Analysis of Multivariate and High Dimensional Data 1st edition by lnge Koch – Ebook PDF Instant Download/Delivery: 1107501768 , 9781107501768
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ISBN 10: 1107501768
ISBN 13: 9781107501768
Author: lnge Koch
‘Big data’ poses challenges that require both classical multivariate methods and contemporary techniques from machine learning and engineering. This modern text equips you for the new world – integrating the old and the new, fusing theory and practice and bridging the gap to statistical learning. The theoretical framework includes formal statements that set out clearly the guaranteed ‘safe operating zone’ for the methods and allow you to assess whether data is in the zone, or near enough. Extensive examples showcase the strengths and limitations of different methods with small classical data, data from medicine, biology, marketing and finance, high-dimensional data from bioinformatics, functional data from proteomics, and simulated data. High-dimension low-sample-size data gets special attention. Several data sets are revisited repeatedly to allow comparison of methods. Generous use of colour, algorithms, Matlab code, and problem sets complete the package. Suitable for master’s/graduate students in statistics and researchers in data-rich disciplines.
Analysis of Multivariate and High Dimensional Data 1st Table of contents:
I CLASSICAL METHODS
1 Multidimensional Data
1.1 Multivariate and High-Dimensional Problems
1.2 Visualisation
1.2.1 Three-Dimensional Visualisation
1.2.2 Parallel Coordinate Plots
1.3 Multivariate Random Vectors and Data
1.3.1 The Population Case
1.3.2 The Random Sample Case
1.4 Gaussian Random Vectors
1.4.1 The Multivariate Normal Distribution and the Maximum Likelihood Estimator
1.4.2 Marginal and Conditional Normal Distributions
1.5 Similarity, Spectral and Singular Value Decomposition
1.5.1 Similar Matrices
1.5.2 Spectral Decomposition for the Population Case
1.5.3 Decompositions for the Sample Case
2 Principal Component Analysis
2.1 Introduction
2.2 Population Principal Components
2.3 Sample Principal Components
2.4 Visualising Principal Components
2.4.1 Scree, Eigenvalue and Variance Plots
2.4.2 Two- and Three-Dimensional PC Score Plots
2.4.3 Projection Plots and Estimates of the Density of the Scores
2.5 Properties of Principal Components
2.5.1 Correlation Structure of X and Its PCs
2.5.2 Optimality Properties of PCs
2.6 Standardised Data and High-Dimensional Data
2.6.1 Scaled and Sphered Data
2.6.2 High-Dimensional Data
2.7 Asymptotic Results
2.7.1 Classical Theory: Fixed Dimension d
2.7.2 Asymptotic Results when d Grows
2.8 Principal Component Analysis, the Number of Components and Regression
2.8.1 Number of Principal Components Based on the Likelihood
2.8.2 Principal Component Regression
3 Canonical Correlation Analysis
3.1 Introduction
3.2 Population Canonical Correlations
3.3 Sample Canonical Correlations
3.4 Properties of Canonical Correlations
3.5 Canonical Correlations and Transformed Data
3.5.1 Linear Transformations and Canonical Correlations
3.5.2 Transforms with Non-Singular Matrices
3.5.3 Canonical Correlations for Scaled Data
3.5.4 Maximum Covariance Analysis
3.6 Asymptotic Considerations and Tests for Correlation
3.7 Canonical Correlations and Regression
3.7.1 The Canonical Correlation Matrix in Regression
3.7.2 Canonical Correlation Regression
3.7.3 Partial Least Squares
3.7.4 The Generalised Eigenvalue Problem
4 Discriminant Analysis
4.1 Introduction
4.2 Classes, Labels, Rules and Decision Functions
