This book presents a variety of advanced statistical methods at a level suitable for advanced undergraduate and graduate students as well as for others interested in familiarizing themselves with these important subjects. It proceeds to illustrate these methods in the context of real-life applications in a variety of areas such as genetics, medicine, and environmental problems.
The book begins in Part I by outlining various data types and by indicating how these are normally represented graphically and subsequently analyzed. In Part II, the basic tools in probability and statistics are introduced with special reference to symbolic data analysis. The most useful and relevant results pertinent to this book are retained. In Part III, the focus is on the tools of machine learning whereas in Part IV the computational aspects of BIG DATA are presented.
This book would serve as a handy desk reference for statistical methods at the undergraduate and graduate level as well as be useful in courses which aim to provide an overview of modern statistics and its applications.
Author(s): Mayer Alvo
Series: Springer Series in the Data Sciences
Publisher: Springer
Year: 2022
Language: English
Pages: 441
City: Cham
Preface
Acknowledgments
Contents
List of Acronyms
List of Nomenclatures
List of Figures
List of Tables
I. Introduction to Big Data
1. Examples of Big Data
1.1. Multivariate Data
1.2. Categorical Data
1.3. Environmental Data
1.4. Genetic Data
1.5. Time Series Data
1.6. Ranking Data
1.7. Social Network Data
1.8. Symbolic Data
1.9. Image Data
II. Statistical Inference for Big Data
2. Basic Concepts in Probability
2.1. Pearson System of Distributions
2.2. Modes of Convergence
2.3. Multivariate Central Limit Theorem
2.4. Markov Chains
3. Basic Concepts in Statistics
3.1. Parametric Estimation
3.2. Hypothesis Testing
3.3. Classical Bayesian Statistics
4. Multivariate Methods
4.1. Matrix Algebra
4.2. Multivariate Analysis as a Generalization of Univariate Analysis
4.2.1. The General Linear Model
4.2.2. One Sample Problem
4.2.3. Two-Sample Problem
4.3. Structure in Multivariate Data Analysis
4.3.1. Principal Component Analysis
4.3.2. Factor Analysis
4.3.3. Canonical Correlation
4.3.4. Linear Discriminant Analysis
4.3.5. Multidimensional Scaling
4.3.6. Copula Methods
5. Nonparametric Statistics
5.1. Goodness-of-Fit Tests
5.2. Linear Rank Statistics
5.3. U Statistics
5.4. Hoeffding's Combinatorial Central Limit Theorem
5.5. Nonparametric Tests
5.5.1. One-Sample Tests of Location
5.5.2. Confidence Interval for the Median
5.5.3. Wilcoxon Signed Rank Test
5.6. Multi-Sample Tests
5.6.1. Two-Sample Tests for Location
5.6.2. Multi-Sample Test for Location
5.6.3. Tests for Dispersion
5.7. Compatibility
5.8. Tests for Ordered Alternatives
5.9. A Unified Theory of Hypothesis Testing
5.9.1. Umbrella Alternatives
5.9.2. Tests for Trend in Proportions
5.10. Randomized Block Designs
5.11. Density Estimation
5.11.1. Univariate Kernel Density Estimation
5.11.2. The Rank Transform
5.11.3. Multivariate Kernel Density Estimation
5.12. Spatial Data Analysis
5.12.1. Spatial Prediction
5.12.2. Point Poisson Kriging of Areal Data
5.13. Efficiency
5.13.1. Pitman Efficiency
5.13.2. Application of Le Cam's Lemmas
5.14. Permutation Methods
6. Exponential Tilting and Its Applications
6.1. Neyman Smooth Tests
6.2. Smooth Models for Discrete Distributions
6.3. Rejection Sampling
6.4. Tweedie's Formula: Univariate Case
6.5. Tweedie's Formula: Multivariate Case
6.6. The Saddlepoint Approximation and Notions of Information
7. Counting Data Analysis
7.1. Inference for Generalized Linear Models
7.2. Inference for Contingency Tables
7.3. Two-Way Ordered Classifications
7.4. Survival Analysis
7.4.1. Kaplan-Meier Estimator
7.4.2. Modeling Survival Data
8. Time Series Methods
