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Chapter 1 Basis of Matrix Calculation
1£®1 Fundamental Concepts
1£®1£®1 Notation
1£®1£®2 ¡°Biggerª²Block¡± Interpretations of Matrix Multiplication
1£®1£®3 Fundamental Linear Algebra
1£®1£®4 Four Fundamental Subspaces of a Matrix
1£®1£®5 Vector Norms
1£®1£®6 Determinants
1£®1£®7 Properties of Determinants
1£®2 The Most Basic Matrix Decomposition
1£®2£®1 Gaussian Elimination
1£®2£®2 The LU Decomposition
1£®2£®3 The LDM Factorization
1£®2£®4 The LDL Decomposition for Symmetric Matrices
1£®2£®5 Cholesky Decomposition
1£®2£®6 Applications and Examples of the Cholesky Decomposition
1£®2£®7 Eigendecomposition
1£®2£®8 Matrix Norms
1£®2£®9 Covariance Matrices
1£®3 Singular Value Decomposition £¨SVD£©
1£®3£®1 Orthogonalization
1£®3£®2 Existence Proof of the SVD
1£®3£®3 Partitioning the SVD
1£®3£®4 Properties and Interpretations of the SVD
1£®3£®5 Relationship between SVD and ED
1£®3£®6 Ellipsoidal Interpretation of the SVD
1£®3£®7 An Interesting Theorem
1£®4 The Quadratic Form
1£®4£®1 Quadratic Form Theory
1£®4£®2 The Gaussian Multiª²Variate Probability Density Function
1£®4£®3 The Rayleigh Quotient

Chapter 2 The Solution of Least Squares Problems
2£®1 Linear Least Squares Estimation
2£®1£®1 Example£º Autoregressive Modelling
2£®1£®2 The Leastª²Squares Solution
2£®1£®3 Interpretation of the Normal Equations
2£®1£®4 Properties of the LS Estimate
2£®1£®5 Linear Leastª²Squares Estimation and the Cramer Rao Lower Bound
2£®2 A Generalized ¡°Pseudoª²Inverse¡± Approach to Solving the Leastª²squares Problem
2£®2£®1 Least Squares Solution Using the SVD
2£®2£®2 Interpretation of the Pseudoª²Inverse

Chapter 3 Principal Component Analysis
3£®1 Introductory Example
3£®2 Theory
3£®2£®1 Taking Linear Combinations
3£®2£®2 Explained Variation
3£®2£®3 PCA as a Model
3£®2£®4 Taking More Components
3£®3 History of PCA
3£®4 Practical Aspects
3£®4£®1 Preprocessing
3£®4£®2 Choosing the Number of Components
3£®4£®3 When Using PCA for Other Purposes
3£®4£®4 Detecting Outliers
References
3£®5 Sklearn PCA
3£®5£®1 Source Code
3£®5£®2 Examples
3£®6 Principal Component Regression
3£®6£®1 Source Code
3£®6£®2 Kª²Fold Crossª²Validation
3£®6£®3 Examples
3£®7 Subspace Methods for Dynamic Model Estimation in PAT Applications
3£®7£®1 Introduction
3£®7£®2 Theory
3£®7£®3 State Space Models in Chemometrics
3£®7£®4 Milk Coagulation Monitoring
3£®7£®5 State Space Based Monitoring
3£®7£®6 Results
3£®7£®7 Concluding remarks
3£®7£®8 Appendix
References

Chapter 4 Partial Least Squares Analysis
4£®1 Basic Concept
4£®1£®1 Partial Least Squares
4£®1£®2 Form of Partial Least Squares
4£®1£®3 PLS Regression
4£®1£®4 Statistic
Reference
4£®2 NIPALS and SIMPLS Algorithm
4£®2£®1 NIPALS
4£®2£®2 SIMPLS
References
4£®3 Programming Method of Standard Partial Least Squares
4£®3£®1 Crossª²validation
4£®3£®2 Procedure of NIPALS
4£®4 Example Application
4£®4£®1 Demo of PLS
4£®4£®2 Corn Dataset
4£®4£®3 Wheat Dataset
4£®4£®4 Pharmaceutical Tablet Dataset
4£®5 Stack Partial Least Squares
4£®5£®1 Introduction
4£®5£®2 Theory of Stack Partial Least Squares
4£®5£®3 Demo of SPLS
4£®5£®4 Experiments
References

Chapter 5 Regularization
5£®1 Regularization
5£®1£®1 Classification
5£®1£®2 Tikhonov Regularization
5£®1£®3 Regularizers for Sparsity
5£®1£®4 Other Uses of Regularization in Statistics and Machine Learning
5£®2 Ridge Regression£º Biased Estimation for Nonorthogonal Problems
5£®2£®1 Properties of Best Linear Unbiased Estimation
5£®2£®2 Ridge Regression
5£®2£®3 The Ridge Trace
5£®2£®4 Mean Square Error Properties of Ridge Regression
5£®2£®5 A General Form of Ridge Regression
5£®2£®6 Relation to Other Work in Regression
5£®2£®7 Selecting a Better Estimate of ?
References
5£®3 Lasso
5£®3£®1 Introduction
5£®3£®2 Theory of the Lasso
References
5£®4 The Example of Ridge Regression and Lasso Regression
5£®4£®1 Example
5£®4£®2 Practical Example
5£®5 Sparse PCA
5£®5£®1 Introduction
5£®5£®2 Motivation and Method Details
5£®5£®3 SPCA for p ¡Ý n and Gene Expression Arrays
5£®5£®4 Demo of SPCA
References

Chapter 6 Transfer Method
6£®1 Calibration Transfer of Spectral Models£Û1£Ý
6£®1£®1 Introduction
6£®1£®2 Calibration Transfer Setting
6£®1£®3 Related Work
6£®1£®4 New or Adapted Methods
6£®1£®5 Standardª²free Alternatives to Methods Requiring Transfer StandardsReferences
6£®2 PLS Subspace Based Calibration Transfer for NIR Quantitative Analysis
6£®2£®1 Calibration Transfer Method
6£®2£®2 Experimental
6£®2£®3 Results and Discussion
6£®2£®4 Conclusion
References
6£®3 Calibration Transfer Based on Affine Invariance for NIR without Standard Samples
6£®3£®1 Theory
6£®3£®2 Experimental
6£®3£®3 Results and Discussion
6£®3£®4 Conclusions