Environmental Data Analysis with MatLab

1: Data analysis with MatLab

• Abstract
• 1.1 Why MatLab?
• 1.2 Getting started with MatLab
• 1.3 Getting organized
• 1.4 Navigating folders
• 1.5 Simple arithmetic and algebra
• 1.6 Vectors and matrices
• 1.7 Multiplication of vectors of matrices
• 1.8 Element access
• 1.9 Representing functions
• 1.10 To loop or not to loop
• 1.11 The matrix inverse
• 1.12 Loading data from a file
• 1.13 Plotting data
• 1.14 Saving data to a file
• 1.15 Some advice on writing scripts
• Problems

2: A first look at data

• Abstract
• 2.1 Look at your data!
• 2.2 More on MatLab graphics
• 2.3 Rate information
• 2.4 Scatter plots and their limitations
• Problems

3: Probability and what it has to do with data analysis

• Abstract
• 3.1 Random variables
• 3.2 Mean, median, and mode
• 3.3 Variance
• 3.4 Two important probability density functions
• 3.5 Functions of a random variable
• 3.6 Joint probabilities
• 3.7 Bayesian inference
• 3.8 Joint probability density functions
• 3.9 Covariance
• 3.10 Multivariate distributions
• 3.11 The multivariate Normal distributions
• 3.12 Linear functions of multivariate data
• Problems

4: The power of linear models

• Abstract
• 4.1 Quantitative models, data, and model parameters
• 4.2 The simplest of quantitative models
• 4.3 Curve fitting
• 4.4 Mixtures
• 4.5 Weighted averages
• 4.6 Examining error
• 4.7 Least squares
• 4.8 Examples
• 4.9 Covariance and the behavior of error
• Problems

5: Quantifying preconceptions

• Abstract
• 5.1 When least square fails
• 5.2 Prior information
• 5.3 Bayesian inference
• 5.4 The product of Normal probability density distributions
• 5.5 Generalized least squares
• 5.6 The role of the covariance of the data
• 5.7 Smoothness as prior information
• 5.8 Sparse matrices
• 5.9 Reorganizing grids of model parameters
• Problems

6: Detecting periodicities

• Abstract
• 6.1 Describing sinusoidal oscillations
• 6.2 Models composed only of sinusoidal functions
• 6.3 Going complex
• 6.4 Lessons learned from the integral transform
• 6.5 Normal curve
• 6.6 Spikes
• 6.7 Area under a function
• 6.8 Time-delayed function
• 6.9 Derivative of a function
• 6.10 Integral of a function
• 6.11 Convolution
• 6.12 Nontransient signals
• Problems

7: The past influences the present

• Abstract
• 7.1 Behavior sensitive to past conditions
• 7.2 Filtering as convolution
• 7.3 Solving problems with filters
• 7.4 An example of an empirically-derived filter
• 7.5 Predicting the future
• 7.6 A parallel between filters and polynomials
• 7.7 Filter cascades and inverse filters
• 7.8 Making use of what you know
• Problems

8: Patterns suggested by data

• Abstract
• 8.1 Samples as mixtures
• 8.2 Determining the minimum number of factors
• 8.3 Application to the Atlantic Rocks dataset
• 8.4 Spiky factors
• 8.5 Weighting of elements
• 8.6 Q-mode factor analysis and spatial clustering
• 8.7 Time-Variable functions
• Problems

9: Detecting correlations among data

• Abstract
• 9.1 Correlation is covariance
• 9.2 Computing autocorrelation by hand
• 9.3 Relationship to convolution and power spectral density
• 9.4 Cross-correlation
• 9.5 Using the cross-correlation to align time series
• 9.6 Least squares estimation of filters
• 9.7 The effect of smoothing on time series
• 9.8 Band-pass filters
• 9.9 Frequency-dependent coherence
• 9.10 Windowing before computing Fourier transforms
• 9.11 Optimal window functions
• Problems

10: Filling in missing data

• Abstract
• 10.1 Interpolation requires prior information
• 10.2 Linear interpolation
• 10.3 Cubic interpolation
• 10.4 Kriging
• 10.5 Interpolation in two-dimensions
• 10.6 Fourier transforms in two dimensions
• Problems

11: “Approximate” is not a pejorative word

• Abstract
• 11.1 The value of approximation
• 11.2 Polynomial approximations and Taylor series
• 11.3 Small number approximations
• 11.4 Small number approximation applied to distance on a sphere
• 11.5 Small number approximation applied to variance
• 11.6 Taylor series in multiple dimensions
• 11.7 Small number approximation applied to covariance
• 11.8 Solving nonlinear problems with iterative least squares
• 11.9 Fitting a sinusoid of unknown frequency
• 11.10 The gradient method
• 11.11 Precomputation of a function and table lookups
• 11.12 Artificial neural networks
• 11.13 Information flow in a neural net
• 11.14 Training a neural net
• 11.15 Neural net for a nonlinear response
• Problems

12: Are my results significant?

• Abstract
• 12.1 The difference is due to random variation!
• 12.2 The distribution of the total error
• 12.3 Four important probability density functions
• 12.4 A hypothesis testing scenario
• 12.5 Chi-squared test for generalized least squares
• 12.6 Testing improvement in fit
• 12.7 Testing the significance of a spectral peak
• 12.8 Bootstrap confidence intervals
• Problems

13: Notes

• Abstract
• Note 1.1 On the persistence of MatLab variables
• Note 2.1 On time
• Note 2.2 On reading complicated text files
• Note 3.1 On the rule for error propagation
• Note 3.2 On the eda_draw() function
• Note 4.1 On complex least squares
• Note 5.1 On the derivation of generalized least squares
• Note 5.2 On MatLab functions
• Note 5.3 On reorganizing matrices
• Note 6.1 On the MatLab atan2() function
• Note 6.2 On the orthonormality of the discrete Fourier data kernel
• Note 6.3 On the expansion of a function in an orthonormal basis
• Note 8.1 On singular value decomposition
• Note 9.1 On coherence
• Note 9.2 On Lagrange multipliers
• Note 11.1 On the chain rule for partial derivatives