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.
3.9 KiB
SWMA: Symmetric Weighted Moving Average
"Take the SMA of an SMA and you get a triangular filter. It is the simplest possible smoothing kernel that has zero phase distortion and no frequency-domain discontinuities. Sometimes simple is exactly what you need."
SWMA applies triangular (symmetric) weights that peak at the center of the window and taper linearly to the edges. For period N, the weight at position i is w(i) = (N/2 + 1) - |i - N/2|, producing a tent-shaped kernel. This is mathematically equivalent to convolving two rectangular windows (SMA of SMA), giving SWMA a frequency response that is the square of the SMA's sinc-like response. The result is smoother than SMA with better sidelobe suppression, at the cost of slightly more lag.
Historical Context
The symmetric (triangular) weighted average is one of the oldest smoothing methods in statistics, predating modern signal processing by centuries. Its equivalence to the double-application of the simple moving average was recognized by Macaulay (1931) in his NBER monograph on time-series smoothing. The TRIMA (Triangular Moving Average) implemented elsewhere in QuanTAlib is the same mathematical operation computed via double SMA composition.
In PineScript, ta.swma refers specifically to the 4-point variant with weights [1, 2, 2, 1]/6, which is a special case of the general symmetric weighted average. QuanTAlib's SWMA generalizes this to arbitrary periods.
The triangular kernel has a natural Bayesian interpretation: if you believe the "true" signal is equally likely to be any value in a window of width N/2, and your observation window is also N/2, the posterior belief about the signal value is triangular. This makes SWMA the optimal Bayesian filter under uniform prior and uniform observation noise assumptions.
Architecture & Physics
1. Weight Computation
For a window of length N with half-width h = (N-1)/2:
w(i) = h + 1 - |i - h|, \quad i = 0, 1, \ldots, N-1
Weights form a triangle peaking at the center. For even N, the peak is a plateau of two equal values.
2. Normalized Weighted Sum
\text{SWMA} = \frac{\sum_{i=0}^{N-1} w(i) \cdot x_{t-i}}{\sum_{i=0}^{N-1} w(i)}
The weight sum equals (h+1)^2 for odd N and h(h+2)+1 for even N.
3. Equivalence to Double SMA
SWMA(N) produces the same output as SMA(M) applied to SMA(M) where M = \lceil N/2 \rceil. This means the streaming implementation can compose two SMA instances for O(1) updates, rather than O(N) convolution.
Mathematical Foundation
The triangular window for length N, with h = (N-1)/2:
w[i] = h + 1 - |i - h|, \quad i = 0, \ldots, N-1
Frequency response:
H_{\text{SWMA}}(f) = H_{\text{SMA}}^2(f) = \left[\frac{\sin(\pi f M)}{\pi f M}\right]^2
where M = \lceil N/2 \rceil. The squared sinc provides:
| Property | SMA | SWMA |
|---|---|---|
| First zero | 1/N |
2/N |
| First sidelobe | -13 dB |
-26 dB |
| Rolloff rate | -6 dB/octave |
-12 dB/octave |
| Passband ripple | Moderate | Low |
Weight sum (closed form):
For odd N = 2m+1: \sum w = (m+1)^2
For even N = 2m: \sum w = m(m+1)
PineScript special case: ta.swma uses N = 4, h = 1.5, weights = [1, 2, 2, 1], \sum w = 6.
Default parameters: period = 4, minPeriod = 2.
Pseudo-code (streaming):
half = (period - 1) / 2.0
sumWV = 0; sumW = 0
for i = 0 to period-1:
w = half + 1 - |i - half|
sumWV += src[i] * w
sumW += w
return sumWV / sumW
Resources
- Macaulay, F.R. (1931). The Smoothing of Time Series. National Bureau of Economic Research. Chapter 3: Moving Averages and Their Properties.
- Oppenheim, A.V. & Schafer, R.W. (2009). Discrete-Time Signal Processing, 3rd ed. Prentice Hall. Section 5.6: The Bartlett (Triangular) Window.
- Murphy, J.J. (1999). Technical Analysis of the Financial Markets. New York Institute of Finance. Chapter 9: Moving Averages.