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Statistical Computing with R
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Statistical Computing with R
by Maria L. Rizzo
Statistical Computing with R
Preliminaries
Preface
Acknowledgements
Chapter 1 Introduction
1.1 Computational Statistics and Statistical Computing
1.2 The R Environment
1.3 Getting Started with R
Syntax
Syntax
1.4 Using the R Online Help System
1.5 Functions
1.6 Arrays, Data Frames, and Lists
Data Frames
Data Frames
Arrays and Matrices
Lists
1.7 Workspace and Files
The Working Directory
The Working Directory
Reading Data from External Files
1.8 Using Scripts
1.9 Using Packages
1.10 Graphics
Colors, plotting symbols, and line types
Colors, plotting symbols, and line types
Table 1.1
Table 1.1
Table 1.2
Table 1.3
Table 1.4
Chapter 2 Probability and Statistics Review
2.1 Random Variables and Probability
Distribution and Density Functions
Distribution and Density Functions
Expectation, Variance, and Moments
Conditional Probability and Independence
Independence
Properties of Expected Value and Variance
2.2 Some Discrete Distributions
Binomial and Multinomial Distribution
Binomial and Multinomial Distribution
Geometric Distribution
Alternative formulation of Geometric distribution
Negative Binomial Distribution
Poisson Distribution
Examples
2.3 Some Continuous Distributions
Normal Distribution
Normal Distribution
Gamma and Exponential Distributions
Chisquare and t
Beta and Uniform Distributions
Lognormal Distribution
Examples
2.4 Multivariate Normal Distribution
The bivariate normal distribution
The bivariate normal distribution
The multivariate normal distribution
2.5 Limit Theorems
Laws of Large Numbers
Laws of Large Numbers
Central Limit Theorem
2.6 Statistics
The empirical distribution function
The empirical distribution function
Bias and Mean Squared Error
Method of Moments
The Likelihood Function
Maximum Likelihood Estimation
2.7 Bayes’ Theorem and Bayesian Statistics
The Law of Total Probability
The Law of Total Probability
Bayes’ Theorem
Bayesian Statistics
2.8 Markov Chains
Chapter 3 Methods for Generating Random Variables
3.1 Introduction
Random Generators of Common Probability Distributions in R
Random Generators of Common Probability Distributions in R
3.2 The Inverse Transform Method
3.2.1 Inverse Transform Method, Continuous Case
3.2.2 Inverse Transform Method, Discrete Case
3.3 The Acceptance-Rejection Method
The Acceptance-Rejection Method
The Acceptance-Rejection Method
3.4 Transformation Methods
3.5 Sums and Mixtures
Convolutions
Convolutions
Mixtures
3.6 Multivariate Distributions
3.6.1 Multivariate Normal Distribution
Method for generating multivariate normal samples
Spectral decomposition method for generating Nd(µ, ∑) samples
SVD Method of generating Nd(µ, Σ) samples
Choleski factorization method of generating Nd(µ, Σ) samples
Comparing Performance of Generators
3.6.2 Mixtures of Multivariate Normals
To generate a random sample from pNd(µ1, Σ1) + (1 − p)Nd(µ2, Σ2)
3.6.3 Wishart Distribution
3.6.4 Uniform Distribution on the d-Sphere
Algorithm to generate uniform variates on the d-Sphere
3.7 Stochastic Processes
Poisson Processes
Poisson Processes
Algorithm for simulating a homogeneous Poisson process on an interval [0, t0] by generating interarrival times.
Nonhomogeneous Poisson Processes
Algorithm for simulating a nonhomogeneous Poisson process on an interval [0, t0] by sampling from a homogeneous Poisson process.
Renewal Processes
Symmetric Random Walk
Algorithm to simulate the state Sn of a symmetric random walk
Packages and Further Reading
Exercises
Figure 3.1
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9
Figure 3.10
Figure 3.11
Table 3.1
Table 3.1
Chapter 4 Visualization of Multivariate Data
4.1 Introduction
4.2 Panel Displays
4.3 Surface Plots and 3D Scatter Plots
4.3.1 Surface plots
Adding elements to a perspective plot
Other functions for graphing surfaces
4.3.2 Three-dimensional scatterplot
4.4 Contour Plots
4.5 Other 2D Representations of Data
4.5.1 Andrews Curves
4.5.2 Parallel Coordinate Plots
4.5.3 Segments, stars, and other representations
4.6 Other Approaches to Data Visualization
Exercises
Figure 4.1
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Table 4.1
Table 4.1
Chapter 5 Monte Carlo Integration and Variance Reduction
5.1 Introduction
5.2 Monte Carlo Integration
5.2.1 Simple Monte Carlo estimator
The standard error of θ^ = 1m∑i=1mg(xi).
