DeWayne R. Derryberry
Basic Data Analysis for Time Series with R
DeWayne R. Derryberry
Basic Data Analysis for Time Series with R
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Written at a readily accessible level, Basic Data Analysis for Time Series with R emphasizes the mathematical importance of collaborative analysis of data used to collect increments of time or space. Balancing a theoretical and practical approach to analyzing data within the context of serial correlation, the book presents a coherent and systematic regression-based approach to model selection. The book illustrates these principles of model selection and model building through the use of information criteria, cross validation, hypothesis tests, and confidence intervals.
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Written at a readily accessible level, Basic Data Analysis for Time Series with R emphasizes the mathematical importance of collaborative analysis of data used to collect increments of time or space. Balancing a theoretical and practical approach to analyzing data within the context of serial correlation, the book presents a coherent and systematic regression-based approach to model selection. The book illustrates these principles of model selection and model building through the use of information criteria, cross validation, hypothesis tests, and confidence intervals.
Focusing on frequency- and time-domain and trigonometric regression as the primary themes, the book also includes modern topical coverage on Fourier series and Akaike s Information Criterion (AIC). In addition, Basic Data Analysis for Time Series with R also features:
Real-world examples to provide readers with practical hands-on experience
Multiple R software subroutines employed with graphical displays
Numerous exercise sets intended to support readers understanding of the core concepts
Specific chapters devoted to the analysis of the Wolf sunspot number data and the Vostok ice core data sets
Focusing on frequency- and time-domain and trigonometric regression as the primary themes, the book also includes modern topical coverage on Fourier series and Akaike s Information Criterion (AIC). In addition, Basic Data Analysis for Time Series with R also features:
Real-world examples to provide readers with practical hands-on experience
Multiple R software subroutines employed with graphical displays
Numerous exercise sets intended to support readers understanding of the core concepts
Specific chapters devoted to the analysis of the Wolf sunspot number data and the Vostok ice core data sets
Produktdetails
- Produktdetails
- Verlag: Wiley & Sons
- 1. Auflage
- Seitenzahl: 320
- Erscheinungstermin: 8. Juli 2014
- Englisch
- Abmessung: 241mm x 159mm x 27mm
- Gewicht: 2821g
- ISBN-13: 9781118422540
- ISBN-10: 1118422546
- Artikelnr.: 40558791
- Verlag: Wiley & Sons
- 1. Auflage
- Seitenzahl: 320
- Erscheinungstermin: 8. Juli 2014
- Englisch
- Abmessung: 241mm x 159mm x 27mm
- Gewicht: 2821g
- ISBN-13: 9781118422540
- ISBN-10: 1118422546
- Artikelnr.: 40558791
DeWayne R. Derryberry, PhD, is Associate Professor in the Department of Mathematics and Statistics at Idaho State University. Dr. Derryberry has published more than a dozen journal articles and his research interests include meta-analysis, discriminant analysis with messy data, time series analysis of the relationship between several cancers, and geographically-weighted regression.
PREFACE xv ACKNOWLEDGMENTS xvii PART I BASIC CORRELATION STRUCTURES 1 RBasics 3 1.1 Getting Started
3 1.2 Special R Conventions
5 1.3 Common Structures
5 1.4 Common Functions
6 1.5 Time Series Functions
6 1.6 Importing Data
7 Exercises
7 2 Review of Regression and More About R 8 2.1 Goals of this Chapter
8 2.2 The Simple(ST) Regression Model
8 2.3 Simulating the Data from a Model and Estimating the Model Parameters in R
9 2.4 Basic Inference for the Model
12 2.5 Residuals Analysis--What Can Go Wrong...
13 2.6 Matrix Manipulation in R
15 Exercises
16 3 The Modeling Approach Taken in this Book and Some Examples of Typical Serially Correlated Data 18 3.1 Signal and Noise
18 3.2 Time Series Data
19 3.3 Simple Regression in the Framework
20 3.4 Real Data and Simulated Data
20 3.5 The Diversity of Time Series Data
21 3.6 Getting Data Into R
24 Exercises
26 4 Some Comments on Assumptions 28 4.1 Introduction
28 4.2 The Normality Assumption
29 4.3 Equal Variance
31 4.4 Independence
31 4.5 Power of Logarithmic Transformations Illustrated
32 4.6 Summary
34 Exercises
34 5 The Autocorrelation Function And AR(1)
AR(2) Models 35 5.1 Standard Models--What are the Alternatives to White Noise?
