Penalty matrix for higher order differences

This function computes the matrix that penalizes the higher order differences.

Usage

P.penalty(tt, P = c(0, 0, 1))

Arguments

tt: vector of the n discretization points or argvals.
P: vector of coefficients with the order of the differences. Default value P=c(0,0,1) penalizes the second order difference.

Value

penalty matrix of size sum(n) x sum(n)

Details

For example, if P=c(0,1,2), the function return the penalty matrix the second order difference of a vector $tt$. That is $$v^T P_j tt= \sum_{i=3} ^{n} (\Delta tt_i) ^2$$ where $$\Delta tt_i= tt_i -2 tt_{i-1} + tt_{i-2}$$ is the second order difference. More details can be found in Kraemer, Boulesteix, and Tutz (2008).

Note

The discretization points can be equidistant or not.

References

N. Kraemer, A.-L. Boulsteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94, 60 - 69. doi:10.1016/j.chemolab.2008.06.009

Author

This version is created by Manuel Oviedo de la Fuente modified the original version created by Nicole Kramer in ppls package.

Examples


P.penalty((1:10)/10,P=c(0,0,1))
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#>  [1,]    1   -2    1    0    0    0    0    0    0     0
#>  [2,]   -2    5   -4    1    0    0    0    0    0     0
#>  [3,]    1   -4    6   -4    1    0    0    0    0     0
#>  [4,]    0    1   -4    6   -4    1    0    0    0     0
#>  [5,]    0    0    1   -4    6   -4    1    0    0     0
#>  [6,]    0    0    0    1   -4    6   -4    1    0     0
#>  [7,]    0    0    0    0    1   -4    6   -4    1     0
#>  [8,]    0    0    0    0    0    1   -4    6   -4     1
#>  [9,]    0    0    0    0    0    0    1   -4    5    -2
#> [10,]    0    0    0    0    0    0    0    1   -2     1
# a more detailed example can be found under script file