# 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](sinema.pine)                       |
| **Signature**    | [sinema_signature](sinema_signature.md) |

- The Sine-Weighted Moving Average (SINEMA) applies sine-wave weighting to data points within the lookback window.
- **Similar:** [ALMA](../alma/alma.md), [BLMA](../blma/blma.md) | **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.