book

Statistical Computing with R

by Maria L. Rizzo

November 2007

Intermediate to advanced

416 pages

8h 45m

English

Chapman and Hall/CRC

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Acknowledgements
1.1 Computational Statistics and Statistical Computing1.2 The R Environment1.3 Getting Started with RSyntax1.4 Using the R Online Help System1.5 Functions1.6 Arrays, Data Frames, and ListsData FramesArrays and MatricesLists1.7 Workspace and FilesThe Working DirectoryReading Data from External Files1.8 Using Scripts1.9 Using Packages1.10 GraphicsColors, plotting symbols, and line typesTable 1.1Table 1.2Table 1.3Table 1.4
2.1 Random Variables and ProbabilityDistribution and Density FunctionsExpectation, Variance, and MomentsConditional Probability and IndependenceIndependenceProperties of Expected Value and Variance2.2 Some Discrete DistributionsBinomial and Multinomial DistributionGeometric DistributionAlternative formulation of Geometric distributionNegative Binomial DistributionPoisson DistributionExamples2.3 Some Continuous DistributionsNormal DistributionGamma and Exponential DistributionsChisquare and tBeta and Uniform DistributionsLognormal DistributionExamples2.4 Multivariate Normal DistributionThe bivariate normal distributionThe multivariate normal distribution2.5 Limit TheoremsLaws of Large NumbersCentral Limit Theorem2.6 StatisticsThe empirical distribution functionBias and Mean Squared ErrorMethod of MomentsThe Likelihood FunctionMaximum Likelihood Estimation2.7 Bayes’ Theorem and Bayesian StatisticsThe Law of Total ProbabilityBayes’ TheoremBayesian Statistics2.8 Markov Chains
3.1 IntroductionRandom Generators of Common Probability Distributions in R3.2 The Inverse Transform Method3.2.1 Inverse Transform Method, Continuous Case3.2.2 Inverse Transform Method, Discrete Case3.3 The Acceptance-Rejection MethodThe Acceptance-Rejection Method3.4 Transformation Methods3.5 Sums and MixturesConvolutionsMixtures3.6 Multivariate Distributions3.6.1 Multivariate Normal DistributionMethod for generating multivariate normal samplesSpectral decomposition method for generating Nd(µ, ∑) samplesSVD Method of generating Nd(µ, Σ) samplesCholeski factorization method of generating Nd(µ, Σ) samplesComparing Performance of Generators3.6.2 Mixtures of Multivariate NormalsTo generate a random sample from pNd(µ1, Σ1) + (1 − p)Nd(µ2, Σ2)3.6.3 Wishart Distribution3.6.4 Uniform Distribution on the d-SphereAlgorithm to generate uniform variates on the d-Sphere3.7 Stochastic ProcessesPoisson ProcessesAlgorithm for simulating a homogeneous Poisson process on an interval [0, t0] by generating interarrival times.Nonhomogeneous Poisson ProcessesAlgorithm for simulating a nonhomogeneous Poisson process on an interval [0, t0] by sampling from a homogeneous Poisson process.Renewal ProcessesSymmetric Random WalkAlgorithm to simulate the state Sn of a symmetric random walkPackages and Further ReadingExercisesFigure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 3.6Figure 3.7Figure 3.8Figure 3.9Figure 3.10Figure 3.11Table 3.1
4.1 Introduction4.2 Panel Displays4.3 Surface Plots and 3D Scatter Plots4.3.1 Surface plotsAdding elements to a perspective plotOther functions for graphing surfaces4.3.2 Three-dimensional scatterplot4.4 Contour Plots4.5 Other 2D Representations of Data4.5.1 Andrews Curves4.5.2 Parallel Coordinate Plots4.5.3 Segments, stars, and other representations4.6 Other Approaches to Data VisualizationExercisesFigure 4.1Figure 4.2Figure 4.3Figure 4.4Figure 4.5Figure 4.6Figure 4.7Figure 4.8Figure 4.9Figure 4.10Table 4.1
5.1 Introduction5.2 Monte Carlo Integration5.2.1 Simple Monte Carlo estimatorThe standard error of θ^ = 1m∑i=1mg(xi).5.2.2 Variance and EfficiencyEfficiency5.3 Variance Reduction5.4 Antithetic Variables5.5 Control Variates5.5.1 Antithetic variate as control variate.5.5.2 Several control variates.5.5.3 Control variates and regression.5.6 Importance SamplingVariance in Importance Sampling5.7 Stratified Sampling5.8 Stratified Importance SamplingExercisesR CodeFigure 5.1
6.1 Introduction6.2 Monte Carlo Methods for Estimation6.2.1 Monte Carlo estimation and standard errorEstimating the standard error of the mean6.2.2 Estimation of MSE6.2.3 Estimating a confidence levelMonte Carlo experiment to estimate a confidence level6.3 Monte Carlo Methods for Hypothesis Tests6.3.1 Empirical Type I error rate6.3.2 Power of a TestMonte Carlo experiment to estimate power of a test against a fixed alternative6.3.3 Power comparisons6.4 Application: “Count Five” Test for Equal VarianceExercisesProjectsFigure 6.1Figure 6.2Figure 6.3Figure 6.4Table 6.1Table 6.2
7.1 The Bootstrap7.1.1 Bootstrap Estimation of Standard Error7.1.2 Bootstrap Estimation of Bias7.2 The JackknifeThe Jackknife Estimate of BiasThe jackknife estimate of standard errorWhen the Jackknife Fails7.3 Jackknife-after-BootstrapJackknife-after-bootstrap: Empirical influence values7.4 Bootstrap Confidence Intervals7.4.1 The Standard Normal Bootstrap Confidence Interval7.4.2 The Basic Bootstrap Confidence Interval7.4.3 The Percentile Bootstrap Confidence Interval7.4.4 The Bootstrap t intervalBootstrap t interval (studentized bootstrap interval)7.5 Better Bootstrap Confidence IntervalsProperties of BCa intervals7.6 Application: Cross ValidationProcedure to estimate prediction error by n-fold (leave-one-out) cross validationExercisesProjectsFigure 7.1Figure 7.2Figure 7.3
8.1 IntroductionPermutation DistributionApproximate permutation test procedure8.2 Tests for Equal DistributionsTwo-sample tests for univariate data8.3 Multivariate Tests for Equal DistributionsNearest neighbor testsEnergy test for equal distributionsComparison of nearest neighbor and energy tests8.4 Application: Distance CorrelationDistance CorrelationPermutation tests of independenceApproximate permutation test procedure for independenceExercisesProjectsFigure 8.1Figure 8.2Figure 8.3Figure 8.4Figure 8.5Table 8.1Table 8.2