8.1. Classical Methods of Analysis
8.2. State Space Modeling
9. Estimating Equations
9.1. Composite Likelihood
9.2. Empirical Likelihood
9.2.1. Application to One-Sample Ranking Problems
9.2.2. Application to Two-Sample Ranking Problems
10. Symbolic Data Analysis
10.1. Introduction
10.2. Some Examples
10.3. Interval Data
10.3.1. Frequency
10.3.2. Sample Mean and Sample Variance
10.3.3. Realization In SODAS
10.4. Multi-nominal Data
10.4.1. Frequency
10.5. Symbolic Regression
10.5.1. Symbolic Regression for Interval Data
10.5.2. Symbolic Regression for Modal Data
10.5.3. Symbolic Regression in SODAS
10.6. Cluster Analysis
10.7. Factor Analysis
10.8. Factorial Discriminant Analysis
10.9. Application to Parkinson's Disease
10.9.1. Data Processing
10.9.2. Result Analysis
10.9.2.1. Viewer
10.9.2.2. Descriptive Statistics
10.9.2.3. Symbolic Regression Analysis
10.9.2.4. Symbolic Clustering
10.9.2.5. Principal Component Analysis
10.9.3. Comparison with Classical Method
10.10. Application to Cardiovascular Disease Analysis
10.10.1. Results of the Analysis
10.10.2. Comparison with the Classical Method
III. Machine Learning for Big Data
11. Tools for Machine Learning
11.1. Regression Models
11.2. Simple Linear Regression
11.2.1. Least Squares Method
11.2.2. Statistical Inference on Regression Coefficients
11.2.3. Verifying the Assumptions on the Error Terms
11.3. Multiple Linear Regression
11.3.1. Multiple Linear Regression Model
11.3.2. Normal Equations
11.3.3. Statistical Inference on Regression Coefficients
11.3.4. Model Fit Evaluation
11.4. Regression in Machine Learning
11.4.1. Optimization for Linear Regression in Machine Learning
11.4.1.1. Gradient Descent
11.4.1.2. Feature Standardization
11.4.1.3. Computing Cost on a Test Set
11.5. Classification Models
11.5.1. Logistic Regression
11.5.1.1. Optimization with Maximal Likelihood for Logistic Regression
11.5.1.2. Statistical Inference
11.5.2. Logistic Regression for Binary Classification
11.5.2.1. Kullback-Leibler Divergence
11.5.3. Logistic Regression with Multiple Response Classes
11.5.4. Regularization for Regression Models in Machine Learning
11.5.4.1. Ridge Regression
11.5.4.2. Lasso Regression
11.5.4.3. The Choice of Regularization Method
11.5.5. Support Vector Machines (SVM)
11.5.5.1. Introduction
11.5.5.2. Finding the Optimal Hyperplane
11.5.5.3. SVM for Nonlinearly Separable Data Sets
11.5.5.4. Illustrating SVM
12. Neural Networks
12.1. Feed-Forward Networks
12.1.1. Motivation
12.1.2. Introduction to Neural Networks
12.1.3. Building a Deep Feed-Forward Network
12.1.4. Learning in Deep Networks
12.1.4.1. Quantitative Model
12.1.4.2. Binary Classification Model
12.1.5. Generalization
12.1.5.1. A Machine Learning Approach to Generalization
12.2. Recurrent Neural Networks
12.2.1. Building a Recurrent Neural Network
12.2.2. Learning in Recurrent Networks
12.2.3. Most Common Design Structures of RNNs
12.2.4. Deep RNN
12.2.5. Bidirectional RNN
12.2.6. Long-Term Dependencies and LSTM RNN
12.2.7. Reduction for Exploding Gradients
12.3. Convolution Neural Networks
12.3.1. Convolution Operator for Arrays
12.3.1.1. Properties of the Convolution Operator
12.3.2. Convolution Layers
12.3.3. Pooling Layers
12.4. Text Analytics
12.4.1. Introduction
12.4.2. General Architecture
IV. Computational Methods for Statistical Inference
13. Bayesian Computation Methods
13.1. Data Augmentation Methods
13.2. Metropolis-Hastings Algorithm
13.3. Gibbs Sampling
13.4. EM Algorithm
13.4.1. Application to Ranking
13.4.2. Extension to Several Populations
13.5. Variational Bayesian Methods
13.5.1. Optimization of the Variational Distribution
13.6. Bayesian Nonparametric Methods
13.6.1. Dirichlet Prior
13.6.2. The Poisson-Dirichlet Prior
13.6.3. Simulation of Bayesian Posterior Distributions
13.6.4. Other Applications
Bibliography
Index
