> *Robert Henderson designed a filter so good that the Australian Bureau of Statistics still uses it a century later. When your smoothing algorithm outlasts empires, you did something right.*
- HEND is a symmetric FIR filter derived from the Henderson (1916) closed-form weight formula, designed to pass cubic polynomial trends without disto...
- **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.
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.
## Historical Context
Robert Henderson published the weight formula in 1916 in the *Transactions of the Actuarial Society of America*, motivated by the need to graduate mortality tables without distorting underlying polynomial trends. The U.S. Census Bureau adopted Henderson filters as the trend-cycle component of the X-11 method (Shiskin, Young, and Musgrave, 1967), where 5, 9, 13, and 23-point Henderson filters became standard choices. The Australian Bureau of Statistics (ABS) uses the 13-point Henderson as its default trend estimator for quarterly national accounts.
Henderson's filter has a unique property among polynomial-preserving smoothers: it minimizes the sum of squared third differences of the filter weights subject to the constraint that polynomials up to degree 3 pass through unchanged. This optimality criterion produces smoother weight sequences than Savitzky-Golay filters of the same polynomial order, at the cost of a fixed (non-configurable) smoothness-fidelity balance.
The requirement for odd period length ($N \geq 5$) stems from the symmetric weight structure. Even-length Henderson filters are mathematically possible but break the centered-symmetry property that guarantees zero phase distortion.
## Architecture & Physics
### 1. Weight Computation (One-Time)
Weights are computed from Henderson's closed-form formula:
where $n = (N+3)/2$ and $k$ ranges from $-(N-1)/2$ to $(N-1)/2$. Weights are normalized to sum to 1.0 after computation.
### 2. Symmetric Convolution
The filter applies as a standard FIR convolution over the circular buffer. Because weights are symmetric ($w(k) = w(-k)$), the implementation can exploit symmetry to halve multiplications, though the normalization step makes this optional.
### 3. Negative Edge Weights
Unlike most window-based averages, Henderson weights are negative at the extremes of the window. This is not a bug; it is the mechanism by which the filter suppresses low-frequency drift that would distort cubic trends. The negative wings act as a gentle high-pass correction.
where $\Delta^3$ is the third-difference operator. The constraints ensure that constant, linear, quadratic, and cubic polynomials are reproduced exactly.
The closed-form solution with $n = (N+3)/2$, $k \in [-(N-1)/2, (N-1)/2]$:
**Frequency response:** The Henderson filter has zeros at specific frequencies determined by the polynomial-preservation constraints. For the 13-point filter, sidelobe attenuation exceeds $-40$ dB.
**Default parameters:**`period = 7` (must be odd, $\geq 5$).
- Henderson, R. (1916). "Note on Graduation by Adjusted Average." *Transactions of the Actuarial Society of America*, 17, 43-48.
- Shiskin, J., Young, A.H., & Musgrave, J.C. (1967). "The X-11 Variant of the Census Method II Seasonal Adjustment Program." Technical Paper 15, U.S. Bureau of the Census.
- Hyndman, R.J. (2011). "Moving Averages." In *International Encyclopedia of Statistical Science*. Springer.
- Kenny, P.B. & Durbin, J. (1982). "Local Trend Estimation and Seasonal Adjustment of Economic and Social Time Series." *JRSS Series A*, 145(1), 1-41.
HEND(N) is a direct FIR convolution using precomputed Henderson weights (computed once at construction). Each `Update()` call pushes one value into the ring buffer and executes a length-N dot product against the weight array. Henderson weights can be negative at edges, so no shortcut reduces the scan.
| Operation | Count | Cost (cycles) | Subtotal |
| :--- | :---: | :---: | :---: |
| Ring buffer push | 1 | 3 | ~3 |
| FIR dot product: N FMA (weight × value + acc) | N | 4 | ~4N |
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).