35 5.2 Autocovariance and Autocorrelation
36 5.3 The acf() Function in R
37 5.4 The First Alternative to White Noise: Autoregressive Errors--AR(1)
AR(2)
40 Exercises
49 6 The Moving Average Models MA(1) And MA(2) 51 6.1 The Moving Average Model
51 6.2 The Autocorrelation for MA(1) Models
51 6.3 A Duality Between MA(l) And AR(m) Models
52 6.4 The Autocorrelation for MA(2) Models
52 6.5 Simulated Examples of the MA(1) Model
52 6.6 Simulated Examples of the MA(2) Model
54 6.7 AR(m) and MA(l) model acf() Plots
54 Exercises
57 PART II ANALYSIS OF PERIODIC DATA AND MODEL SELECTION 7 Review of Transcendental Functions and Complex Numbers 61 7.1 Background
61 7.2 Complex Arithmetic
62 7.3 Some Important Series
63 7.4 Useful Facts About Periodic Transcendental Functions
64 Exercises
64 8 The Power Spectrum and the Periodogram 65 8.1 Introduction
65 8.2 A Definition and a Simplified Form for p(f )
66 8.3 Inverting p(f ) to Recover the Ck Values
66 8.4 The Power Spectrum for Some Familiar Models
68 8.5 The Periodogram
a Closer Look
72 8.6 The Function spec.pgram() in R
75 Exercises
77 9 Smoothers
The Bias-Variance Tradeoff
and the Smoothed Periodogram 79 9.1 Why is Smoothing Required?
79 9.2 Smoothing
Bias
and Variance
79 9.3 Smoothers Used in R
80 9.4 Smoothing the Periodogram for a Series With a Known and Unknown Period
85 9.5 Summary
87 Exercises
87 10 A Regression Model for Periodic Data 89 10.1 The Model
89 10.2 An Example: The NYC Temperature Data
91 10.3 Complications 1: CO2 Data
93 10.4 Complications 2: Sunspot Numbers
94 10.5 Complications 3: Accidental Deaths
96 10.6 Summary
96 Exercises
96 11 Model Selection and Cross-Validation 98 11.1 Background
98 11.2 Hypothesis Tests in Simple Regression
99 11.3 A More General Setting for Likelihood Ratio Tests
101 11.4 A Subtlety Different Situation
104 11.5 Information Criteria
106 11.6 Cross-validation (Data Splitting): NYC Temperatures
108 11.7 Summary
112 Exercises
113 12 Fitting Fourier series 115 12.1 Introduction: More Complex Periodic Models
115 12.2 More Complex Periodic Behavior: Accidental Deaths
116 12.3 The Boise River Flow data
121 12.4 Where Do We Go from Here?
124 Exercises
124 13 Adjusting for AR(1) Correlation in Complex Models 125 13.1 Introduction
125 13.2 The Two-Sample t-Test--UNCUT and Patch-Cut Forest
125 13.3 The Second Sleuth Case--Global Warming
A Simple Regression
132 13.4 The Semmelweis Intervention
138 13.5 The NYC Temperatures (Adjusted)
142 13.6 The Boise River Flow Data: Model Selection With Filtering
147 13.7 Implications of AR(1) Adjustments and the "Skip" Method
151 13.8 Summary
152 Exercises
153 PART III COMPLEX TEMPORAL STRUCTURES 14 The Backshift Operator
the Impulse Response Function
and General ARMA Models 159 14.1 The General ARMA Model
159 14.2 The Backshift (Shift
Lag) Operator
161 14.3 The Impulse Response Operator--Intuition
164 14.4 Impulse Response Operator
g(B)--Computation
165 14.5 Interpretation and Utility of the Impulse Response Function
167 Exercises
167 15 The Yule-Walker Equations and the Partial Autocorrelation Function 169 15.1 Background
169 15.2 Autocovariance of an ARMA(m
l) Model
169 15.3 AR(m) and the Yule-Walker Equations
