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R Programming for Simulation and Monte Carlo Methods focuses on using R software to program probabilistic simulations, often called Monte Carlo Simulations. Typical simplified "real-world" examples include simulating the probabilities of a baseball player having a 'streak' of twenty sequential season games with 'hits-at-bat' or estimating the likely total number of taxicabs in a strange city when one observes a certain sequence of numbered cabs pass a particular street corner over a 60 minute period. In addition to detailing half a dozen (sometimes amusing) 'real-world' extended example applications, the course also explains in detail how to use existing R functions, and how to write your own R functions, to perform simulated inference estimates, including likelihoods and confidence intervals, and other cases of stochastic simulation. Techniques to use R to generate different characteristics of various families of random variables are explained in detail. The course teaches skills to implement various approaches to simulate continuous and discrete random variable probability distribution functions, parameter estimation, Monte-Carlo Integration, and variance reduction techniques. The course partially utilizes the Comprehensive R Archive Network (CRAN) spuRs package to demonstrate how to structure and write programs to accomplish mathematical and probabilistic simulations using R statistical software.
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|Section 1: Review of Vectors, Matrices, Lists and Functions|
Install R and RStudio
Review: Vectors, Matrices, Lists (part 1)
Review: Vectors, Matrices, Lists (part 2)Preview
Sequences and Replications (part 1)
Sequences and Replications (part 2)Preview
Sort and Order
Creating a Matrix (part 1)
Using Matrices (part 2)
List Structures and Horsekicks (part 1)
Dpois() Function and Horsekicks (part 2)
Sampling from a Dataframe
Section 1 Exercises
|Section 2: Simulation Examples: Tossing a Coin|
R Expressions Exercises Answers (part 1)
R Expressions Exercises Answers (part 2)
Introduction to Simulation: A Game of Tossing a Coin (part 1)Preview
Introduction to Simulation: A Game of Tossing a Coin (part 2)
Write a Simulation Function (part 1)
Write a Simulation Function (part 2)
Continue Coin Tossing Simulation (part 3)
Continue Coin Tossing Simulation (part 4)
|Section 3: Simulation Examples: Returning Checked Hats|
A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation. A good example of a random permutation is the shuffling of a deck of cards: this is ideally a random permutation of the 52 cards.
Random Permutations: Hat Problem (part 2 )
Random Permutations: Hat Problem (part 3)
Random Permutations: Hat Problem (part 4)
Random Permutations: Hat Problem (part 5)Preview
Random Permutations: Hat Problem (part 6)
Checking Hats Exercise
|Section 4: Simulation Examples: Collecting Baseball Cards and "Streaky" Behavior|
Solution to Checking Hats Exercise
Collecting Baseball Cards Simulation (part 1)
Collecting Baseball Cards Simulation (part 2)Preview
Collecting Baseball Cards Simulation (part 3)
Collecting Baseball Cards Simulation (part 4)
Collecting Quarters Exercise
Collecting State Quarters Exercise Solution
"Streaky" Baseball Batting Behavior (part 1)
"Streaky" Baseball Batting Behavior (part 2)
"Streaky" Baseball Batting Behavior (part 3)
"Streaky" Behavior Exercise
|Section 5: Monte Carlo Methods for Inference|
Solution to "Streaky" Behavior Exercise
Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other mathematical methods. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution.
Sleepless in Seattle (part 1)Preview
Sleepless in Seattle (part 2)
Statistical inference is the process of deducing properties of an underlying distribution by analysis of data. Inferential statistical analysis infers properties about a population: this includes testing hypotheses and deriving estimates. The population is assumed to be larger than the observed data set; in other words, the observed data is assumed to be sampled from a larger population.
Applying Monte Carlo Methods to Inference (part 2)
Applying Monte Carlo Methods to Inference (part 3)
Applying Monte Carlo Methods to Inference (part 4)
Applying Monte Carlo Methods to Inference (part 5)
Comparing Estimators: The Taxi Problem (part 1)
Comparing Estimators: The Taxi Problem (part 2)
Late to Class Again ? Exercise
|Section 6: Stochastic Simulation and Random Variable Generation|
Late to Class Again Exercise Solution
A stochastic simulation is a simulation that traces the evolution of variables that can change stochastically (randomly) with certain probabilities.
In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be a categorical distribution; experiments whose sample space is encoded by discrete random variables, where the distribution can be specified by a probability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.
