Symmetric Henderson weights — henderson

Weights of the symmetric Henderson moving average of a given odd length, from the closed formula.

Usage

henderson_weights(length)

Arguments

length: Odd integer >= 3, number of terms.

Value

A numeric vector of length weights, for offsets -p..p.

Details

With length = 2p + 1 and m = p + 2, the weight at offset $j = -p, \dots, p$ is $$w_j = \frac{315\,[(m-1)^2 - j^2][m^2 - j^2][(m+1)^2 - j^2] [3m^2 - 16 - 11j^2]}{8m(m^2-1)(4m^2-1)(4m^2-9)(4m^2-25)}.$$ The weights sum to one, are symmetric, reproduce polynomials of degree up to three and minimise the sum of squares of the third differences of the weights (Henderson 1916; Kenny and Durbin 1982). For length = 3 the filter is the identity (0, 1, 0).

Examples

w <- henderson_weights(13)
round(w, 5)
#>  [1] -0.01935 -0.02786  0.00000  0.06549  0.14736  0.21434  0.24006  0.21434
#>  [9]  0.14736  0.06549  0.00000 -0.02786 -0.01935
sum(w)
#> [1] 1