Visualizing Statistical Models And Concepts

Front Cover
CRC Press, 2002 M06 14 - 280 pages
Examines classic algorithms, geometric diagrams, and mechanical principles for enhancing visualization of statistical estimation procedures and mathematical concepts in physics, engineering, and computer programming.
 

Selected pages

Contents

Introduction
1
12 The Role of Geometrical Models in Statistics
3
13 The Analogy Implicit in Some Statistical Nomenclature
5
14 A Simple Mechanical Model for the Arithmetic Mean
8
15 Mechanical Models in Science
10
References
11
Abstract Geometrical and Mechanical Representations
13
212 Hypothesis Tests Based on Differences in Energy Levels
15
55 Elemental Set Characterisation of Solutions to Fitting Problems
125
552 Elemental Set Characterisation of the Weighted Least Squares Solution
127
553 Explicit Characterisation of the Weighted Least Squares Solution hi the TwoDimensional Case
128
554 The Lpnorm Criterion
130
555 Elemental Set Approximations
131
56 Prediction Space Representation
132
562 The OneDimensional Fitting Problem
136
563 Higher Dimensional Models
139

22 Geometrical Representations in Prediction Space
16
221 Prediction Space Representation
17
222 A Geometrical Analogy for the Correlation Coefficient
18
223 Planes and Hyperplanes in Prediction Space
19
23 Observation Space Representation
21
24 Parameter Space Representation
22
25 Line Fitting by Eye
25
References
28
Mechanical Models for Multidimensional Medians
29
312 Balance of Forces in OneDimensional Medians
33
32 Mechanical Models for TwoDimensional Medians
35
322 Geometrical Solution in Parameter Space
36
323 Vectorial Representation of Forces
37
324 Particular Solutions
42
325 Algorithm for the Mediancentre
43
326 Nonunit Weights
45
33 Linear Curvilinear and Regional Constraints
46
332 Regional Constraints
49
333 Hypothesis Tests
50
34 Three Further Generalisations of the Mechanical Model
52
343 Transportation Network with Multiple Nodes
53
35 Mechanical Models in Sums of Areas
55
352 Ojas Spatial Median
58
353 Functional Approximation
59
References
60
Method of Least Squared Deviations
63
412 Balance of Forces in the OneDimensional Case
65
413 Mechanical Model in the TwoDimensional Case
66
414 Balance of Forces in the TwoDimensional Case
67
415 The Modulus of the Parameter Estimates
68
42 Linear Curvilinear and Regional Constraints
70
43 Simple Linear Regression
76
432 Mechanical Model and Balance of Forces in the Conventional Case
78
433 Mechanical Model in the Orthogonal Case
80
434 Balance of Forces in the Orthogonal Case
82
44 Moduli of Parameter Estimates
83
442 Oblique Transformation
84
443 Relative Equilibrium
86
452 Adding Observations
87
454 Ridge Regression
88
46 Linear Constraints and Hypothesis Tests
89
462 Hypothesis Test
90
463 Slope Test
92
47 Inertial Models
93
48 Matrix Representation
94
Method of Least Absolute Deviation
97
512 Mechanical Models for HigherDimensional Medians
100
52 Simple Linear Regression
101
522 Orthogonal Line Fitting Problem
106
523 Linear Constraints
107
524 Hypothesis Tests
108
53 Parameter Space Representation and Duality
110
532 Perpendicular Models in Parameter Space
111
533 Mechanical Models for Directional Data
114
542 Edgeworths Double Median Method
115
543 Edgeworths Iterative Method
120
545 Linear Programming Formulation
121
546 Applications to Voting in Committees
122
References
140
Minimax Absolute Deviation Method
143
612 Mechanical Model Based on Strings and Blocks
145
613 Parameter Space Representation
146
62 Simple Linear Regression
147
622 Influential Observations
150
623 Geometrical Representation in OneDimensional Parameter Space
151
624 Geometrical Representation in TwoDimensional Parameter Space
152
625 Simplified Geometrical Solution in TwoDimensional Parameter Space
156
626 Linear Programming Formulation
158
63 Geometrical Representation of the Harmonic Model
159
References
160
Method of Least Median of Squared Deviations
163
712 One and TwoDimensional Means
164
713 Parameter Space Representation
167
72 Simple Linear Regression
169
722 Influential Observations
170
724 An Alternative Geometrical Solution
172
726 Nonlinear Programming
174
73 Generalisation to Minimum Volume Median Ellipsoids and Ellipsoidal Cylinders
175
Mechanical Models of Metric Graphs
177
82 Metric Graphs
178
83 Pairwise Preference Orderings
181
84 Transitive Preference Orderings
183
85 ThreeDimensional Models
186
86 FourDimensional Models
188
87 Further Generalisations
189
References
190
Categorical Data Analysis
191
92 Hydrostatic Models 921 Static Model
194
922 Balance of Forces
196
923 Potential Energy Analysis
197
924 Hypothesis Test of Independence
198
93 Gas Pressure Models
199
932 Potential Energy Analysis
201
933 Hypothesis Tests of Independence
202
934 Analogy with the Chemical Balance Model
203
94 TwoWay Tables
204
95 Hydrostatic Models in Economics
206
Method of Averages and Curve Fitting by Splines
207
1012 Multidimensional Means
208
102 The Method of Averages
209
103 Smoothing by Linear and Cubic Splines
212
1032 Cubic Splines
215
104 Multidimensional Medians
217
References
218
Multivariate Generalisations of the Method of Least Squares
219
1122 Fitting Elliptical Contours
223
1123 Fitting a Circle
224
113 Procrustes Rotation
226
1132 Fitting Spherical Data
227
114 Multidimensional Scaling
229
115 Concluding Remarks
231
List of Figures
233
List of Tables
241
Name Index
243
Subject Index
245
Copyright

Other editions - View all

Visualizing Statistical Models And Concepts
R.W. Farebrother,Michael Schyns
Limited preview - 2002
Visualizing Statistical Models And Concepts
R.W. Farebrother,Michael Schyns
Limited preview - 2002
Visualizing Statistical Models and Concepts
R. W. Farebrother,Michael Schyns
No preview available - 2020
View all »

Common terms and phrases

absolute deviations problem Analysis arbitrary line arbitrary point arithmetic mean axis ba-plane balance of forces bar chart Cartesian plane centroid choose values circle clockwise context convex hull corresponding defined determine direction discussed edited equation equilibrium estimates Farebrother fitted line force acting generalised geometrical given points hexagonal horizontal plane hyperplane identify illustrated in Figure ith observation L1-norm least sum likelihood function line fitting problem line of balance line segments linear programming mechanical model median of squares mediancentre method middlemost minimax minimise the sum modulus Observation space representation obtained optimal value optimality criterion optimality function orthogonal orthogonal least squares P₁ pair of parallel parameter space pass position potential energy function Procrustes q-dimensional regression represented rotate Section set of observations slope squared deviations Statistical straight line strings Subsection sum of absolute tion two-dimensional Typical contour unconstrained unit weights variables vector vertical weighted least squares xy-plane y-axis y₁ zero

References to this book

About the author (2002)

R. W. Farebrother and Michael Schyns

Bibliographic information

TitleVisualizing Statistical Models And Concepts
Volume 166 of Statistics, textbooks and monographs
AuthorsR.W. Farebrother, Michael Schyns
PublisherCRC Press, 2002
ISBN0203908996, 9780203908990
Length280 pages
  
Export CitationBiBTeX EndNote RefMan