4.3 Linear Discriminant Rules
4.3.1 Fisher’s Discriminant Rule for the Population
4.3.2 Fisher’s Discriminant Rule for the Sample
4.3.3 Linear Discrimination for Two Normal Populations or Classes
4.4 Evaluation of Rules and Probability of Misclassification
4.4.1 Boundaries and Discriminant Regions
4.4.2 Evaluation of Discriminant Rules
4.5 Discrimination under Gaussian Assumptions
4.5.1 Two and More Normal Classes
4.5.2 Gaussian Quadratic Discriminant Analysis
4.6 Bayesian Discrimination
4.6.1 Bayes Discriminant Rule
4.6.2 Loss and Bayes Risk
4.7 Non-Linear, Non-Parametric and Regularised Rules
4.7.1 Nearest-Neighbour Discrimination
4.7.2 Logistic Regression and Discrimination
4.7.3 Regularised Discriminant Rules
4.7.4 Support Vector Machines
4.8 Principal Component Analysis, Discrimination and Regression
4.8.1 Discriminant Analysis and Linear Regression
4.8.2 Principal Component Discriminant Analysis
4.8.3 Variable Ranking for Discriminant Analysis
Problems for Part I
II FACTORS AND GROUPINGS
5 Norms, Proximities, Features and Dualities
5.1 Introduction
5.2 Vector and Matrix Norms
5.3 Measures of Proximity
5.3.1 Distances
5.3.2 Dissimilarities
5.3.3 Similarities
5.4 Features and Feature Maps
5.5 Dualities for X and XT
6 Cluster Analysis
6.1 Introduction
6.2 Hierarchical Agglomerative Clustering
6.3 k-Means Clustering
6.4 Second-Order Polynomial Histogram Estimators
6.5 Principal Components and Cluster Analysis
6.5.1 k-Means Clustering for Principal Component Data
6.5.2 Binary Clustering of Principal Component Scores and Variables
6.5.3 Clustering High-Dimensional Binary Data
6.6 Number of Clusters
6.6.1 Quotients of Variability Measures
6.6.2 The Gap Statistic
6.6.3 The Prediction Strength Approach
6.6.4 Comparison of k-Statistics
7 Factor Analysis
7.1 Introduction
7.2 Population k-Factor Model
7.3 Sample k-Factor Model
7.4 Factor Loadings
7.4.1 Principal Components and Factor Analysis
7.4.2 Maximum Likelihood and Gaussian Factors
7.5 Asymptotic Results and the Number of Factors
7.6 Factor Scores and Regression
7.6.1 Principal Component Factor Scores
7.6.2 Bartlett and Thompson Factor Scores
7.6.3 Canonical Correlations and Factor Scores
7.6.4 Regression-Based Factor Scores
7.6.5 Factor Scores in Practice
7.7 Principal Components, Factor Analysis and Beyond
8 Multidimensional Scaling
8.1 Introduction
8.2 Classical Scaling
8.2.1 Classical Scaling and Principal Coordinates
8.2.2 Classical Scaling with Strain
8.3 Metric Scaling
8.3.1 Metric Dissimilarities and Metric Stresses
8.3.2 Metric Strain
8.4 Non-Metric Scaling
8.4.1 Non-Metric Stress and the Shepard Diagram
8.4.2 Non-Metric Strain
8.5 Data and Their Configurations
8.5.1 HDLSS Data and the X and XT Duality
8.5.2 Procrustes Rotations
8.5.3 Individual Differences Scaling
8.6 Scaling for Grouped and Count Data
8.6.1 Correspondence Analysis
8.6.2 Analysis of Distance
8.6.3 Low-Dimensional Embeddings
Problems for Part II
III NON-GAUSSIAN ANALYSIS
9 Towards Non-Gaussianity
9.1 Introduction
9.2 Gaussianity and Independence
9.3 Skewness, Kurtosis and Cumulants
9.4 Entropy and Mutual Information
9.5 Training, Testing and Cross-Validation
9.5.1 Rules and Prediction
9.5.2 Evaluating Rules with the Cross-Validation Error
10 Independent Component Analysis
10.1 Introduction
10.2 Sources and Signals
10.2.1 Population Independent Components
10.2.2 Sample Independent Components
10.3 Identification of the Sources
10.4 Mutual Information and Gaussianity
10.4.1 Independence, Uncorrelatedness and Non-Gaussianity
10.4.2 Approximations to the Mutual Information
