Monte Carlo: Concepts, Algorithms, and Applications
Contents
1. Introduction
- About This Book
- Available Software
- What This Book Does Not Contain
- Conventions
2. Estimating Volume and Count
- Volume
- Error and Sample Size Considerations
- Confidence Intervals
- Exploiting Regional Bounds
- Relative Error
- Network Reliability
- Multivariable Integration
- Exploiting Function Bounds
- Exploiting ParameterBounds
- Restricting the Sampling Region
- Reducing the Sampling Dimension
- Counting Problems
- Sensitivity Analysis
- Simultaneous Confidence Intervals
- Ratio Estimation
- Sequential Estimation
3. Generating Samples
- Independence and Dependence
- Inverse Transform Method
- Cutpoint Method
- Composition Method
- Alias Method
- Acceptance-Rejection Method
- Ratio-of-Uniforms Method
- Exact-Approximation Method
- Algorithms for Selected Distributions
- Exponential Distribution
- Normal Distribution
- Lognormal Distribution
- Cauchy Distribution
- Gamma Distribution
- Beta Distribution
- Student's t Distribution
- Snedecor's F Distribution
- Revisiting the Ratio-of-Uniforms Method
- Poisson Distribution
- Binomial Distribution
- Hypergeometric Distribution
- Geometric Distribution
- Negative Binomial Distribution
- Multivariate Normal Distribution
- Multinomial Distribution
- Order Statistics
- Sampling Without Replacement and Permutations
- Points in and on a Simplex
- Points in and on a Hyperellipsoid
- Bernoulli Trials
- Sampling from a Changing Probability Table
- Random Spanning Trees
4. Increasing Efficiency
- Importance Sampling
- Control Variates
- Stratified Sampling
- Inducing Correlation
- Conditonal Monte Carlo
5. Random Tours
- Markov Processes
- Random Walk
- Markov Time
- Score Processes
- Neutron Transport
- Buffer Exceedance on a Production Line
- Fredholm Equations of the Second Kind
- Catastrophic Failure
- First Passage Time
- Random StoppingTime
- Generating Random Points from a Target Distribution
- Generating a Uniformly Distributed Point on a Finite Set
- Generating All Coordinates in a Bounded Region on Each Step
- Metropolis Method
- Sampling Coordinates One at a Time
- Markov Random Fields
- Gibbs Sampling
- Simulated Annealing
- Bayesian Posterior Distributions
- Edge Effects
- Time to Stationarity
- Spectral Structure
- Bounds on Error
- Varying Initial Conditions
- Random Walk on a Hypercube
- Conductance
- More About a Random Walk on a Hypercube
- An Alternative Error Bound for Stationarity
- Sampling from a Hyperrectangular Grid
- Sampling from a Hyperrectangle
- Product Estimator
- Estimating the Volume of a Convex Body
- Estimating the Permanent
- Coupling
- Strong Markov Time
- Strong Stationary Dual Process
- Thresholds
6. Designing and Analyzing Sample Paths
- Problem Context
- A First Approach to Computing Confidence Intervals
- Warm-Up Analysis
- Choosing a "Good" Initial State or a "Good" Initial Distribution
- Strictly Stationary Stochastic Processes
- Optimal Choice of Sample Path Length t and Number of
Replications n
- Estimating Required Sample Path Length
- Characterizing Convergence
- An Alternative View of the Variance of the Sample Mean
- Batch Means Method
- Batch Means Analysis Programs
- Regenerative Processes
- Selecting an Optimal Acceptance Scheme for Metropolis Sampling
7. Generating Pseudorandom Numbers
- Linear Recurrence Generators
- Prime Modulus Generators
- Power-of-Two Modulus Generators
- Mixed Congruential Generators
- Implementation and Portability
- Apparent Randomness
- Spectral Test
- Minimal Number of Parallel Hyperplanes
- Distance Between Points
- Discrepancy
- Beyer Quotient
- Empirical Assessments
- Combining Linear Congruential Generators
- j-Step Linear Recurrence
- Feedback Shift Register Generators
- Generalized Feedback Shift Register Generators
- Nonlinear Generators