Outline

ATMS 552: Objective Methods of Data Analysis

-   D. L. Hartmann   -

1. Review of Basic Statistics

1.1 Some fundamental statistical quantities

1.2 Probability concepts and laws

1.3 Probability distributions

1.4 The Normal distribution

1.5 Testing for statistical significance using the Normal distribution

1.6 Hypothesis testing

1.7 Combinatorics and the Binomial and Hypergeometric Distributions

1.8 Non-Parametric Statistical Tests

1.9 A Priori, A Posteriori and Degrees of Freedom

2. Compositing or superposed epoch analysis

2.1 Steps in the compositing process

2.2 Evaluating compositing studies

3. Regression

3.1 Linear least-square curve fitting

3.2 Generalized normal equations

3.2.1 Use of Singular Value Decomposition in regression

3.3 Theory of correlation

3.4 Sampling theory of correlation

3.5 Multiple regression

4. Matrix Methods in Data Analysis: EOF, SVD, etc.

4.1 Principle component analysis

4.2 Empirical orthogonal functions

4.3 Statistical prediction with principle components

4.4 Factor analysis

4.5 How many factors should be retained

4.6 Rotation of EOF's and factors

4.7 Eight physical variables example

4.8 Singular value decomposition

4.9 Cluster analysis

5. Objective Analysis of Observations onto a Regular Grid

5.1 Polynomial fitting methods

5.2 The correction method

5.3 Optimum interpolation - Simple theory

5.4 Optimum interpolation - Applied

6. Time or Space Series Analysis

6.1 Autocorrelation

6.1.1 The autocorrelation function

6.1.2 Red noise

6.1.3 Statistical prediction and red noise

6.1.4 White noise

6.1.5 Degrees of freedom and autocorrelation

6.1.6 Verification of forecast models

6.2 Harmonic Analysis

6.2.1 Discrete Fourier Transform

6.2.2 The power spectrum

6.2.3 Methods of computing power spectra

6.2.4 Plotting power spectra

6.2.5 Data windows and window carpentry

6.2.6 Statistical significance of spectral peaks

6.2.7 Experimental red-noise spectra

6.3 Cross Spectrum Analysis

6.3.1 Complex exponential expansion

6.3.2 Spectral Coherence and Phase relations

6.4 Mixed Space-Time Spectral Analysis

6.5 Singular Spectral Analysis

7. Filtering of Time and Space Series

7.1 Introduction

7.2 Filtering

7.2.1 Fourier method

7.2.2 Weighting method

7.3 The response function

7.4 Inverse Problem - Filter design

7.5 Construction of symmetric nonrecursive filters

7.4.1 Fourier determination of filter weights

7.4.2 Lanzcos smoothing of filter weights

7.4 Recursive Filters

7.4.1 Time shifting theorem

7.4.2 Response function for general linear filters

7.4.3 A simple recursive filter

7.4.4 Impulse response of a recursive filter

7.5 Practical construction of recursive filters

8. Kalman Filtering

8.1 Combining estimates of vectors

8.1.1 Scalar example

8.1.2 Vector operations form

8.2 Sequential estimation

8.3 Kalman filter application to mapping of satellite data

9. Wavelets

9.1 Wavelet Types

9.2 The Haar Wavelet

9.3 Daubechies Wavelet Filter Coefficients

9.4 Morlet Wavelets and Fourier Analysis

10. Generalized Statistical Models