Quantitative Methods of Data Analysis for the Physical Sciences and Engineering

This book provides a thorough and comprehensive coverage of most of the new and important quantitative methods of data analysis for graduate students and practitioners. In recent years, data analysis methods have exploded alongside advanced computing power, and it is critical to understand such methods to get the most out of data, and to extract signal from noise. The book excels in explaining difficult concepts through simple explanations and detailed explanatory illustrations. Most unique is the focus on confidence limits for power spectra and their proper interpretation, something rare or completely missing in other books. Likewise, there is a thorough discussion of how to assess uncertainty via use of Expectancy, and the easy to apply and understand Bootstrap method. The book is written so that descriptions of each method are as self-contained as possible. Many examples are presented to clarify interpretations, as are user tips in highlighted boxes Contents......Page 3 Preface......Page 8 --- Fundamentals......Page 10 Data Nomenclature......Page 11 Representing Discrete Data & Functions as Vectors......Page 13 Data Limits......Page 14 Data Errors......Page 16 Practical Issues......Page 20 Probability Theory......Page 23 Definitions......Page 24 Probability......Page 26 Univariate Distributions......Page 27 Multivariate Distributions......Page 35 Moments of Random Variables......Page 39 Common Distributions & their Moments......Page 58 Take-Home Points......Page 67 Questions......Page 68 Estimation......Page 70 Estimating the Distribution......Page 74 Point Estimates......Page 77 Principle of Maximum Likelihood (An Important Principle)......Page 84 Interval Estimates......Page 88 Hypothesis Testing......Page 94 Sample-Based Distributions......Page 104 Take-Home Points......Page 108 Questions......Page 109 --- Fitting Curves to Data......Page 110 Interpolation......Page 111 Piecewise Continuous Interpolants......Page 116 Continuous Interpolants......Page 130 Take-Home Points......Page 132 Questions......Page 133 Introduction......Page 134 Functional Form of the Curve......Page 135 De fi ning “ Best ” Fit......Page 136 Determining Parameter Values for a Best-Fit Curve......Page 144 Orthogonal Fitting of a Straight Line......Page 164 Assessing Uncertainty in Optimal Parameter Values......Page 165 Assessing the Fit of the Best-Fit Curve......Page 178 Questions......Page 182 Weighted Curve Fits......Page 184 Constrained Fits......Page 191 Regression/Calibration......Page 199 Correlation Coef fi cient......Page 201 Take-Home Points......Page 206 Questions......Page 207 --- Sequential Data Fundamentals......Page 211 Serial Products......Page 212 Statistical Considerations......Page 214 Characterizing a Random Process......Page 216 Convolution......Page 227 Serial Cross-Covariance and Cross-Correlation......Page 239 Take-Home Points......Page 254 Questions......Page 255 Fourier Series......Page 257 Periodic Functions......Page 258 Fourier Series......Page 270 Questions......Page 275 Discrete Periodic Data......Page 276 Discrete Sine and Cosine Transforms......Page 287 Continuous Sine and Cosine Transforms......Page 293 The Fourier Transform......Page 294 Fourier Transform of Non-Periodic Data......Page 301 Fourier Transform Properties......Page 306 Fourier Transform Theorems......Page 316 Fast Fourier Transform......Page 324 Take-Home Points......Page 325 Questions......Page 326 Fourier Sampling Theory......Page 327 Sampling Theorem......Page 328 Relationship between Discrete and Continuous Transform......Page 343 Other Sampling Considerations......Page 352 Questions......Page 353 Spectral Analysis......Page 355 Noise in the Spectrum......Page 356 More Stable Estimates of the Fourier Coef fi cients......Page 362 Spectral Estimation in Practice......Page 396 Bootstrap Testing with Time Series......Page 405 Take-Home Points......Page 408 Questions......Page 409 Joint PDF Moments in the Time Domain......Page 411 Frequency Domain Estimation of the ccf......Page 419 Statistical Considerations......Page 424 Take-Home Points......Page 428 Questions......Page 429 Filtering & Deconvolution......Page 430 Frequency Domain Representation......Page 432 Special Types of Filters......Page 435 Other Low-Pass Filter Shapes......Page 438 Practical Considerations......Page 441 Exact (Deterministic) Deconvolution......Page 442 Best-Fit Deconvolution......Page 451 Take-Home Points......Page 459 Questions......Page 460 Linear Parametric Modeling......Page 461 Discrete Linear Stochastic Process Models......Page 463 Model Identi fi cation and Solution......Page 474 Parameter Estimation......Page 481 Parametric Spectral Estimation......Page 484 Questions......Page 496 Time Series References......Page 497 Introduction......Page 500 Eigenvector Analysis......Page 504 Principal Components (PC)......Page 515 Singular Spectrum Analysis (SSA)......Page 529 Questions......Page 538 A1.2 De fi nitions......Page 540 A1.3 Basic Matrix Operations......Page 544 A1.4 Special Matrix Products......Page 550 A1.5 Matrix “ Division ” : Inverse Matrix......Page 553 A1.6 Useful Properties......Page 558 A1.7 More Advanced Topics......Page 559 A1.8 Statistical Topics......Page 572 A1.9 Matrix References......Page 575 A2.2 Classi fi cation of Errors......Page 577 A2.3 Expectance as Variance......Page 579 A2.4 Bootstrap......Page 594 A2.5 Expectance Versus Bootstrap......Page 600 Refs......Page 601 Index......Page 604 "This book provides a thorough and comprehensive coverage of most of the new and important quantitative methods of data analysis for graduate students and practitioners. In recent years, data analysis methods have exploded alongside advanced computing power, and it is critical to understand such methods to get the most out of data, and to extract signal from noise. The book excels in explaining difficult concepts through simple explanations and detailed explanatory illustrations. Most unique is the focus on confidence limits for power spectra and their proper interpretation, something rare or completely missing in other books. Likewise, there is a thorough discussion of how to assess uncertainty via use of Expectancy, and the easy to apply and understand Bootstrap method. The book is written so that descriptions of each method are as self-contained as possible."--Résumé de l'éditeur