CS 351 IT 351 Lecture 05 Modelling and

CS 351/ IT 351 Lecture 05 Modelling and Simulation Technologies Dr. Jim Holten

Errors • Sources of Errors • Characterizing Errors • Using Error Bounds • Interpreting Error Implications

Sources of Errors • Input Values (measurements) • Machine Inaccuracies • Algorithm Inaccuracies • Bad models

Measurement Errors • Measurement granularity • Accuracy ==> Error intervals • Types of measurements

Machine Errors: Representation • Float: 7 decimal places, E+/-38, or subnormal E-45 • Double – 16 decimal places, E +/-308, or subnormal E-324

Machine Errors: Representation • Equality comparisons • Overflow • Underflow

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

Algorithm Sources of Errors • Inaccurate representation of real world • Inaccurate representation of ideal world • Computational errors

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

Ideal Model to Implementation • Machine errors in number representations • Machine errors in arithmetic calculations • Results in even worse implementation model values

Computational Errors • Numerical calculation of math functions • Numerical Integration • Numerical differentiation • Techniques used determine the error behavior

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.

Bad Models • Wrong equations • Wrong numerical methods • Details gone awry • All irrationally affect results.

Characterizing Errors • Error Forms • Error propagation effects on error forms • Limitations versus needs

Error Forms • Error probability distributions • The normal distribution • Error bounds • Generalized error estimation functions

Error Probability Distributions • Measurement error characteristics • Calculation error characteristics • Introduced algorithmic error terms

Measurement Error Characteristics • Discrete sample on a number line • Spacing determines “range” for each measurement point • Actual value may be anywhere in that range

Calculation Error Characteristics • Round-off • Divide by near-zero • Divide by zero • Algorithm inaccuracies

Algorithmic Error Characteristics • Depends on the algorithms/solvers used • Depends on the problem size • Depends on inter-submodel data sharing patterns and volume

Error Normal Distributions • Easy to characterize • Propagates nicely through linear stages • Useless for nonlinearities, special conditions • Not always a good fit

Error Bounds • Not commonly used • Easy to represent (+/-error magnitude) • Can be propagated through nonlinear calculations • Still awkward for some calculations

Generalized Distributions • Not commonly used • Easy to represent (histograms) • Propagated through nonlinear calculations • Awkward, histograms for each variable

Propagating an Error Distribution • Highly dependent on the distribution and the calculations being performed. • Generally only linear operations give easily predictable algebraic results.

Error Bounds • Expected value, +/-error magnitude • Propagation Through Calculations • More complex forms may be needed

Unhandled Error Implications • Misinterpretation of results • Misplaced confidences • “Chicken Little” and “The Boy Who Cried 'Wolf'”