QuanTAlib/lib/trends_FIR/swma/Swma.md

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.

Property	Value
Category	Trend (FIR MA)
Inputs	Source (close)
Parameters	period (default 4)
Outputs	Single series (Swma)
Output range	Tracks input
Warmup	period bars
PineScript	swma.pine
Signature	swma_signature

• SWMA applies triangular (symmetric) weights that peak at the center of the window and taper linearly to the edges.
• Similar: WMA, SMA | Trading note: Symmetric-Weighted MA; bell-shaped weight profile centered on middle. Reduces end-point bias.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

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.

Performance Profile

Operation Count (Streaming Mode)

SWMA(N) is an O(N) FIR convolution using symmetric triangular weights (ascending then descending). Weights are precomputed at construction and normalized to sum = 1. The triangular shape gives the center bar the highest weight.

Operation	Count	Cost (cycles)	Subtotal
Ring buffer push	1	3	~3
FIR dot product: N FMA	N	4	~4N
Total	N + 1	—	~(4N + 3) cycles

O(N) per bar. For default N = 14: ~59 cycles. Triangular weights are strictly positive — numerically clean. WarmupPeriod = N.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
FIR convolution	Yes	VFMADD231PD; all-positive weights
Symmetric triangular window	Yes	Fold: only ⌈N/2⌉ unique weights; halves FMA count
Cross-bar independence	Yes	4 output bars per AVX2 pass

Symmetric folding reduces the effective FMA count to ⌈N/2⌉. For N = 14: 7 FMAs per bar. AVX2 batch throughput: ~N/8 cycles per bar. Among the windowed FIR filters, SWMA has the fewest effective operations due to its simple triangular shape.