Simulation and Random Variable Generation (part 2)
Simulation and Random Variable Generation (part 3)
Simulating Discrete Random Variables (part 1)Preview
Simulating Discrete Random Variables (part 2)
Simulating Discrete Random Variables (part 3)
The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line (which can be computed using the tools ofcalculus), and one computes the x-intercept of this tangent line (which is easily done with elementary algebra). This x-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated.
Root Finding: Newton-Raphson Technique (part 2)
Create Random Variables Exercise
|Section 7: Inverse and General Transforms|
Create Random Variables Exercise Solution (part 1)
Create Random Variables Exercise Solution (part 2)
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, golden rule,) is a basic method for pseudo-random number sampling, i.e. for generating sample numbers at random from any probability distribution given its cumulative distribution function (cdf).
The basic idea is to uniformly sample a number between 0 and 1, interpreted as a probability, and then return the largest number from the domain of the distribution such that . For example, imagine that is the standardnormal distribution (i.e. with mean 0, standard deviation 1). Then if we choose , we would return 0, because 50% of the probability of a normal distribution occurs in the region where . Similarly, if we choose , we would return 1.95996...; if we choose , we would return 2.5758...; if we choose , we would return 4.7534243...; if we choose , we would return 4.891638...; if we choose , we would return 8.1258906647...; if we choose , we would return 8.2095361516... etc. Essentially, we are randomly choosing a proportion of the area under the curve and returning the number in the domain such that exactly this proportion of the area occurs to the left of that number. Intuitively, we are unlikely to choose a number in the tails because there is very little area in them: We'd have to pick a number very close to 0 or 1.
Inverse Transforms (part 2)
General Transformations (part 1)
General Transformations (part 2)
In mathematics, rejection sampling is a basic technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm" and is a type of Monte Carlo method. The method works for any distribution in with a density.
Rejection sampling is based on the observation that to sample a random variable one can sample uniformly from the region under the graph of its density function.
Accept-Reject Method (part 2)
Accept-Reject Methods (part 3)
Random Variable (Poisson) Exercise 2
|Section 8: Simulating Numerical Integration|
Random Variable Exercise Solution (part 1)
Random Variable Exercise Solution (part 2)
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations.
Introduction to Simulating Numerical Integration (part 2)
Simpson's Rule for Trapezoidal Approximation
Simulating Numerical Integration (part 1)
Simulating Numerical Integration (part 2)Preview
More on Simpson's Rule
Simpson's Rule with phi Functions
Phi Functions Exercises
Hit and Miss (part 1)
Hit and Miss (part 2)
|Section 9: Permutation Tests|
Phi Functions (Numerical Integration) Exercise Solution
In statistics, resampling is any of a variety of methods for doing one of the following:
Common resampling techniques include bootstrapping, jackknifing and permutation tests.
Permutation Tests on a Distribution: Chckwts Example (part 2)
Permutation Tests on a Distribution: Chckwts Example (part 3)
Permutation Tests on a Distribution: Chckwts Example (part 4)
Finish Permutation Tests and an Exercise
|Section 10: Simulation Case Studies: Seed Dispersal|
Dr. Geoffrey Hubona held full-time tenure-track, and tenured, assistant and associate professor faculty positions at 3 major state universities in the Eastern United States from 1993-2010. In these positions, he taught dozens of various statistics, business information systems, and computer science courses to undergraduate, master's and Ph.D. students. He earned a Ph.D. in Business Administration (Information Systems and Computer Science) from the University of South Florida (USF) in Tampa, FL (1993); an MA in Economics (1990), also from USF; an MBA in Finance (1979) from George Mason University in Fairfax, VA; and a BA in Psychology (1972) from the University of Virginia in Charlottesville, VA. He was a full-time assistant professor at the University of Maryland Baltimore County (1993-1996) in Catonsville, MD; a tenured associate professor in the department of Information Systems in the Business College at Virginia Commonwealth University (1996-2001) in Richmond, VA; and an associate professor in the CIS department of the Robinson College of Business at Georgia State University (2001-2010). He is the founder of the Georgia R School (2010-2014) and of R-Courseware (2014-Present), online educational organizations that teach research methods and quantitative analysis techniques. These research methods techniques include linear and non-linear modeling, multivariate methods, data mining, programming and simulation, and structural equation modeling and partial least squares (PLS) path modeling. Dr. Hubona is an expert of the analytical, open-source R software suite and of various PLS path modeling software packages, including SmartPLS. He has published dozens of research articles that explain and use these techniques for the analysis of data, and, with software co-development partner Dean Lim, has created a popular cloud-based PLS software application, PLS-GUI.