 47.3D = ?)

  7. CS 351/ IT 351 Machine Errors • Divide by zero (+/- Inf), or divide zero by zero (NaN) • Propagate “bad” values • Worst-case scenarios, not seen as errors • Near zero results of add or subtract • Near zero denominator

  8. CS 351/ IT 351 Algorithm Sources of Errors • Inaccurate representation of real world • Inaccurate representation of ideal world • Computational errors

  9. CS 351/ IT 351 Real World to Ideal Model • Math Models are Idealistic • Real world has many perturbations • Statistical estimates are only “best fit” to observed measurements • Results in an inaccurate ideal model

  10. CS 351/ IT 351 Ideal Model to Implementation • Machine errors in number representations • Machine errors in arithmetic calculations • Results in even worse implementation model values

  11. CS 351/ IT 351 Computational Errors • Numerical calculation to approximate math functions • Numerical Integration • Numerical differentiation • Techniques used determine the error behaviors

  12. CS 351/ IT 351 Controllable Errors • Understanding 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.

  13. CS 351/ IT 351 What Gives Bad Models? • Wrong equations • Wrong numerical methods • Details gone awry • All irrationally affect results.

  14. CS 351/ IT 351 Characterizing Errors • Error Forms (Probability Distributions?) • Error propagation effects on error forms • Limitations versus needs

  15. CS 351/ IT 351 Error Characterizationss • Error probability distributions • The normal distribution • Zoo of common other distributions • Arbitrary distributions • Error bounds • Generalized error estimation functions • Enumerated values and “false negatives”

  16. CS 351/ IT 351 Error Probability Distributions • Measurement error characteristics • Calculation error characteristics • Introduced algorithmic error terms

  17. CS 351/ IT 351 Measurement ErrorCharacteristics • Discrete sample on a number line • Spacing determines “range” for each measurement point • Actual value may be anywhere in that range

  18. CS 351/ IT 351 Calculation ErrorCharacteristics • Round-off • Divide by near-zero • Divide by zero • Algorithm inaccuracies

  19. CS 351/ IT 351 Algorithmic ErrorCharacteristics • Depends on the algorithms/solvers used • Depends on the problem size • Depends on inter-submodel data sharing patterns and volume

  20. CS 351/ IT 351 Errors: Normal Distributions • Easy to characterize • Propagates nicely through linear stages • Useless for nonlinearities, special conditions • Not always a good fit

  21. CS 351/ IT 351 Errors:Generalized Distributions • Not commonly used • Easy to represent (histograms into PDFs) • Propagate through nonlinear calculations? • Awkward: histograms, PDFs, CDFs for each variable

  22. CS 351/ IT 351 Errors: Bounded • Not commonly used • Easy to represent (+/-error magnitude) • Can be propagated through nonlinear calculations • Still awkward for some calculations

  23. CS 351/ IT 351 Errors: Propagating a Distribution • Highly dependent on the distribution and the calculations being performed. • Generally only linear operations give easily predictable algebraic results. • Others require piecewise approximations

  24. CS 351/ IT 351 Error Bounds • Expected value, +/-error magnitude, or min/max • Propagates through calculations? • More complex forms may be needed after propagation – bounded piecewise linear distributions

  25. CS 351/ IT 351 Errors: Unhandled Implications • Misinterpretation of results • Misplaced confidences • “Chicken Little”, “The Boy Who Cried 'Wolf'”, and ignored real consequences

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