QuanTAlib/lib/trends_FIR/sinema/Sinema.md

SINEMA: Sine-Weighted Moving Average

Nature doesn't do straight lines, and neither should your weights.

Property	Value
Category	Trend (FIR MA)
Inputs	Source (close)
Parameters	period
Outputs	Single series (Sinema)
Output range	Tracks input
Warmup	period bars
PineScript	sinema.pine
Signature	sinema_signature

• The Sine-Weighted Moving Average (SINEMA) applies sine-wave weighting to data points within the lookback window.
• Similar: ALMA, BLMA | Complementary: Cycle indicators | Trading note: Sine-weighted MA; half-sine kernel for naturally smooth bell-shaped weights.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Sine-Weighted Moving Average (SINEMA) applies sine-wave weighting to data points within the lookback window. Weights follow the formula w_i = \sin(\pi \cdot (i+1) / N), creating a smooth bell-shaped distribution that emphasizes middle values while gracefully tapering at the edges. Unlike SMA's uniform weighting or WMA's linear ramp, sine weighting provides a natural transition that reduces high-frequency noise while preserving mid-frequency trends.

Historical Context

Sine-weighted smoothing emerges from signal processing, where windowing functions shape the frequency response of filters. The sine window (also called the cosine window when phase-shifted) is a member of the generalized cosine window family. Its application to financial moving averages provides a middle ground between the harsh cutoff of rectangular windows (SMA) and the aggressive center-weighting of triangular windows (TRIMA).

Architecture & Physics

1. Weight Calculation

For a period N, the weight at position i (0-indexed) is:

w_i = \sin\left(\frac{\pi \cdot (i+1)}{N}\right)

This produces a half-sine wave: weights start small, peak at the center, and taper back down. For period 5: weights ≈ [0.588, 0.951, 1.0, 0.951, 0.588].

2. Normalization

The weighted average normalizes by the sum of weights:

\text{SINEMA}_t = \frac{\sum_{i=0}^{N-1} P_{t-i} \cdot w_i}{\sum_{i=0}^{N-1} w_i}

3. Warmup Adaptation

During warmup (fewer than N values), weights are recalculated for the current buffer size k:

w_i^{(k)} = \sin\left(\frac{\pi \cdot (i+1)}{k}\right)

This ensures smooth output from the first bar rather than waiting for a full window.

Mathematical Foundation

Weight Distribution

The sine weight function produces:

• Symmetric weighting: Equal emphasis on equidistant past values
• Smooth edges: No abrupt transitions at window boundaries
• Peak at center: Maximum weight at position \lfloor N/2 \rfloor

Frequency Response

As an FIR filter, SINEMA has linear phase response (no phase distortion) but O(N) complexity per bar in streaming mode. The sine window provides moderate side-lobe suppression (~23 dB), better than rectangular (SMA) but less than Hamming or Blackman windows.

Performance Profile

Operation Count (Streaming Mode, Scalar)

Operation	Count	Cost (cycles)	Subtotal
ADD	N	1	N
MUL	N	3	3N
DIV	1	15	15
Total	2N+1	—	~4N+15 cycles

Pre-calculated weights eliminate sin() calls in steady state.

Batch Mode (SIMD)

The batch calculation uses stackalloc for buffers ≤256 elements and ArrayPool for larger periods. SIMD vectorization is limited due to the weighted sum's data dependency, but memory locality is optimized.

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Exact weighted mean calculation
Timeliness	4/10	Moderate lag (~N/3 due to center weighting)
Overshoot	0/10	Never exceeds input data range
Smoothness	7/10	Smoother than SMA; less prone to drop-off jumps

Validation

SINEMA is not implemented in standard technical analysis libraries.

Library	Status	Notes
TA-Lib	N/A	Not implemented
Skender	N/A	Not implemented
Tulip	N/A	Not implemented
Ooples	N/A	Not implemented
PineScript	✅	Reference implementation matches

Validation tests verify:

• Sine weight mathematical correctness
• Constant input produces constant output
• Batch/Streaming/Span mode consistency
• Output bounded by input range
• Warmup weight adaptation

Common Pitfalls

1. O(N) Complexity: Unlike SMA's O(1) running sum, SINEMA requires O(N) operations per bar. For very long periods (>500), consider whether the smoothness benefits justify the cost.

2. Warmup Behavior: The adaptive warmup recalculates weights for partial buffers. This produces valid output from bar 1 but with different effective weighting than steady state.

3. Weight Pre-calculation: Weights are computed once at construction. Changing the period requires a new indicator instance.

4. NaN Propagation: A single NaN in the window corrupts the result. QuanTAlib substitutes the last valid value to prevent this.

5. Memory: Each instance stores a pre-calculated weight array of size N. For many concurrent indicators with large periods, memory adds up.

References

• Harris, F. J. (1978). "On the use of windows for harmonic analysis with the discrete Fourier transform." Proceedings of the IEEE, 66(1), 51-83.
• Oppenheim, A. V., & Schafer, R. W. (2010). Discrete-Time Signal Processing (3rd ed.). Pearson.