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:
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.
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.
