Wavelet Transforms in Time Series Analysis

Wavelet Transforms in time SeriesAnalysisAndrew TangbornGlobal Modeling and Assimilation Office, Goddard Space Flight fourier What is a Wavelet ?3. Continuous and discrete Wavelet Transforms4. Construction of Wavelets through dilation Example - Haar wavelets6. Daubechies Compactly Supported Data compression, efficient Soft Continuous Transform - Morlet Wavelet10. Applications to approximating error correlationsFourier Transforms A good way to understand how wavelets work and why they are useful is bycomparing them with fourier Transforms . The fourier Transform converts a time Series into the frequency domain:Continuous Transformof a function f(x): f( ) = Z f(x)e i xdxwhere f( ) represents thestrengthof the function at frequency , where Transformof a function f(x): f(k) = Z f(x)e ikxdxwherekis a discrete discrete dataf(xj),j= 1,..,N fk=NXj=1fje( i2 (k 1)(j 1)/N) The Fast fourier Transform (FFT) iso(NlogN) : Single Frequency Signalf(t) = sin(2 t)012345678910 1 Coefficient Imaginary CoefficientFourier Transform discrete fourier Transform (DFT) locates the single frequency and re-flection.

• The Fourier Transform converts a time series into the frequency domain: Continuous Transform of a function f(x): fˆ(ω) = Z∞ −∞ f(x)e−iωxdx where fˆ(ω) represents the strength of the function at frequency ω, where ω is continuous. Discrete Transform of a function f(x): fˆ(k) = Z∞ −∞ f(x)e−ikxdx where kis a discrete ...

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  Series, Time, Discrete, Time series, Fourier

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