Notes on Wavelets- Sandra Chapman (MPAGS: Time series analysis) Wavelets Recall: we can choose ! f (t ) as basis on which we expand, ie: y(t ) = " y f (t ) = "G f ! f (t ) f f ! f may be orthogonal – chosen for "appropriate" properties. # This is equivalent to the transform: y(t ) = $ G( f )!( f ,t )d f "# We have discussed !( f ,t ) = e2!ift for the Fourier transform. Now choose different Kernel- in particular to achieve space-time localization. Main advantage- offers complete space-time localization (which may deal with issues of non- stationarity) whilst retaining scale invariant property of 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 frequencies, and a long time interval at low frequencies; i.e. simple Wavelet can in principle be constructed as a band- pass Fourier process. A subset of wavelets are orthogonal (energy preserving c.f Parseval theorem) and have inverse transforms. Finite time domain DFT Wavelet- note scale parameter s 1 Notes on Wavelets- Sandra Chapman (MPAGS: Time series analysis) So at its simplest, a wavelet transform is simply a collection of windowed band pass filters applied to the Fourier transform- and this is how wavelet transforms are often computed (as in Matlab). However we will want to impose some desirable properties, invertability (orthogonality) and completeness. Continuous Fourier transform: " m 1 T / 2 2!ifmt !2!ifmt x(t ) = Sme , fm = Sm = x(t )e dt # "!T / 2 m=!" T T ! i( n!m)x with orthogonality: e dx = 2!"mn "!! " x(t ) = # S( f )e2!ift df continuous Fourier transform pair: !" " S( f ) = # x(t )e!2!ift dt !" Continuous Wavelet transform: $ W !,a = x t " * t dt ( ) % ( ) ! ,a ( ) #$ 1 $ & $ ) da x(t) = ( W (!,a)"! ! ,a d! + 2 C % % a " 0 '#$ * 1 $ t # " ' Where the mother wavelet is ! " ,a (t) = ! & ) where ! is the shift parameter and a a % a ( is the scale (dilation) parameter (we can generalize to have a scaling function a(t)). Here, * is the complex conjugate and ! is the dual of the mother wavelet. For the transform ! ! pair to ‘work’ we need an orthogonality condition (recall Fourier transform pair): " " 1 & t '! % ) & t ! % ) $ $! d%da = , (t ! t ') # # a3 '( a *+ '( a *+ 0 !" This defines the dual. A subset of ‘admissible’ (information/energy preserving) wavelets are: !! = C "1! ! with 2 # " C = df where " is the FT of ! ! $ 0 f is a positive constant that depends on the chosen wavelet. 2 Notes on Wavelets- Sandra Chapman (MPAGS: Time series analysis) Properties of (all) wavelets: # $ ! (u)du = 0 "# vanishes at infinity and integrates to 1 – energy preserving # $ ! 2 (u)du = 1 "# Choice of the dilation and shift parameter- we want to ‘tile’ the frequency, time domain in a complete way (again, this partitioning can be considered as a filter/convolution process). choose the dilation and translation parameters to completely cover the domain and have self- similar property: p p ap = 2 ,! pq = 2 q 1 # t " 2 p q& so that ! t = ! pq ( ) p % p ( 2 $ 2 ' p (actually any ap = a0 can be used, general practice is a0 = 2 ; also one can have a(t)) . 3 Notes on Wavelets- Sandra Chapman (MPAGS: Time series analysis) Making time discrete: 1 N "1 x = S e2!imk / N k N!t # m Discrete Fourier Transform: m=0 N "1 "2!ikm/ N Sm = !t# xk e k=0 N !t * % k $ m( Wm, j = " xk# ' * a0 k =0 & a j ) Discrete Wavelet Transform (DWT): j a0 1 a j = a0 f j = a j !t As above, this is a convolution- therefore is realized as a set of (Fourier) band pass filters. Some examples of mother wavelets: (Note- Daubechies family of mother wavelets and dilation (scaling) functions- property that it is always zero outside a fixed time doman) ‘order’ of a filter refers to the degree of the approximating polynomial in frequency space- must exceed that of the signal. 4 Notes on Wavelets- Sandra Chapman (MPAGS: Time series analysis) Power spectrum estimation: Cone of influence: defined as the e-folding time for the autocorrelation of wavelet power at each scale. Estimate of 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 of the sunspot number, with (normalized) frequency=1/scale 5 Notes on Wavelets- Sandra Chapman (MPAGS: Time series analysis) Note the self- similar nature of the scaling/dilation of the wavelet transform results in uniform binning in log space of frequency (Fourier has uniform binning in linear frequency space). We can define a level of significance (confidence level) w.r.t. a white noise process - details in Torrence and Compo (1998). 6 .