Textbook

# Statistics for Imaging, Optics, and Photonics

A vivid, hands-on discussion of the statistical methods in imaging, optics, and photonics applications

In the field of imaging science, there is a growing need for students and practitioners to be equipped with the necessary knowledge and tools to carry out quantitative analysis of data. Providing a self-contained approach that is not too heavily statistical in nature, Statistics for Imaging, Optics, and Photonics presents necessary analytical techniques in the context of real examples from various areas within the field, including remote sensing, color science, printing, and astronomy.

Bridging the gap between imaging, optics, photonics, and statistical data analysis, the author uniquely concentrates on statistical inference, providing a wide range of relevant methods. Brief introductions to key probabilistic terms are provided at the beginning of the book in order to present the notation used, followed by discussions on multivariate techniques such as:

• Linear regression models, vector and matrix algebra, and random vectors and matrices

• Multivariate statistical inference, including inferences about both mean vectors and covariance matrices

• Principal components analysis

• Canonical correlation analysis

• Discrimination and classification analysis for two or more populations and spatial smoothing

• Cluster analysis, including similarity and dissimilarity measures and hierarchical and nonhierarchical clustering methods

Intuitive and geometric understanding of concepts is emphasized, and all examples are relatively simple and include background explanations. Computational results and graphs are presented using the freely available R software, and can be replicated by using a variety of software packages. Throughout the book, problem sets and solutions contain partial numerical results, allowing readers to confirm the accuracy of their approach; and a related website features additional resources including the book's datasets and figures.

Statistics for Imaging, Optics, and Photonics is an excellent book for courses on multivariate statistics for imaging science, optics, and photonics at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for professionals working in imaging, optics, and photonics who carry out data analyses in their everyday work.

