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Measurement, mathematics and new quantification theory / Shizuhiko Nishisato.

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 dis...

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Bibliographic Details
Online Access: Full Text (via Springer)
Main Author: Nishisato, Shizuhiko, 1935- (Author)
Format: Electronic eBook
Language:English
Published: Singapore : Springer, [2023]
Series:Behaviormetrics ; v. 16.
Subjects:
Predicate calculus.
Table of Contents:
  • 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

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