Nontechnical introduction to wavelets Continuous wavelet transforms Fourier versus wavelets - examples A major part of economic time series analysis is done in the time or frequency domain separately. Wavelet analysis combines these two fundamental approaches allowing study of the time series in the time-frequency domain. Wavelets allow us to study the frequency components of time series without losing the time information. Decomposition using wavelets shows dynamics of the process on various scales. Wavelets have significant advantages over basic Fourier analysis when the object under study is locally stationary and inhomogeneous. Recent research: Long memory estimators (Fay et. al. 2009 JoE, Nielsen and Frederiksen 2005), Unit root tests, cointegration, fractional cointegration (Fan Gencay 2010, Fan and Whitcher 2003), Wavelet coherence dynamic correlations (Vacha Barunik 2012), Quadratic variation and covariation.
2 QF II L 8 Wavelets.nb First used in geophysics (Jumps that were not isolated) The Fourier transform cannot detected since its bases are not compact. Haar wavelet has compact support
QF II L 8 Wavelets.nb 3 A wavelet is a real-valued square integrable function ψ L 2 (R) A wavelet has two control parameters, u and s. The location parameter u determines the exact position of the wavelet and the scale parameter s defines how the wavelet is stretched or dilated. Scale has an inverse relation to frequency; thus lower (higher) scale means a more (less) compressed wavelet, which is able to detect higher (lower) frequencies of a time series. Figure is from: Cazelles, Bernard, et al. Wavelet analysis of ecological time series. Oecologia 156.2 (2008): 287-304. Admissibility condition - wavelet does not have a zero frequency component, therefore it has zero mean. It also ensures reconstruction of a time series from its wavelet transform.
4 QF II L 8 Wavelets.nb The wavelet is usually normalized to have unit energy. It Implies that the wavelet makes some excursion away from zero. Proposed by Ingrid Daubechies. Derived with the criterion of a compactly supported function with the maximum number of vanishing moments. Longer supports than Haar Number of vanishing moments is half the length of the filter D(4) improves frequency domain characteristics of Haar.
QF II L 8 Wavelets.nb 5 The continuous wavelet transform Wx (u, s) is obtained by projecting a specific wavelet ψ(.) onto the examined time series x(t), i.e., We decompose and then subsequently perfectly reconstruct a time series x(t): The continuous wavelet transform preserves the energy of the examined time series,
6 QF II L 8 Wavelets.nb plots the transform coefficients as rows of colorized rectangles, in which large absolute values are shown darker and each subsequent row corresponds to different wavelet index specifications.
QF II L 8 Wavelets.nb 7 = - [ ] = [ ] [ { } ] 1.0 0.5 200 400 600 800 1000-0.5-1.0 scale 1 2 3 4 5 6 7 8 9 200 400 600 800 1000 time
8 QF II L 8 Wavelets.nb = [ π ] < [ π ] < [ π ] < [ π ] [ ] = [ [ ] { }] [ ] 1.0 0.5-0.5-1.0 200 400 600 800 1000 1 2 3 4 5 6 7 8 9 200 400 600 800 1000 The wavelet power spectrum defined as Wx (u, s) 2 measures the local variance of the time series x(t) at various scales s. We get energy (variance) decomposition with a good time localization of the time series under study. We test the statistical significance of the peaks in the power spectrum against a null hypothesis that the examined time series is generated by a white noise process. Statistically significant areas at the 5% significance level are bordered by a black bold line.
QF II L 8 Wavelets.nb 9 The Fourier analysis is convenient when the time series under study stationary. Financial and economic time series usually exhibit volatility clustering, abrupt changes, trends. Wavelets can deal with time varying characteristics of real time series. Wavelets are ideal for studying nonstationary times series. Using wavelets we do not lose time information. Problem with forecasting
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12 QF II L 8 Wavelets.nb Aguiar, Luís Francisco, and Maria Joana Soares. The Continuous Wavelet Transform: A Primer. No. 23/2010. NIPE-Universidade do Minho, 2010. Mallat, S. (1989) A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 674693. Ramsey, J.B. (2002) Wavelets in economics and finance: Past and future. Studies in Nonlinear Dynamics & Econometrics 3, 129. Daubechies, I. (1992) Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM. Fan, Y. (2003) On the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes. IEEE Transactions on Information Theory 49, 516521. Percival, D.B. (1995) On estimation of the wavelet variance. Biometrica 82, 619631. Percival, D.B. and H.O. Mofjeld (1997) Analysis of subtidal coastal sea level fluctuations using wavelets. Journal of the American Statistical Association 92, 868880.
QF II L 8 Wavelets.nb 13 Percival, D.B. and A.T. Walden (2000) Wavelet Methods for Time Series Analysis. Cambridge Press.