IEE 475: Simulating Stochastic Systems

In this lecture, we review estimating absolute performance from simulation, with focus on choosing the number of necessary replications of transient simulations of terminating systems. The lecture starts by overviewing point estimation, bias, and different types of point estimators. This includes an overview of quantile estimation and how to use quantile estimation to use simulations as null-hypothesis-prediction generators. We the introduce interval estimation with confidence intervals and prediction intervals. Confidence intervals, which are visualizations of t-tests, provide an alternative way to choose the number of required replications without doing a formal power analysis.Interactive widgets referenced in this lecture include:Concept Explorer: Confidence Intervals and Variance Reduction Techniques (covers prediction intervals as well)Concept Explorer: Output Analyzer (fully functional statistical analysis tool)Concept Explorer: Statistical Power AnalysisConcept Explorer: Input Analyzer (fully functional Input Modeling tool)

In this lecture, we introduce the estimation of absolute performance measures in simulation – effectively shifting our focus from validating input models to validating and making inferences about simulation outputs. Most of this lecture is a review of statistics and reasons for the assumptions for various parametric and non-exact non-parametric methods. We also introduce a few more advanced statistical topics, such as non-parametric methods and special high-power tests for normality. We then switch to focusing on simulations and their outputs, starting with the definition of terminating and non-terminating systems as well as the related transient and steady-state simulations. We will pick up next time with discussing details related to performance measures (and methods) for transient simulations next time and steady-state simulations after that. Our goal was to discuss the difference between point estimation and interval estimation for simulation, but we will hold off to discuss that topic in the next lecture.The following widgets are referenced in this lecture:Concept Explorer: Input Analyzer (fully functional Input Modeling tool)Concept Explorer: Output Analyzer (fully functional statistical analysis tool)Concept Explorer: Statistical Power AnalysisConcept Explorer: Multiple Comparisons Explorer

In this lecture, we review statistical fundamentals – such as the origins of the t-test, the meaning of type-I and type-II error (and alternative terminology for both, such as false positive rate and false negative rate) and the connection to statistical power (sensitivity). We review the Receiver Operating Characteristic (ROC) curve and give a qualitative description of where it gets its shape in a hypothesis test. We close with a validation example (from Lecture H) where we use a power analysis on a one-sample t-test to help justify whether we have gathered enough data to trust that a simulation model is a good match for reality when it has a similar mean output performance to the real system. Peppered throughout the lecture are also comments about why normality is required for t-tests, why there is a minimum expected count for chi-squared tests, and how to avoid statistical inference issues when making multiple comparisons. The lecture also discusses how nonparametric tests have fewer assumptions than parametric tests but tend to pay a cost in terms of statistical power.The following tools and concept explorers are referred to in this lecture:Concept Explorer: Statistical Power AnalysisConcept Explorer: Output Analyzer (fully functional statistical analysis tool)Concept Explorer: Input Analyzer (fully functional Input Modeling tool)

At the start of this lecture, we review statistical topics and fitting techniques from Unit G (particularly Lecture G3, on goodness of fit). In particular, we review hypothesis testing fundamentals (type-I error, type-II error, statistical power, sensitivity, false positive rate, true negative rate, receiver operating characteristic, ROC, alpha, beta) and then go into examples of using Chi-squared and Kolmogorov–Smirnov tests for goodness of fit for arbitrary distributions. We also introduce Anderson–Darling (for flexibility and higher power) and Shapiro–Wilk (for high-powered normality testing).We then pivot to formally defining simulation verification, validation, and calibration and then introducing techniques that incorporate rigorous statistical tools into the validation and calibration process. We focus specifically on the use of the t-test (for confirming that populations of simulation data are consistent with the mean behaviors from the real systems they are meant to represent) and the power analysis (for understanding the conditions when a failure to detect a difference between simulation and real system allows for inferring that the simulation is sufficiently close to the real system).This course makes some reference to the:Concept Explorer: Input Analyzer (fully functional Input Modeling tool)Concept Explorer: Output Analyzer (fully functional statistical analysis tool)

