Statistics for Imaging, Optics, and Photonics

Other Topics in Experimental Design, 124Supplement 4A. Vector and Matrix Algebra, 125Vectors, 125Matrices, 127Eigenvalues and Eigenvectors of Matrices, 130Spectral Decomposition of Matrices, 130Positive Definite Matrices, 131A Square Root Matrix, 131Supplement 4B. Random Vectors and Matrices, 132Sphering, 1345 Fundamentals of Multivariate Statistics 1375.1 Introduction, 1375.2 The Multivariate Random Sample, 1395.3 Multivariate Data Visualization, 1435.4 The Geometry of the Sample, 1485.4.1 The Geometric Interpretation of the Sample Mean, 1485.4.2 The Geometric Interpretation of the Sample Standard Deviation, 1495.4.3 The Geometric Interpretation of the Sample Correlation Coefficient, 1505.5 The Generalized Variance, 1515.6 Distances in the p-Dimensional Space, 1595.7 The Multivariate Normal (Gaussian) Distribution, 1635.7.1 The Definition and Properties of the Multivariate Normal Distribution, 1635.7.2 Properties of the Mahalanobis Distance, 1666 Multivariate Statistical Inference 1736.1 Introduction, 1736.2 Inferences About a Mean Vector, 1736.2.1 Testing the Multivariate Population Mean, 1736.2.2 Interval Estimation for the Multivariate Population Mean, 1756.2.3 T2 Confidence Regions, 1796.3 Comparing Mean Vectors from Two Populations, 1836.3.1 Equal Covariance Matrices, 1846.3.2 Unequal Covariance Matrices and Large Samples, 1856.3.3 Unequal Covariance Matrices and Samples Sizes Not So Large, 1866.4 Inferences About a Variance-Covariance Matrix, 1876.5 How to Check Multivariate Normality, 1887 Principal Component Analysis 1937.1 Introduction, 1937.2 Definition and Properties of Principal Components, 1957.2.1 Definition of Principal Components, 1957.2.2 Finding Principal Components, 1967.2.3 Interpretation of Principal Component Loadings, 2007.2.4 Scaling of Variables, 2077.3 Stopping Rules for Principal Component Analysis, 2097.3.1 Fair-Share Stopping Rules, 2107.3.2 Large-Gap Stopping Rules, 2137.4 Principal Component Scores, 2177.5 Residual Analysis, 2207.6 Statistical Inference in Principal Component Analysis, 2277.6.1 Independent and Identically Distributed Observations, 2277.6.2 Imaging Related Sampling Schemes, 2287.7 Further Reading, 2388 Canonical Correlation Analysis 2418.1 Introduction, 2418.2 Mathematical Formulation, 2428.3 Practical Application, 2458.4 Calculating Variability Explained by Canonical Variables, 2468.5 Canonical Correlation Regression, 2518.6 Further Reading, 256Supplement 8A. Cross-Validation, 2569 Discrimination and Classification - Supervised Learning 2619.1 Introduction, 2619.2 Classification for Two Populations, 2649.2.1 Classification Rules for Multivariate Normal Distributions, 2679.2.2 Cross-Validation of Classification Rules, 2779.2.3 Fisher's Discriminant Function, 2809.3 Classification for Several Populations, 2849.3.1 Gaussian Rules, 2849.3.2 Fisher's Method, 2869.4 Spatial Smoothing for Classification, 2919.5 Further Reading, 29310 Clustering - Unsupervised Learning 29710.1 Introduction, 29710.2 Similarity and Dissimilarity Measures, 29810.2.1 Similarity and Dissimilarity Measures for Observations, 29810.2.2 Similarity and Dissimilarity Measures for Variables and Other Objects, 30410.3 Hierarchical Clustering Methods, 30410.3.1 Single Linkage Algorithm, 30510.3.2 Complete Linkage Algorithm, 31210.3.3 Average Linkage Algorithm, 31510.3.4 Ward Method, 31910.4 Nonhierarchical Clustering Methods, 32010.4.1 K-Means Method, 32010.5 Clustering Variables, 32310.6 Further Reading, 325Appendix A Probability Distributions 329Appendix B Data Sets 349Appendix C Miscellanea 355References 365Index 371