Mathematical Gnostics: Advanced Data Analysis for Research and Engineering Practice

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Introduction
Author Biography
1. Introductory Kindergarten
	1.1. Elemental Notions
		1.1.1. Abelian Group
		1.1.2. Variability
		1.1.3. Morphism and Invariant
		1.1.4. Vector Space
		1.1.5. Matrices
		1.1.6. Probability Distribution
	1.2. Sources of Inspiration for Mathematical Gnostics
		1.2.1. Theory of General Systems
		1.2.2. Theory of Measurement
		1.2.3. Geometries
		1.2.4. Maxwell's Contributions
		1.2.5. Relativistic Physics
		1.2.6. Thermodynamics
		1.2.7. Matrix Algebra
	1.3. Conclusions
2. Axioms
	2.1. Axioms of the Data Model
	2.2. Applications of Axiom 1
	2.3. Data Aggregation as the Second Gnostic Axiom
	2.4. Conclusions
3. Introduction to Non-Standard Thought
	3.1. Paradigm
	3.2. Statistical Paradigms
	3.3. Statistical Data Weighing
	3.4. Non-Statistical Paradigms of Uncertainty
	3.5. On the Need of an Alternative to Statistics
	3.6. Principles of Advanced Data Analysis
	3.7. The Gnostic Concept
	3.8. Conclusions
4. Quantification
	4.1. Ideal Quantification
	4.2. Real Quantification
	4.3. Conclusions
5. Estimation and Ideal Gnostic Cycle
	5.1. A Game with Nature
	5.2. Double Numbers
	5.3. Gnostic Data Characteristics
	5.4. The Ideal Gnostic Cycle
	5.5. Information Perpetuum Mobile?
	5.6. Existence and Uniqueness of the Ideal Gnostic Cycle
	5.7. Conclusions
6. Geometry
	6.1. A Historical Dispute on Robustness of Statistics
	6.2. Distance as a Problem
	6.3. Additivity in Data Aggregation
		6.3.1. Statistical Mean Value and Data Weighting
	6.4. Double Robustness
	6.5. The Curvature of the Space of Uncertain Data
	6.6. Three Geometries
	6.7. Conclusions
7. Aggregation
	7.1. Why the Least Squares Method (Frequently) Works
	7.2. Aggregation of Uncertain Data
	7.3. The Second Axiom
	7.4. Conclusions
8. Thermodynamics of Uncertain Data
	8.1. Thermodynamic Interpretation of Gnostic Data Characteristics
	8.2. Maxwell's Demon
	8.3. Entropy ↔ Information Conversion
	8.4. Albert Perez's Information
	8.5. Statistical Interpretation of Gnostic Data Characteristics
	8.6. Between Mediocristan and Extremistan
	8.7. Conclusions
9. Kernel Estimation
	9.1. Parzen's Estimating Kernel
	9.2. Gnostic Kernel
	9.3. Scale Parameters
	9.4. Conclusions
10. Probability Distribution Functions
	10.1. Probabilities
	10.2. Data Domains
	10.3. Tasks Solvable by Distribution Functions
	10.4. The Estimating Local Distribution
	10.5. Quantifying Distributions
	10.6. Empirical Distribution Function and the Fit
	10.7. Some Applications of Distribution Functions
		10.7.1. Revealing Historical Information
		10.7.2. Hypotheses Testing
		10.7.3. A Large Survey of Chemical Pollutants
	10.8. The Homogeneity Problem
	10.9. Conclusions
11. Applications of Local Distributions
	11.1. Enrichment of the EGDF-Analysis
	11.2. Revealing Inner Structure of a Data Sample
	11.3. Marginal Analysis
	11.4. Information Capability of Data
	11.5. Interval Analysis
	11.6. Diversity of Samples
	11.7. Conclusions
12. On the Notion of Normality
	12.1. Normality of Data
		12.1.1. Statistical Approach
		12.1.2. Empirical Way in Clinical Practice
		12.1.3. Similarity-Based Reference Values in Economy
		12.1.4. Fuzzy-Set Approach
		12.1.5. Automatic Warning and Emergency Systems
	12.2. Requirements to Ideal Estimation of Bounds of Normality
	12.3. Elements of Gnostic Solution of the Normality Problem in a One-Dimensional Analysis
	12.4. Critics on the Identity Gaussian ≡ Normal
		12.4.1. Re-definition of Normality
		12.4.2. On a Still Daydreamed Research Project BONUS
	12.5. Conclusions
13. Applications of Global Distribution Functions
	13.1. Global Distribution Function
	13.2. Comparison of Global with Local Distribution
	13.3. Two Didactic Stories
	13.4. Conclusions
14. Data Censoring
	14.1. Uncensored Data
	14.2. Left-Censored Data
	14.3. Right-Censored Data
