ELEMENTS OF COMPUTER & I.T. Contents Chapter-1 Introduction to statistical computing Early, classical, and modern concerns Computation in different areas of statistics Applications Theory of statistical methods Numerical statistical methods Graphical statistical methods Meta-methods: strategies for data analysis Statistical meta-theory Different kinds of computation in statistics Numerical computation Graphical computation Symbolic computation Computing environments Theoretical computation Statistics in different areas of computer science Some notes on the history of statistical computing Chapter -2 Basic numerical methods Floating-point arithmetic Rounding error and error analysis Algorithms for moment computations Error analysis for means Computing the variance Diagnostics and conditioning Inner products and general moment computations Floating-point standards Chapter-3 numerical linear algebra Multiple regression analysis The least-squares problem and orthogonal transformations Householder transformations Computing with householder transformations Decomposing Interpreting the results from householder transformations Other orthogonalization methods Givens rotations Solving linear systems The cholesky factorization The SWEEP operator The elementary SWEEP operator The matrix SWEEP operator Stepwise regression The constant term Collinearity and conditioning Collinearity Tolerance The singular-value decomposition Conditioning and numerical stability Generalized inverses Regression diagnostics Residuals and fitted values Leverage Other case diagnostics Variable diagnostics Conditioning diagnostics Other diagnostic methods Regression updating Updating the inverse matrix Updating matrix factorizations Other regression updating Principal components and eigenproblems Eigenvalues and eigenvectors Generalized eigenvalues and eigenvectors Population principal components Principal components of data Solving eigenproblems The symmetric QR algorithm The power method The QR algorithm with origin shifts The implicit QR algorithm The golub-reinsch singular value decomposition Householder-golub-reinsch singular value decomposition The SVD of a bidiagonal matrix The chan modification to the SVD The generalized eigenproblem Jacobi methods Generalizations of least-squares regression GLM: the general linear model WLS: weighted least squares GLS: generalized least squares GLIM: generalized least squares Iteratively reweighted least squares Iterative methods The Jacobi iteration Gauss-seidel iteration Other iterative methods Additional topics and further reading Lp regression Robust regression Subset selection All-subsets regression Stepwise regression Other stepwise methods Chapter-4 Nonlinear statistical methods Maximum likelihood estimation General notions The MLE and its standard error: the scalar case The MLE and the information matrix: the vector case Solving f =0 scalar Simple iteration Newton-raphson The secant method (regula falsi) Bracketing methods Starting values and convergence criteria Some statistical examples A multinomial problem Poisson regression Solving =0: the vector case Generalizations of univariate methods Newton-raphson Newton-like methods Discrete newton methods Generalized secant methods Rescaled simple iteration Quasi-Newton methods Nonlinear gauss-seidel iteration Some statistical examples Example: binomial/Poisson mixture Example: poisson regression Example: logistic regression Obtaining the hessian matrix Optimization methods Grid search Linear search strategies Golden-section search Local polynomial approximation Successive approximation Selecting the step direction Newton steps Steepest descent Levenberg-marquardt adjustment Quasi-newton steps Conjugate-gradient methods Some practical considerations Nonlinear least squares and the gauss-newton method Iteratively reweighted least squares Constrained optimization Linear programming Least squares with linear equality constraints Linear regression with linear equality constraints Nonquadratic programming with linear constraints Nonlinear constraints Estimation diagnostics Computer-intensive methods Nonparametric and semi parametric regression Projection-selection regression Additive spline models Alternating conditional expectations Generalized additive models Missing data: The EM algorithm Time-series analysis Conditional likelihood estimation for ARMA models The method of backward forecasts Factorization methods Kalman filter methods A note on standard errors Chapter-5 numerical integration and approximation