Transcription of Wavelet Transforms in Time Series Analysis
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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.
• It is also a tool for decomposing a signal by location and frequency. Consider the Fourier transform: A signal is only decomposed into its frequency components. No information is extracted about location and time. • What happens when applying a Fourier transform to a signal that has a time varying frequency? 0 2 4 6 8 10 12 14 16 18 20 ...
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