Notes on Wavelets- Sandra Chapman (MPAGS: Time series analysis) # $ ( ) = G f. y t

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1 Wavelets Recall: we can choose! t ) as basis on which we expand, ie: ) = y t ) = G! t ) y t! may be orthogonal chosen or appropriate properties. This is equivalent to the transorm: ) = G y t )!,t )d 2 it We have discussed!, t ) = e! or the Fourier transorm. Now choose dierent Kernel- in particular to achieve space-time localization. Main advantage- oers complete space-time localization which may deal with issues o nonstationarity) whilst retaining scale invariant property o the!. First, why not just use windowed) short time DFT to achieve space-time localization? Wavelets- we can optimize i.e. have a short time interval at high requencies, and a long time interval at low requencies; i.e. simple Wavelet can in principle be constructed as a band- pass Fourier process. A subset o wavelets are orthogonal energy preserving c. Parseval theorem) and have inverse transorms. Finite time domain DFT Wavelet- note scale parameter s 1

2 So at its simplest, a wavelet transorm is simply a collection o windowed band pass ilters applied to the Fourier transorm- and this is how wavelet transorms are oten computed as in Matlab). However we will want to impose some desirable properties, invertability orthogonality) and completeness. Continuous Fourier transorm: ) = S m e 2!i m t x t, m = m T m=! with orthogonality:! e i n!m)x!! S m = 1 T dx = 2! mn continuous Fourier transorm pair: Continuous Wavelet transorm: ) = S x t! T / 2 x t )e!2!i m t dt!t / 2 )e 2!it d S ) = x t )e!2!it dt! ) = x t W!,a x t) = 1 C ) *!,a t)dt 0 & ' )!!,a d! W!,a ) da + * a 2 Where the mother wavelet is!,a t) = 1 a! t ' & a ) where! is the shit parameter and a is the scale dilation) parameter we can generalize to have a scaling unction at)). Here,! * is the complex conjugate and! is the dual o the mother wavelet. For the transorm pair to work we need an orthogonality condition recall Fourier transorm pair): 1 & t '! ) a 3 ' a * +! & t! ) ' a * + dda =, t! t ' 0! ) This deines the dual. A subset o admissible inormation/energy preserving) wavelets are:! = C! 1! with 2 C! = d where is the FT o! 0 is a positive constant that depends on the chosen wavelet. 2

3 Properties o all) wavelets:! u)du = 0! 2 u)du = 1 vanishes at ininity and integrates to 1 energy preserving Choice o the dilation and shit parameter- we want to tile the requency, time domain in a complete way again, this partitioning can be considered as a ilter/convolution process). choose the dilation and translation parameters to completely cover the domain and have selsimilar property: so that! pq t a p = 2 p,! pq = 2 p q ) = 1 2 p! t 2 p q actually any a p = a 0 p can be used, general practice is a 0 = 2 ; also one can have at)). 2 p & ' 3

4 Making time discrete: Discrete Fourier Transorm: x k = 1 N!t S m =!t N 1 k=0 N 1 m=0 S m e 2!imk / N x k e 2!ikm/ N Discrete Wavelet Transorm DWT): W m, j =!t a 0 a j = a 0 j N k m x k * ' k =0 & a j j = a 0 a j 1!t As above, this is a convolution- thereore is realized as a set o Fourier) band pass ilters. * ) Some examples o mother wavelets: Note- Daubechies amily o mother wavelets and dilation scaling) unctions- property that it is always zero outside a ixed time doman) order o a ilter reers to the degree o the approximating polynomial in requency space- must exceed that o the signal. 4

5 Power spectrum estimation: Cone o inluence: deined as the e-olding time or the autocorrelation o wavelet power at each scale. Estimate o Power Spectral Density PSD): in the same manner as integrating a Fourier spectrogram across time to obtain an averaged PSD one can integrate the wavelet scaleogram across time: Mortlet wavelet PSD o the sunspot number, with normalized) requency=1/scale 5

6 Note the sel- similar nature o the scaling/dilation o the wavelet transorm results in uniorm binning in log space o requency Fourier has uniorm binning in linear requency space). We can deine a level o signiicance conidence level) w.r.t. a white noise process - details in Torrence and Compo 1998). 6