Wavelet Analysis of Time Series

Standard spectral tools (like the Fourier transform) tell you which frequencies are present in a series overall — say, "there's a strong weekly cycle" — but they answer this as a single global summary, blind to when that cycle was strong. If a stock's return series has a pronounced weekly pattern during calm markets but that pattern vanishes entirely during a crisis month, a plain Fourier decomposition mashes both regimes together into one "average" frequency picture, hiding exactly the kind of regime change a trader most wants to see. Wavelet analysis solves this by decomposing a series into different time scales (fast wiggles versus slow trends) while keeping track of where in time each scale's activity actually occurred.

An analogy: a musical score versus a single sustained chord

A Fourier transform of an entire song tells you which notes appear overall, roughly how strongly — but it can't tell you the verse was in a minor key and the chorus switched to major. A musical score, by contrast, tells you exactly which notes play at exactly which moments. Wavelet analysis is closer to reading the score: instead of one global answer to "what frequencies are present," it gives a two-dimensional map — frequency (or scale) on one axis, time on the other — showing which scales of movement were active at which moments, letting you see a market's "melody" change from a slow drift in January to fast, choppy oscillation in March.

The mechanics, one symbol at a time

A wavelet transform correlates the series with a small, localized wave-shaped function (the "mother wavelet") $\psi$, stretched and shifted across the data:

$$W(s, \tau) = \int y(t)\, \frac{1}{\sqrt{s}}\, \psi\!\left(\frac{t-\tau}{s}\right) dt,$$

where $s$ is the scale (how stretched the wavelet is — large $s$ picks up slow, low-frequency movement; small $s$ picks up fast, high-frequency wiggles), $\tau$ is the time location (where the wavelet is centered), and $y(t)$ is the observed series. In plain English: $W(s, \tau)$ measures how strongly a wiggle of a given size (scale $s$) is present around a given moment in time ($\tau$) — unlike a Fourier component, which is a wave stretching across the entire series with no notion of "where." Plotting $|W(s,\tau)|^2$ (the wavelet power) across both $s$ and $\tau$ produces a scalogram — a heatmap showing exactly which scales were active exactly when, letting a trend that dominates for three months and then vanishes show up clearly as a localized bright patch rather than being smeared across the whole timeline.

Worked example 1: reading a scalogram

Suppose a daily equity return series is decomposed by wavelet, and the scalogram shows strong power at the "weekly" scale (roughly 5-day cycles) concentrated in January and February, fading to near-zero from March through June, then reappearing weakly in a "monthly" scale (roughly 20-day cycles) starting in July. A Fourier periodogram over the full six months might report modest average power at both the weekly and monthly frequencies, understating how strong the weekly effect actually was in early months and completely hiding that it disappeared afterward and was replaced by a different, slower rhythm. The wavelet view tells a trader: "the weekly-rebalancing-driven pattern from Q1 is gone by Q2 — don't keep trading on it."

Worked example 2: variance by scale

Suppose a wavelet variance decomposition of a volatility series attributes total variance as: scale 1 (2-4 day cycles) = 15%, scale 2 (4-8 day) = 35%, scale 3 (8-16 day) = 30%, scale 4 (16-32 day) = 20%. This tells you the bulk of the series' variability (65%) lives in the 4-to-16-day range — not in ultra-fast daily noise, nor in slow monthly drift. If a risk model were built using only daily-frequency inputs it could be missing where most of the actual variance concentrates, guiding a modeler toward sampling or smoothing choices matched to where the signal actually lives, roughly a 1-to-3-week horizon in this example.

What this means in practice

Wavelet analysis is used in finance to detect regime changes in cyclical behavior, to build multi-horizon risk decompositions (how much of a portfolio's risk comes from fast noise versus slow trends), and to denoise series while preserving structure at the scale of interest. It complements the periodogram (see The Periodogram and Spectral Density) precisely where that tool's blindness to time-location becomes a real limitation — non-stationary series where the relevant cycles themselves change over the sample.