Wavelet-Domain Regime Detection with the MODWT

"The market changed" is the kind of statement that is obviously true and almost useless. Changed how fast? Changed in the minute-scale chop, or in the multi-day trend? A single rolling volatility number cannot tell you, because it sums all of those timescales into one figure and throws the decomposition away.

The maximal-overlap discrete wavelet transform (MODWT) keeps the decomposition. This post is about why that decomposition is the right coordinate system for regime detection, and how MorphIQ Labs uses it.

Why not just the DWT

The ordinary discrete wavelet transform decimates: each scale has half the samples of the one below it. That is efficient, but for time-series analysis it has two costs we are not willing to pay — the transform is not shift-invariant (sliding the window by one bar reshuffles the coefficients), and the coefficients are not aligned in time with the events that produced them.

The MODWT drops the decimation. Every scale keeps the full sample count $N$. You give up orthogonality and the $O(N)$ flop count of the DWT for an $O(N\log N)$ transform that is shift-invariant and time-aligned. For research on a return series — where we constantly slide windows and want a coefficient at time $t$ to mean something at time $t$ — that is the right trade.

The decomposition

Let $X_t$ be a return series. The MODWT to level $J$ produces, at each scale $j = 1, \dots, J$, a vector of wavelet coefficients $\widetilde{W}_{j,t}$ capturing variation at physical scale $\tau_j = 2^{j-1}$, plus one vector of scaling coefficients $\widetilde{V}_{J,t}$ holding everything slower than $\tau_J$.

The property that makes this worth doing is an exact additive decomposition of the sample variance. With no approximation,

$$\hat{\sigma}_X^2 \;=\; \sum_{j=1}^{J} \hat{\nu}_X^2(\tau_j) \;+\; \hat{\nu}_V^2$$

where the wavelet variance at scale $\tau_j$ is

$$\hat{\nu}_X^2(\tau_j) \;=\; \frac{1}{N_j}\sum_{t} \widetilde{W}_{j,t}^{\,2}$$

Total risk is partitioned, scale by scale, with the pieces summing back to the whole. That is the sentence that does the work: a regime is no longer a vague mood, it is a shape of the vector $\bigl(\hat\nu_X^2(\tau_1), \dots, \hat\nu_X^2(\tau_J)\bigr)$.

Regimes as energy ratios

Define the energy share at scale $j$ over a rolling window:

$$\pi_{j,t} \;=\; \frac{\hat{\nu}_X^2(\tau_j)}{\sum_{i=1}^{J}\hat{\nu}_X^2(\tau_i)}$$

The vector $\boldsymbol{\pi}_t = (\pi_{1,t}, \dots, \pi_{J,t})$ lives on the simplex and is scale-of-volatility invariant — multiply every return by a constant and $\boldsymbol{\pi}_t$ does not move. It describes the texture of the market, not its loudness. Two readings tend to matter:

• High-frequency-dominated — energy concentrated in $\tau_1, \tau_2$. The fast scales are loud relative to the slow ones: choppy, mean-reverting, microstructure-heavy tape. Short-horizon signals are favored; trend-following bleeds.
• Low-frequency-dominated — energy concentrated in $\tau_{J-1}, \tau_J$. A persistent, directional regime where slow scales carry the variance. Trend signals have room; fast mean-reversion is fighting the drift.

A scalar that compresses this is the high-to-low energy ratio

$$R_t \;=\; \frac{\sum_{j \le j^\star}\hat{\nu}_X^2(\tau_j)} {\sum_{j > j^\star}\hat{\nu}_X^2(\tau_j)}$$

with the split scale $j^\star$ chosen for the instrument's bar size. Crossings of $\log R_t$ through a hysteresis band give discrete regime labels without the flip-flopping a single threshold produces.

Putting an error bar on it

A regime label with no confidence interval is a guess wearing a lab coat. The wavelet variance has tractable sampling theory: for a Gaussian process the estimator is approximately

$$\hat{\nu}_X^2(\tau_j) \;\overset{\cdot}{\sim}\; \text{Normal}\!\left(\nu_X^2(\tau_j),\; \frac{2\,A_j}{N_j}\right)$$

where $A_j$ is an integral of the squared spectral density over the scale's octave band. In practice we estimate the variance of the estimator directly, either from the autocovariance of the squared wavelet coefficients or by a block bootstrap over $\widetilde{W}_{j,t}$, and propagate it into a confidence band on $\boldsymbol{\pi}_t$.

Practical notes

A few things that matter once this leaves the notebook:

• Filter choice. The Daubechies least-asymmetric LA(8) filter is the default — long enough for clean scale separation, short enough that boundary contamination stays contained.
• Boundary coefficients. Some MODWT coefficients near each end depend on circular wrapping. They are excluded from the variance sums; $N_j$ counts only interior coefficients. Pretending otherwise quietly biases the most recent — and most decision-relevant — estimates.
• Determinism. The whole transform is a fixed sequence of FIR convolutions. Given the same returns it returns the same coefficients bit-for-bit, which is what lets the regime label sit inside the Anvil event log and replay exactly.

Where this goes

Energy shares are a descriptive statistic. The open work is the dynamics: model $\boldsymbol{\pi}_t$ as a hidden-state process so the system anticipates a regime turn rather than confirming it after the fact. A Dirichlet state-space model on the simplex is the current candidate. That, and the link between wavelet scales and the volatility-surface term structure from the Ferro risk work, are the next two write-ups.