There is a growing need for statistical analysis
of data in imaging, optics, and photonics applications. Although
there is a vast literature explaining statistical methods needed
for such applications, there are two major difficulties for practitioners
using these statistical resources. The first difficulty is that
most statistical books are written in a formal statistical and mathematical
language, which an occasional user of statistics may find difficult
to understand. The second difficulty is that the needed material
is scattered among many statistical books.
The purpose of this book is to bridge the gap between
imaging, optics, and photonics, and statistics and data analysis.
The statistical techniques are explained in the context of real
examples from remote sensing, color science, printing, astronomy,
and other related disciplines. I emphasize intuitive and geometric
understanding of concepts and provide many (198) graphs for their
illustration.
The scope of the material is very broad. It starts
with rudimentary data analysis and ends with sophisticated multivariate
statistical methods. Necessarily, the presentation is brief and
does not cover all aspects of the discussed methods. I concentrate
on teaching the skills of statistical thinking, and providing the
tools needed the most in imaging, optics, and photonics. There are
many details in this book that you will not find in most statistical
textbooks. They enhance the reader’s understanding and answer
the usual questions asked by students of the subject.
In the following, I provide brief descriptions of
the book chapters and list some highlights that are either original
contributions, or interesting applications, or other features that
distinguish this book from other books on statistics. The listed
highlights are not intended as the list of the most important topics.
You can find a full table of contents by clicking here .
Chapter 1. An “appetizer”
chapter with examples of data and types of analyses performed in
this book
Chapter 2. A brief
review of descriptive statistics, graphs, and probability for readers
with some prior experience with those topics. Some distinguishing
features:
- In Example 2.3, I combine the use of a histogram and
a definition of a connected set in order to solve a practical
imaging problem.
-
Property 2.3 is an original contribution.
-
Section 2.8 includes some examples that shed a new light
on the properties of kurtosis.
Chapter 3. A brief review of inferential
statistics (confidence intervals and hypothesis testing) for readers
with some prior experience with those topics. Some distinguishing
features:
-
For the Ryan-Joiner test of normality presented in Section
3.6.4, correct critical values are provided (in Appendix C.4). Some
books and software provide very imprecise critical values for that
test.
-
In Section 3.7, I explain the importance of the sample size
in outlier detection. This topic is often explained incorrectly
in other sources. Many software packages use incorrect methods for
assessing outliers.
-
Sections 3.8 and 3.9 provide a brief overview of Monte Carlo
simulations and bootstrapping.
Chapter 4. A brief review of linear
regression models and design and analysis of experiments. Supplement
4A provides basics of matrix algebra, and Supplement 4B discusses
random vectors. Some distinguishing features:
-
In Section 4.2.7, I advocate use of externally studentized
residuals and provide guidelines on the outlier detection. The outlier
threshold should be at least 3.5, but usually much higher. The outlier
detection in regression is often done incorrectly in other sources,
including some statistical software.
-
In Section 4.3 (p. 113),I discuss the scientific rule of
changing one factor at a time in experiments. I explain what goes
wrong when this rule is applied too rigidly, resulting in the so-called
one-factor-at-a-time experiments. I then show a better implementation
of this rule in the full factorial designs.
Chapter 5. Introduction to fundamental
concepts of multivariate statistics. Some distinguishing features:
In Section 5.5, I introduce k-dimensional generalized sample
variance, particularly useful for small samples from high-dimensional
data
-
In Section 5.7.2, I discuss what happens with two- and three-sigma
rules in multidimensional spaces and how this relates to the so-called
“curse of dimensionality.” I also provide formulas and
graphs for detection of multivariate outliers with the use of the
Mahalanobis distance.
Chapter 6. Statistical inference
for one and two multivariate samples. The case of more than two
samples is covered when testing equality of the variance-covariance
matrices. Some distinguishing features:
-
Section 6.2.2 provides a thorough coverage of Bonferroni
confidence intervals, including some illustrative comparisons to
the single confidence intervals.
-
Example 6.4 and Figure 6.3 illustrate the relationship between
the univariate confidence intervals and the elliptical confidence
regions.
Chapter 7. Principal component
analysis. Some distinguishing features:
-
A new tool of impact plots is introduced in Section 7.2.3
-
Some new stopping rules that are particularly suited for
high-dimensional data are introduced in Section 7.3.
-
Residual analysis of imaging spectral data (Section 7.5)
reveals spatial structure in high-order principal components that
are often believed to contain only noise.
-
Section 7.6.2 shows examples of statistical inference in
sampling schemes relevant in imaging data. I also explain why the
classic statistical inference based on the independent and identically
distributed sampling schemes is typically not suitable for imaging
data.
Chapter 8. Canonical Correlation
Analysis. Some distinguishing features:
-
Section 8.5 covers the canonical correlation regression,
which is rarely discussed in statistical books.
-
Supplement 8A explains the concept of cross-validation.
Chapter 9. Discrimination and Classification
– Supervised Learning. Some distinguishing features:
-
Calculation of misclassification rates on p. 271 is very
illustrative for educational purposes.
-
Section 9.4 shows how to use the spatial information to
enhance a classification rule.
Chapter 10. Clustering –
Unsupervised Learning. Some distinguishing features:
-
In Section 10.2.1, I give some guidelines on how to decide
which Lp metric is most appropriate in a given context.
-
In Section 10.3.1, I discuss the minimum spanning tree and
the minimum forest, and then I use those tools in graphical representation
of clusters.
Appendix A. A list of probability distributions
with a brief description of their properties. Tabulation of some
distributions is also provided.
Appendix B. Background information
about 14 case studies used throughout the book.
Appendix C. Miscellaneous topics:
-
Singular values decomposition
-
An alternative approach to canonical correlation analysis
-
List of abbreviations and mathematical symbols used throughout
the book
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