Measurement, mathematics and new quantification theory - Nishisato, Shizuhiko; Springer (2023)
The purpose of this book is to thoroughly prepare diverse areas of researchers in quantification theory. As is well known, quantification theory has attracted the attention of a countless number of researchers, some mathematically oriented and others not, but all of them are experts in their own disciplines. Quantifying non-quantitative (qualitative) data requires a variety of mathematical and statistical strategies, some of which are quite complicated. Unlike many books on quantification theory, the current book places more emphasis on preliminary requisites of mathematical tools than on details of quantification theory. As such, the book is primarily intended for readers whose specialty is outside mathematical sciences. The book was designed to offer non-mathematicians a variety of mathematical tools used in quantification theory in simple terms. Once all the preliminaries are fully discussed, quantification theory is then introduced in the last section as a simple application of those mathematical procedures fully discussed so far. The book opens up further frontiers of quantification theory as simple applications of basic mathematics
Intro
Preface
Acknowledgments
Contents
Part I Measurement
1 Information for Analysis
1.1 An Overview
1.2 Introduction
1.2.1 Fundamental Arithmetic Operations
1.2.2 Data Types
1.3 Stevens' Theory of Measurement
1.3.1 Nominal Measurement
1.3.2 Ordinal Measurement
1.3.3 Interval Measurement
1.3.4 Ratio Measurement
1.4 Concluding Remarks on Measurement
1.5 Task of Quantification Theory
References
2 Data Analysis and Likert Scale
2.1 Two Examples of Uninformative Reports
2.1.1 Number of COVID Patients
2.1.2 Number of Those Vaccinated 2.2 Likert Scale, a Popular but Misused Tool
2.2.1 How Does Likert Scale Work?
2.2.2 Warnings on Inappropriate Use of Likert Scale
References
Part II Mathematics
3 Preliminaries
3.1 An Overview
3.2 Series and Limit
3.2.1 Examples from Quantification Theory
3.3 Differentiation
3.4 Derivative of a Function of One Variable
3.5 Derivative of a Function of a Function
3.6 Partial Derivative
3.7 Differentiation Formulas
3.8 Maximum and Minimum Value of a Function
3.9 Lagrange Multipliers
3.9.1 Example 1
3.9.2 Example 2
References
4 Matrix Calculus 4.1 Different Forms of Matrices
4.1.1 Transpose
4.1.2 Rectangular Versus Square Matrix
4.1.3 Symmetric Matrix
4.1.4 Diagonal Matrix
4.1.5 Vector
4.1.6 Scaler Matrix and Identity Matrix
4.1.7 Idempotent Matrix
4.2 Simple Operations
4.2.1 Addition and Subtraction
4.2.2 Multiplication
4.2.3 Scalar Multiplication
4.2.4 Determinant
4.2.5 Inverse
4.2.6 Hat Matrix
4.2.7 Hadamard Product
4.3 Linear Dependence and Linear Independence
4.4 Rank of a Matrix
4.5 System of Linear Equations
4.6 Homogeneous Equations and Trivial Solution 4.7 Orthogonal Transformation
4.8 Rotation of Axes
4.9 Characteristic Equation of the Quadratic Form
4.10 Eigenvalues and Eigenvectors
4.10.1 Example: Canonical Reduction
4.11 Idempotent Matrices
4.12 Projection Operator
4.12.1 Example 1: Maximal Correlation
4.12.2 Example 2: General Decomposition Formula
References
5 Statistics in Matrix Notation
5.1 Mean
5.2 Variance-Covariance Matrix
5.3 Correlation Matrix
5.4 Linear Regression
5.5 One-Way Analysis of Variance
5.6 Multiway Analysis of Variance
5.7 Discriminant Analysis 5.8 Principal Component Analysis
References
6 Multidimensional Space
6.1 Introduction
6.2 Pierce's Description
6.2.1 Pythagorean Theorem
6.2.2 The Cosine Law
6.2.3 Young-Householder Theorem
6.2.4 Eckart-Young Theorem
6.2.5 Chi-Square Distance
6.3 Distance in Multidimensional Space
6.4 Correlation in Multidimensional Space
References
Part III A New Look at Quantification Theory
7 General Introduction
7.1 An Overview
7.2 Historical Background and Reference Books
7.3 First Step
7.3.1 Assignment of Unknown Numbers
Includes bibliographic reference
Keywords: Predicate Calculus