This book synthesizes those techniques from numerical analysis, algorithms, data structures, and optimization theory mostcommonly employed in statistics and machine learning. We provide concrete applications of these methods by giving complete reference implementations for a large set of the most commonly used statistical estimators. The goal is to provide a self-contained textbook explaining the inner algorithmic workings of statistical estimators.
Autorentext
Taylor Arnold is an assistant professor of statistics at the University of Richmond. His work at the intersection of computer vision, natural language processing, and digital humanities has been supported by multiple grants from the National Endowment for the Humanities (NEH) and the American Council of Learned Societies (ACLS). His first book, Humanities Data in R, was published in 2015.
Michael Kane is an assistant professor of biostatistics at Yale University. He is the recipient of grants from the National Institutes of Health (NIH), DARPA, and the Bill and Melinda Gates Foundation. His R package bigmemory won the Chamber's prize for statistical software in 2010.
Bryan Lewis is an applied mathematician and author of many popular R packages, including irlba, doRedis, and threejs.
Klappentext
A Computational Approach to Statistical Learning gives a novel introduction to predictive modeling by focusing on the algorithmic and numeric motivations behind popular statistical methods. The text contains annotated code to over 80 original reference functions. These functions provide minimal working implementations of common statistical learning algorithms. Every chapter concludes with a fully worked out application that illustrates predictive modeling tasks using a real-world dataset.
The text begins with a detailed analysis of linear models and ordinary least squares. Subsequent chapters explore extensions such as ridge regression, generalized linear models, and additive models. The second half focuses on the use of general-purpose algorithms for convex optimization and their application to tasks in statistical learning. Models covered include the elastic net, dense neural networks, convolutional neural networks (CNNs), and spectral clustering. A unifying theme throughout the text is the use of optimization theory in the description of predictive models, with a particular focus on the singular value decomposition (SVD). Through this theme, the computational approach motivates and clarifies the relationships between various predictive models.
Taylor Arnold is an assistant professor of statistics at the University of Richmond. His work at the intersection of computer vision, natural language processing, and digital humanities has been supported by multiple grants from the National Endowment for the Humanities (NEH) and the American Council of Learned Societies (ACLS). His first book, Humanities Data in R, was published in 2015.
Michael Kane is an assistant professor of biostatistics at Yale University. He is the recipient of grants from the National Institutes of Health (NIH), DARPA, and the Bill and Melinda Gates Foundation. His R package bigmemory won the Chamber's prize for statistical software in 2010.
Bryan Lewis is an applied mathematician and author of many popular R packages, including irlba, doRedis, and threejs.
Inhalt
1. Introduction
Computational approach
Statistical learning
Example
Prerequisites
How to read this book
Supplementary materials
Formalisms and terminology
Exercises
2. Linear Models
Introduction
Ordinary least squares
The normal equations
Solving least squares with the singular value decomposition
Directly solving the linear system
(?) Solving linear models with orthogonal projection
(?) Sensitivity analysis
(?) Relationship between numerical and statistical error
Implementation and notes
Application: Cancer incidence rates
Exercises
3. Ridge Regression and Principal Component Analysis
Variance in OLS
Ridge regression
(?) A Bayesian perspective
Principal component analysis
Implementation and notes
Application: NYC taxicab data
Exercises
4. Linear Smoothers
Non-linearity
Basis expansion
Kernel regression
Local regression
Regression splines
(?) Smoothing splines
(?) B-splines
Implementation and notes
Application: US census tract data
Exercises
5. Generalized Linear Models
Classification with linear models
Exponential families
Iteratively reweighted GLMs
(?) Numerical issues
(?) Multi-class regression
Implementation and notes
Application: Chicago crime prediction
Exercises
6. Additive Models
Multivariate linear smoothers
Curse of dimensionality
Additive models
(?) Additive models as linear models
(?) Standard errors in additive models
Implementation and notes
Application: NYC flights data
Exercises
7. Penalized Regression Models
Variable selection
Penalized regression with the `- and `-norms
Orthogonal data matrix
Convex optimization and the elastic net
Coordinate descent
(?) Active set screening using the KKT conditions
(?) The generalized elastic net model
Implementation and notes
Application: Amazon product reviews
Exercises
8. Neural Networks
Dense neural network architecture
Stochastic gradient descent
Backward propagation of errors
Implementing backpropagation
Recognizing hand written digits
(?) Improving SGD and regularization
(?) Classification with neural networks
(?) Convolutional neural networks
Implementation and notes
Application: Image classification with EMNIST
Exercises
9. Dimensionality Reduction
Unsupervised learning
Kernel functions
Kernel principal component analysis
Spectral clustering
t-Distributed stochastic neighbor embedding (t-SNE)
Autoencoders
Implementation and notes
Application: Classifying and visualizing fashion MNIST
Exercises
10. Computation in Practice
Reference implementations
Sparse matrices
Sparse generalized linear models
Computation on row chunks
Feature hashing
Data quality issues
Implementation and notes
Application
Exercises
A Matrix Algebra
A Vector spaces
A Matrices
A Other useful matrix decompositions
B Floating Point Arithmetic and Numerical Computation
B Floating point arithmetic
B Numerical sources of error
B Computational effort