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Miha Kralj
3eae9a76fe
- Implemented Sdchannel class for calculating standard deviation channels based on linear regression. - Added detailed documentation for SDCHANNEL, including overview, calculation methods, and interpretation. - Updated project files to include new numerics library components in Channels and Volatility projects.
225 lines
8.0 KiB
Markdown
225 lines
8.0 KiB
Markdown
# DECAYCHANNEL: Decay Min-Max Channel
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> "Yesterday's high matters less today. Tomorrow, it matters even less. Decay channels know this."
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Decay Min-Max Channel (DECAYCHANNEL) tracks the highest high and lowest low like Donchian, then applies exponential decay toward the midpoint. Fresh extremes snap the bands outward; time compresses them inward. The result: channels that respect recent price action while gradually forgetting stale levels. This implementation uses true half-life mathematics—50% convergence over the period length—ensuring predictable decay behavior across all timeframes.
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## Historical Context
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Traditional Donchian Channels treat all extremes within the lookback window equally. A high from 19 bars ago has the same influence as a high from 1 bar ago. This works for breakout detection but creates artificial support/resistance levels that persist until they mechanically exit the window.
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Traders noticed this rigidity. A 20-day high from exactly 20 days ago shouldn't matter as much as one from 5 days ago. Various "adaptive channel" approaches emerged in the 1990s-2000s, but most used arbitrary decay rates or complex volatility weighting.
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DECAYCHANNEL takes a simpler approach: pure exponential decay with mathematically defined half-life. The decay constant $\lambda = \ln(2) / \text{period}$ guarantees that bands converge 50% toward the midpoint over exactly one period. After two periods: 75%. After three: 87.5%. No tuning parameters, no volatility lookups—just consistent, predictable decay.
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## Architecture & Physics
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DECAYCHANNEL consists of four interconnected components that balance extreme tracking with temporal decay.
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### 1. Extreme Tracking (Highest/Lowest)
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Internal Highest and Lowest indicators maintain the actual max/min over the period:
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$$
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H_t^{raw} = \max_{i=0}^{n-1}(High_{t-i})
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$$
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$$
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L_t^{raw} = \min_{i=0}^{n-1}(Low_{t-i})
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$$
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These raw values constrain the decayed bands—the upper band can never exceed the actual highest high, and the lower band can never go below the actual lowest low.
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### 2. Decay Timers
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Separate counters track how long since each band was reset by a new extreme:
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$$
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\tau_U = \text{bars since } High_t = H_t^{raw}
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$$
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$$
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\tau_L = \text{bars since } Low_t = L_t^{raw}
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$$
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When price makes a new extreme, the corresponding timer resets to zero. Otherwise, it increments each bar.
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### 3. Exponential Decay Engine
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The decay rate uses the half-life formula:
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$$
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\lambda = \frac{\ln(2)}{\text{period}}
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$$
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For each bar, compute the decay factor based on elapsed time:
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$$
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d_U = 1 - e^{-\lambda \cdot \tau_U}
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$$
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$$
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d_L = 1 - e^{-\lambda \cdot \tau_L}
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$$
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At $\tau = 0$ (new extreme), $d = 0$ (no decay). At $\tau = \text{period}$, $d = 0.5$ (half decayed).
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### 4. Midpoint Convergence
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Bands decay toward the current midpoint, not toward price:
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$$
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M_t = \frac{U_{t-1} + L_{t-1}}{2}
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$$
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$$
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U_t = U_{t-1} - d_U \cdot (U_{t-1} - M_t)
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$$
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$$
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L_t = L_{t-1} + d_L \cdot (M_t - L_{t-1})
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$$
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Finally, constrain to actual extremes:
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$$
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U_t = \max(U_t, H_t^{raw})
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$$
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$$
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L_t = \min(L_t, L_t^{raw})
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$$
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## Mathematical Foundation
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### Half-Life Derivation
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Exponential decay follows:
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$$
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V(t) = V_0 \cdot e^{-\lambda t}
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$$
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For half-life $t_{1/2}$ where $V(t_{1/2}) = \frac{V_0}{2}$:
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$$
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\frac{V_0}{2} = V_0 \cdot e^{-\lambda t_{1/2}}
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$$
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$$
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\lambda = \frac{\ln(2)}{t_{1/2}}
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$$
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Setting $t_{1/2} = \text{period}$ gives the implementation's decay constant.
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### Convergence Schedule
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| Elapsed Time | Decay Factor | Remaining Distance |
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| :--- | :---: | :---: |
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| 0 bars | 0% | 100% |
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| period/2 bars | 29.3% | 70.7% |
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| period bars | 50% | 50% |
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| 2×period bars | 75% | 25% |
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| 3×period bars | 87.5% | 12.5% |
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### Middle Band Calculation
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The output middle band is the average of the decayed upper and lower bands:
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$$
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Middle_t = \frac{U_t + L_t}{2}
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$$
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This differs from the convergence midpoint (which uses previous bar's values) to avoid feedback loops.
