H21 - as a linear combination of sinusoids , the idea...

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H21 Wavelet-based time series analysis One (unconventional) method of describing a function () f x is via a series expansion. Suppose that the domain of f is [0 . Then ,1] 1 () () jj j f xx θϕ = = , where 1 0 () () jj f xx d θϕ = x . () j x ϕ are the elements of the orthonormal system, and the j θ are called the Fourier coefficients . A system of functions is called orthonormal if 1 0 () () 0 ij xx d x ϕϕ = for ij and 1 2 0 (( ) ) 1 j x dx ϕ = for all j . In practice, a truncated (finite) orthogonal series (or so-called partial sum 1 () () J Jj j j f xx θϕ = = is used to approximate f . The parameter is called the cutoff J The classical orthonormal trigonometric Fourier system is defined by 02 1 2 () :1 , () : 2s i n ( 2 ) , () : 2c o s ( 2 ) , 1 ,2 , . . . jj j j x j ϕϕ π ϕ π == = = . () f x is [0, 2 ] π , 1/ 2 () ( 2 ) k k f xc π −− =−∞ = e . Example 1
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Unformatted text preview: as a linear combination of sinusoids , the idea underlying wavelet analysis is to represent the function as a linear combination of wavelets . In Fourier analysis, each sinusoid is associated with a particular frequency ω . In contrast, each wavelet is associated with two independent variables, time and scale t τ , so that a wavelet is essentially nonzero only inside a particular interval, [ , ] t t − + . Within the interval, a wavelet looks like a "small wave" centered at t . By expanding a time series into wavelets, one can learn how it varies on particular scales across time. Haar wavelets Example 2 How a Haar system approximates the functions from Example 1. Example 3 Northern Hemisphere temperature series. Wavelet Variance...
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This note was uploaded on 09/23/2009 for the course MATH 200 taught by Professor Gur during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.

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