QuanTAlib/lib/trends_IIR/gdema/Gdema.md

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.

Property	Value
Category	Trend (IIR MA)
Inputs	Source (close)
Parameters	period (default 10), vfactor (default 1.0)
Outputs	Single series (Gdema)
Output range	Tracks input
Warmup	period bars
PineScript	gdema.pine
Signature	gdema_signature

• GDEMA extends the standard DEMA (Double Exponential Moving Average) with a tunable gain factor v that controls the aggressiveness of lag compensa...
• Similar: DEMA, T3 | Complementary: Signal crossovers | Trading note: Generalized DEMA; tunable volume factor between EMA and DEMA behavior.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

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.