10.5 Estimation ofthe Mixing Matrix
10.5.1 An Estimating Function Approach
10.5.2 Properties of Estimating Functions
10.6 Non-Gaussianity and Independence in Practice
10.6.1 Independent Component Scores and Solutions
10.6.2 Independent Component Solutions for Real Data
10.6.3 Performance of J for Simulated Data
10.7 Low-Dimensional Projections of High-Dimensional Data
10.7.1 Dimension Reduction and Independent Component Scores
10.7.2 Properties of Low-Dimensional Projections
10.8 Dimension Selection with Independent Components
11 Projection Pursuit
11.1 Introduction
11.2 One-Dimensional Projections and Their Indices
11.2.1 Population Projection Pursuit
11.2.2 Sample Projection Pursuit
11.3 Projection Pursuit with Two- and Three-Dimensional Projections
11.3.1 Two-Dimensional Indices: QE, QC and QU
11.3.2 Bivariate Extension by Removal of Structure
11.3.3 A Three-Dimensional Cumulant Index
11.4 Projection Pursuit in Practice
11.4.1 Comparison of Projection Pursuit and Independent Component Analysis
11.4.2 From a Cumulant-Based Index to FastICA Scores
11.4.3 The Removal of Structure and FastICA
11.4.4 Projection Pursuit: A Continuing Pursuit
11.5 Theoretical Developments
11.5.1 Theory Relating to QR
11.5.2 Theory Relating to QU and QD
11.6 Projection Pursuit Density Estimation and Regression
11.6.1 Projection Pursuit Density Estimation
11.6.2 Projection Pursuit Regression
12 Kernel and More Independent Component Methods
12.1 Introduction
12.2 Kernel Component Analysis
12.2.1 Feature Spaces and Kernels
12.2.2 Kernel Principal Component Analysis
12.2.3 Kernel Canonical Correlation Analysis
12.3 Kernel Independent Component Analysis
12.3.1 The F-Correlation and Independence
12.3.2 Estimating the F-Correlation
12.3.3 Comparison of Non-Gaussian and Kernel Independent Components Approaches
12.4 Independent Components from Scatter Matrices (aka Invariant Coordinate Selection)
12.4.1 Scatter Matrices
12.4.2 Population Independent Components from Scatter Matrices
12.4.3 Sample Independent Components from Scatter Matrices
12.5 Non-Parametric Estimation of Independence Criteria
12.5.1 A Characteristic Function View of Independence
12.5.2 An Entropy Estimator Based on Order Statistics
12.5.3 Kernel Density Estimation of the Unmixing Matrix
13 Feature Selection and Principal Component Analysis Revisited
13.1 Introduction
13.2 Independent Components and Feature Selection
13.2.1 Feature Selection in Supervised Learning
13.2.2 Best Features and Unsupervised Decisions
13.2.3 Test of Gaussianity
13.3 Variable Ranking and Statistical Learning
13.3.1 Variable Ranking with the Canonical Correlation Matrix C
13.3.2 Prediction with a Selected Number of Principal Components
13.3.3 Variable Ranking for Discriminant Analysis Based on C
13.3.4 Properties of the Ranking Vectors of the Naive C when d Grows
13.4 Sparse Principal Component Analysis
13.4.1 The Lasso, SCoTLASS Directions and Sparse Principal Components
13.4.2 Elastic Nets and Sparse Principal Components
13.4.3 Rank One Approximations and Sparse Principal Components
13.5 (In)Consistency of Principal Components as the Dimension Grows
13.5.1 (In)Consistency for Single-Component Models
13.5.2 Behaviour of the Sample Eigenvalues, Eigenvectors and Principal Component Scores
13.5.3 Towards a General Asymptotic Framework for Principal Component Analysis
Problems for Part III
Bibliography
Author Index
Subject Index
Data Index
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Tags: lnge Koch, Dimensional Data, Analysis of Multivariate