QuanTAlib/lib/trends_FIR/hend/Hend.md

HEND: Henderson Moving Average

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.

Property	Value
Category	Trend (FIR MA)
Inputs	Source (close)
Parameters	period (default 7)
Outputs	Single series (Hend)
Output range	Tracks input
Warmup	period bars
PineScript	hend.pine
Signature	hend_signature

• 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, TSF | 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.

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:

w(k) = \frac{315 \left[(n-1)^2 - k^2\right]\left[n^2 - k^2\right]\left[(n+1)^2 - k^2\right]\left[3n^2 - 16 - 11k^2\right]}{8n(n^2-1)(4n^2-1)(4n^2-9)(4n^2-25)}

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.

Mathematical Foundation

The Henderson filter minimizes:

\min_{\{w_k\}} \sum_{k} (\Delta^3 w_k)^2 \quad \text{subject to} \quad \sum_{k} k^j w_k = \delta_{j0}, \quad j = 0, 1, 2, 3

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]:

w(k) = \frac{315 \cdot \left[(n-1)^2 - k^2\right]\left[n^2 - k^2\right]\left[(n+1)^2 - k^2\right]\left[3n^2 - 16 - 11k^2\right]}{8n(n^2-1)(4n^2-1)(4n^2-9)(4n^2-25)}

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).

Pseudo-code (streaming):

// One-time weight computation
half = (period - 1) / 2
n = (period + 3) / 2
for k = -half to half:
    w[k] = 315 * ((n-1)²-k²) * (n²-k²) * ((n+1)²-k²) * (3n²-16-11k²)
           / [8n(n²-1)(4n²-1)(4n²-9)(4n²-25)]
normalize(w)

// Per-bar convolution
buffer.push(price)
if count < period: return price
result = Σ buffer[j] * w[j] for j = 0..period-1
return result

Resources

• 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.

Performance Profile

Operation Count (Streaming Mode)

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
Total	N + 1	—	~(4N + 3) cycles

O(N) per bar. For default N = 7 (5-term odd period): ~31 cycles. For N = 23 (common seasonal use): ~95 cycles. WarmupPeriod = N.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
FIR dot product per bar	Yes	VFMADD231PD with weight array; 4 doubles/cycle
Weight array (precomputed, static)	Yes	Loaded once into registers
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).