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E-raamat: Analysis of Variance for Functional Data

(National University of Singapore)
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Analysis of Variance for Functional DataSuurem pilt
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"Preface Functional data analysis has been a popular statistical research topic for the last three decades. Functional data are often obtained via observing a number of subjects over time, space or other continua densely. They are frequently collected from various research areas, including audiology, biology, children's growth studies, ergonomics, environmentology, me- teorology, and women's health studies among others in the form of curves, surfaces, or other complex objects. Statistical inference for functional data generally refers to estimation and hypothesis testing about functional data. There are at least two monographs available in the literature which are devoted in estimation and classi

Despite research interest in functional data analysis in the last three decades, few books are available on the subject. Filling this gap, Analysis of Variance for Functional Data presents up-to-date hypothesis testing methods for functional data analysis. The book covers the reconstruction of functional observations, functional ANOVA, functional linear models with functional responses, ill-conditioned functional linear models, diagnostics of functional observations, heteroscedastic ANOVA for functional data, and testing equality of covariance functions. Although the methodologies presented are designed for curve data, they can be extended to surface data.

Useful for statistical researchers and practitioners analyzing functional data, this self-contained book gives both a theoretical and applied treatment of functional data analysis supported by easy-to-use MATLAB® code. The author provides a number of simple methods for functional hypothesis testing. He discusses pointwise, L2-norm-based, F-type, and bootstrap tests.

Assuming only basic knowledge of statistics, calculus, and matrix algebra, the book explains the key ideas at a relatively low technical level using real data examples. Each chapter also includes bibliographical notes and exercises. Real functional data sets from the text and MATLAB codes for analyzing the data examples are available for download from the author’s website.

Arvustused

" a focused presentation of functional ANOVA and linear function-on-scalar regression problems using the smooth first approach to estimation and inference. I would recommend this book to anyone interested in theoretical developments and hypothesis testing in this commonly encountered class of problems." Jeff Goldsmith, Journal of the American Statistical Association, March 2014

