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.
4.9 KiB
APZ: Adaptive Price Zone
APZ constructs a volatility-adaptive envelope using double-smoothed exponential moving averages with an aggressive smoothing factor derived from \sqrt{\text{period}}, making it significantly faster than standard EMA-based channels. The center line is a double-EMA of price; the band width is a double-EMA of the high-low range, scaled by a multiplier. Designed specifically for mean-reversion trading in non-trending markets, APZ identifies overbought/oversold extremes where price is likely to reverse rather than continue. A closing price outside the zone signals an immediate overshoot, not a breakout.
Historical Context
Lee Leibfarth created the Adaptive Price Zone and published it in Technical Analysis of Stocks & Commodities (September 2006) under the article "Trading With An Adaptive Price Zone." Leibfarth recognized that most indicators fail in choppy, range-bound markets: trend followers get whipsawed, and oscillators saturate at extremes. APZ fills this gap by adapting its bandwidth dynamically to statistical noise, allowing traders to fade extremes in consolidation phases.
The critical design decision is the square-root smoothing factor: \alpha = 2 / (\sqrt{P} + 1). For a period of 20, \sqrt{20} \approx 4.47, producing \alpha \approx 0.365, which behaves like an EMA of period $\sim$3.5. This makes APZ extremely responsive compared to a standard 20-period EMA (\alpha = 0.095). The double-smoothing (EMA of EMA) adds some lag back, but the net result is still far faster than conventional approaches. The compound warmup compensator e = \beta^{2t} (where \beta = 1 - \alpha) ensures accurate values from bar 1 without the typical EMA initialization bias.
Architecture & Physics
1. Aggressive Smoothing Factor
\alpha = \frac{2}{\sqrt{P} + 1}, \quad \beta = 1 - \alpha
2. Center Line (Double-Smoothed EMA of Price)
First EMA:
\text{EMA1}_t = \alpha \cdot x_t + \beta \cdot \text{EMA1}_{t-1}
Second EMA (double-smoothing):
\text{Center}_t = \alpha \cdot \text{EMA1}_t + \beta \cdot \text{Center}_{t-1}
3. Adaptive Range (Double-Smoothed EMA of High-Low)
R_t = H_t - L_t
\text{EMA1R}_t = \alpha \cdot R_t + \beta \cdot \text{EMA1R}_{t-1}
\text{SmoothRange}_t = \alpha \cdot \text{EMA1R}_t + \beta \cdot \text{SmoothRange}_{t-1}
4. Band Construction
\text{Width}_t = F \cdot \text{SmoothRange}_t
\text{Upper}_t = \text{Center}_t + \text{Width}_t
\text{Lower}_t = \text{Center}_t - \text{Width}_t
5. Warmup Compensation
To eliminate EMA initialization bias, a compound compensator tracks the accumulated decay:
e_t = \beta^2 \cdot e_{t-1}, \quad e_0 = 1
During warmup (e > 10^{-10}):
\text{Center}_t^* = \frac{\text{Center}_t}{1 - e_t}, \quad \text{SmoothRange}_t^* = \frac{\text{SmoothRange}_t}{1 - e_t}
6. Complexity
O(1) per bar: 4 EMA updates (2 for price, 2 for range), plus band arithmetic. The square root is computed once at initialization. No buffers required.
Mathematical Foundation
Parameters
| Parameter | Description | Default | Constraint |
|---|---|---|---|
period |
Input period (P); \sqrt{P} used for smoothing |
20 | > 0 |
multiplier |
Band width factor (F) |
2.0 | > 0 |
source |
Input price series | close |
Effective EMA Period
The actual smoothing period experienced by the double-EMA is much shorter than the input period:
P_{\text{effective}} = \sqrt{P} \approx \frac{2}{\alpha} - 1
For P = 20: P_{\text{eff}} \approx 4.47. For P = 100: P_{\text{eff}} \approx 10.
Pseudo-code
function APZ(source, high, low, period, multiplier):
validate: period > 0, multiplier > 0
alpha = 2 / (√period + 1)
beta = 1 - alpha
// Double-smoothed EMA of price
ema1_price = alpha * source + beta * ema1_price
center = alpha * ema1_price + beta * center
// Double-smoothed EMA of range
range = high - low
ema1_range = alpha * range + beta * ema1_range
smooth_range = alpha * ema1_range + beta * smooth_range
// Warmup compensator
e *= beta²
if e > 1e-10:
compensator = 1 / (1 - e)
center *= compensator
smooth_range *= compensator
// Bands
width = multiplier * smooth_range
upper = center + width
lower = center - width
return [center, upper, lower]
Output Interpretation
| Output | Description |
|---|---|
center |
Double-smoothed EMA of price (fast center line) |
upper |
Center + scaled adaptive range (overbought zone) |
lower |
Center - scaled adaptive range (oversold zone) |
Resources
- Leibfarth, L. "Trading With An Adaptive Price Zone." Technical Analysis of Stocks & Commodities, September 2006. (Original APZ specification)
- Mulloy, P. "Smoothing Data with Less Lag." Technical Analysis of Stocks & Commodities, February 1994. (Double-smoothed EMA theory)