170 15.4 The Partial Autocorrelation Plot
174 15.5 The Spectrum For Arma Processes
175 15.6 Summary
177 Exercises
178 16 Modeling Philosophy and Complete Examples 180 16.1 Modeling Overview
180 16.2 A Complex Periodic Model--Monthly River Flows
Furnas 1931-1978
185 16.3 A Modeling Example--Trend and Periodicity: CO2 Levels at Mauna Lau
193 16.4 Modeling Periodicity with a Possible Intervention--Two Examples
198 16.5 Periodic Models: Monthly
Weekly
and Daily Averages
205 16.6 Summary
207 Exercises
207 PART IV SOME DETAILED AND COMPLETE EXAMPLES 17 Wolf's Sunspot Number Data 213 17.1 Background
213 17.2 Unknown Period ==> Nonlinear Model
214 17.3 The Function nls() in R
214 17.4 Determining the Period
216 17.5 Instability in the Mean
Amplitude
and Period
217 17.6 Data Splitting for Prediction
220 17.7 Summary
226 Exercises
226 18 An Analysis of Some Prostate and Breast Cancer Data 228 18.1 Background
228 18.2 The First Data Set
229 18.3 The Second Data Set
232 Exercises
243 19 Christopher Tennant/Ben Crosby Watershed Data 245 19.1 Background and Question
245 19.2 Looking at the Data and Fitting Fourier Series
246 19.3 Averaging Data
248 19.4 Results
250 Exercises
250 20 Vostok Ice Core Data 251 20.1 Source of the Data
251 20.2 Background
252 20.3 Alignment
253 20.4 A Nä?ve Analysis
256 20.5 A Related Simulation
259 20.6 An AR(1) Model for Irregular Spacing
265 20.7 Summary
269 Exercises
270 Appendix A Using Datamarket 273 A.1 Overview
273 A.2 Loading a Time Series in Datamarket
277 A.3 Respecting Datamarket Licensing Agreements
280 Appendix B AIC is PRESS! 281 B.1 Introduction
281 B.2 PRESS
281 B.3 Connection to Akaike's Result
282 B.4 Normalization and R2
282 B.5 An example
283 B.6 Conclusion and Further Comments
283 Appendix C A 15-Minute Tutorial on Nonlinear Optimization 284 C.1 Introduction
284 C.2 Newton's Method for One-Dimensional Nonlinear Optimization
284 C.3 A Sequence of Directions
Step Sizes
and a Stopping Rule
285 C.4 What Could Go Wrong?
285 C.5 Generalizing the Optimization Problem
286 C.6 What Could Go Wrong--Revisited
286 C.7 What Can be Done?
287 REFERENCES 291 INDEX 293
3 1.2 Special R Conventions
5 1.3 Common Structures
5 1.4 Common Functions
6 1.5 Time Series Functions
6 1.6 Importing Data
7 Exercises
7 2 Review of Regression and More About R 8 2.1 Goals of this Chapter
8 2.2 The Simple(ST) Regression Model
8 2.3 Simulating the Data from a Model and Estimating the Model Parameters in R
9 2.4 Basic Inference for the Model
12 2.5 Residuals Analysis--What Can Go Wrong...
13 2.6 Matrix Manipulation in R
15 Exercises
16 3 The Modeling Approach Taken in this Book and Some Examples of Typical Serially Correlated Data 18 3.1 Signal and Noise
18 3.2 Time Series Data
19 3.3 Simple Regression in the Framework
20 3.4 Real Data and Simulated Data
20 3.5 The Diversity of Time Series Data
21 3.6 Getting Data Into R
24 Exercises
26 4 Some Comments on Assumptions 28 4.1 Introduction
28 4.2 The Normality Assumption
29 4.3 Equal Variance
31 4.4 Independence
31 4.5 Power of Logarithmic Transformations Illustrated
32 4.6 Summary
34 Exercises
34 5 The Autocorrelation Function And AR(1)
AR(2) Models 35 5.1 Standard Models--What are the Alternatives to White Noise?