Newton-cotes methods Riemann integrals The trapezoidal rule Simpson’s rule General Newton-cotes rules Extended rules Romberg integration Improper integration Integrands with singularities at the end points Integration over infinite intervals Gaussian quadrature Gauss-legendre rules Orthogonal polynomials On computing Gaussian quadrature rules Other gauss-like integration rules Patterson-kronrod rules Automatic and adaptive quadrature Interpolating splines Characterization of spline functions Representations for spline functions Truncated power functions Piecewise-polynomial representations B-splines Choosing an interpolating spline Computing and evaluating an interpolating spline Computing with truncated power functions Cubic splines based on B-splines Monte Carlo integration Simple monte carol Variance reduction A hybrid method Number-theoretic methods Multiple integrals Iterated integrals and products rules General multivariate regions Adaptive partitioning methods Monte carlo methods Gaussian orthant probabilities Bayesian computations Exploring a multivariate posterior density Some computational approaches Laplace’s method Gauss-hermite quadrature The tanner-Wong method of data augmentation The Tierney-kadane –Laplace method General approximation methods Cumulative distribution functions Tail areas Percent points Methods of approximation Series approximation Continued fractions Polynomial approximation Rational approximation Tail-areas and inverse cdf’s for common distributions The normal distribution Normal tail areas Normal quantiles The X2 distribution The F distribution Student ‘s t distribution Other distributions Chapter-6 Smoothing and density estimation Histograms and related density estimators The simple histogram A native density estimator Kernel estimators Nearest-neighbor estimates Computational considerations Linear smoothers Running means Kernel smoothers Running lines General linear smoothers Spline smoothing Smoothing splines Regression splines Multivariate spline smoothing Nonlinear smoothers LOWESS Super smoother Running medians Other methods Choosing the smoothing parameter Applications and extensions Robust smoothing Smoothing on circles and spheres Smoothing periodic time series Estimating functions with discontinuities Hazard estimation
ELEMENTS OF COMPUTER & I.T.
Contents
Chapter-1 Introduction to statistical computing
Early, classical, and modern concerns
Computation in different areas of statistics
Applications
Theory of statistical methods
Numerical statistical methods
Graphical statistical methods
Meta-methods: strategies for data analysis
Statistical meta-theory
Different kinds of computation in statistics
Numerical computation
Graphical computation
Symbolic computation
Computing environments
Theoretical computation
Statistics in different areas of computer science
Some notes on the history of statistical computing
Chapter -2 Basic numerical methods
Floating-point arithmetic
Rounding error and error analysis
Algorithms for moment computations
Error analysis for means
Computing the variance
Diagnostics and conditioning
Inner products and general moment computations
Floating-point standards
Chapter-3 numerical linear algebra
Multiple regression analysis
The least-squares problem and orthogonal transformations
Householder transformations
Computing with householder transformations
Decomposing
Interpreting the results from householder transformations
Other orthogonalization methods
Givens rotations
Solving linear systems
The cholesky factorization
The SWEEP operator
The elementary SWEEP operator
The matrix SWEEP operator
Stepwise regression
The constant term
Collinearity and conditioning
Collinearity
Tolerance
The singular-value decomposition
Conditioning and numerical stability
Generalized inverses
Regression diagnostics
Residuals and fitted values
Leverage
Other case diagnostics
Variable diagnostics
Conditioning diagnostics
Other diagnostic methods
Regression updating
Updating the inverse matrix
Updating matrix factorizations
Other regression updating
Principal components and eigenproblems
Eigenvalues and eigenvectors
Generalized eigenvalues and eigenvectors
Population principal components
Principal components of data
Solving eigenproblems
The symmetric QR algorithm
The power method
The QR algorithm with origin shifts
The implicit QR algorithm
The golub-reinsch singular value decomposition
Householder-golub-reinsch singular value decomposition
The SVD of a bidiagonal matrix
The chan modification to the SVD
The generalized eigenproblem
Jacobi methods
