Software issues in wavelet analysis of financial data - Numerical ...

Numerical Algorithms GroupMathematics and technology for optimized performanceSoftware Issues in Wavelet Analysis ofFinancial DataRobert TongResults Matter. Trust NAG.

Overview Why Use Wavelets? Wavelet transforms Multi-Resolution Analysis Software implementations and algorithms Choosing a wavelet method Some ApplicationsResults Matter. Trust NAG.2

Wavelet Transformsx(t),Decompose time series, by convolution with dilated and translated motherwavelet,ψ (t ) Continuous (CWT) Discrete (DWT)dψ( uu , s, s )( t )∞= ∫− ∞=1x ( t ) ψ⎛ψ ⎜⎝e.g. Morlet wavelet,ψ ( t)=Results Matter. Trust NAG.s1e2πtu , s−s− ikt −te2( t ) dtu/ 2⎞⎟⎠,Filter pair: G – high passH – low passwith D – down-sampling(Hx)k= ∑ hn−nk4xn

Wavelet Transforms (2) CWT requires:∫ ∞ −∞ψ ( t)dt= 0∫ ∞ −∞| ψ ( t)|2 dt= 1 DWT (orthogonal filter pair) requires:∑ h nh n + 2 j=n0,Results Matter. Trust NAG.∑2h n= 1,∑ g g n n+2 j= 0, ∑2g n= 1,nnngnn= ( −1)h1−n∑ h g n n+j= 0 2n5

Wavelet Functions1Morlet (k=5)0.2Daubechies:DB40.150.50.10.0500-0.5-0.05-0.1-1-8 -6 -4 -2 0 2 4 6 8Gaussian 3 rd derivative-0.15DB120 200 400 600 800 100010.10.50.0500-0.5-0.05-1-4 -2 0 2 4Results Matter. Trust NAG.-0.10 200 400 600 800 10006

Multi-Resolution AnalysisDiscrete Wavelet Transformx(t)Gx | 2d 1d 2Hx | 2Gs 1 | 2Hs 1 | 2Gs 2 | 2Results Matter. Trust NAG.Hs 2 | 27

Stationary DWT (SDWT)Note: DWT is NOT translation invariantchoosing odd entries in series, in place of even ones when downsamplinggives a different orthogonal transformationDWT MRA – choice of shift at each level gives multiple possible setsof coefficientsSDWT – NO down-sampling,pad filters with zeros in MRAincludes all DWT MRA possibilitiestranslation invariant, but increases storagecan relate wavelet coefficients to dataResults Matter. Trust NAG.9

Translation invariant SDWTx(t)d 1x(t)d 2d 3d 4s 4Results Matter. Trust NAG.10

Choosing a Wavelet Method CWTContinuous transformVisualise as surface DWT/MRADiscrete, multiresolutionEfficient storage of signal Matching PursuitAdapt basis to dataAt each level of MRA choosewaveform to minimise residualpassed to next level SDWTTranslation invariantNo down-samplingResults Matter. Trust NAG.11

CWT: how can quantitative information be obtained?Results Matter. Trust NAG.12

CWT (2) Find common normalisation for wavelet spectrum – ensurewavelet has unit energy at each scaleChoice of wavelet:non-orthogonal is useful for time series, but highly redundantChoice of scales:can use arbitrary set of scales to show structureCone of influence:for finite length series defines where edge effects occurRelate to Fourier frequencye.g. see Torrence and Compo (1998)Results Matter. Trust NAG.13

Software implementation and algorithms Reproducibility is desirable –algorithms precisely defined to allow independent implementations –Taswell (1998), c.f. Buckheit and Donoho (1995) Edge effects –contaminate ends of transform for finite signals – various end conditionsused to reduce their effect: periodic extension, reflection, zero-padding … CWT –implement quadrature and convolution in time domain orelse convolution with Fourier Transform of wavelet in frequency domainParallel implementationResults Matter. Trust NAG.14

Implementation and algorithms (2) Definition of forward and inverse transforms –DWT orthogonality conditions allow for different choices of forward andinverse transforms Pre-processing of data –data may need cleaning, interpolation to produce homogeneous series, …Results Matter. Trust NAG.15

Applications De-noising Identifying seasonality Self-similarity Prediction Estimation of varianceResults Matter. Trust NAG.16

De-NoisingTransform data into wavelet domainApply thresholding – suppress smallest coefficientsTransform backUse:DWT for efficient storage – SDWT to align with dataDe-noised data can help modelling of underlying structuree.g. Capobianco (1997) – analysis of Nikkei indexDonoho and Johnstone (1998)Results Matter. Trust NAG.17

Identifying Seasonality Apply SDWT to data Wavelet detail coefficients at a given level capture a particular rangeof frequencies Identify detail coefficients carrying seasonal periodicity Filter out seasonal effectse.g. Gencay et al. (2001) for application to FX returnsResults Matter. Trust NAG.18

Self-similarityScalogram represents energy of series in the wavelet coefficientse.g. Jamdee and Los (2004) use Morlet wavelet CWT for analysis of interestrate series and calculation of Hurst exponentPredictionApply backward looking wavelet MRA to time seriesUse wavelet coefficients as input to a neural network model forpredictione.g. Renaud et al. (2004)Results Matter. Trust NAG.19

Estimation of VarianceFor DWT coefficients wSDWT coefficients w s||2 2x || = || w || =||sw||2Since,2|| x || ∝ Var(x)The wavelet transform provides an alternative representation of thevariance (Percival, 1995)Results Matter. Trust NAG.20

Summary1) Wavelet transforms provide a rigorous framework for data analysisin time and frequency2) Software implementations vary in their choices of transformdefinitions3) Wavelet analysis provides an important tool for determining thestructure of time series arising in financeResults Matter. Trust NAG.21