35 5.2 Autocovariance and Autocorrelation
36 5.3 The acf() Function in R
37 5.4 The First Alternative to White Noise: Autoregressive Errors--AR(1)
AR(2)
40 Exercises
49 6 The Moving Average Models MA(1) And MA(2) 51 6.1 The Moving Average Model
51 6.2 The Autocorrelation for MA(1) Models
51 6.3 A Duality Between MA(l) And AR(m) Models
52 6.4 The Autocorrelation for MA(2) Models
52 6.5 Simulated Examples of the MA(1) Model
52 6.6 Simulated Examples of the MA(2) Model
54 6.7 AR(m) and MA(l) model acf() Plots
54 Exercises
57 PART II ANALYSIS OF PERIODIC DATA AND MODEL SELECTION 7 Review of Transcendental Functions and Complex Numbers 61 7.1 Background
61 7.2 Complex Arithmetic
62 7.3 Some Important Series
63 7.4 Useful Facts About Periodic Transcendental Functions
64 Exercises
64 8 The Power Spectrum and the Periodogram 65 8.1 Introduction
65 8.2 A Definition and a Simplified Form for p(f )
66 8.3 Inverting p(f ) to Recover the Ck Values
66 8.4 The Power Spectrum for Some Familiar Models
68 8.5 The Periodogram
a Closer Look
72 8.6 The Function spec.pgram() in R
75 Exercises
77 9 Smoothers
The Bias-Variance Tradeoff
and the Smoothed Periodogram 79 9.1 Why is Smoothing Required?
79 9.2 Smoothing
Bias
and Variance
79 9.3 Smoothers Used in R
80 9.4 Smoothing the Periodogram for a Series With a Known and Unknown Period
85 9.5 Summary
87 Exercises
87 10 A Regression Model for Periodic Data 89 10.1 The Model
89 10.2 An Example: The NYC Temperature Data
91 10.3 Complications 1: CO2 Data
93 10.4 Complications 2: Sunspot Numbers
94 10.5 Complications 3: Accidental Deaths
96 10.6 Summary
96 Exercises
96 11 Model Selection and Cross-Validation 98 11.1 Background
98 11.2 Hypothesis Tests in Simple Regression
99 11.3 A More General Setting for Likelihood Ratio Tests
101 11.4 A Subtlety Different Situation
104 11.5 Information Criteria
106 11.6 Cross-validation (Data Splitting): NYC Temperatures
108 11.7 Summary
112 Exercises
113 12 Fitting Fourier series 115 12.1 Introduction: More Complex Periodic Models
115 12.2 More Complex Periodic Behavior: Accidental Deaths
116 12.3 The Boise River Flow data
121 12.4 Where Do We Go from Here?
124 Exercises
124 13 Adjusting for AR(1) Correlation in Complex Models 125 13.1 Introduction
125 13.2 The Two-Sample t-Test--UNCUT and Patch-Cut Forest
125 13.3 The Second Sleuth Case--Global Warming
A Simple Regression
132 13.4 The Semmelweis Intervention
138 13.5 The NYC Temperatures (Adjusted)
142 13.6 The Boise River Flow Data: Model Selection With Filtering
147 13.7 Implications of AR(1) Adjustments and the "Skip" Method
151 13.8 Summary
152 Exercises
153 PART III COMPLEX TEMPORAL STRUCTURES 14 The Backshift Operator
the Impulse Response Function
and General ARMA Models 159 14.1 The General ARMA Model
159 14.2 The Backshift (Shift
Lag) Operator
161 14.3 The Impulse Response Operator--Intuition
164 14.4 Impulse Response Operator
g(B)--Computation
165 14.5 Interpretation and Utility of the Impulse Response Function
167 Exercises
167 15 The Yule-Walker Equations and the Partial Autocorrelation Function 169 15.1 Background
169 15.2 Autocovariance of an ARMA(m
l) Model
169 15.3 AR(m) and the Yule-Walker Equations
170 15.4 The Partial Autocorrelation Plot
174 15.5 The Spectrum For Arma Processes
175 15.6 Summary
177 Exercises
178 16 Modeling Philosophy and Complete Examples 180 16.1 Modeling Overview
180 16.2 A Complex Periodic Model--Monthly River Flows
Furnas 1931-1978
185 16.3 A Modeling Example--Trend and Periodicity: CO2 Levels at Mauna Lau
193 16.4 Modeling Periodicity with a Possible Intervention--Two Examples
198 16.5 Periodic Models: Monthly
Weekly
and Daily Averages
205 16.6 Summary
207 Exercises
207 PART IV SOME DETAILED AND COMPLETE EXAMPLES 17 Wolf's Sunspot Number Data 213 17.1 Background
213 17.2 Unknown Period ==> Nonlinear Model
214 17.3 The Function nls() in R
214 17.4 Determining the Period
216 17.5 Instability in the Mean
Amplitude
and Period
217 17.6 Data Splitting for Prediction
220 17.7 Summary
226 Exercises
226 18 An Analysis of Some Prostate and Breast Cancer Data 228 18.1 Background
228 18.2 The First Data Set
229 18.3 The Second Data Set
232 Exercises
243 19 Christopher Tennant/Ben Crosby Watershed Data 245 19.1 Background and Question
