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# lecture17 - Lecture 17: Spectral Analysis Steven Skiena...

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Lecture 17: Spectral Analysis Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794–4400 http://www.cs.sunysb.edu/ skiena

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Spectral Analysis Certain phenomena of financial (and other) time series data is best revealed in the frequency domain , or equivalently represented by their spectra . A duality transform is a one-to-one mathematical function that takes a mathematical object of type-1 and maps it to an equivalent type-2 mathematical object. Sample duality relations are point-line duality in compu- tational geometry, and Laplace transforms used solving differential equations. Such transforms are useful if there are interesting algorithms and tools for manipulating data of type 2.
Fourier Transforms Perhaps the most useful duality transform known is the Fourier transform for representing time-series data as the sums of sine and cosine functions. Its wide applicability is due to the existence of Fast Fourier Transform algorithm or FFT which computes what seems like an inherently quadratic function in O ( n lg n ) time.

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Filtering via the FFT On the left, we construct a time series of points sampled from a sine function with added random noise. On the right we take the Fourier transform of this series, plotting the coefficients of the resulting sine functions: 50 100 150 200 250 -1.5 -1 -0.5 0.5 1 1.5 50 100 150 200 250 2 4 6 8
Filtering – By eliminating undesirable high- and/or low- frequency components (i.e. dropping some of the sine functions) and taking an inverse Fourier transform to get us back into the time domain, we can filter a function to remove noise and other artifacts. Compression – A smoothed function less information than a noisy function, while retaining a similar appearance. We can perform lossy compression by eliminating the coefficients of sine functions that contribute relatively little to the function.

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A Fourier Transform Tale We add white noise to points sampled from the sum of two trigonometric series:
A Periodic Function, with White Noise The behavior of the sampled function remains apparent even in the presence of noise:

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## lecture17 - Lecture 17: Spectral Analysis Steven Skiena...

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