In this lecture, we aim to finish our coverage of Input Modeling, where the focus of this lecture is on parameter estimation and assessing goodness of fit. We start by identifying different shape parameters (including location, scale, and rate) as quantities that need to be estimated. Before getting to their estimation, we also discuss summary statistics (sample mean and sample variance) and maximum likelihood estimation (MLE), with an example for a point estimate of the rate of an exponential. We then discuss how AIC and BIC provide modern ways of ranking models by MLE that also account for the number of parameters in that candidate model. After covering these model selection topics, we transition to a discussion of goodness of fit. That first requires a review of the fundamentals of hypothesis testing. We discuss type-I error, p-values, type-II error, effect sizes, and statistical power. We discuss the dangers of using p-values at very large sample sizes (where small p-values are not meaningful) and at very small sample sizes (where large p-values are not meaningful). We give some examples of this applied to best-of-7 sports tournaments and voting. Finally, we close by re-introducing the chi-squared goodness of fit test for larger sample sizes and the  Kolmogorov–Smirnov (K–S, higher powered) goodness-of-fit test (and a relative, Anderson-Darling, and a distribution specific alternative for testing normality, Shapiro–Wilk) for smaller sample sizes. What we do not finish in the coverage of goodness-of-fit here will be covered in the next lecture. Interactive widgets relevant to this lecture include:https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/input_analyzer.htmlhttps://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/prob_models.htmlhttps://tpavlic.github.io/asu-simulating-stochastic-systems/power_analysis/power_explorer.htmlhttps://tpavlic.github.io/asu-simulating-stochastic-systems/output_analysis/output_analyzer/output_analyzer.html

In this lecture, we continue discussing the choice of input models in stochastic simulation. Here, we pivot from talking about data collection to selection of the broad family of probabilistic distributions that may be a good fit for data. We start with an example where a histogram leads us to introduce additional input models into a flow chart. The rest of the lecture is about choosing models based on physical intuition and the shape of the sampled data (e.g., the shape of histograms). We close with a discussion of probability plots – Q-Q plots and P-P plots, as are used with "fat-pencil tests" – as a good tool for justifying the choice of a family for a certain data set. The next lecture will go over the actual estimation of the parameters for the chosen families and how to quantitatively assess goodness of fit. Concept explorers relevant to content in this lecture:https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/prob_models.htmlhttps://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/input_analyzer.html

In this lecture, we introduce the detailed process of input modeling. Input models are probabilistic models that introduce variation in simulation models of systems. Those input models must be chosen to match statistical distributions in data. Over this unit, we cover collection of data for this process, choice of probabilistic families to fit to these data, and then optimized parameter choice within those families and evaluation of fit with goodness of fit. In this lecture, we discuss issues related to data collection.The Input Analyzer referenced and used in this lecture can be found at:https://tpavlic.github.io/asu-simulating-stochastic-systems/input_modeling/input_analyzer.html

During this lecture, we review the topics covered up to this point in the course as preparation for the upcoming midterm exam. Students are encouraged to bring their own questions to class so that we can focus on the topics that students feel like they need the most help with.

In this lecture, we review pseudo-random number generation and then introduce random-variate generation by way of inverse-transform sampling. In particular, we start with a review of the two most important properties of a pseudo-random number generator (PRNG), uniformity and independence, and discuss statistically rigorous methods for testing for these two properties. For uniformity, we focus on a Chi-square/Chi-squared test for larger numbers of samples and a Kolmogorov–Smirnov (KS) test for smaller numbers of samples. For independence, we discuss autocorrelation tests and runs test, and then we demonstrate a runs above-and-below-the-mean test. We then shift to discussing inverse-transform sampling for continuous random variates and discrete random variates and how the resulting random-variate generators might be implemented in a tool like Rockwell Automation's Arena.Concept explorers relevant to content in this lecture:Concept Explorer: Pseudo-Random Number Generators and AI WatermarkingConcept Explorer: Probability Distributions for Input Modeling and StatisticsConcept Explorer: Input Modeling (fully functional Input Analyzer)

In this lecture, we first cover some discrete distributions (and the Poisson process) that we ran out of time for during the previous lecture. We then launch into a discussion of how to generate pseudo-random numbers distributed uniformly between 0 and 1 (which are necessary for us to easily generate random variates of any distribution). We talk about the two most important properties of a pseudo-random number generator (PRNG), uniformity and independence. We then talk about desirable properties. Some examples are given of some early PRNG's, and then we introduce the linear congruential generator (LCG) and its variants (like the Combined Linear Congruential Generator, CLCG), which represent a much more modern PRNG that has a number of good properties. We close with a discussion of tests of uniformity. We will continue this discussion and add on tests for independence during next lecture (which will primarily cover random-VARIATE generation).Students may wish to follow up by checking out the Pseudo-Random Number Generators and AI Watermarking concept explorer, which covers LCG's, CLCG's, and the more modern MRG approach to PRNG. It also covers methods for AI watermarking of stochastic outputs, like the ones that are now used in large language models like Gemini and Claude.