245 19.2 Looking at the Data and Fitting Fourier Series
246 19.3 Averaging Data
248 19.4 Results
250 Exercises
250 20 Vostok Ice Core Data 251 20.1 Source of the Data
251 20.2 Background
252 20.3 Alignment
253 20.4 A Nä?ve Analysis
256 20.5 A Related Simulation
259 20.6 An AR(1) Model for Irregular Spacing
265 20.7 Summary
269 Exercises
270 Appendix A Using Datamarket 273 A.1 Overview
273 A.2 Loading a Time Series in Datamarket
277 A.3 Respecting Datamarket Licensing Agreements
280 Appendix B AIC is PRESS! 281 B.1 Introduction
281 B.2 PRESS
281 B.3 Connection to Akaike's Result
282 B.4 Normalization and R2
282 B.5 An example
283 B.6 Conclusion and Further Comments
283 Appendix C A 15-Minute Tutorial on Nonlinear Optimization 284 C.1 Introduction
284 C.2 Newton's Method for One-Dimensional Nonlinear Optimization
284 C.3 A Sequence of Directions
Step Sizes
and a Stopping Rule
285 C.4 What Could Go Wrong?
285 C.5 Generalizing the Optimization Problem
286 C.6 What Could Go Wrong--Revisited
286 C.7 What Can be Done?
287 REFERENCES 291 INDEX 293
PREFACE xv ACKNOWLEDGMENTS xvii PART I BASIC CORRELATION STRUCTURES 1 RBasics 3 1.1 Getting Started
3 1.2 Special R Conventions
5 1.3 Common Structures
5 1.4 Common Functions
6 1.5 Time Series Functions
6 1.6 Importing Data
7 Exercises
7 2 Review of Regression and More About R 8 2.1 Goals of this Chapter
8 2.2 The Simple(ST) Regression Model
8 2.3 Simulating the Data from a Model and Estimating the Model Parameters in R
9 2.4 Basic Inference for the Model
12 2.5 Residuals Analysis--What Can Go Wrong...
13 2.6 Matrix Manipulation in R
15 Exercises
16 3 The Modeling Approach Taken in this Book and Some Examples of Typical Serially Correlated Data 18 3.1 Signal and Noise
18 3.2 Time Series Data
19 3.3 Simple Regression in the Framework
20 3.4 Real Data and Simulated Data
20 3.5 The Diversity of Time Series Data
21 3.6 Getting Data Into R
24 Exercises
26 4 Some Comments on Assumptions 28 4.1 Introduction
28 4.2 The Normality Assumption
29 4.3 Equal Variance
31 4.4 Independence
31 4.5 Power of Logarithmic Transformations Illustrated
32 4.6 Summary
34 Exercises
34 5 The Autocorrelation Function And AR(1)
AR(2) Models 35 5.1 Standard Models--What are the Alternatives to White Noise?
35 5.2 Autocovariance and Autocorrelation
36 5.3 The acf() Function in R
37 5.4 The First Alternative to White Noise: Autoregressive Errors--AR(1)
AR(2)
40 Exercises
49 6 The Moving Average Models MA(1) And MA(2) 51 6.1 The Moving Average Model
51 6.2 The Autocorrelation for MA(1) Models
51 6.3 A Duality Between MA(l) And AR(m) Models
52 6.4 The Autocorrelation for MA(2) Models
52 6.5 Simulated Examples of the MA(1) Model
52 6.6 Simulated Examples of the MA(2) Model
54 6.7 AR(m) and MA(l) model acf() Plots
54 Exercises
57 PART II ANALYSIS OF PERIODIC DATA AND MODEL SELECTION 7 Review of Transcendental Functions and Complex Numbers 61 7.1 Background
61 7.2 Complex Arithmetic
62 7.3 Some Important Series
63 7.4 Useful Facts About Periodic Transcendental Functions
64 Exercises
64 8 The Power Spectrum and the Periodogram 65 8.1 Introduction
65 8.2 A Definition and a Simplified Form for p(f )
66 8.3 Inverting p(f ) to Recover the Ck Values
66 8.4 The Power Spectrum for Some Familiar Models
68 8.5 The Periodogram
a Closer Look
72 8.6 The Function spec.pgram() in R
75 Exercises
77 9 Smoothers
The Bias-Variance Tradeoff
and the Smoothed Periodogram 79 9.1 Why is Smoothing Required?
