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Extends previous chapters with consideration of unequal-interval data and use of periodogram approaches, Schuster and Lomb-Scargle periodograms. Periodograms as examples of least-squares spectral analysis (LSSA) and criteria for statistical significance (p-value). Development of statistical effect-size and significance ideas from periodograms and correlation-regression approaches and application to DFT/FFT power spectra coefficients. Variance spectrum and proportion of variance explained by DFT/FFT frequency components, statistical significance (p-value) of DFT/FFT frequency components.
Starts with concept of a frequency function as an ordered list of Fourier series coefficients. Summary of derivation of Fourier transform from complex Fourier series and development of frequency-function ideas. Introduction of the Dirac delta function as the Fourier transform of a sinusoid to set context for characteristics of the discrete Fourier transform (DFT) spectrum. Intuitive derivation of DFT from Fourier series of piecewise-approximated sampled continuous data. Symmetry and linearity of Fourier transform and DFT. Frequency resolution of DFT, sampling theorem and Nyquist frequency. Fast Fourier Transform (FFT) as optimised DFT algorithms.
Standard form of forward and inverse Fast Fourier Transform (FFT). Sets up a systematic approach for generating frequency indices and calibrated frequencies for FFT spectra. Develops systematic approach for generating and interpreting amplitude and power spectra as vectors in a complete FFT output data-table with frequency indices. Worked examples using real time-series as typify ‘clean’ and ‘noisy’ data, data with single and multiple frequencies, data with trends.
States the context of the material in the book, i.e. Fourier and closely related techniques for the detection of periodic features in time-series data. States the assumed prior knowledge, i.e. broadly, mathematics and statistics as studied in latter-year mathematics courses at secondary/high school and as accompany many university undergraduate science courses.
Review of correlation and simple linear regression. Introduction to lagged (cross-) correlation for identifying recurrent and periodic features in common between pairs of time-series, statistical evidence of possible causal relationships. Introduction to (lagged) autocorrelation for identifying recurrent and periodic features in time-series. Use of correlation and simple linear regression for statistical comparison of time-series to reference datasets, with focus on periodic (sinusoidal) reference datasets. Interpretation of statistical effect-size and significance (p-value).
Starts with statement of the real Fourier series and orthogonality conditions. Develops the complex Fourier series and introduces the concept of negative frequency, with emphasis as a mathematical convenience, and symmetry of positive-negative frequency coefficients. Concepts of symmetry and linearity. Discrete frequencies and nature of discrete frequency spectrum as comprised of specific fundamental and harmonic frequencies.
Stationarity, stationary and non-stationary data in the context of time-series analysis and frequency content. Introduction to spectrograms as a means of investigating time-frequency relationships in data. Time and frequency resolution and uncertainty principle. Worked examples with stationary and non-stationary time-series.
How the FFT interprets features with frequencies below the fundamental frequency, windowing. How the FFT interprets features with frequencies above the Nyquist frequency, aliasing. How the FFT interprets features with frequencies intermediate between FFT harmonic frequencies, ‘tuning’ and padding time-series, use of dummy-data with specific frequency content. Basic approaches to replacement of missing data. Worked examples for uncomplicated data, data with intermediate frequencies, tuning datasets.
Overview of key identifying features of noise as can typically occur in geoscience time-series. Categorisation according to noise colour; white, red and blue noise. Consideration of autocorrelation and autoregression, power spectral density and power-law. Worked red-noise example to illustrate.
Time-series analysis is used to identify and quantify periodic features in datasets and has many applications across the geosciences, from analysing weather data, to solid-Earth geophysical modelling. This intuitive introduction provides a practical 'how-to' guide to basic Fourier theory, with a particular focus on Earth system applications. The book starts with a discussion of statistical correlation, before introducing Fourier series and building to the fast Fourier transform (FFT) and related periodogram techniques. The theory is illustrated with numerous worked examples using R datasets, from Milankovitch orbital-forcing cycles to tidal harmonics and exoplanet orbital periods. These examples highlight the key concepts and encourage readers to investigate more advanced time-series techniques. The book concludes with a consideration of statistical effect size and significance. This useful book is ideal for graduate students and researchers in the Earth system sciences who are looking for an accessible introduction to time-series analysis.
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