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## Performance Profile
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### Operation Count (Streaming Mode, Scalar)
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Per-bar cost including internal Highest/Lowest updates:
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| ADD/SUB | 8 | 1 | 8 |
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| MUL | 4 | 3 | 12 |
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| DIV | 1 | 15 | 15 |
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| EXP | 2 | 50 | 100 |
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| CMP/MAX/MIN | 6 | 1 | 6 |
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| **Total** | **21** | — | **~141 cycles** |
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**Breakdown:**
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- Lambda: precomputed at construction (0 cycles per bar)
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- Midpoint: 1 ADD + 1 DIV = 16 cycles
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- Decay factors (×2): 2 MUL + 2 EXP + 2 SUB = 106 cycles
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- Band updates: 2 MUL + 2 SUB = 8 cycles
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- Constraint checks: 4 CMP = 4 cycles
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- Internal Highest/Lowest: ~8 cycles (amortized O(1))
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**Dominant cost:** EXP operations at 71% of total cycles.
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### Batch Mode (512 values, SIMD/FMA)
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| Operation | Scalar Ops | SIMD Benefit | Notes |
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| :--- | :---: | :---: | :--- |
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| Decay calculation | 2 | Limited | Sequential dependency on timers |
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| Band update | 4 | 2× via FMA | `band - decay × (band - mid)` |
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| Max/Min constraint | 4 | 1× | Comparison-based |
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**Batch efficiency (512 bars):**
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| Mode | Cycles/bar | Total (512 bars) | Improvement |
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| :--- | :---: | :---: | :---: |
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| Scalar streaming | 141 | 72,192 | — |
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| FMA-optimized | ~135 | ~69,120 | **~4%** |
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Limited improvement due to:
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1. **EXP dominates**: 100 of 141 cycles are exponential operations (not SIMD-friendly in scalar mode)
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2. **Timer dependency**: Each bar's decay factor depends on its timer value
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3. **State coupling**: Upper/lower bands depend on previous bar's midpoint
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 10/10 | Mathematically exact exponential decay |
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| **Timeliness** | 8/10 | Immediate response to new extremes |
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| **Overshoot** | 6/10 | New extremes reset decay, can spike bands |
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| **Smoothness** | 7/10 | Exponential decay provides smooth convergence between resets |
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| **Adaptivity** | 8/10 | Channels naturally tighten during consolidation |
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## Validation
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| Library | Status | Notes |
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| :--- | :---: | :--- |
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| **TA-Lib** | N/A | Not implemented |
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| **Skender** | N/A | Not implemented |
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| **Tulip** | N/A | Not implemented |
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| **Ooples** | N/A | Not implemented |
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| **Internal** | ✅ | Four-mode consistency verified (streaming, batch, span, event) |
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DECAYCHANNEL is a QuanTAlib-specific indicator with no external reference implementations.
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## Common Pitfalls
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1. **Decay Rate Confusion**: The period parameter controls half-life, not full decay. At period=100, bands are 50% decayed after 100 bars, not fully converged. For near-complete convergence (>95%), allow 4-5× the period.
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2. **Constraint Snap-Back**: When the actual highest high drops (because an old extreme exits the Highest window), the upper band can snap downward even mid-decay. This is intentional—decayed bands never exceed actual extremes.
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3. **Initialization Period**: DECAYCHANNEL needs `period` bars to establish meaningful extremes before decay becomes relevant. IsHot reflects this warmup requirement.
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4. **Timer State Management**: Using `isNew=false` for bar correction requires restoring both the band values and the decay timers. The implementation handles this via state snapshots, but improper use corrupts both.
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5. **Midpoint Targeting**: Bands decay toward the channel midpoint, not toward current price. In strong trends, this means the trailing band decays toward a point that may be far from price, creating asymmetric behavior.
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6. **Memory Overhead**: Each instance maintains two Highest/Lowest indicators plus decay state. For period=100, budget ~1.6 KB per instance for the internal monotonic deques plus ~64 bytes for state.
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7. **Exponential Sensitivity**: Small period values create aggressive decay. At period=10, bands are 50% converged after just 10 bars. For most applications, period≥50 provides more stable channels.
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## References
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- Murphy, J. J. (1999). *Technical Analysis of the Financial Markets*. New York Institute of Finance.
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- Kaufman, P. J. (2013). *Trading Systems and Methods* (5th ed.). John Wiley & Sons.
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- Press, W. H., et al. (2007). *Numerical Recipes: The Art of Scientific Computing* (3rd ed.). Cambridge University Press. [Exponential decay mathematics]
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