79 9.2 Smoothing
Bias
and Variance
79 9.3 Smoothers Used in R
80 9.4 Smoothing the Periodogram for a Series With a Known and Unknown Period
85 9.5 Summary
87 Exercises
87 10 A Regression Model for Periodic Data 89 10.1 The Model
89 10.2 An Example: The NYC Temperature Data
91 10.3 Complications 1: CO2 Data
93 10.4 Complications 2: Sunspot Numbers
94 10.5 Complications 3: Accidental Deaths
96 10.6 Summary
96 Exercises
96 11 Model Selection and Cross-Validation 98 11.1 Background
98 11.2 Hypothesis Tests in Simple Regression
99 11.3 A More General Setting for Likelihood Ratio Tests
101 11.4 A Subtlety Different Situation
104 11.5 Information Criteria
106 11.6 Cross-validation (Data Splitting): NYC Temperatures
108 11.7 Summary
112 Exercises
113 12 Fitting Fourier series 115 12.1 Introduction: More Complex Periodic Models
115 12.2 More Complex Periodic Behavior: Accidental Deaths
116 12.3 The Boise River Flow data
121 12.4 Where Do We Go from Here?
124 Exercises
124 13 Adjusting for AR(1) Correlation in Complex Models 125 13.1 Introduction
125 13.2 The Two-Sample t-Test--UNCUT and Patch-Cut Forest
125 13.3 The Second Sleuth Case--Global Warming
A Simple Regression
132 13.4 The Semmelweis Intervention
138 13.5 The NYC Temperatures (Adjusted)
142 13.6 The Boise River Flow Data: Model Selection With Filtering
147 13.7 Implications of AR(1) Adjustments and the "Skip" Method
151 13.8 Summary
152 Exercises
153 PART III COMPLEX TEMPORAL STRUCTURES 14 The Backshift Operator
the Impulse Response Function
and General ARMA Models 159 14.1 The General ARMA Model
159 14.2 The Backshift (Shift
Lag) Operator
161 14.3 The Impulse Response Operator--Intuition
164 14.4 Impulse Response Operator
g(B)--Computation
165 14.5 Interpretation and Utility of the Impulse Response Function
167 Exercises
167 15 The Yule-Walker Equations and the Partial Autocorrelation Function 169 15.1 Background
169 15.2 Autocovariance of an ARMA(m
l) Model
169 15.3 AR(m) and the Yule-Walker Equations
170 15.4 The Partial Autocorrelation Plot
174 15.5 The Spectrum For Arma Processes
175 15.6 Summary
177 Exercises
178 16 Modeling Philosophy and Complete Examples 180 16.1 Modeling Overview
180 16.2 A Complex Periodic Model--Monthly River Flows
Furnas 1931-1978
185 16.3 A Modeling Example--Trend and Periodicity: CO2 Levels at Mauna Lau
193 16.4 Modeling Periodicity with a Possible Intervention--Two Examples
198 16.5 Periodic Models: Monthly
Weekly
and Daily Averages
205 16.6 Summary
207 Exercises
207 PART IV SOME DETAILED AND COMPLETE EXAMPLES 17 Wolf's Sunspot Number Data 213 17.1 Background
213 17.2 Unknown Period ==> Nonlinear Model
214 17.3 The Function nls() in R
214 17.4 Determining the Period
216 17.5 Instability in the Mean
Amplitude
and Period
217 17.6 Data Splitting for Prediction
220 17.7 Summary
226 Exercises
226 18 An Analysis of Some Prostate and Breast Cancer Data 228 18.1 Background
228 18.2 The First Data Set
229 18.3 The Second Data Set
232 Exercises
243 19 Christopher Tennant/Ben Crosby Watershed Data 245 19.1 Background and Question