5.2.2 Variance and Efficiency
Efficiency
5.3 Variance Reduction
5.4 Antithetic Variables
5.5 Control Variates
5.5.1 Antithetic variate as control variate.
5.5.2 Several control variates.
5.5.3 Control variates and regression.
5.6 Importance Sampling
Variance in Importance Sampling
Variance in Importance Sampling
5.7 Stratified Sampling
5.8 Stratified Importance Sampling
Exercises
R Code
Figure 5.1
Figure 5.1
Chapter 6 Monte Carlo Methods in Inference
6.1 Introduction
6.2 Monte Carlo Methods for Estimation
6.2.1 Monte Carlo estimation and standard error
Estimating the standard error of the mean
6.2.2 Estimation of MSE
6.2.3 Estimating a confidence level
Monte Carlo experiment to estimate a confidence level
6.3 Monte Carlo Methods for Hypothesis Tests
6.3.1 Empirical Type I error rate
6.3.2 Power of a Test
Monte Carlo experiment to estimate power of a test against a fixed alternative
6.3.3 Power comparisons
6.4 Application: “Count Five” Test for Equal Variance
Exercises
Projects
Figure 6.1
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Table 6.1
Table 6.1
Table 6.2
Chapter 7 Bootstrap and Jackknife
7.1 The Bootstrap
7.1.1 Bootstrap Estimation of Standard Error
7.1.2 Bootstrap Estimation of Bias
7.2 The Jackknife
The Jackknife Estimate of Bias
The Jackknife Estimate of Bias
The jackknife estimate of standard error
When the Jackknife Fails
7.3 Jackknife-after-Bootstrap
Jackknife-after-bootstrap: Empirical influence values
7.4 Bootstrap Confidence Intervals
7.4.1 The Standard Normal Bootstrap Confidence Interval
7.4.2 The Basic Bootstrap Confidence Interval
7.4.3 The Percentile Bootstrap Confidence Interval
7.4.4 The Bootstrap t interval
Bootstrap t interval (studentized bootstrap interval)
7.5 Better Bootstrap Confidence Intervals
Properties of BCa intervals
Properties of BCa intervals
7.6 Application: Cross Validation
Procedure to estimate prediction error by n-fold (leave-one-out) cross validation
Procedure to estimate prediction error by n-fold (leave-one-out) cross validation
Exercises
Projects
Figure 7.1
Figure 7.1
Figure 7.2
Figure 7.3
Chapter 8 Permutation Tests
8.1 Introduction
Permutation Distribution
Permutation Distribution
Approximate permutation test procedure
8.2 Tests for Equal Distributions
Two-sample tests for univariate data
Two-sample tests for univariate data
8.3 Multivariate Tests for Equal Distributions
Nearest neighbor tests
Nearest neighbor tests
Energy test for equal distributions
Comparison of nearest neighbor and energy tests
8.4 Application: Distance Correlation
Distance Correlation
Distance Correlation
Permutation tests of independence
Approximate permutation test procedure for independence
Exercises
Projects
Figure 8.1
Figure 8.1
Figure 8.2
Figure 8.3
Figure 8.4
Figure 8.5
Table 8.1
Table 8.1
Table 8.2
Chapter 9 Markov Chain Monte Carlo Methods
9.1 Introduction
9.1.1 Integration problems in Bayesian inference
9.1.2 Markov Chain Monte Carlo Integration
9.2 The Metropolis-Hastings Algorithm
9.2.1 Metropolis-Hastings Sampler
9.2.2 The Metropolis Sampler
9.2.3 Random Walk Metropolis
9.2.4 The Independence Sampler
9.3 The Gibbs Sampler
9.4 Monitoring Convergence
9.4.1 The Gelman-Rubin Method
9.5 Application: Change Point Analysis
Exercises
R Code
Code for Figure 9.3 on page 255
Code for Figure 9.3 on page 255
Code for Figures 9.4(a) on page 259 and 9.4(b) on page 259
Code for Figure 9.11 on page 276
Code for Figure 9.12 on page 276
Figure 9.1
Figure 9.1
Figure 9.2
Figure 9.3
Figure 9.4
Figure 9.5
Figure 9.6
Figure 9.7
Figure 9.8
Figure 9.9
Figure 9.10
Figure 9.11
Figure 9.12
Table 9.1
Table 9.1
Chapter 10 Probability Density Estimation
10.1 Univariate Density Estimation
10.1.1 Histograms
Sturges’ Rule
Scott’s Normal Reference Rule
Freedman-Diaconis Rule
10.1.2 Frequency Polygon Density Estimate
10.1.3 The Averaged Shifted Histogram
10.2 Kernel Density Estimation
Boundary kernels
Boundary kernels
10.3 Bivariate and Multivariate Density Estimation
10.3.1 Bivariate Frequency Polygon
3D Histogram
10.3.2 Bivariate ASH
10.3.3 Multidimensional kernel methods
10.4 Other Methods of Density Estimation
Exercises
R Code
Code to generate data as shown in Table 10.1 on page 289.
Code 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 10.1
Figure 10.1
Figure 10.2
Figure 10.3
Figure 10.4
Figure 10.5
Figure 10.6
Figure 10.7
Figure 10.8
Figure 10.9
Figure 10.10
Figure 10.11
Figure 10.12
Figure 10.13
Table 10.1
Table 10.1
Table 10.2
Chapter 11 Numerical Methods in R
11.1 Introduction
Computer representation of real numbers
Computer representation of real numbers
Evaluating Functions
11.2 Root-finding in One Dimension
Bisection method
Bisection method
Brent’s method
11.3 Numerical Integration
11.4 Maximum Likelihood Problems
11.5 One-dimensional Optimization
11.6 Two-dimensional Optimization
11.7 The EM Algorithm
11.8 Linear Programming – The Simplex Method
11.9 Application: Game Theory
Exercises
Figure 11.1
Figure 11.1
Figure 11.2
Figure 11.3
Figure 11.4
Table 11.1
Table 11.1
Appendix A Notation
Appendix B Working with Data Frames and Arrays
B.1 Resampling and Data Partitioning
B.1.1 Using the boot function
B.1.2 Sampling without replacement
B.2 Subsetting and Reshaping Data
B.2.1 Subsetting Data
B.2.2 Stacking/Unstacking Data
B.2.3 Merging Data Frames
B.2.4 Reshaping Data
B.3 Data Entry and Data Analysis
B.3.1 Manual Data Entry
B.3.2 Recoding Missing Values
B.3.3 Reading and Converting Dates
B.3.4 Importing/exporting .csv files
B.3.5 Examples of data entry and analysis
Stacked data entry
Extracting statistics and estimates from fitted models
Create data frame in stacked layout
References
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