Historical Coefficient Fields: A Controlled Evaluation of Geometric Market-Memory Representations for Financial Forecasting

Technical Summary: Historical Coefficient Fields and Geometric Market Memory

Problem Statement
Financial forecasting typically relies on chronological representations of history, such as lagged prices, rolling statistics, or latent sequences learned by deep learning models. However, market states that are distant in time may be structurally similar, while consecutive sessions may differ significantly. This observation motivates the hypothesis that organizing historical data by structural similarity in a feature space, rather than by calendar time, might improve directional forecasting. The paper investigates whether representing the recent history of a financial instrument as a geometric field of normalized coefficient vectors yields predictive advantages over standard chronological baselines.

Methodology
The authors propose the Historical Coefficient Field (HCF), a geometric representation of market memory. The methodology is defined as follows:

1. Input Representation: Each trading session is encoded as an open-relative coefficient vector ($C_t$), consisting of normalized geometric ratios (upper, lower, and vertical excursions relative to the opening price). This ensures scale invariance.
2. Field Construction: The HCF is defined as a point cloud $F_t^{(N)} = (H_t^{(N)}, d)$, where $H_t^{(N)}$ is the set of the $N$ most recent coefficient vectors and $d$ is a Euclidean metric. The organization of this cloud, rather than its chronological index, is the object of study.
3. Geometric Descriptors: To summarize the cloud's structure, a set of seven causal descriptors is computed over a strictly past rolling window:
   • Local density ($\rho$)
   • Persistence ($P$)
   • Dispersion ($D$)
   • Trajectory stability ($S$)
   • Nearest-analogue distance ($\eta$)
   • Partition entropy ($H$)
   • Geometric energy ($E$)
   • Note: The authors explicitly note that geometric energy is algebraically redundant ($E = D^2$) and that dispersion, stability, and energy form a highly correlated cluster.
4. Evaluation Protocol: A strictly controlled, representation-only protocol is employed to isolate the effect of the input representation.
   • Fixed Variables: The learning algorithms (Logistic Regression, Random Forest, Gradient Boosting), the walk-forward data splits (expanding window, minimum 750 sessions, refit every 252), and the target variables are held constant.
   • Variable: Only the input representation changes. Four representations are tested:
     • R1: Lagged returns (10 lags).
     • R2: Rolling statistics (multi-horizon return, volatility, momentum).
     • R3: Raw open-relative coefficient vector ($C_t$).
     • R4: HCF ($C_t$ + geometric descriptors).
   • Assets: Four public instruments with varying price scales: Gold (GC=F), S&P 500 (^GSPC), EUR/USD (6E=F), and Bitcoin (BTC-USD).
   • Targets: (1) The sign of the next-session return (directional forecasting); (2) Volatility expansion (whether the next session's normalized range exceeds its 21-session median).
   • Baseline: Performance is measured against the majority-class base rate (the accuracy of always predicting the more frequent direction) to prevent trend-driven inflation of accuracy metrics.

Key Results
The empirical evaluation yields a robust negative result:

• Directional Forecasting: No representation, across any of the three models or four instruments, meaningfully exceeds the majority-class base rate (averaged at 52.0%). All accuracy figures fall within the 50.9–52.4% range, consistent with random noise.
• Impact of Geometric Descriptors: The proposed HCF representation (R4) does not improve upon the raw coefficient vector (R3). In fact, adding the geometric descriptors slightly degrades directional accuracy (e.g., 51.6% vs. 52.4% for Logistic Regression on average).
• Descriptor Redundancy: Correlation analysis reveals that the proposed descriptor set is not independent. Dispersion, trajectory stability, and geometric energy are mutually correlated at 0.95–0.98, effectively collapsing three descriptors into a single "overall spread" quantity.
• Volatility Expansion Target: On the more predictable target of volatility expansion (where signal exists and all representations beat the 51.4% base rate), the geometric representations (R3 and R4) are inferior to simple rolling statistics (R2), which achieve ~57% accuracy. The geometric organization is not merely redundant but actively underperforms standard features where signal is present.
• Robustness: The negative results hold across different history window lengths ($N \in \{126, 252, 504\}$ sessions).

Key Contributions
The paper frames its contributions not as a new predictive theory, but as a rigorous empirical correction and a methodological template:

1. Definition of HCF: A formal definition of the Historical Coefficient Field and a compact set of geometric descriptors, explicitly positioned against established concepts like delay-coordinate embedding, analog forecasting, and kernel density estimation.
2. Controlled Evaluation Protocol: A "representation-only" experimental design that holds models and data splits fixed, ensuring that any performance differences are attributable solely to the input representation.
3. Demonstrated Negative Result: The honest reporting that, on the tested assets and protocol, the geometric organization of market memory carries no directional information beyond simple baselines. The paper argues that a more elaborate geometric bookkeeping of a signal-free representation does not create signal.

Significance and Claims
The authors claim significance primarily through the rigor of the negative result and the methodological discipline applied:

• Prevention of Unverified Theory: The study prevents the construction of unverifiable geometric "theories" based on representations that carry no directional signal. It establishes that geometric reorganization of market memory is not a shortcut to predictability.
• Evaluation Standard: The paper supplies a reusable, controlled, base-rate-referenced, and model-independent template for assessing future representations of financial history.
• Modest Scope: The authors explicitly state that their result is a statement about this specific representation family on these specific tasks. They do not claim a universal impossibility of geometric forecasting but argue that any future geometric representation must first pass this same controlled standard before richer structures are built upon it.

In summary, the paper concludes that while the geometric organization of market memory is a valid conceptual framework, it fails to provide predictive value for directional forecasting or volatility expansion compared to simple chronological baselines and rolling statistics.