245 19.2 Looking at the Data and Fitting Fourier Series
246 19.3 Averaging Data
248 19.4 Results
250 Exercises
250 20 Vostok Ice Core Data 251 20.1 Source of the Data
251 20.2 Background
252 20.3 Alignment
253 20.4 A Nä?ve Analysis
256 20.5 A Related Simulation
259 20.6 An AR(1) Model for Irregular Spacing
265 20.7 Summary
269 Exercises
270 Appendix A Using Datamarket 273 A.1 Overview
273 A.2 Loading a Time Series in Datamarket
277 A.3 Respecting Datamarket Licensing Agreements
280 Appendix B AIC is PRESS! 281 B.1 Introduction
281 B.2 PRESS
281 B.3 Connection to Akaike's Result
282 B.4 Normalization and R2
282 B.5 An example
283 B.6 Conclusion and Further Comments
283 Appendix C A 15-Minute Tutorial on Nonlinear Optimization 284 C.1 Introduction
284 C.2 Newton's Method for One-Dimensional Nonlinear Optimization
284 C.3 A Sequence of Directions
Step Sizes
and a Stopping Rule
285 C.4 What Could Go Wrong?
285 C.5 Generalizing the Optimization Problem
286 C.6 What Could Go Wrong--Revisited
286 C.7 What Can be Done?
287 REFERENCES 291 INDEX 293
3 1.2 Special R Conventions
5 1.3 Common Structures
5 1.4 Common Functions
6 1.5 Time Series Functions
6 1.6 Importing Data
7 Exercises
7 2 Review of Regression and More About R 8 2.1 Goals of this Chapter
8 2.2 The Simple(ST) Regression Model
8 2.3 Simulating the Data from a Model and Estimating the Model Parameters in R
9 2.4 Basic Inference for the Model
12 2.5 Residuals Analysis--What Can Go Wrong...
13 2.6 Matrix Manipulation in R
15 Exercises
16 3 The Modeling Approach Taken in this Book and Some Examples of Typical Serially Correlated Data 18 3.1 Signal and Noise
18 3.2 Time Series Data
19 3.3 Simple Regression in the Framework
20 3.4 Real Data and Simulated Data
20 3.5 The Diversity of Time Series Data
21 3.6 Getting Data Into R
24 Exercises
26 4 Some Comments on Assumptions 28 4.1 Introduction
28 4.2 The Normality Assumption
29 4.3 Equal Variance
31 4.4 Independence
31 4.5 Power of Logarithmic Transformations Illustrated
32 4.6 Summary
34 Exercises
34 5 The Autocorrelation Function And AR(1)
AR(2) Models 35 5.1 Standard Models--What are the Alternatives to White Noise?
35 5.2 Autocovariance and Autocorrelation
36 5.3 The acf() Function in R
37 5.4 The First Alternative to White Noise: Autoregressive Errors--AR(1)
AR(2)
40 Exercises
49 6 The Moving Average Models MA(1) And MA(2) 51 6.1 The Moving Average Model
51 6.2 The Autocorrelation for MA(1) Models
51 6.3 A Duality Between MA(l) And AR(m) Models
52 6.4 The Autocorrelation for MA(2) Models
52 6.5 Simulated Examples of the MA(1) Model
52 6.6 Simulated Examples of the MA(2) Model
54 6.7 AR(m) and MA(l) model acf() Plots
54 Exercises
57 PART II ANALYSIS OF PERIODIC DATA AND MODEL SELECTION 7 Review of Transcendental Functions and Complex Numbers 61 7.1 Background
61 7.2 Complex Arithmetic
62 7.3 Some Important Series
63 7.4 Useful Facts About Periodic Transcendental Functions
64 Exercises
64 8 The Power Spectrum and the Periodogram 65 8.1 Introduction
65 8.2 A Definition and a Simplified Form for p(f )
66 8.3 Inverting p(f ) to Recover the Ck Values
66 8.4 The Power Spectrum for Some Familiar Models
68 8.5 The Periodogram
a Closer Look
72 8.6 The Function spec.pgram() in R
75 Exercises
77 9 Smoothers
The Bias-Variance Tradeoff
and the Smoothed Periodogram 79 9.1 Why is Smoothing Required?
79 9.2 Smoothing
Bias
and Variance
79 9.3 Smoothers Used in R
80 9.4 Smoothing the Periodogram for a Series With a Known and Unknown Period
85 9.5 Summary
87 Exercises
87 10 A Regression Model for Periodic Data 89 10.1 The Model
89 10.2 An Example: The NYC Temperature Data
91 10.3 Complications 1: CO2 Data
93 10.4 Complications 2: Sunspot Numbers
94 10.5 Complications 3: Accidental Deaths
96 10.6 Summary
96 Exercises
96 11 Model Selection and Cross-Validation 98 11.1 Background
98 11.2 Hypothesis Tests in Simple Regression
99 11.3 A More General Setting for Likelihood Ratio Tests
101 11.4 A Subtlety Different Situation
104 11.5 Information Criteria
106 11.6 Cross-validation (Data Splitting): NYC Temperatures
108 11.7 Summary
112 Exercises
113 12 Fitting Fourier series 115 12.1 Introduction: More Complex Periodic Models
115 12.2 More Complex Periodic Behavior: Accidental Deaths
116 12.3 The Boise River Flow data
121 12.4 Where Do We Go from Here?
