Mathematical Gnostics: Advanced Data Analysis for Research and Engineering Practice

343 Pages·2023·14.586 MB·English

by  Pavel Kovanic

Mathematical Gnostics
The book describes the theoretical principles of nonstatistical methods of data
analysis but without going deep into complex mathematics. The emphasis is laid
on presentation of solved examples of real data either from authors’ laboratories
or from open literature. The examples cover wide range of applications such as
quality assurance and quality control, critical analysis of experimental data, com-
parison of data samples from various sources, robust linear regression as well
as various tasks from financial analysis. The examples are useful primarily for
chemical engineers including analytical/quality laboratories in industry, design-
ers of chemical and biological processes.
Features:
• Exclusive title on Mathematical Gnostics with multidisciplinary applications,
and specific focus on chemical engineering.
• Clarifies the role of data space metrics including the right way of aggregation
of uncertain data.
• Brings a new look on the data probability, information, entropy and thermo-
dynamics of data uncertainty.
• Enables design of probability distributions for all real data samples including
smaller ones.
• Includes data for examples with solutions with exercises in R or Python.
The book is aimed for Senior Undergraduate Students, Researchers, and Profes-
sionals in Chemical/Process Engineering, Engineering Physics, Stats, Math-
ematics, Materials, Geotechnical, Civil Engineering, Mining, Sales, Marketing
and Service, and Finance.

Mathematical Gnostics
Advanced Data Analysis for
Research and Engineering Practice
Pavel Kovanic
First edition published 2023
by CRC Press
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ISBN: 978-1-138-33923-1 (hbk)
ISBN: 978-1-032-42351-7 (pbk)
ISBN: 978-0-429-44119-6 (ebk)
DOI: 10.1201/9780429441196
Typeset in Latin Modern
by KnowledgeWorks Global Ltd.
Contents
Preface xi
Introduction xv
Author Biography xix
1 Introductory Kindergarten 1
1.1 Elemental Notions . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Abelian Group . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Variability . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Morphism and Invariant . . . . . . . . . . . . . . . . . 3
1.1.4 Vector Space . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.5 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.6 Probability Distribution . . . . . . . . . . . . . . . . . 6
1.2 Sources of Inspiration for Mathematical Gnostics . . . . . . . 6
1.2.1 Theory of General Systems . . . . . . . . . . . . . . . 7
1.2.2 Theory of Measurement . . . . . . . . . . . . . . . . . 7
1.2.3 Geometries . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 Maxwell’s Contributions . . . . . . . . . . . . . . . . . 8
1.2.5 Relativistic Physics. . . . . . . . . . . . . . . . . . . . 8
1.2.6 Thermodynamics . . . . . . . . . . . . . . . . . . . . . 8
1.2.7 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Axioms 11
2.1 Axioms of the Data Model . . . . . . . . . . . . . . . . . . . 11
2.2 Applications of Axiom 1 . . . . . . . . . . . . . . . . . . . . 12
2.3 Data Aggregation as the Second Gnostic Axiom . . . . . . . 14
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Introduction to Non-Standard Thought 15
3.1 Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Statistical Paradigms . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Statistical Data Weighing . . . . . . . . . . . . . . . . . . . . 18
3.4 Non-Statistical Paradigms of Uncertainty . . . . . . . . . . . 19
3.5 On the Need of an Alternative to Statistics . . . . . . . . . . 20
3.6 Principles of Advanced Data Analysis . . . . . . . . . . . . . 23
v
vi Contents
3.7 The Gnostic Concept . . . . . . . . . . . . . . . . . . . . . . 25
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Quantification 29
4.1 Ideal Quantification . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Real Quantification . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Estimation and Ideal Gnostic Cycle 35
5.1 A Game with Nature . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Double Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Gnostic Data Characteristics . . . . . . . . . . . . . . . . . . 36
5.4 The Ideal Gnostic Cycle . . . . . . . . . . . . . . . . . . . . . 39
5.5 Information Perpetuum Mobile? . . . . . . . . . . . . . . . . 40
5.6 Existence and Uniqueness of the Ideal Gnostic Cycle . . . . 40
