Applications of Recurrent Neural Networks to Optimization Problems
Alaeddin Malek · InTech eBooks · 2008
IntroductionFourier methods are commonplace in the Earth Sciences and have greatly enhanced our understanding and forecast capabilities for cyclical phenomena that recur on interannual (e.g.El Niño, Pacific Decadal Oscillation) to millennial scales (e.g.Milankovitch cycles).Nowadays most low level programming languages (C, Fortran) have math libraries that include the fast Fourier transform algorithm (FFT) and nearly all abstract programs (Python, Octave/Matlab, IDL, R) provide an array of Fourier functions for scripting sophisticated signal processing routines.Whether your interest as a practicing Earth scientist is in Fourier transformation for efficient data manipulation, or for problems where the Fourier transform or its power spectrum is needed for direct analysis, you have probably found no shortage of relevant literature.Nonetheless, you may also have found some difficulty in making sense of which Fourier methods to implement for your particular analysis idea, and how to appropriately apply them.This chapter will serve you as a basic guide for unraveling some of the complicated implementations of discrete-time power spectrum analysis using direct language and supplementary Matlab/Octave routines using both observed and modeled data.This chapter assumes that you have a certain task to accomplish, and therefore it is designed to teach you how to set up an approach appropriate for Fourier analysis, and also to advise you of potential pitfalls and limitations in Fourier analysis.The reader need not have prior exposure to signal processing methodologies, but should have a solid base in mathematics, probability theory and more importantly the issues related to your analysis data so that the significances of cycles within your data can be rationally interpreted.If you find yourself lost by the terminology I recommend you familiarize yourself with basic treatments of discrete-time systems, for example Press et al. (1992) or Cadzow (1973). The body of this chapter is split into three sections, Preprocessing data, Single Series Spectrum Analysis, and Multiseries Spectral Analysis. Step-by-step examples aregiven on the analysis of a variety of freely accessible earth science datasets covering atmospheric science, biosphere-atmosphere carbon cycling, climate modeling, and paleodiversity as well as some example implementations of Markov chain Monte Carlo routines for computing statistical significances.Each section contains direct explanations with ready to deploy example code that you are free to use for your own investigations.Supplementary code can be accessed online from ftp://ftp.climatemodeling.org/pub/esg/.