List of Figures
xv
List of Tables
xix
Preface xxi
1 Introduction
1(18)
1.1 Functional Data
1(1)
1.2 Motivating Functional Data
1(15)
1.2.1 Progesterone Data
2(1)
1.2.2 Berkeley Growth Curve Data
3(3)
1.2.3 Nitrogen Oxide Emission Level Data
6(1)
1.2.4 Canadian Temperature Data
6(3)
1.2.5 Audible Noise Data
9(2)
1.2.6 Left-Cingulum Data
11(1)
1.2.7 Orthosis Data
11(2)
1.2.8 Ergonomics Data
13(3)
1.3 Why Is Functional Data Analysis Needed?
16(1)
1.4 Overview of the Book
17(1)
1.5 Implementation of Methodologies
17(1)
1.6 Options for Reading This Book
18(1)
1.7 Bibliographical Notes
18(1)
2 Nonparametric Smoothers for a Single Curve
19(28)
2.1 Introduction
19(1)
2.2 Local Polynomial Kernel Smoothing
20(8)
2.2.1 Construction of an LPK Smoother
20(2)
2.2.2 Two Special LPK Smoothers
22(2)
2.2.3 Selecting a Good Bandwidth
24(2)
2.2.4 Robust LPK Smoothing
26(2)
2.3 Regression Splines
28(7)
2.3.1 Truncated Power Basis
29(1)
2.3.2 Regression Spline Smoother
30(1)
2.3.3 Knot Locating and Knot Number Selection
30(4)
2.3.4 Robust Regression Splines
34(1)
2.4 Smoothing Splines
35(5)
2.4.1 Smoothing Spline Smoothers
35(1)
2.4.2 Cubic Smoothing Splines
36(1)
2.4.3 Smoothing Parameter Selection
37(3)
2.5 P-Splines
40(4)
2.5.1 P-Spline Smoothers
40(1)
2.5.2 Smoothing Parameter Selection
41(3)
2.6 Concluding Remarks and Bibliographical Notes
44(3)
3 Reconstruction of Functional Data
47(36)
3.1 Introduction
47(2)
3.2 Reconstruction Methods
49(12)
3.2.1 Individual Function Estimators
49(1)
3.2.2 Smoothing Parameter Selection
50(1)
3.2.3 LPK Reconstruction
50(4)
3.2.4 Regression Spline Reconstruction
54(3)
3.2.5 Smoothing Spline Reconstruction
57(2)
3.2.6 P-Spline Reconstruction
59(2)
3.3 Accuracy of LPK Reconstructions
61(7)
3.3.1 Mean and Covariance Function Estimation
63(2)
3.3.2 Noise Variance Function Estimation
65(1)
3.3.3 Effect of LPK Smoothing
66(1)
3.3.4 A Simulation Study
66(2)
3.4 Accuracy of LPK Reconstruction in FLMs
68(6)
3.4.1 Coefficient Function Estimation
69(1)
3.4.2 Significance Tests of Covariate Effects
70(2)
3.4.3 A Real Data Example
72(2)
3.5 Technical Proofs
74(6)
3.6 Concluding Remarks and Bibliographical Notes
80(1)
3.7 Exercises
81(2)
4 Stochastic Processes
83(46)
4.1 Introduction
83(1)
4.2 Stochastic Processes
83(9)
4.2.1 Gaussian Processes
85(1)
4.2.2 Wishart Processes
86(2)
4.2.3 Linear Forms of Stochastic Processes
88(1)
4.2.4 Quadratic Forms of Stochastic Processes
88(3)
4.2.5 Central Limit Theorems for Stochastic Processes
91(1)
4.3 Type Mixtures
92(8)
4.3.1 Cumulants
93(1)
4.3.2 Distribution Approximation
94(6)
4.4 F-Type Mixtures
100(7)
4.4.1 Distribution Approximation
101(6)
4.5 One-Sample Problem for Functional Data
107(12)
4.5.1 Pointwise Tests
109(2)
4.5.2 L2-Norm-Based Test
111(3)
4.5.3 F-Type Test
114(1)
4.5.4 Bootstrap Test
115(1)
4.5.5 Numerical Implementation
116(3)
4.5.6 Effect of Resolution Number
119(1)
4.6 Technical Proofs
119(7)
4.7 Concluding Remarks and Bibliographical Notes
126(1)
4.8 Exercises
127(2)
5 ANOVA for Functional Data
129(68)
5.1 Introduction
129(1)
5.2 Two-Sample Problem
129(13)
5.2.1 Pivotal Test Function
133(1)
5.2.2 Methods for Two-Sample Problems
134(8)
5.3 One-Way ANOVA
142(22)
5.3.1 Estimation of Group Mean and Covariance Functions
145(3)
5.3.2 Main-Effect Test
148(12)
5.3.3 Tests of Linear Hypotheses
160(4)
5.4 Two-Way ANOVA
164(26)
5.4.1 Estimation of Cell Mean and Covariance Functions
166(3)
5.4.2 Main and Interaction Effect Functions
169(2)
5.4.3 Tests of Linear Hypotheses
171(7)
5.4.4 Balanced Two-Way ANOVA with Interaction
178(6)
5.4.5 Balanced Two-Way ANOVA without Interaction
184(6)
5.5 Technical Proofs
190(5)
5.6 Concluding Remarks and Bibliographical Notes
195(1)
5.7 Exercises
196(1)
6 Linear Models with Functional Responses
197(44)
6.1 Introduction
197(1)
6.2 Linear Models with Time-Independent Covariates
197(24)
6.2.1 Coefficient Function Estimation
200(2)