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Preface xiii

1 Introduction 1

1.1 Who Should Read This Book, 6

1.2 How This Book is Organized, 6

1.3 How to Read This Book and Learn from It, 7

1.4 Note for Instructors, 8

1.5 Book Web Site, 9

2 Fundamentals of Statistics 11

2.1 Statistical Thinking, 11

2.2 Data Format, 13

2.3 Descriptive Statistics, 14

2.3.1 Measures of Location, 14

2.3.2 Measures of Variability, 16

2.4 Data Visualization, 17

2.4.1 Dot Plots, 17

2.4.2 Histograms, 19

2.4.3 Box Plots, 23

2.4.4 Scatter Plots, 24

2.5 Probability and Probability Distributions, 26

2.5.1 Probability and Its Properties, 26

2.5.2 Probability Distributions, 30

2.5.3 Expected Value and Moments, 33

2.5.4 Joint Distributions and Independence, 34

2.5.5 Covariance and Correlation, 38

2.6 Rules of Two and Three Sigma, 40

2.7 Sampling Distributions and the Laws of Large Numbers, 41

2.8 Skewness and Kurtosis, 44

3 Statistical Inference 51

3.1 Introduction, 51

3.2 Point Estimation of Parameters, 53

3.2.1 Definition and Properties of Estimators, 53

3.2.2 The Method of the Moments and Plug-In Principle, 56

3.2.3 The Maximum Likelihood Estimation, 57

3.3 Interval Estimation, 60

3.4 Hypothesis Testing, 63

3.5 Samples From Two Populations, 71

3.6 Probability Plots and Testing for Population Distributions, 73

3.6.1 Probability Plots, 74

3.6.2 Kolmogorov–Smirnov Statistic, 75

3.6.3 Chi-Squared Test, 76

3.6.4 Ryan–Joiner Test for Normality, 76

3.7 Outlier Detection, 77

3.8 Monte Carlo Simulations, 79

3.9 Bootstrap, 79

4 Statistical Models 85

4.1 Introduction, 85

4.2 Regression Models, 85

4.2.1 Simple Linear Regression Model, 86

4.2.2 Residual Analysis, 94

4.2.3 Multiple Linear Regression and Matrix Notation, 96

4.2.4 Geometric Interpretation in an n-Dimensional Space, 99

4.2.5 Statistical Inference in Multiple Linear Regression, 100

4.2.6 Prediction of the Response and Estimation of the Mean Response, 104

4.2.7 More on Checking the Model Assumptions, 107

4.2.8 Other Topics in Regression, 110

4.3 Experimental Design and Analysis, 111

4.3.1 Analysis of Designs with Qualitative Factors, 116

4.3.2 Other Topics in Experimental Design, 124

Supplement 4A. Vector and Matrix Algebra, 125

Vectors, 125

Matrices, 127

Eigenvalues and Eigenvectors of Matrices, 130

Spectral Decomposition of Matrices, 130

Positive Definite Matrices, 131

A Square Root Matrix, 131

Supplement 4B. Random Vectors and Matrices, 132

Sphering, 134

5 Fundamentals of Multivariate Statistics 137

5.1 Introduction, 137

5.2 The Multivariate Random Sample, 139

5.3 Multivariate Data Visualization, 143

5.4 The Geometry of the Sample, 148

5.4.1 The Geometric Interpretation of the Sample Mean, 148

5.4.2 The Geometric Interpretation of the Sample Standard Deviation, 149

5.4.3 The Geometric Interpretation of the Sample Correlation Coefficient, 150

5.5 The Generalized Variance, 151

5.6 Distances in the p-Dimensional Space, 159

5.7 The Multivariate Normal (Gaussian) Distribution, 163

5.7.1 The Definition and Properties of the Multivariate Normal Distribution, 163

5.7.2 Properties of the Mahalanobis Distance, 166

6 Multivariate Statistical Inference 173

6.1 Introduction, 173

6.2 Inferences About a Mean Vector, 173

6.2.1 Testing the Multivariate Population Mean, 173

6.2.2 Interval Estimation for the Multivariate Population Mean, 175

6.2.3 T2 Confidence Regions, 179

6.3 Comparing Mean Vectors from Two Populations, 183

6.3.1 Equal Covariance Matrices, 184

6.3.2 Unequal Covariance Matrices and Large Samples, 185

6.3.3 Unequal Covariance Matrices and Samples Sizes Not So Large, 186

6.4 Inferences About a Variance–Covariance Matrix, 187

6.5 How to Check Multivariate Normality, 188

7 Principal Component Analysis 193

7.1 Introduction, 193

7.2 Definition and Properties of Principal Components, 195

7.2.1 Definition of Principal Components, 195

7.2.2 Finding Principal Components, 196

7.2.4 Scaling of Variables, 207

7.3 Stopping Rules for Principal Component Analysis, 209

7.3.1 Fair-Share Stopping Rules, 210

7.3.2 Large-Gap Stopping Rules, 213

7.4 Principal Component Scores, 217

7.5 Residual Analysis, 220

7.6 Statistical Inference in Principal Component Analysis, 227

7.6.1 Independent and Identically Distributed Observations, 227

7.6.2 Imaging Related Sampling Schemes, 228

8 Canonical Correlation Analysis 241

8.1 Introduction, 241

8.2 Mathematical Formulation, 242

8.3 Practical Application, 245

8.4 Calculating Variability Explained by Canonical Variables, 246

8.5 Canonical Correlation Regression, 251

Supplement 8A. Cross-Validation, 256

9 Discrimination and Classification – Supervised Learning 261

9.1 Introduction, 261

9.2 Classification for Two Populations, 264

9.2.1 Classification Rules for Multivariate Normal Distributions, 267

9.2.2 Cross-Validation of Classification Rules, 277

9.2.3 Fisher’s Discriminant Function, 280

9.3 Classification for Several Populations, 284

9.3.1 Gaussian Rules, 284

9.3.2 Fisher’s Method, 286

9.4 Spatial Smoothing for Classification, 291

10 Clustering – Unsupervised Learning 297

10.1 Introduction, 297

10.2 Similarity and Dissimilarity Measures, 298

10.2.1 Similarity and Dissimilarity Measures for Observations, 298

10.2.2 Similarity and Dissimilarity Measures for Variables and Other Objects, 304

10.3 Hierarchical Clustering Methods, 304

10.3.4 Ward Method, 319

10.4 Nonhierarchical Clustering Methods, 320

10.4.1 K-Means Method, 320

10.5 Clustering Variables, 323

Appendix A Probability Distributions 329

Appendix B Data Sets 349

Appendix C Miscellanea 355

References 365

Index 371

Author Information

PETER BAJORSKI, PhD, is Associate Professor in the Graduate Statistics Department at Rochester Institute of Technology, where he is also a core member of the graduate program faculty at the Center for Imaging Science. The author of numerous published articles on statistics and imaging, Dr. Bajorski's areas of statistical expertise include regression techniques, multivariate analysis, design of experiments, nonparametric methods, and visualization methods. A senior member of the IEEE and SPIE, his research in imaging includes unmixing and target detection in spectral images.

## Available Versions

Statistics for Imaging, Optics, and Photonics
by Peter Bajorski
ISBN 978-0-470-50945-6
October 2011
Hardcover, 408 pages