9.1 Introduction9.1.1 Integration problems in Bayesian inference9.1.2 Markov Chain Monte Carlo Integration9.2 The Metropolis-Hastings Algorithm9.2.1 Metropolis-Hastings Sampler9.2.2 The Metropolis Sampler9.2.3 Random Walk Metropolis9.2.4 The Independence Sampler9.3 The Gibbs Sampler9.4 Monitoring Convergence9.4.1 The Gelman-Rubin Method9.5 Application: Change Point AnalysisExercisesR CodeCode for Figure 9.3 on page 255Code for Figures 9.4(a) on page 259 and 9.4(b) on page 259Code for Figure 9.11 on page 276Code for Figure 9.12 on page 276Figure 9.1Figure 9.2Figure 9.3Figure 9.4Figure 9.5Figure 9.6Figure 9.7Figure 9.8Figure 9.9Figure 9.10Figure 9.11Figure 9.12Table 9.1
10.1 Univariate Density Estimation10.1.1 HistogramsSturges’ RuleScott’s Normal Reference RuleFreedman-Diaconis Rule10.1.2 Frequency Polygon Density Estimate10.1.3 The Averaged Shifted Histogram10.2 Kernel Density EstimationBoundary kernels10.3 Bivariate and Multivariate Density Estimation10.3.1 Bivariate Frequency Polygon3D Histogram10.3.2 Bivariate ASH10.3.3 Multidimensional kernel methods10.4 Other Methods of Density EstimationExercisesR CodeCode to generate data as shown in Table 10.1 on page 289.Code to plot the histograms in Figure 10.4 on page 294.Code to plot Figure 10.6 on page 298.Code to plot kernels in Figure 10.7 on page 299.Figure 10.1Figure 10.2Figure 10.3Figure 10.4Figure 10.5Figure 10.6Figure 10.7Figure 10.8Figure 10.9Figure 10.10Figure 10.11Figure 10.12Figure 10.13Table 10.1Table 10.2
11.1 IntroductionComputer representation of real numbersEvaluating Functions11.2 Root-finding in One DimensionBisection methodBrent’s method11.3 Numerical Integration11.4 Maximum Likelihood Problems11.5 One-dimensional Optimization11.6 Two-dimensional Optimization11.7 The EM Algorithm11.8 Linear Programming – The Simplex Method11.9 Application: Game TheoryExercisesFigure 11.1Figure 11.2Figure 11.3Figure 11.4Table 11.1
B.1 Resampling and Data PartitioningB.1.1 Using the boot functionB.1.2 Sampling without replacementB.2 Subsetting and Reshaping DataB.2.1 Subsetting DataB.2.2 Stacking/Unstacking DataB.2.3 Merging Data FramesB.2.4 Reshaping DataB.3 Data Entry and Data AnalysisB.3.1 Manual Data EntryB.3.2 Recoding Missing ValuesB.3.3 Reading and Converting DatesB.3.4 Importing/exporting .csv filesB.3.5 Examples of data entry and analysisStacked data entryExtracting statistics and estimates from fitted modelsCreate data frame in stacked layout

Overview

Computational statistics and statistical computing are two areas that employ computational, graphical, and numerical approaches to solve statistical problems, making the versatile R language an ideal computing environment for these fields. One of the first books on these topics to feature R, Statistical Computing with R covers the traditiona