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Geometry 43
6.1 A Historical Dispute on Robustness of Statistics . . . . . . . 43
6.2 Distance as a Problem . . . . . . . . . . . . . . . . . . . . . . 45
6.3 Additivity in Data Aggregation . . . . . . . . . . . . . . . . 47
6.3.1 Statistical Mean Value and Data Weighting . . . . . . 48
6.4 Double Robustness . . . . . . . . . . . . . . . . . . . . . . . 49
6.5 The Curvature of the Space of Uncertain Data . . . . . . . . 50
6.6 Three Geometries . . . . . . . . . . . . . . . . . . . . . . . . 51
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Aggregation 55
7.1 Why the Least Squares Method (Frequently) Works . . . . . 56
7.2 Aggregation of Uncertain Data . . . . . . . . . . . . . . . . . 57
7.3 The Second Axiom . . . . . . . . . . . . . . . . . . . . . . . 59
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8 Thermodynamics of Uncertain Data 61
8.1 Thermodynamic Interpretation of Gnostic Data Characteristics 61
8.2 Maxwell’s Demon . . . . . . . . . . . . . . . . . . . . . . . . 63
8.3 Entropy ↔ Information Conversion . . . . . . . . . . . . . . 64
8.4 Albert Perez’s Information . . . . . . . . . . . . . . . . . . . 65
8.5 Statistical Interpretation of Gnostic Data Characteristics . . 67
8.6 Between Mediocristan and Extremistan . . . . . . . . . . . . 70
8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Contents vii
9 Kernel Estimation 73
9.1 Parzen’s Estimating Kernel . . . . . . . . . . . . . . . . . . . 73
9.2 Gnostic Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.3 Scale Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10 Probability Distribution Functions 79
10.1 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
10.2 Data Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 80
10.3 Tasks Solvable by Distribution Functions . . . . . . . . . . . 82
10.4 The Estimating Local Distribution . . . . . . . . . . . . . . . 84
10.5 Quantifying Distributions . . . . . . . . . . . . . . . . . . . . 85
10.6 Empirical Distribution Function and the Fit . . . . . . . . . 86
10.7 Some Applications of Distribution Functions . . . . . . . . . 90
10.7.1 Revealing Historical Information . . . . . . . . . . . . 90
10.7.2 Hypotheses Testing. . . . . . . . . . . . . . . . . . . . 95
10.7.3 A Large Survey of Chemical Pollutants . . . . . . . . 96
10.8 The Homogeneity Problem . . . . . . . . . . . . . . . . . . . 99
10.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11 Applications of Local Distributions 107
11.1 Enrichment of the EGDF-Analysis . . . . . . . . . . . . . . . 107
11.2 Revealing Inner Structure of a Data Sample . . . . . . . . . 108
11.3 Marginal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 109
11.4 Information Capability of Data . . . . . . . . . . . . . . . . . 112
11.5 Interval Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 113
11.6 Diversity of Samples . . . . . . . . . . . . . . . . . . . . . . . 115
11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12 On the Notion of Normality 119
12.1 Normality of Data . . . . . . . . . . . . . . . . . . . . . . . . 119
12.1.1 Statistical Approach . . . . . . . . . . . . . . . . . . . 120
12.1.2 Empirical Way in Clinical Practice . . . . . . . . . . . 121
12.1.3 Similarity-Based Reference Values in Economy . . . . 122
12.1.4 Fuzzy-Set Approach . . . . . . . . . . . . . . . . . . . 123
12.1.5 Automatic Warning and Emergency Systems . . . . . 124
12.2 Requirements to Ideal Estimation of Bounds of Normality . 124
12.3 Elements of Gnostic Solution of the Normality Problem in a
One-Dimensional Analysis . . . . . . . . . . . . . . . . . . . 125
12.4 Critics on the Identity Gaussian ≡ Normal . . . . . . . . . . 127
12.4.1 Re-definition of Normality . . . . . . . . . . . . . . . . 127
12.4.2 On a Still Daydreamed Research Project BONUS . . . 129
12.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
viii Contents
13 Applications of Global Distribution Functions 135
13.1 Global Distribution Function . . . . . . . . . . . . . . . . . . 135
13.2 Comparison of Global with Local Distribution . . . . . . . . 139
13.3 Two Didactic Stories . . . . . . . . . . . . . . . . . . . . . . 140
13.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
14 Data Censoring 143
14.1 Uncensored Data . . . . . . . . . . . . . . . . . . . . . . . . . 143
14.2 Left-Censored Data . . . . . . . . . . . . . . . . . . . . . . . 144
14.3 Right-Censored Data . . . . . . . . . . . . . . . . . . . . . . 146
14.4 Interval Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
