Nonparametric Functional Data Analysis: Theory and Practice (Google eBook)

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Springer Science & Business Media, Nov 22, 2006 - Business & Economics - 278 pages
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Modern apparatuses allow us to collect samples of functional data, mainly curves but also images. On the other hand, nonparametric statistics produces useful tools for standard data exploration. This book links these two fields of modern statistics by explaining how functional data can be studied through parameter-free statistical ideas. This book starts from theoretical foundations including functional nonparametric modeling, description of the mathematical framework, construction of the statistical methods, and statements of their asymptotic behaviors. It proceeds to computational issues including R and S-PLUS routines. Several functional datasets in chemometrics, econometrics, and pattern recognition are used to emphasize the wide scope of nonparametric functional data analysis in applied sciences. The companion Web site includes R and S-PLUS routines, command lines for reproducing examples presented in the book, and the functional datasets. Rather than set application against theory, this book is really an interface of these two features of statistics. A special effort has been made in writing this book to accommodate several levels of reading. The computational aspects are oriented toward practitioners whereas open problems emerging from this new field of statistics will attract Ph.D. students and academic researchers. Finally, this book is also accessible to graduate students starting in the area of functional statistics.
  

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Contents

922 Median
129
923 Mode
130
93 Measuring Heterogeneity
131
941 How to Build a Partitioning Heterogeneity Index?
132
943 Classification Algorithm
134
RS+ Routines
135
96 Theoretical Advances on the Functional Mode
137
961 Hypotheses on the Distribution
138

22 Speech Recognition Data
15
23 Electricity Consumption Data
17
232 The Forecasting Problematic
18
What is a WellAdapted Space for Functional Data?
21
32 SemiMetrics as Explanatory Tool
22
34 SemiMetrics in Practice
28
a New Way to Build SemiMetrics
30
343 Semimetrics Based on Derivatives
32
35 R and S+ Implementations
33
a WellAdapted Framework
35
Local Weighting of Functional Variables
36
411 Real Case
38
412 Multivariate Case
39
413 Functional Case
41
42 Local Weighting and Small Ball Probabilities
42
43 A Few Basic Theoretical Advances
43
Nonparametric Prediction from Functional Data
45
Functional Nonparametric Prediction Methodologies
48
52 Various Approaches to the Prediction Problem
50
53 Functional Nonparametric Modelling for Prediction
52
54 Kernel Estimators
55
Some Selected Asymptotics
61
62 Almost Complete Convergence
62
622 Conditional Median Estimation
66
623 Conditional Mode Estimation
70
624 Conditional Quantile Estimation
76
63 Rates of Convergence
79
632 Conditional Median Estimation
80
633 Conditional Mode Estimation
87
634 Conditional Quantile Estimation
90
635 Complements on Conditional Distribution Estimation
92
64 Discussion Bibliography and Open Problems
93
642 Going Back to Finite Dimensional Setting
94
643 Some Tracks for the Future
95
Computational Issues
99
711 Prediction via Regression
100
712 Prediction via Functional Conditional Quantiles
103
713 Prediction via Functional Conditional Mode
104
72 Predicting Fat Content From Spectrometric Curves
105
722 Functional Prediction in Action
106
73 Conclusion
107
Nonparametric Classification of Functional Data
110
Functional Nonparametric Supervised Classification
113
82 Method
114
83 Computational Issues
116
832 Automatic Selection of the kNN Parameter
117
RS+ Routines
118
84 Functional Nonparametric Discrimination in Action
119
842 Chemometric Data
122
86 Additional Bibliography and Comments
123
Functional Nonparametric Unsupervised Classification
125
92 Centrality Notions for Functional Variables
127
97 The Kernel Functional Mode Estimator
140
aco Convergence
141
aco Convergence
144
974 Comments and Bibliography
145
98 Conclusions
146
Nonparametric Methods for Dependent Functional Data
150
Mixing Nonparametric and Functional Statistics
153
a Short Overview
154
Some General Considerations
155
104 Mixing and Nonparametric Functional Statistics
156
Some Selected Asymptotics
158
112 Prediction with Kernel Regression Estimator
160
1122 Complete Convergence Properties
161
1123 An Application to the Geometrically Mixing Case
163
1124 An Application to the Arithmetically Mixing Case
166
113 Prediction with Functional Conditional Quantiles
167
1132 Complete Convergence Properties
168
1133 Application to the Geometrically Mixing Case
171
1134 Application to the Arithmetically Mixing Case
175
114 Prediction with Conditional Mode
177
1142 Complete Convergence Properties
178
1143 Application to the Geometrically Mixing Case
183
1144 Application to the Arithmetically Mixing Case
184
115 Complements on Conditional Distribution Estimation
185
1152 Rates of Convergence
187
116 Nonparametric Discrimination of Dependent Curves
189
1162 Complete Convergence Properties
190
117 Discussion
192
1173 Some Open Problems
193
Application to Continuous Time Processes Prediction
195
122 Functional Approach to Time Series Prediction
197
123 Computational Issues
198
1242 The Forecasted Electrical Consumption
200
1243 Conclusions
201
Conclusions
202
Small Ball Probabilities and Semimetrics
203
132 The Role of Small Ball Probabilities
206
133 Some Special Infinite Dimensional Processes
207
1332 Exponentialtype Processes
209
1333 Links with Semimetric Choice
212
134 Back to the Onedimensional Setting
214
135 Back to the Multi but Finite Dimensional Setting
219
a Crucial Parameter
223
Some Perspectives
224
Some Probabilistic Tools
225
A1 Almost Complete Convergence
228
A2 Exponential Inequalities for Independent rrv
233
A3 Inequalities for Mixing rrv
235
References
239
Index
255
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