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Check out the new article: Regime Discovery by Structure: Implementing Toeplitz Inverse Covariance Clustering (TICC).

This article presents a full MQL5 pipeline for Toeplitz Inverse Covariance Clustering: stacked observations, ADMM‑based graphical lasso with block‑Toeplitz constraints, and dynamic‑programming regime assignment. It separates structure from volatility and conditions away the shared USD leg. The fitted regimes are displayed causally as a non-repainting ribbon with a dependency graph for the active state.

Take five FX majors: EURUSD, GBPUSD, USDJPY, AUDUSD and USDCHF. Their correlation matrix holds large numbers almost everywhere and almost no information, because four of the five quote the same currency on one side. The narrower question is whether EURUSD still tells us anything about USDCHF once we know what the other three did on this bar. That is conditional dependence, and the precision matrix answers it.

Write the covariance matrix of the returns as S and its inverse as theta. The off-diagonal entries of theta convert into partial correlations:

p_ij = -theta_ij / sqrt( theta_ii * theta_jj )

That is the correlation between i and j with every other symbol's linear influence removed: scale free, and exactly zero under conditional independence. Those zeros are the point, and they are also the fragility: sampling noise leaves a directly inverted covariance with small non-zero numbers everywhere. The graphical lasso fixes it with an L1 penalty on the off-diagonal entries, so an absent edge claims conditional independence rather than weak correlation.

Fig. 1. The same five symbols under correlation and under partial correlation. Conditioning strips out most of the shared dollar leg and leaves a sparse graph.

Conditioning is linear. A common factor such as the dollar is shrunk, not deleted: with four other USD pairs in the basket it is pinned down well, so what remains of it is small, but nonlinear co-movement and anything driven from outside the basket stay in the estimate.

All of that concerns a single instant. What separates one regime from another is largely which symbol moves first, which needs a change of representation.

Author: Muhammad Minhas Qamar