Generalizations of least-squares regression
GLM: the general linear model
WLS: weighted least squares
GLS: generalized least squares
GLIM: generalized least squares
Iteratively reweighted least squares
Iterative methods
The Jacobi iteration
Gauss-seidel iteration
Other iterative methods
Additional topics and further reading
Lp regression
Robust regression
Subset selection
All-subsets regression
Other stepwise methods
Chapter-4 Nonlinear statistical methods
Maximum likelihood estimation
General notions
The MLE and its standard error: the scalar case
The MLE and the information matrix: the vector case
Solving f =0 scalar
Simple iteration
Newton-raphson
The secant method (regula falsi)
Bracketing methods
Starting values and convergence criteria
Some statistical examples
A multinomial problem
Poisson regression
Solving =0: the vector case
Generalizations of univariate methods
Newton-like methods
Discrete newton methods
Generalized secant methods
Rescaled simple iteration
Quasi-Newton methods
Nonlinear gauss-seidel iteration
Example: binomial/Poisson mixture
Example: poisson regression
Example: logistic regression
Obtaining the hessian matrix
Optimization methods
Grid search
Linear search strategies
Golden-section search
Local polynomial approximation
Successive approximation
Selecting the step direction
Newton steps
Steepest descent
Levenberg-marquardt adjustment
Quasi-newton steps
Conjugate-gradient methods
Some practical considerations
Nonlinear least squares and the gauss-newton method
Constrained optimization
Linear programming
Least squares with linear equality constraints
Linear regression with linear equality constraints
Nonquadratic programming with linear constraints
Nonlinear constraints
Estimation diagnostics
Computer-intensive methods
Nonparametric and semi parametric regression
Projection-selection regression
Additive spline models
Alternating conditional expectations
Generalized additive models
Missing data: The EM algorithm
Time-series analysis
Conditional likelihood estimation for ARMA models
The method of backward forecasts
Factorization methods
Kalman filter methods
A note on standard errors
Chapter-5 numerical integration and approximation
Newton-cotes methods
Riemann integrals
The trapezoidal rule
Simpson’s rule
General Newton-cotes rules
Extended rules
Romberg integration
Improper integration
Integrands with singularities at the end points
Integration over infinite intervals
Gaussian quadrature
Gauss-legendre rules
Orthogonal polynomials
On computing Gaussian quadrature rules
Other gauss-like integration rules
Patterson-kronrod rules
Automatic and adaptive quadrature
Interpolating splines
Characterization of spline functions
Representations for spline functions
Truncated power functions
Piecewise-polynomial representations
B-splines
Choosing an interpolating spline
Computing and evaluating an interpolating spline
Computing with truncated power functions
Cubic splines based on B-splines
Monte Carlo integration
Simple monte carol
Variance reduction
A hybrid method
Number-theoretic methods
Multiple integrals
Iterated integrals and products rules
General multivariate regions
Adaptive partitioning methods
Monte carlo methods
Gaussian orthant probabilities
Bayesian computations
Exploring a multivariate posterior density
Some computational approaches
Laplace’s method
Gauss-hermite quadrature
The tanner-Wong method of data augmentation
The Tierney-kadane –Laplace method
General approximation methods
Cumulative distribution functions
Tail areas
Percent points
Methods of approximation
Series approximation
Continued fractions
Polynomial approximation
Rational approximation
Tail-areas and inverse cdf’s for common distributions
The normal distribution
Normal tail areas
Normal quantiles
The X2 distribution
The F distribution
Student ‘s t distribution
Other distributions
Chapter-6 Smoothing and density estimation
Histograms and related density estimators
The simple histogram
A native density estimator
Kernel estimators
Nearest-neighbor estimates
Computational considerations
Linear smoothers
Running means
Kernel smoothers
Running lines
General linear smoothers
Spline smoothing
Smoothing splines
Regression splines
Multivariate spline smoothing
Nonlinear smoothers
LOWESS
Super smoother
Running medians
Other methods
Choosing the smoothing parameter
Applications and extensions
Robust smoothing
Smoothing on circles and spheres
Smoothing periodic time series
Estimating functions with discontinuities
Hazard estimation
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