0.1 Wavelet Analysis
Fourier-based tools (spectral density, periodogram) assume the frequency content of a series is constant over time (stationarity). Many real series — financial volatility, climate signals, EEG — have time-varying frequency content, motivating wavelet analysis, which decomposes a signal jointly in time and scale (frequency).
Given a mother wavelet \(\psi\) (a localized, oscillating, mean-zero function), the continuous wavelet transform of \(x(t)\) is \[ W(a,b) = \frac{1}{\sqrt{a}}\int x(t)\,\overline{\psi\!\left(\frac{t-b}{a}\right)}\,dt, \] where \(a>0\) is the scale (inverse frequency) and \(b\) is the time location. \(|W(a,b)|^2\) is the wavelet power spectrum, showing how power at scale \(a\) evolves through time — a time-localized analogue of the periodogram.
Discrete wavelet transform (DWT). Using dyadic scales \(a=2^j\) and an orthonormal wavelet basis, the DWT decomposes a series of length \(n\) into detail coefficients at each scale \(j\) (capturing variation at resolution \(2^j\)) plus a smooth approximation, analogous to a multiresolution version of Time Series Decomposition.
Applications. Detecting and localizing structural breaks or regime changes (complementing Regime-Switching Models), denoising, multiresolution analysis of realized volatility, and time-frequency analysis of macroeconomic and climate series.