Recap

The central message of this course something like the following:

• The world is more complex and more variable than can be handled by the kind of simple mathematical models that we learn in algebra and calculus.
• Simulation gives us a way to model these complex and variable systems. We can use those models to understand the real system, to predict its behavior, and to test its behavior in alternative conditions.
• Because the world is variable, things may differ from our predictions merely by chance. Statistics gives us a way to test whether something is statistically significant.

Variability

Here's an example I thought of recently, thanks to a conversation with a student.

• Suppose that a country has proven oil reserves of 30,000 million barrels.
• Suppose the country withdraws 600 million barrels a year.
• It's trivial to compute that the oil will run out in 50 years.
• But what if each of those numbers is not handed to us by an omniscient oracle, but is instead a point estimate of an unknown quantity.
• We can describe the variability using concepts such as
  • distribution
  • mean, median, and mode
  • variance, standard deviation, interquartile range
• We can estimate the distribution of the ratio by simulation:

  oil-depletion.mox

  Here's another, related model.

  ratio-distribution.mox

Probability

We learned about what it means for something to be random: we may not be able to predict the exact value, but we can predict the distribution -- the probability of different outcomes.

We learned about lots of different kinds of distributions:

• Discrete: uniform integer, binomial, negative binomial, poisson,
• Continuous: uniform real, triangular, exponential, Gaussian, power-law
• empirical table

We learned about concepts like:

• independence
• conditional probability
• Bayes' Rule
• the law of large numbers
• the Central Limit Theorem

We also learned some concepts and fallacies:

• the Gambler's fallacy
• the "hot hands" fallacy
• the birthday paradox
• the Monty Hall problem

Statistics

We learned about point estimates and confidence intervals

We learned about the importance of using statistical techniques to avoid being fooled by chance occurrences.

We learned the fundamental logic behind all statistical tests:

1. Plan your experiment, including your null and alternative hypotheses, rejection regions (one or two tails) and what you will measure.
2. Run the experiment and measure something.
3. Determine the probability of the result under the null hypothesis
4. Reject the null hypothesis if the probability is sufficiently small.
   • 0.05 = significant
   • 0.01 = highly significant

Standard Tests

We learned the following standard specific statistical tests

• z-test
• two-sample t-test
• chi-square test
• linear regression

Bootstrap

We learned some applications of the idea of bootstrapping, an innovative statistical technique that substitutes large amounts of computation for dubious assumptions.

• The bootstrap can be used to compute confidence intervals for any kind of measurement.
• The bootstrap can be used to test differences between samples, as a replacement for the t-test

Choosing Tests

Many standard tests make certain assumptions and if the assumptions aren't met, the test may be invalid.

Tests can be wrong in two ways: a type I error and a type II error.

When we have a choice of tests, choose the most powerful

Avoid multiple tests in a single experiment.

Simulation Concepts

We modeled different kinds of systems.

Continuous or Process Simulation

Model a continuous process by means of discrete steps.

Examples include compound interest, birth/death, predator/prey, cooling, falling, lake pollution, global heat.

• The step size needs to be small enough to capture all the important "wiggles" of the process
• The step size shouldn't be so small that simulations take forever to compute
• Sometimes, we need to "spread out" an addition to a holding tank over the course of a fundamental time unit (not the same thing as a time step).

Discrete Event Simulation

Model a process as a sequence of states, where transitions between states are events such as the arrival or departure of an item. Time steps are variable, depending on the events.

Examples: queueing systems, such as grocery stores, banks, call centers, and the like. Also, home repair, cpu scheduling, fast food restaurants, computer networks.

Simulation Implementation

We learned about scheduling events, where the simulation system must know the next event for each item or each block, so that it can choose the earliest. Choosing the earliest can be made more efficient by using a priority queue such as a heap.

We learned about random variate generation:

• uniform [0,1) variables, using the linear congruence method
• transformed variables, such as the exponential
• variables computed by rejection

Simulation, Probability and Statistics

Simulations rely on probability theory in generating random numbers that control or influence the behavior of the model

Simulations produce point estimates or distributions, which must then be compared using statistical techniques.

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