Finding Hidden Patterns in Datasets: Table of Contents
In December 2024, I decided to put my doctoral dissertation (from 1994) online as responsive web pages. This is in progress; sections will be uploaded as they are completed.
Here's a pdf of my entire dissertation (about 13.5 Mb) which you can view or download:
Original title: Detecting Hidden Periodicities in Discrete-Domain Data
Table of Contents
- 1.1 The Purpose of This Dissertation
- 1.2 Notation
- 1.3 Periodicity:
- 1.4 More on Periodic Functions
- 1.5 Using Identified Periodic Components for Prediction
- 1.6 Sinusoids
- 1.7 Historical Contributions in the Search for Hidden Periodicities
- 2.1 Random Behavior:
- 2.2 Tests for Specific Conjectured Components: Linear Least-Squares Approximation
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2.3 Computer Application Considerations for Linear Least-Squares:
- Condition Numbers (in progress)
- Discrete Orthogonal Functions
- 2.4 Nonlinear Least-Squares Approximation:
- 2.5 Cubic Spline Interpolation
- 2.6 Discrete Fourier Series and the Periodogram
- 2.7 The Periodogram, via the Discrete Fourier Transform
MATLAB Implementations
GNU Octave is free software that is largely compatible with MATLAB. As I get the dissertation online, I will test all provided code with Octave and indicate any change(s) that must be made.
- Arranging Time Values in Increasing Order; Locating Repeated Time Values in a Data Set; Checking For a Uniform Time List; Supplying Alternative Indexing of a List
- Reshaping Procedure to Test for Relatively Prime Cycles
- Sums of Sinusoids with Close Periods, Amplitudes, Phase Shifts
- Applying the Turning Point Test
- Computing the Probability that the Martingale Algorithm will Stop in less than or equal to $\,N\,$ steps, beginning with a series of length $\,m$
- Linear Least-Squares Approximation
- Converting Functions to Discrete Orthogonal Functions
- Nonlinear Least-Squares, Gradient Method
- A Genetic Algorithm
- Polynomial and Spline Interpolation
- Discrete Fourier Series and the Periodogram
- Finding the Periodogram, using a Fast Fourier Transform
- Applying a Nonrecursive Filter
- Finding the Transfer Function for a Symmetric Nonrecursive Filter
- Finding a Symmetric Nonrecursive Filter Corresponding to a Desired Transfer Function
- Deleting ‘Not A Number’ Entries in a Matrix
The Symbol $\,\bigstar$
The symbol $\,\bigstar\,$ is used in this dissertation to mark occasional digressions that are important theoretically, but may go beyond the assumed prerequisites. Such material can be skipped over without any loss of continuity.
