## Bayesian Spectrum Analysis and Parameter Estimation

Download Bayesian Spectrum Analysis and Parameter Estimation by G. Larry Bretthorst, It is primarily a research document on the application of probability theory to the parameter estimation problem. The folks that are curious about this material are physicists, economists, and engineers who have to deal with data on a daily basis; consequently, we've got included a great deal of introductory and tutorial material. someone with the equivalent of the mathematics background needed for the graduatelevel study of physics should be able to follow the material contained in this book, without not without effort.

CONTENTS-

1 INTRODUCTION
1.1 Historical Perspective
1.2 Method of Calculation

2 SINGLE STATIONARY SINUSOID PLUS NOISE
2.1 The Model
2.2 The Likelihood Function
2.3 Elimination of Nuisance Parameters
2.4 Resolving Power
2.5 The Power Spectral Density ^p
2.6 Wolf 's Relative Sunspot Numbers

3 THE GENERAL MODEL EQUATION PLUS NOISE
3.1 The Likelihood Function
3.2 The Orthonormal Model Equations
3.3 Elimination of the Nuisance Parameters
3.4 The Bessel Inequality
3.5 An Intuitive Picture
3.6 A Simple Diagnostic Test

4 ESTIMATING THE PARAMETERS
4.1 The Expected Amplitudes (Aj)
4.2 The Second Posterior Moments (AjAk)
4.3 The Estimated Noise Variance h2i
4.4 The Signal-To-Noise Ratio
4.5 Estimating the {w} Parameters
4.6 The Power Spectral Density

5 MODEL SELECTION
5.2 The Relative Probability of Model fj
5.3 One More Parameter
5.4 What is a Good Model?

6 SPECTRAL ESTIMATION
6.1 The Spectrum of a Single Frequency
6.1.1 The \Student t-Distribution"
6.1.2 Example { Single Harmonic Frequency
6.1.3 The Sampling Distribution of the Estimates
6.1.4 Violating the Assumptions { Robustness
6.1.5 Nonuniform Sampling
6.2 A Frequency with Lorentzian Decay
6.2.1 The \Student t-Distribution"
6.2.2 Accuracy Estimates
6.2.3 Example { One Frequency with Decay
6.3 Two Harmonic Frequencies
6.3.1 The \Student t-Distribution"
6.3.2 Accuracy Estimates
6.3.3 More Accuracy Estimates
6.3.4 The Power Spectral Density
6.3.5 Example { Two Harmonic Frequencies
6.4 Estimation of Multiple Stationary Frequencies
6.5 The \Student t-Distribution"
6.5.1 Example { Multiple Stationary Frequencies
6.5.2 The Power Spectral Density
6.5.3 The Line Power Spectral Density
6.6 Multiple Nonstationary Frequency Estimation

7 APPLICATIONS
7.1 NMR Time Series
7.2 Corn Crop Yields
7.3 Another NMR Example
7.4 Wolf 's Relative Sunspot Numbers
7.4.1 Orthogonal Expansion of the Relative SunspotNumbers
7.4.2 Harmonic Analysis of the Relative Sunspot Numbers
7.4.3 The Sunspot Numbers in Terms of Harmonically Related Frequencies
7.4.4 Chirp in the Sunspot Numbers
7.5 Multiple Measurements
7.5.1 The Averaging Rule
7.5.2 The Resolution Improvement
7.5.3 Signal Detection
7.5.4 The Distribution of the Sample Estimates
7.5.5 Example { Multiple Measurements

8 SUMMARY AND CONCLUSIONS
8.1 Summary
8.2 Conclusions
A Choosing a Prior Probability
B Improper Priors as Limits
C Removing Nuisance Parameters
D Uninformative Prior Probabilities
E Computing the \Student t-Distribution"