CS 351/ IT 351 Modeling and Simulation Technologies Errors In Models Dr. Jim Holten

CS 351/ IT 351 Errors in Models 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 Granularity accuracy ==> Error intervals Types of measurements

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)

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 D = ?)

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

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” to observed measurements Results in an 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 to approximate math functions Numerical Integration Numerical differentiation Techniques used determine the error behaviors

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 What Gives Bad Models? Wrong equations Wrong numerical methods Details gone awry All irrationally affect results.

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

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”

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

CS 351/ IT 351 Measurement Error Characteristics 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 Error Characteristics Round-off Divide by near-zero Divide by zero Algorithm inaccuracies

CS 351/ IT 351 Algorithmic Error Characteristics Depends on the algorithms/solvers used Depends on the problem size Depends on inter-submodel data sharing patterns and volume

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

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

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

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

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

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