124 Exercises
124 13 Adjusting for AR(1) Correlation in Complex Models 125 13.1 Introduction
125 13.2 The Two-Sample t-Test--UNCUT and Patch-Cut Forest
125 13.3 The Second Sleuth Case--Global Warming
A Simple Regression
132 13.4 The Semmelweis Intervention
138 13.5 The NYC Temperatures (Adjusted)
142 13.6 The Boise River Flow Data: Model Selection With Filtering
147 13.7 Implications of AR(1) Adjustments and the "Skip" Method
151 13.8 Summary
152 Exercises
153 PART III COMPLEX TEMPORAL STRUCTURES 14 The Backshift Operator
the Impulse Response Function
and General ARMA Models 159 14.1 The General ARMA Model
159 14.2 The Backshift (Shift
Lag) Operator
161 14.3 The Impulse Response Operator--Intuition
164 14.4 Impulse Response Operator
g(B)--Computation
165 14.5 Interpretation and Utility of the Impulse Response Function
167 Exercises
167 15 The Yule-Walker Equations and the Partial Autocorrelation Function 169 15.1 Background
169 15.2 Autocovariance of an ARMA(m
l) Model
169 15.3 AR(m) and the Yule-Walker Equations
170 15.4 The Partial Autocorrelation Plot
174 15.5 The Spectrum For Arma Processes
175 15.6 Summary
177 Exercises
178 16 Modeling Philosophy and Complete Examples 180 16.1 Modeling Overview
180 16.2 A Complex Periodic Model--Monthly River Flows
Furnas 1931-1978
185 16.3 A Modeling Example--Trend and Periodicity: CO2 Levels at Mauna Lau
193 16.4 Modeling Periodicity with a Possible Intervention--Two Examples
198 16.5 Periodic Models: Monthly
Weekly
and Daily Averages
205 16.6 Summary
207 Exercises
207 PART IV SOME DETAILED AND COMPLETE EXAMPLES 17 Wolf's Sunspot Number Data 213 17.1 Background
213 17.2 Unknown Period ==> Nonlinear Model
214 17.3 The Function nls() in R
214 17.4 Determining the Period
216 17.5 Instability in the Mean
Amplitude
and Period
217 17.6 Data Splitting for Prediction
220 17.7 Summary
226 Exercises
226 18 An Analysis of Some Prostate and Breast Cancer Data 228 18.1 Background
228 18.2 The First Data Set
229 18.3 The Second Data Set
232 Exercises
243 19 Christopher Tennant/Ben Crosby Watershed Data 245 19.1 Background and Question
245 19.2 Looking at the Data and Fitting Fourier Series
246 19.3 Averaging Data
248 19.4 Results
250 Exercises
250 20 Vostok Ice Core Data 251 20.1 Source of the Data
251 20.2 Background
252 20.3 Alignment
253 20.4 A Nä?ve Analysis
256 20.5 A Related Simulation
259 20.6 An AR(1) Model for Irregular Spacing
265 20.7 Summary
269 Exercises
270 Appendix A Using Datamarket 273 A.1 Overview
273 A.2 Loading a Time Series in Datamarket
277 A.3 Respecting Datamarket Licensing Agreements
280 Appendix B AIC is PRESS! 281 B.1 Introduction
281 B.2 PRESS
281 B.3 Connection to Akaike's Result
282 B.4 Normalization and R2
282 B.5 An example
283 B.6 Conclusion and Further Comments
283 Appendix C A 15-Minute Tutorial on Nonlinear Optimization 284 C.1 Introduction
284 C.2 Newton's Method for One-Dimensional Nonlinear Optimization
284 C.3 A Sequence of Directions
Step Sizes
and a Stopping Rule
285 C.4 What Could Go Wrong?
285 C.5 Generalizing the Optimization Problem
286 C.6 What Could Go Wrong--Revisited
286 C.7 What Can be Done?
287 REFERENCES 291 INDEX 293