Modeling Randomness

How to Use This Deck

This deck is a checklist, not a summary. Every slide is phrased as something you should be able to state, derive, or compute without looking it up.

• Question formats: multiple choice, true/false, matching, fill-in-the-blank, multi-slot fill-in-the-blank, and numeric answers.
• Work the Drills slides by hand first; the answers are in the appendix at the end.
• The Common Traps slides are built from the actual wrong answers — read them last.

What This Chapter Covers

Objectives — Random Numbers

You should be able to:

• Describe the KSL’s generator and quote its period, stream count, and sub-stream length
• Define seed, stream, and sub-stream, and explain what determines each
• Explain why a re-run of a simulation reproduces its results exactly
• State what each stream-control method does
• Define common random numbers and the antithetic variate of \(U\)

Objectives — Random Variates

You should be able to:

• Give the parameters of the common KSL random variable classes
• Explain how DEmpiricalRV and MixtureRV are constructed
• Name the classes that implement truncation, shifting, and acceptance-rejection
• State which generation method the KSL uses for almost all distributions
• Distinguish the immutable rvariable classes from the mutable Distribution classes
• Name the four default PDFModeler scoring models and describe each

The KSL Random Number Generator

Based on L’Ecuyer et al. (2002) — the combination of two multiple recursive generators (MRGs).

Know these as powers of ten: the quiz asks for the exponent alone (57, 19, 22).

Seeds, Streams, and Reproducibility

• A seed is the starting value that determines a stream.
• If a simulation always produces the same results when re-run, the most likely cause is that the generator starts with the same seed each time. This is a feature: it is what makes debugging and controlled comparison possible.
• Two separately constructed RNStreamProvider instances use the same default underlying seeds, so they produce identical sequences of streams.
• A stream produced by crnInstance() is a clone; it is not managed by the provider and does not count as one of the provider’s streams.
• A random variable constructed without a stream number is automatically given a fresh, unique stream from the default provider.

Stream Control Methods

KSLRandom and the Provider

• KSLRandom is the convenient single entry point. It wraps a default RNStreamProvider and offers a wide range of random-variate-generation methods.
• KSLRandom.nextRNStream() calls the underlying default provider to create a new random number stream.

Common Random Numbers and Antithetics

• Common Random Numbers (CRN) is a variance reduction technique in which different experiments use the same random numbers, so as to block out the effect of randomness when comparing them.
• It is not a way to ensure independence, and it is not a goodness-of-fit test.
• The antithetic variate of a pseudo-random number \(U\) is \(1 - U\).

Random Variable Classes — Parameters

Discrete Random Variable Classes

Degenerate and Empirical Random Variables

• ConstantRV — a degenerate mass on a single value that cannot be changed after construction.
• VConstantRV — the same, but the value can be changed after construction.
• DEmpiricalRV is constructed from an array of values plus an array of cumulative distribution probabilities, with the last element equal to 1.0.
• MixtureRV’s second argument is likewise an array representing the CDF over the component random variables, last element 1.0.

Shaping an Existing Distribution

For almost all distributions the KSL generates variates by the inverse transform method, using exact inverses where available and numerical approximations otherwise.

Truncated and Shifted — The Algorithms

Truncation to \([a,b]\). Generate \(U \sim U(0,1)\), then

\[W = F(a) + \bigl(F(b) - F(a)\bigr)\,U, \qquad X = F^{-1}(W)\]

Shift by \(\delta\). If \(X\) has CDF \(F\) and density \(f\), then \(X + \delta\) has density

\[g(x) = f(x - \delta)\]

Generate \(X\) by any preferred method and add \(\delta\).

Acceptance-Rejection

The algorithm requires three inputs: the target PDF \(f(x)\), the proposal distribution \(w(x)\), and the majorizing constant \(c\).

Chapter example. \(f(x) = \tfrac34 (1 - x^2)\) on \([-1,1]\) with majorizing function \(g(x) = 3/4\):

\[c = \int_{-1}^{1} \tfrac34\,dx\]

Be able to evaluate that integral and state the resulting acceptance probability.

Mutability and Best Practice

• The classes in ksl.utilities.random.rvariable create immutable random variables — their parameters cannot be changed after construction.
• The Distribution classes (e.g. Binomial, Normal) do permit parameter changes, and can create random variables from their current parameter values.
• The value property returns a newly generated random value each time it is accessed.
• sample() on SampleIfc returns an array of a specified number of generated values.
• Best practice: create the random variable object outside the generation loop.

Distribution Fitting — PDFModeler

The four default scoring models for continuous distribution recommendation are BIC, AD, CVM, and QQC.

Two things get asked about them:

• Which four are the defaults. KS is a KSL goodness-of-fit statistic, but it is not one of the four default scoring models.
• Which direction is better. Lower AIC and lower BIC are preferred.

The Scoring Metrics

The Fitting Framework Classes

Why Equal-Probability Break Points?

For chi-squared goodness-of-fit testing the KSL uses equal-probability break points rather than arbitrary histogram bins.

Reason: each interval then has approximately the same expected number of observations, which reduces the test’s sensitivity to the choice of intervals.

Drills

Work these by hand. Answers follow at the end of the deck.

1. The L’Ecuyer generator’s period is approximately \(3.1 \times 10^{57}\). Expressed as a power of ten, what is the exponent? What is the exponent for the number of streams?
2. Using the inverse transform for an exponential distribution with mean \(\mu = 1.333333\) and \(U = 0.7\), compute \(X = -\mu \ln(1-U)\) to four decimal places.
3. For \(f(x) = \tfrac34(1-x^2)\) on \([-1,1]\) with \(g(x) = 3/4\), compute the majorizing constant \(c = \int_{-1}^{1} g(x)\,dx\).
4. For the shifted negative binomial with \(r = 3\), \(p = 0.3\), generated by convolution from \(u = 0.35,\ 0.64,\ 0.14\) using \(X_i = 1 + \lfloor \ln(1-u)/\ln(1-p) \rfloor\), what is the resulting integer sum?

Common Traps — Streams

• KSLRandom.nextRNStream() creates a new stream; it does not return the next \(U\).
• A crnInstance() clone is not one of the provider’s managed streams.
• Two separately constructed providers do give identical stream sequences — they share the same default seeds.
• resetStartSubstream() returns to the start of the current substream; advanceToNextSubStream() moves to the next one. Do not swap them.

Common Traps — Random Variables

• NormalRV and LognormalRV take mean and variance, not mean and standard deviation.
• DEmpiricalRV and MixtureRV take CDF arrays ending at 1.0, not PMF arrays.
• ConstantRV is the immutable one; VConstantRV is the mutable one.
• The rvariable classes are immutable; the Distribution classes are mutable.
• Create the random variable outside the generation loop, not inside it.
• Lower AIC and BIC are better.

Appendix — Drill Answers

1. 57 and 19. (Sub-stream length is \(7.6 \times 10^{22}\), exponent 22.)
2. \(X = -1.333333 \ln(0.3) = -1.333333(-1.203973) = \mathbf{1.6053}\)
3. \(c = \tfrac34 \cdot 2 = \mathbf{1.5}\), so \(P_a = 1/c = 2/3\).
4. With \(\ln(0.7) = -0.356675\): \(u=0.35 \to X_1 = 2\); \(u=0.64 \to X_2 = 3\); \(u=0.14 \to X_3 = 1\). Sum \(= \mathbf{6}\).