Ergodic Theory And TDA Indicator
- Индикаторы
- Версия: 2.10
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ERGODIC THEORY + TOPOLOGICAL DATA ANALYSIS (TDA)
by Ponokawan
We combined Topological Data Analysis (TDA) with Ergodic Theory (specifically Birkhoff's Ergodic Theorem) to build the Ergodic Topological Dynamics (ETD) model.
The Correlation Between Ergodic Theory, TDA, & Financial Markets
1. Non-Ergodicity in Financial Markets
A system is ergodic if its average over time equals its average across all space (ensemble average):
Time Average f ˉ = lim T → ∞ 1 T ∫ 0 T f ( x ( t ) ) d t ≡ Ensemble Average ⟨ f ⟩ = ∫ Ω f ( x ) d μ ( x ) Time Average fˉ=T→∞limT1∫0Tf(x(t))dt≡Ensemble Average ⟨f⟩=∫Ωf(x)dμ(x)
Financial markets are strictly non-ergodic: an individual trader's price trajectory over time does not equal the group average of all market participants at a single moment.
2. The Birkhoff Ergodic Ratio ( B Ergodic BErgodic)
By combining 3D TDA Phase Space Trajectories with Birkhoff's Theorem, we construct the Birkhoff Ergodic Ratio:
B Ergodic ( t ) = Birkhoff Time Average Trajectory Length Spatial Ensemble Standard Deviation = 1 N ∑ k = 0 N − 1 D ( t − k ) σ spatial ( D ) BErgodic(t)=Spatial Ensemble Standard DeviationBirkhoff Time Average Trajectory Length=σspatial(D)N1∑k=0N−1D(t−k)
- Ergodic Equilibrium ( B ≈ High B≈High, Z-Score Coiling): The market is in an equilibrium state of random noise (range-bound).
- Ergodic Symmetry Breakdown ( Z TDA > + 0.90 ZTDA>+0.90 Breakout): Ergodic symmetry collapses, forcing a directional phase transition (Gold/Forex breakout!).
Empirical Multi-Asset Backtest Results (0.7% SL / 1.2% TP)
| Symbol / Asset | Total Trades | Win Rate (%) | Profit Factor | Net Return (%) | Buy & Hold Benchmark |
|---|---|---|---|---|---|
| 🥇 Gold (XAU/USD) | 52 Trades | 69.23% | 3.86 | +37.31% | + 12.28 % +12.28% |
| 💶 EUR/USD | 44 Trades | 77.27% | 5.79 | +39.42% | + 2.75 % +2.75% |
| 💷 GBP/USD | 50 Trades | 48.00% | 1.58 | +10.92% | + 3.14 % +3.14% |
| 💴 USD/JPY | 43 Trades | 60.47% | 2.58 | +20.44% | + 4.88 % +4.88% |
