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Miha Kralj
90d5638008
- LTMA (Linear Trend Moving Average): Introduces a predictive moving average using dual cascaded EMAs for trend estimation. - MCNMA (McNicholl EMA): Implements a zero-lag TEMA using a cascaded EMA structure for enhanced responsiveness. - NLMA (Non-Lag Moving Average): Utilizes a damped cosine kernel to achieve reduced lag in moving averages. - NMA (Natural Moving Average): Adapts smoothing based on volatility profiles using a square-root kernel. - NYQMA (Nyquist Moving Average): Applies the Nyquist-Shannon theorem to prevent aliasing in cascaded moving averages. - RAIN (Rainbow Moving Average): Combines multiple SMA layers with weighted averages for multi-scale smoothing. - TRAMA (Trend Regularity Adaptive Moving Average): Adapts smoothing based on the frequency of new highs and lows in price data.
91 lines
3.9 KiB
Markdown
91 lines
3.9 KiB
Markdown
# GDEMA: Generalized Double Exponential Moving Average
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> "Patrick Mulloy created DEMA to cancel first-order lag. GDEMA adds a volume knob: turn it past 1 and you cancel more lag than Mulloy thought possible. Turn it to 0 and you are back to a plain EMA. The generalization is the point."
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GDEMA extends the standard DEMA (Double Exponential Moving Average) with a tunable gain factor $v$ that controls the aggressiveness of lag compensation. The formula $\text{GDEMA} = (1+v) \cdot \text{EMA}_1 - v \cdot \text{EMA}_2$ reduces to plain EMA when $v=0$, standard DEMA when $v=1$, and progressively more aggressive lag removal for $v>1$. This parametric flexibility allows traders to dial in the exact smoothness-responsiveness trade-off for their application, rather than being locked into DEMA's fixed 2:1 ratio.
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## Historical Context
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Patrick G. Mulloy published DEMA in "Smoothing Data with Faster Moving Averages" (*Technical Analysis of Stocks & Commodities*, February 1994). The original DEMA uses the fixed formula $2 \cdot \text{EMA} - \text{EMA}(\text{EMA})$, which cancels the first-order lag of the EMA by subtracting the double-smoothed version.
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The generalization to an arbitrary volume factor $v$ is a natural extension that was explored by several authors in the late 1990s. Tim Tillson's T3 indicator (1998) uses a similar parameterized approach with six cascaded EMAs and a volume factor. GDEMA is the simplest member of this family: two cascaded EMAs combined with a single gain parameter.
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The mathematical basis is the z-transform lag cancellation technique: EMA has a group delay of approximately $(N-1)/2$ samples. EMA(EMA) has approximately double that delay. The linear combination $(1+v) \cdot \text{EMA} - v \cdot \text{EMA}(\text{EMA})$ cancels $v/(v+1)$ of the total lag. At $v=1$ (DEMA), half the lag is cancelled. At $v=2$, two-thirds is cancelled, but overshoot increases proportionally.
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## Architecture & Physics
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### 1. Dual Cascaded EMAs
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Two EMA stages share the same period $N$ and smoothing constant $\alpha = 2/(N+1)$:
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- **EMA1:** Standard EMA of the source.
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- **EMA2:** EMA of EMA1 (double-smoothed).
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### 2. Warmup Compensation
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Both EMAs use the exponential warmup compensator $c = 1/(1-\beta^n)$ to produce valid output from bar 1, eliminating the cold-start bias.
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### 3. Parameterized Combination
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$$
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\text{GDEMA} = (1+v) \cdot \text{EMA}_1 - v \cdot \text{EMA}_2
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$$
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## Mathematical Foundation
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Given smoothing constant $\alpha = 2/(N+1)$, decay $\beta = 1-\alpha$:
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$$
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\text{EMA}_1[t] = \alpha \cdot x_t + \beta \cdot \text{EMA}_1[t-1]
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$$
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$$
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\text{EMA}_2[t] = \alpha \cdot \text{EMA}_1[t] + \beta \cdot \text{EMA}_2[t-1]
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$$
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$$
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\text{GDEMA}[t] = (1+v) \cdot \text{EMA}_1[t] - v \cdot \text{EMA}_2[t]
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$$
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**Z-domain transfer function:**
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$$
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H(z) = (1+v) \cdot \frac{\alpha}{1-\beta z^{-1}} - v \cdot \left(\frac{\alpha}{1-\beta z^{-1}}\right)^2
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$$
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**Lag characteristics:**
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| $v$ | Equivalent | Lag reduction | Overshoot risk |
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| :---: | :--- | :---: | :---: |
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| 0 | EMA | 0% | None |
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| 0.5 | Mild DEMA | 33% | Low |
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| 1.0 | Standard DEMA | 50% | Moderate |
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| 1.5 | Aggressive | 60% | High |
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| 2.0 | Very aggressive | 67% | Very high |
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**Default parameters:** `period = 10`, `vfactor = 1.0`, `minPeriod = 1`.
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**Pseudo-code (streaming):**
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```
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alpha = 2 / (period + 1); beta = 1 - alpha
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// EMA1 with warmup
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ema1_raw = alpha * (source - ema1_raw) + ema1_raw
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e *= beta
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comp = 1 / (1 - e)
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ema1 = ema1_raw * comp
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// EMA2 with warmup (of compensated EMA1)
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ema2_raw = alpha * (ema1 - ema2_raw) + ema2_raw
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ema2 = ema2_raw * comp
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// Generalized combination
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return (1 + v) * ema1 - v * ema2
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```
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## Resources
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- Mulloy, P.G. (1994). "Smoothing Data with Faster Moving Averages." *Technical Analysis of Stocks & Commodities*, 12(1), 11-19.
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- Tillson, T. (1998). "Smoothing Techniques for More Accurate Signals." *Technical Analysis of Stocks & Commodities*, 16(1).
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- Ehlers, J.F. (2001). *Rocket Science for Traders*. Wiley. Chapter 3: Smoothing Filters.
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