6.2.2 Properties of the Estimators
202(2)
6.2.3 Multiple Correlation Coefficient
204(1)
6.2.4 Comparing Two Nested FLMs
205(6)
6.2.5 Significance of All the Non-Intercept Coefficient Functions
211(1)
6.2.6 Significance of a Single Coefficient Function
212(2)
6.2.7 Tests of Linear Hypotheses
214(4)
6.2.8 Variable Selection
218(3)
6.3 Linear Models with Time-Dependent Covariates
221(11)
6.3.1 Estimation of the Coefficient Functions
221(1)
6.3.2 Compare Two Nested FLMs
222(6)
6.3.3 Tests of Linear Hypotheses
228(4)
6.4 Technical Proofs
232(4)
6.5 Concluding Remarks and Bibliographical Notes
236(2)
6.6 Exercises
238(3)
7 Ill-Conditioned Functional Linear Models
241(32)
7.1 Introduction
241(4)
7.2 Generalized Inverse Method
245(14)
7.2.1 Estimability of Regression Coefficient Functions
245(2)
7.2.2 Methods for Finding Estimable Linear Functions
247(5)
7.2.3 Estimation of Estimable Linear Functions
252(1)
7.2.4 Tests of Testable Linear Hypotheses
253(6)
7.3 Reparameterization Method
259(2)
7.3.1 The Methodology
259(1)
7.3.2 Determining the Reparameterization Matrices
259(1)
7.3.3 Invariance of the Reparameterization
260(1)
7.3.4 Tests of Testable Linear Hypotheses
261(1)
7.4 Side-Condition Method
261(5)
7.4.1 The Methodology
261(1)
7.4.2 Methods for Specifying the Side-Conditions
262(1)
7.4.3 Invariance of the Side-Condition Method
263(1)
7.4.4 Tests of Testable Linear Hypotheses
264(2)
7.5 Technical Proofs
266(5)
7.6 Concluding Remarks and Bibliographical Notes
271(1)
7.7 Exercises
271(2)
8 Diagnostics of Functional Observations
273(34)
8.1 Introduction
273(3)
8.2 Residual Functions
276(3)
8.2.1 Raw Residual Functions
276(1)
8.2.2 Standardized Residual Functions
277(1)
8.2.3 Jackknife Residual Functions
277(2)
8.3 Functional Outlier Detection
279(12)
8.3.1 Standardized Residual-Based Method
279(4)
8.3.2 Jackknife Residual-Based Method
283(2)
8.3.3 Functional Depth-Based Method
285(6)
8.4 Influential Case Detection
291(1)
8.5 Robust Estimation of Coefficient Functions
292(1)
8.6 Outlier Detection for a Sample of Functions
293(5)
8.6.1 Residual Functions
293(1)
8.6.2 Functional Outlier Detection
294(4)
8.7 Technical Proofs
298(1)
8.8 Concluding Remarks and Bibliographical Notes
298(1)
8.9 Exercises
299(8)
9 Heteroscedastic ANOVA for Functional Data
307(44)
9.1 Introduction
307(1)
9.2 Two-Sample Behrens-Fisher Problems
308(10)
9.2.1 Estimation of Mean and Covariance Functions
310(2)
9.2.2 Testing Methods
312(6)
9.3 Heteroscedastic One-Way ANOVA
318(12)
9.3.1 Estimation of Group Mean and Covariance Functions
320(2)
9.3.2 Heteroscedastic Main-Effect Test
322(7)
9.3.3 Tests of Linear Hypotheses under Heteroscedasticity
329(1)
9.4 Heteroscedastic Two-Way ANOVA
330(14)
9.4.1 Estimation of Cell Mean and Covariance Functions
335(2)
9.4.2 Tests of Linear Hypotheses under Heteroscedasticity
337(7)
9.5 Technical Proofs
344(4)
9.6 Concluding Remarks and Bibliographical Notes
348(1)
9.7 Exercises
348(3)
10 Test of Equality of Covariance Functions
351(18)
10.1 Introduction
351(1)
10.2 Two-Sample Case
351(5)
10.2.1 Pivotal Test Function
352(2)
10.2.2 Testing Methods
354(2)
10.3 Multi-Sample Case
356(8)
10.3.1 Estimation of Group Mean and Covariance Functions
358(2)
10.3.2 Testing Methods
360(4)
10.4 Technical Proofs
364(2)
10.5 Concluding Remarks and Bibliographical Notes
366(1)
10.6 Exercises
366(3)
Bibliography 369(12)
Index 381
Jin-Ting Zhang is an associate professor in the Department of Statistics and Applied Probability at the National University of Singapore. He has published extensively and has served on the editorial boards of several international statistical journals. He is the coauthor of Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effect Modelling Approaches and the coeditor of Advances in Statistics: Proceedings of the Conference in Honor of Professor Zhidong Bai on His 65th Birthday.
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