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Notes on - Sandra Chapman (MPAGS: 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 . 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 at high , and a long time interval at low frequencies; i.e. simple 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 DFT

Wavelet- note s

1 Notes on Wavelets- Sandra Chapman (MPAGS: Time series analysis)

So at its simplest, a 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: " T / 2 2 if t m 1 2 if t ! m ! ! m x(t ) = # Sme , fm = Sm = x(t )e dt T "!T / 2 m=!" 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 , time domain in a complete way (again, this partitioning can be considered as a filter/ 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 estimation:

Cone of influence: defined as the e-folding time for the of wavelet at each scale.

Estimate of Power (PSD): in the same manner as integrating a Fourier 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 process - details in Torrence and Compo (1998).

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