Environmental Data Analysis with MatLab (eBook)

Front Cover 1
Environmental Data Analysis with MatLab 4
Copyright 5
Dedication 6
Preface 8
Advice on scripting for beginners 14
Contents 16
Chapter 1: Data analysis with MatLab 20
1.1. Why MatLab? 20
1.2. Getting started with MatLab 22
1.3. Getting organized 22
1.4. Navigating folders 23
1.5. Simple arithmetic and algebra 24
1.6. Vectors and matrices 26
1.7. Multiplication of vectors of matrices 26
1.8. Element access 27
1.9. To loop or not to loop 28
1.10. The matrix inverse 30
1.11. Loading data from a file 30
1.12. Plotting data 31
1.13. Saving data to a file 32
1.14. Some advice on writing scripts 32
Problems 34
Chapter 2: A first look at data 36
2.1. Look at your data! 36
2.2. More on MatLab graphics 43
2.3. Rate information 47
2.4. Scatter plots and their limitations 49
Problems 52
Chapter 3: Probability and what it has to do with data analysis 54
3.1. Random variables 54
3.2. Mean, median, and mode 56
3.3. Variance 60
3.4. Two important probability density functions 61
3.5. Functions of a random variable 63
3.6. Joint probabilities 65
3.7. Bayesian inference 67
3.8. Joint probability density functions 68
3.9. Covariance 71
3.10. Multivariate distributions 73
3.11. The multivariate Normal distributions 73
3.12. Linear functions of multivariate data 76
Problems 79
Chapter 4: The power of linear models 80
4.1. Quantitative models, data, and model parameters 80
4.2. The simplest of quantitative models 82
4.3. Curve fitting 83
4.4. Mixtures 86
4.5. Weighted averages 87
4.6. Examining error 90
4.7. Least squares 93
4.8. Examples 95
4.9. Covariance and the behavior of error 98
Problems 100
Chapter 5: Quantifying preconceptions 102
5.1. When least square fails 102
5.2. Prior information 103
5.3. Bayesian inference 105
5.4. The product of Normal probability density distributions 107
5.5. Generalized least squares 109
5.6. The role of the covariance of the data 111
5.7. Smoothness as prior information 112
5.8. Sparse matrices 114
5.9. Reorganizing grids of model parameters 117
Problems 120
Chapter 6: Detecting periodicities 122
6.1. Describing sinusoidal oscillations 122
6.2. Models composed only of sinusoidal functions 124
6.3. Going complex 131
6.4. Lessons learned from the integral transform 133
6.5. Normal curve 134
6.6. Spikes 135
6.7. Area under a function 137
6.8. Time-delayed function 137
6.9. Derivative of a function 139
6.10. Integral of a function 139
6.11. Convolution 140
6.12. Nontransient signals 141
Problems 143
Chapter 7: The past influences the present 146
7.1. Behavior sensitive to past conditions 146
7.2. Filtering as convolution 150
7.3. Solving problems with filters 151
7.4. Predicting the future 158
7.5. A parallel between filters and polynomials 159
7.6. Filter cascades and inverse filters 161
7.7. Making use of what you know 164
Problems 166
Chapter 8: Patterns suggested by data 168
8.1. Samples as mixtures 168
8.2. Determining the minimum number of factors 170
8.3. Application to the Atlantic Rocks dataset 174
8.4. Spiky factors 175
8.5. Time-Variable functions 179
Problems 182
Chapter 9: Detecting correlations among data 186
9.1. Correlation is covariance 186
9.2. Computing autocorrelation by hand 192
9.3. Relationship to convolution and power spectral density 192
9.4. Cross-correlation 193
9.5. Using the cross-correlation to align time series 195
9.6. Least squares estimation of filters 197
9.7. The effect of smoothing on time series 199
9.8. Band-pass filters 203
9.9. Frequency-dependent coherence 207
9.10. Windowing before computing Fourier transforms 214
9.11. Optimal window functions 215
Problems 220
Chapter 10: Filling in missing data 222
10.1. Interpolation requires prior information 222
10.2. Linear interpolation 224
10.3. Cubic interpolation 225
10.4. Kriging 227
10.5. Interpolation in two-dimensions 229
10.6. Fourier transforms in two dimensions 232
Problems 234
Chapter 11: Are my results significant? 236
11.1. The difference is due to random variation! 236
11.2. The distribution of the total error 237
11.3. Four important probability density functions 239
11.4. A hypothesis testing scenario 241
11.5. Testing improvement in fit 247
11.6. Testing the significance of a spectral peak 248
11.7. Bootstrap confidence intervals 253
Problems 257
Chapter 12: Notes 258
Note 1.1. On the persistence of MatLab variables 258
Note 2.1. On time 259
Note 2.2. On reading complicated text files 260
Note 3.1. On the rule for error propagation 261
Note 3.2. On the eda_draw() function 261
Note 4.1. On complex least squares 262
Note 5.1. On the derivation of generalized least squares 264
Note 5.2. On MatLab functions 264
Note 5.3. On reorganizing matrices 265
Note 6.1. On the MatLab atan2() function 265
Note 6.2. On the orthonormality of the discrete Fourier data kernel 265
Note 8.1. On singular value decomposition 266
Note 9.1. On coherence 267
Note 9.2. On Lagrange multipliers 268
Index 270