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lib/trends_FIR/hend/Hend.md
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| **Signature** | [hend_signature](hend_signature.md) |
- HEND is a symmetric FIR filter derived from the Henderson (1916) closed-form weight formula, designed to pass cubic polynomial trends without disto...
- Parameterized by `period` (default 7).
- Output range: Tracks input.
- Requires `period` bars of warmup before first valid output (IsHot = true).
- **Similar:** [LSMA](../lsma/lsma.md), [TSF](../tsf/Tsf.md) | **Complementary:** StdDev | **Trading note:** Henderson MA; used by Australian Bureau of Statistics. Optimal for extracting smooth trend from noisy data.
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
HEND is a symmetric FIR filter derived from the Henderson (1916) closed-form weight formula, designed to pass cubic polynomial trends without distortion while maximally suppressing irregular noise. Used as the core smoother in the X-11 and X-13ARIMA-SEATS seasonal adjustment frameworks by statistical agencies worldwide, HEND achieves the theoretically optimal trade-off between smoothness (measured by the sum of squared third differences of the weights) and fidelity for cubic trends. Weights can be negative at the edges, giving the filter a bandpass-like property that sharpens trend-cycle extraction.
@@ -117,4 +115,4 @@ O(N) per bar. For default N = 7 (5-term odd period): ~31 cycles. For N = 23 (com
| Negative-weight handling | Yes | No special treatment needed; signed FMA handles negatives |
| Cross-bar independence | Yes | Each bar's output is independent; full outer-loop vectorization |
With AVX2, 4 bars can be processed simultaneously (each is an N-tap dot product). Total batch throughput: ~N/4 cycles per bar for large series. For N = 23 and 1000-bar batch: ~5750 cycles vs ~95000 scalar — approximately 16.5× speedup (memory-bound at larger N).
With AVX2, 4 bars can be processed simultaneously (each is an N-tap dot product). Total batch throughput: ~N/4 cycles per bar for large series. For N = 23 and 1000-bar batch: ~5750 cycles vs ~95000 scalar — approximately 16.5× speedup (memory-bound at larger N).