- GDEMA extends the standard DEMA (Double Exponential Moving Average) with a tunable gain factor $v$ that controls the aggressiveness of lag compensa...
- Parameterized by `period` (default 10), `vfactor` (default 1.0).
- Output range: Tracks input.
- Requires `period` bars of warmup before first valid output (IsHot = true).
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
> "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 $\beta = 1-\alpha$:
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).
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.