	14.4. Interval Data
	14.5. On an Unknown Limit of Detection
	14.6. Examples of Surviving
	14.7. Non-Standard Application of Data Censoring
		14.7.1. Data and Psychology
		14.7.2. Three Aspects of Data Interpretation
	14.8. Conclusions
15. Gnostic Thermodynamic Analysis of Data Uncertainty
	15.1. Gnostic Data Calibration
		15.1.1. Real Data for Examples
	15.2. Data Calibration
		15.2.1. LS-Optimal Numerical Operators
	15.3. Calibration of the NIST12 Data
	15.4. Calibration of the NIST37 Data
	15.5. Conclusions
16. Robust Estimation of a Constant
	16.1. Gnostic Data Aggregation Principle Used in Estimation
	16.2. Scale Parameter
	16.3. More on the Gnostic Data Aggregation
		16.3.1. Example
		16.3.2. Example of Robust Estimation of the Mean of Multiplicative Data
		16.3.3. Robust Estimation of the Mean of Simulated Data
	16.4. Conclusions
17. Measuring the Data Uncertainty
	17.1. Shortly on the Standard Approach
	17.2. The Need of Objective Measuring the Variability
	17.3. The Triplication of the Mean Values
	17.4. The Need of a Unit of Uncertainty
	17.5. The Error of a Mean
	17.6. Examples
		17.6.1. Swiss Fertility and Socioeconomic Indicators (1888) Data
		17.6.2. Financial Statement Analysis
		17.6.3. Weather Parameters
		17.6.4. An Important Medical Parameter
		17.6.5. Non-homogeneous Data
		17.6.6. Parameters of Uncertainty
	17.7. Discussion on Different Means
		17.7.1. Re-definition of Variance
	17.8. Conclusions
18. Homo- or Heteroscedastic Data
	18.1. Decision Making
	18.2. Examples
	18.3. Conclusion
19. Gnostic Multidimensional Regression Models
	19.1. Formulation of the Robust Regression Problem
	19.2. Additive and Multiplicative Regression Models
	19.3. Comparison of Robust Regression Models
		19.3.1. Statistical Methods for Comparison
		19.3.2. Robust Regression in Mathematical Gnostics
		19.3.3. Data for Comparison
		19.3.4. Criteria for Evaluation of Methods
		19.3.5. Results of Comparison
		19.3.6. Discussion of the Results
	19.4. The Explicit and Implicit Regression Models
	19.5. Examples
	19.6. Homogeneity of an MD-Model
	19.7. An Important Multidimensional Model
	19.8. Applications of the Robust Regression Models
	19.9. Conclusions
20. Data Filtering
	20.1. Filtering
	20.2. Total Data Variability and Its Components
	20.3. Filtering by Regression
	20.4. Filtering Effect of Proper Data Aggregation
	20.5. Improving the Matrix Quality
	20.6. Cleaning of Matrices
	20.7. Conclusions
21. Decision Making in Mathematical Gnostics
	21.1. Datacratic Decision Making in Mathematical Gnostics
	21.2. Conclusions
22. Comparisons
	22.1. Comparisons of Measurement of Toxicity
	22.2. Comparison of Measurement of Concentration of Cannabinoids
		22.2.1. Comparison of Multiplicative Errors
	22.3. Requirements to the Advanced Comparison
	22.4. Preparing Data for Analysis
	22.5. Analysis of Measurement Errors
		22.5.1. Characterization of Data
		22.5.2. Comparison by Parameters
	22.6. Conclusions
23. Advanced Production Quality Control
	23.1. Exploratory Analysis
	23.2. Automation of the Exploratory Analysis
	23.3. On the Necessity of Data Inspection
	23.4. Data Certification
	23.5. Example of Advanced Quality Control
	23.6. Homogeneity and Outliers
	23.7. Estimation of Left-Censored Data
	23.8. Data Certification and Interval Analysis
	23.9. Comparison of Laboratories
	23.10. Conclusions
24. Robust Correlation
	24.1. Correlation via Distribution Functions
	24.2. Correlations by Means of Regression
	24.3. Correlation and Filtering
	24.4. Autocorrelations
	24.5. Conclusions
25. General Relations
	25.1. Relations Considered in Mathematical Gnostics
	25.2. Robust Curve Fitting
	25.3. The Experimental Mathematics
	25.4. Visualization of a Matrix
	25.5. Critical Points
		25.5.1. Relations in Biology
		25.5.2. Relations in Technology
		25.5.3. Relations in Meteorology
		25.5.4. Auto-Relations
	25.6. Conclusions
Bibliography
Index