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Modelling and Simulation Technologies Dr. Jim Holten. CS 351/ IT 351 Lecture 05. Errors. Sources of Errors Characterizing Errors Using Error Bounds Interpreting Error Implications. Sources of Errors. Input Values (measurements) Machine Inaccuracies Algorithm Inaccuracies Bad models.
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Modelling and Simulation Technologies Dr. Jim Holten CS 351/ IT 351Lecture 05
CS 351/ IT 351 Errors • Sources of Errors • Characterizing Errors • Using Error Bounds • Interpreting Error Implications
CS 351/ IT 351 Sources of Errors • Input Values (measurements) • Machine Inaccuracies • Algorithm Inaccuracies • Bad models
CS 351/ IT 351 Measurement Errors • Measurement granularity • Accuracy ==> Error intervals • Types of measurements
CS 351/ IT 351 Machine Errors: Representation • Float: 7 decimal places, E+/-38, or subnormal E-45 • Double – 16 decimal places, E +/-308, or subnormal E-324
CS 351/ IT 351 Machine Errors: Representation • Equality comparisons • Overflow • Underflow
CS 351/ IT 351 Machine Errors • Divide by zero, or divide zero by zero • Propagate “bad” values • Worst-case scenarios, not seen as errors • Near zero results of add or subtract • Near zero denominator
CS 351/ IT 351 Algorithm Sources of Errors • Inaccurate representation of real world • Inaccurate representation of ideal world • Computational errors
CS 351/ IT 351 Real World to Ideal Model • Math Models are Idealistic • Real world has many perturbations • Statistical estimates are only “best fit” • Results in inaccurate ideal model
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
CS 351/ IT 351 Computational Errors • Numerical calculation of math functions • Numerical Integration • Numerical differentiation • Techniques used determine the error behavior
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.
CS 351/ IT 351 Bad Models • Wrong equations • Wrong numerical methods • Details gone awry • All irrationally affect results.
CS 351/ IT 351 Characterizing Errors • Error Forms • Error propagation effects on error forms • Limitations versus needs
CS 351/ IT 351 Error Forms • Error probability distributions • The normal distribution • Error bounds • Generalized error estimation functions
CS 351/ IT 351 Error Probability Distributions • Measurement error characteristics • Calculation error characteristics • Introduced algorithmic error terms
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
CS 351/ IT 351 Calculation ErrorCharacteristics • Round-off • Divide by near-zero • Divide by zero • Algorithm inaccuracies
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
CS 351/ IT 351 Error Normal Distributions • Easy to characterize • Propagates nicely through linear stages • Useless for nonlinearities, special conditions • Not always a good fit
CS 351/ IT 351 Error Bounds • Not commonly used • Easy to represent (+/-error magnitude) • Can be propagated through nonlinear calculations • Still awkward for some calculations
CS 351/ IT 351 Generalized Distributions • Not commonly used • Easy to represent (histograms) • Propagated through nonlinear calculations • Awkward, histograms for each variable
CS 351/ IT 351 Propagating an Error Distribution • Highly dependent on the distribution and the calculations being performed. • Generally only linear operations give easily predictable algebraic results.
CS 351/ IT 351 Error Bounds • Expected value, +/-error magnitude • Propagation Through Calculations • More complex forms may be needed
CS 351/ IT 351 Unhandled Error Implications • Misinterpretation of results • Misplaced confidences • “Chicken Little” and “The Boy Who Cried 'Wolf'”