Opis: Basic Data Analysis for Time Series with R - DeWayne Derryberry

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 setsPREFACE 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 2.2.1 Ordinary Least Squares, 8 2.2.2 Properties of OLS Estimates, 9 2.2.3 Matrix Representation of the Problem, 9 2.3 Simulating the Data from a Model and Estimating the Model Parameters in R, 9 2.3.1 Simulating Data, 9 2.3.2 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 2.6.1 Introduction, 15 2.6.2 OLS the Hard Way, 15 2.6.3 Some Other Matrix Commands, 16 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 3.6.1 Overview, 24 3.6.2 The Diskette and the scan() and ts() Functions--New York City Temperatures, 25 3.6.3 The Diskette and the read.table() Function--The Semmelweis Data, 25 3.6.4 Cut and Paste Data to a Text Editor, 26 Exercises, 26 4 Some Comments on Assumptions 28 4.1 Introduction, 28 4.2 The Normality Assumption, 29 4.2.1 Right Skew, 30 4.2.2 Left Skew, 30 4.2.3 Heavy Tails, 30 4.3 Equal Variance, 31 4.3.1 Two-Sample t-Test, 31 4.3.2 Regression, 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 5.2.1 Stationarity, 36 5.2.2 A Note About Conditions, 36 5.2.3 Properties of Autocovariance, 36 5.2.4 White Noise, 37 5.2.5 Estimation of the Autocovariance and Autocorrelation, 37 5.3 The acf() Function in R, 37 5.3.1 Background, 37 5.3.2 The Basic Code for Estimating the Autocovariance, 38 5.4 The First Alternative to White Noise: Autoregressive Errors--AR(1), AR(2), 40 5.4.1 Definition of the AR(1) and AR(2) Models, 40 5.4.2 Some Preliminary Facts, 40 5.4.3 The AR(1) Model Autocorrelation and Autocovariance, 41 5.4.4 Using Correlation and Scatterplots to Illustrate the AR(1) Model, 41 5.4.5 The AR(2) Model Autocorrelation and Autocovariance, 41 5.4.6 Simulating Data for AR(m) Models, 42 5.4.7 Examples of Stable and Unstable AR(1) Models, 44 5.4.8 Examples of Stable and Unstable AR(2) Models, 46 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 7.2.1 The Number i, 62 7.2.2 Complex Conjugates, 62 7.2.3 The Magnitude of a Complex Number, 62 7.3 Some Important Series, 63 7.3.1 The Geometric and Some Transcendental Series, 63 7.3.2 A Rationale for Euler's Formula, 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 8.4.1 White Noise, 68 8.4.2 The Spectrum for AR(1) Models, 68 8.4.3 The Spectrum for AR(2) Models, 70 8.5 The Periodogram, a Closer Look, 72 8.5.1 Why is the Periodogram Useful?, 72 8.5.2 Some Na¨yve Code for a Periodogram, 72 8.5.3 An Example--The Sunspot Data, 74 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 9.3.1 The R Function lowess(), 81 9.3.2 The R Function smooth.spline(), 82 9.3.3 Kernel Smoothers in spec.pgram(), 83 9.4 Smoothing the Periodogram for a Series With a Known and Unknown Period, 85 9.4.1 Period Known, 85 9.4.2 Period Unknown, 86 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 10.2.1 Fitting a Periodic Function, 91 10.2.2 An Outlier, 92 10.2.3 Refitting the Model with the Outlier Corrected, 92 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 11.6.1 Explained Variation, R2, 108 11.6.2 Data Splitting, 108 11.6.3 Leave-One-Out Cross-Validation, 110 11.6.4 AIC as Leave-One-Out Cross-Validation, 112 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 12.2.1 Fourier Series Structure, 116 12.2.2 R Code for Fitting Large Fourier Series, 116 12.2.3 Model Selection with AIC, 117 12.2.4 Model Selection with Likelihood Ratio Tests, 118 12.2.5 Data Splitting, 119 12.2.6 Accidental Deaths--Some Comment on Periodic Data, 120 12.3 The Boise River Flow data, 121 12.3.1 The Data, 121 12.3.2 Model Selection with AIC, 122 12.3.3 Data Splitting, 123 12.3.4 The Residuals, 123 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 13.2.1 The Sleuth Data and the Question of Interest, 125 13.2.2 A Simple Adjustment for t-Tests When the Residuals Are AR(1), 128 13.2.3 A Simulation Example, 129 13.2.4 Analysis of the Sleuth Data, 131 13.3 The Second Sleuth Case--Global Warming, A Simple Regression, 132 13.3.1 The Data and the Question, 132 13.3.2 Filtering to Produce (Quasi-)Independent Observations, 133 13.3.3 Simulated Example--Regression, 134 13.3.4 Analysis of the Regression Case, 135 13.3.5 The Filtering Approach for the Logging Case, 136 13.3.6 A Few Comments on Filtering, 137 13.4 The Semmelweis Intervention, 138 13.4.1 The Data, 138 13.4.2 Why Serial Correlation?, 139 13.4.3 How This Data Differs from the Patch/Uncut Case, 139 13.4.4 Filtered Analysis, 140 13.4.5 Transformations and Inference, 142 13.5 The NYC Temperatures (Adjusted), 142 13.5.1 The Data and Prediction Intervals, 142 