14.5 On an Unknown Limit of Detection . . . . . . . . . . . . . . 150
14.6 Examples of Surviving . . . . . . . . . . . . . . . . . . . . . . 151
14.7 Non-Standard Application of Data Censoring . . . . . . . . . 153
14.7.1 Data and Psychology. . . . . . . . . . . . . . . . . . . 153
14.7.2 Three Aspects of Data Interpretation . . . . . . . . . 154
14.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
15 Gnostic Thermodynamic Analysis of Data Uncertainty 159
15.1 Gnostic Data Calibration . . . . . . . . . . . . . . . . . . . . 159
15.1.1 Real Data for Examples . . . . . . . . . . . . . . . . . 159
15.2 Data Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 163
15.2.1 LS-Optimal Numerical Operators . . . . . . . . . . . . 164
15.3 Calibration of the NIST12 Data . . . . . . . . . . . . . . . . 164
15.4 Calibration of the NIST37 Data . . . . . . . . . . . . . . . . 167
15.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
16 Robust Estimation of a Constant 171
16.1 Gnostic Data Aggregation Principle Used in Estimation . . . 171
16.2 Scale Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 172
16.3 More on the Gnostic Data Aggregation . . . . . . . . . . . . 173
16.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 174
16.3.2 ExampleofRobustEstimationoftheMeanof
Multiplicative Data . . . . . . . . . . . . . . . . . . . 176
16.3.3 Robust Estimation of the Mean of Simulated Data . . 177
16.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
17 Measuring the Data Uncertainty 181
17.1 Shortly on the Standard Approach . . . . . . . . . . . . . . . 181
17.2 The Need of Objective Measuring the Variability . . . . . . . 183
17.3 The Triplication of the Mean Values . . . . . . . . . . . . . . 183
17.4 The Need of a Unit of Uncertainty . . . . . . . . . . . . . . . 184
17.5 The Error of a Mean . . . . . . . . . . . . . . . . . . . . . . 186
17.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
17.6.1 Swiss Fertility and Socioeconomic Indicators (1888)
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Contents ix
17.6.2 Financial Statement Analysis . . . . . . . . . . . . . . 188
17.6.3 Weather Parameters . . . . . . . . . . . . . . . . . . . 189
17.6.4 An Important Medical Parameter. . . . . . . . . . . . 190
17.6.5 Non-homogeneous Data . . . . . . . . . . . . . . . . . 191
17.6.6 Parameters of Uncertainty . . . . . . . . . . . . . . . . 194
17.7 Discussion on Different Means . . . . . . . . . . . . . . . . . 196
17.7.1 Re-definition of Variance. . . . . . . . . . . . . . . . . 196
17.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
18 Homo- or Heteroscedastic Data 199
18.1 Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . 199
18.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
18.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
19 Gnostic Multidimensional Regression Models 205
19.1 Formulation of the Robust Regression Problem . . . . . . . . 207
19.2 Additive and Multiplicative Regression Models . . . . . . . . 214
19.3 Comparison of Robust Regression Models . . . . . . . . . . . 215
19.3.1 Statistical Methods for Comparison . . . . . . . . . . 215
19.3.2 Robust Regression in Mathematical Gnostics . . . . . 216
19.3.3 Data for Comparison . . . . . . . . . . . . . . . . . . . 218
19.3.4 Criteria for Evaluation of Methods . . . . . . . . . . . 219
19.3.5 Results of Comparison . . . . . . . . . . . . . . . . . . 220
19.3.6 Discussion of the Results . . . . . . . . . . . . . . . . 221
19.4 The Explicit and Implicit Regression Models . . . . . . . . . 222
19.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
19.6 Homogeneity of an MD-Model . . . . . . . . . . . . . . . . . 229
19.7 An Important Multidimensional Model . . . . . . . . . . . . 230
19.8 Applications of the Robust Regression Models . . . . . . . . 233
19.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
20 Data Filtering 235
20.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
20.2 Total Data Variability and Its Components . . . . . . . . . . 235
20.3 Filtering by Regression . . . . . . . . . . . . . . . . . . . . . 236
20.4 Filtering Effect of Proper Data Aggregation . . . . . . . . . 237
20.5 Improving the Matrix Quality . . . . . . . . . . . . . . . . . 238
20.6 Cleaning of Matrices . . . . . . . . . . . . . . . . . . . . . . . 240
20.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
21 Decision Making in Mathematical Gnostics 243
21.1 Datacratic Decision Making in Mathematical Gnostics . . . . 244
21.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246