5.2 KiB
GDEMA: Generalized Double Exponential Moving Average
"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."
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.
Historical Context
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.
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.
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.
Architecture & Physics
1. Dual Cascaded EMAs
Two EMA stages share the same period N and smoothing constant \alpha = 2/(N+1):
- EMA1: Standard EMA of the source.
- EMA2: EMA of EMA1 (double-smoothed).
2. Warmup Compensation
Both EMAs use the exponential warmup compensator c = 1/(1-\beta^n) to produce valid output from bar 1, eliminating the cold-start bias.
3. Parameterized Combination
\text{GDEMA} = (1+v) \cdot \text{EMA}_1 - v \cdot \text{EMA}_2
Mathematical Foundation
Given smoothing constant \alpha = 2/(N+1), decay \beta = 1-\alpha:
\text{EMA}_1[t] = \alpha \cdot x_t + \beta \cdot \text{EMA}_1[t-1]
\text{EMA}_2[t] = \alpha \cdot \text{EMA}_1[t] + \beta \cdot \text{EMA}_2[t-1]
\text{GDEMA}[t] = (1+v) \cdot \text{EMA}_1[t] - v \cdot \text{EMA}_2[t]
Z-domain transfer function:
H(z) = (1+v) \cdot \frac{\alpha}{1-\beta z^{-1}} - v \cdot \left(\frac{\alpha}{1-\beta z^{-1}}\right)^2
Lag characteristics:
v |
Equivalent | Lag reduction | Overshoot risk |
|---|---|---|---|
| 0 | EMA | 0% | None |
| 0.5 | Mild DEMA | 33% | Low |
| 1.0 | Standard DEMA | 50% | Moderate |
| 1.5 | Aggressive | 60% | High |
| 2.0 | Very aggressive | 67% | Very high |
Default parameters: period = 10, vfactor = 1.0, minPeriod = 1.
Pseudo-code (streaming):
alpha = 2 / (period + 1); beta = 1 - alpha
// EMA1 with warmup
ema1_raw = alpha * (source - ema1_raw) + ema1_raw
e *= beta
comp = 1 / (1 - e)
ema1 = ema1_raw * comp
// EMA2 with warmup (of compensated EMA1)
ema2_raw = alpha * (ema1 - ema2_raw) + ema2_raw
ema2 = ema2_raw * comp
// Generalized combination
return (1 + v) * ema1 - v * ema2
Resources
- Mulloy, P.G. (1994). "Smoothing Data with Faster Moving Averages." Technical Analysis of Stocks & Commodities, 12(1), 11-19.
- Tillson, T. (1998). "Smoothing Techniques for More Accurate Signals." Technical Analysis of Stocks & Commodities, 16(1).
- Ehlers, J.F. (2001). Rocket Science for Traders. Wiley. Chapter 3: Smoothing Filters.
Performance Profile
Operation Count (Streaming Mode)
GDEMA(N, v) runs two cascaded EMA stages. The output is (1+v)×EMA₁ - v×EMA₂ — a linear combination with precomputed coefficient _onePlusV. Both EMAs use bias-compensated warmup (E factor).
| Operation | Count | Cost (cycles) | Subtotal |
|---|---|---|---|
| EMA₁: FMA(α, src, decay×ema1) | 1 | 4 | ~4 |
| Bias factor update E₁ | 1 | 3 | ~3 |
| EMA₂: FMA(α, ema1, decay×ema2) | 1 | 4 | ~4 |
| Bias factor update E₂ | 1 | 3 | ~3 |
| Output: FMA(onePlusV, ema1, −v×ema2) | 1 | 4 | ~4 |
| Total | 5 | — | ~18 cycles |
O(1) per bar. Two FMAs for EMA stages, one FMA for the combination. Fastest of the multi-stage EMA indicators. WarmupPeriod = N.
Batch Mode (SIMD Analysis)
| Operation | Vectorizable? | Notes |
|---|---|---|
| EMA₁ pass | No | Recursive IIR |
| EMA₂ pass (depends on EMA₁ output) | No | Sequential dependency on EMA₁ series |
| Output combination (1+v)×E1 − v×E2 | Yes | VFNMADD231PD across bar series once EMA passes complete |
Both EMA passes are recursive IIR. The final linear combination is vectorizable after the two EMA sweeps. Net batch speedup: minimal (~1.1×) since combination is only 3 of 18 cycles.