// PAPER 02 — PHASE-SPACE RECONSTRUCTION
TED-PE
Thompson-Enhanced Derivative Pattern Engine.
Price oscillators as generators of a four-dimensional kinematic phase space — velocity, acceleration, jerk, snap. Historical analogs matched by weighted geometry, aggregated by similarity, tuned by a Bayesian bandit. Live. Drag the cloud.
// drag to rotate · d⁴ = color
Each point is a historical bar mapped into the reconstructed phase space. The amber trail is the active state vector. When the engine fires, it pulls the nearest historical analogs out of the cloud — the matches flash brighter than the rest.
Abstract
Standard financial technical analysis is primarily dogmatic and linear, assuming static thresholds that fail under non-stationary regimes. This paper introduces the Thompson-Enhanced Derivative Pattern Engine (TED-PE), which treats price oscillators as generators of a multi-dimensional kinematic phase space by extracting the first four temporal derivatives — Velocity (d¹), Acceleration (d²), Jerk (d³), Snap (d⁴). A causal, Gaussian-weighted Tompson Low-Pass Filter controls the noise amplification inherent in numerical differentiation, and rolling Z-score normalization makes the axes comparable. Historical structural analogs are located via a weighted Euclidean nearest-neighbor search in ℝ⁴ and aggregated into a similarity-weighted ensemble. Analog outcomes are combined into an Expected Value estimate — probability-weighted average win minus probability-weighted average loss — rather than a raw win rate. Dimension weights are set by a Bayesian Multi-Armed Bandit with an explicit forgetting factor. This document states plainly where the model's independence assumptions and validation gaps remain open.01The Dogma Problem
Classical indicators were designed in the mid-to-late 20th century for low-frequency, human-brokered markets. An RSI of 70 on a 5-minute chart of a high-beta crypto-asset during institutional accumulation is a completely different structural state than an RSI of 70 during low-volatility weekend retail drift. And by smoothing with trailing averages, traditional indicators introduce phase delay — rendering signals late at exactly the moments that matter.
The alternative is to study the rate of change of the market's internal momentum: transition from static horizontal lines to kinematic phase-space reconstruction.
02The Physics of Momentum — A Reversal in Five Phases
When a trend exhausts, the reversal is not instantaneous. It progresses sequentially through higher-order derivatives long before price or moving averages confirm:
The high-order derivative sequence contains the structural fingerprint of the reversal — before the baseline indicator crosses its threshold.
03Mathematical Engineering
3.1 — The Noise Problem
Each successive finite difference amplifies high-frequency noise exponentially:
By the fourth difference, a raw signal is typically overwhelmed by white noise. The filter is therefore not optional.
3.2 — Causal Gaussian-Weighted Filtering (Tompson Filter)
3.3 — Rolling Z-Score Normalization
Different indicators print on different scales; their derivatives cannot be compared natively. Each derivative is normalized:
04Phase-Space Reconstruction
The four normalized derivatives form a state vector in ℝ⁴:
The engine scans a historical search window for the closest states — a weighted Euclidean distance:
Distance converts to a similarity score, and the top-C analogs beyond a similarity threshold are aggregated:
Rather than a plain top-C average, each analog's contribution is weighted by its similarity — and outcomes are combined into an Expected Value, not a win rate:
05Self-Optimization — The Bayesian Bandit
Each derivative arm maintains a Beta posterior, updated by sign-of-derivative vs. subsequent price direction, with an explicit forgetting factor so stale regimes decay rather than accumulate:
The sampled/expected probabilities normalize to sum to 4.0, dynamically re-weighting the ωₖ in the distance metric.
// Live laboratory — analog matching in ℝ⁴
● LIVE
A synthetic oscillator replays in phase space. The active state (amber) drags its analog trail behind it; the C nearest historical states (cyan) flash as the engine finds them, and their projected returns roll the Expected Value readout. Adjust the search window and analog count and watch the ensemble change character.
06Live Evidence
Production output from the TED-PE indicator on TrendSpider — structural inflections clustering at swing highs and lows, with the four-derivative cascade in the sub-pane. Chart screenshot forthcoming.
07System Parameters
| System Parameter | Evaluated Value / Setting |
|---|---|
| Base Indicator | Commodity Channel Index (CCI) |
| Indicator Length | 12 bars |
| Filter | Tompson Low-Pass Filter (period 7) |
| Bayesian Bandit | Enabled (Z-score lookback 50) |
| Search Window | 1000 bars |
| Analog Matches | 7 analogs |
| Similarity Threshold | 0.60 |
| Projection Horizon | 5 bars |
08Known Limitations & Assumptions
- The Tompson filter is causal, not zero-lag; it carries ≈ (L−1)/2 bars of group delay.
- "Thompson Sampling" is a Normal-moment-matched approximation to a Beta draw — a deterministic posterior-mean mode is offered as an alternative.
- Derivative axes and bandit arms both assume independence that the kinematic-cascade premise itself contradicts. Mahalanobis distance and correlated-arm bandits are natural, unimplemented extensions.
- Analog sample sizes are typically small; win-rate/EV estimates are high-variance.
- No out-of-sample or walk-forward validation has been conducted — descriptive observations only.
- Transaction costs, slippage, and spread are not modeled in the EV or TP/SL calculations.
09Conclusion
TED-PE replaces dogmatic thresholds with geometry: a market state is a point in phase space, and a signal is a measured distance to the states that mattered before. The framework is honest about its approximations — the corrected filter causality, the moment-matched sampler, the independence assumptions — and the live implementation ships the full parameter surface on the store.
// NEXT FRAMEWORK
From phase space to material state.
CFE treats support and resistance as latent barriers with fatigue, damage, and healing — a level engine with a stress test built in.