13.5.2 The AR(1) Prediction Model, 144 13.5.3 A Simulation to Evaluate These Formulas, 144 13.5.4 Application to NYC Data, 146 13.6 The Boise River Flow Data: Model Selection With Filtering, 147 13.6.1 The Revised Model Selection Problem, 147 13.6.2 Comments on R2 and R2 pred, 147 13.6.3 Model Selection After Filtering with a Matrix, 148 13.7 Implications of AR(1) Adjustments and the "Skip" Method, 151 13.7.1 Adjustments for AR(1) Autocorrelation, 151 13.7.2 Impact of Serial Correlation on p-Values, 152 13.7.3 The "skip" Method, 152 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 14.1.1 The Mathematical Formulation, 159 14.1.2 The arima.sim() Function in R Revisited, 159 14.1.3 Examples of ARMA(m,l) Models, 160 14.2 The Backshift (Shift, Lag) Operator, 161 14.2.1 Definition of B, 161 14.2.2 The Stationary Conditions for a General AR(m) Model, 161 14.2.3 ARMA(m,l) Models and the Backshift Operator, 162 14.2.4 More Examples of ARMA(m,l) Models, 162 14.3 The Impulse Response Operator--Intuition, 164 14.4 Impulse Response Operator, g(B)--Computation, 165 14.4.1 Definition of g(B), 165 14.4.2 Computing the Coefficients, gj., 165 14.4.3 Plotting an Impulse Response Function, 166 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 15.2.1 A Preliminary Result, 169 15.2.2 The Autocovariance Function for ARMA(m,l) Models, 170 15.3 AR(m) and the Yule--Walker Equations, 170 15.3.1 The Equations, 170 15.3.2 The R Function ar.yw() with an AR(3) Example, 171 15.3.3 Information Criteria-Based Model Selection Using ar.yw(), 173 15.4 The Partial Autocorrelation Plot, 174 15.4.1 A Sequence of Hypothesis Tests, 174 15.4.2 The pacf() Function--Hypothesis Tests Presented in a 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 16.1.1 The Algorithm, 180 16.1.2 The Underlying Assumption, 180 16.1.3 An Example Using an AR(m) Filter to Model MA(3), 181 16.1.4 Generalizing the "Skip" Method, 184 16.2 A Complex Periodic Model--Monthly River Flows, Furnas 1931--1978, 185 16.2.1 The Data, 185 16.2.2 A Saturated Model, 186 16.2.3 Building an AR(m) Filtering Matrix, 187 16.2.4 Model Selection, 189 16.2.5 Predictions and Prediction Intervals for an AR(3) Model, 190 16.2.6 Data Splitting, 191 16.2.7 Model Selection Based on a Validation Set, 192 16.3 A Modeling Example--Trend and Periodicity: CO2 Levels at Mauna Lau, 193 16.3.1 The Saturated Model and Filter, 193 16.3.2 Model Selection, 194 16.3.3 How Well Does the Model Fit the Data?, 197 16.4 Modeling Periodicity with a Possible Intervention--Two Examples, 198 16.4.1 The General Structure, 198 16.4.2 Directory Assistance, 199 16.4.3 Ozone Levels in Los Angeles, 202 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 17.6.1 The Approach, 220 17.6.2 Step 1--Fitting One Step Ahead, 222 17.6.3 The AR Correction, 222 17.6.4 Putting it All Together, 223 17.6.5 Model Selection, 223 17.6.6 Predictions Two Steps Ahead, 224 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 18.3.1 Background and Questions, 232 18.3.2 Outline of the Statistical Analysis, 233 18.3.3 Looking at the Data, 233 18.3.4 Examining the Residuals for AR(m) Structure, 235 18.3.5 Regression Analysis with Filtered Data, 238 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 19.2.1 The Structure of the Data, 246 19.2.2 Fourier Series Fits to the Data, 246 19.2.3 Connecting Patterns in Data to Physical Processes, 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 20.3.1 Need for Alignment, and Possible Issues Resulting from Alignment, 253 20.3.2 Is the Pattern in the Temperature Data Maintained?, 254 20.3.3 Are the Dates Closely Matched?, 254 20.3.4 Are the Times Equally Spaced?, 255 20.4 A Na¨yve Analysis, 256 20.4.1 A Saturated Model, 256 20.4.2 Model Selection, 258 20.4.3 The Association Between CO2 and Temperature Change, 258 20.5 A Related Simulation, 259 20.5.1 The Model and the Question of Interest, 259 20.5.2 Simulation Code in R, 260 20.5.3 A Model Using all of the Simulated Data, 261 20.5.4 A Model Using a Sample of 283 from the Simulated Data, 262 20.6 An AR(1) Model for Irregular Spacing, 265 20.6.1 Motivation, 265 20.6.2 Method, 266 20.6.3 Results, 266 20.6.4 Sensitivity Analysis, 267 20.6.5 A Final Analysis, Well Not Quite, 268 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

Szczegóły: Basic Data Analysis for Time Series with R - DeWayne Derryberry

Tytuł: Basic Data Analysis for Time Series with R
Autor: DeWayne Derryberry
Producent: John Wiley
ISBN: 9781118422540
Rok produkcji: 2014
Ilość stron: 320
Oprawa: Twarda
Waga: 0.66 kg

Recenzje: Basic Data Analysis for Time Series with R - DeWayne Derryberry