# my.bspline: my.bspline In pendensity: Density Estimation with a Penalized Mixture Approach

## Description

Integrates the normal B-Spline Base to a value of one. The dimension of the base depends on the input of number of knots 'k' and of the order of the B-Spline base 'q'.

## Usage

 `1` ```my.bspline(h, q, knots.val, y, K, plot.bsp) ```

## Arguments

 `h` if equidistant knots are just (default in pendensity()), h is the distance between two neighbouring knots `q` selected order of the B-Spline base `knots.val` selected values for the knots `y` values of the response variable y `K` the number of knots K for the construction of the base `plot.bsp` Indicator variable TRUE/FALSE if the integrated B-Spline base should be plotted

## Details

Firstly, the function constructs the B-Spline base to the given number of knots 'K' and the given locations of the knots 'knots.val\\$val. Due to the recursive construction of the B-Spline, for all orders greater than 2, the dimension of the B-Spline base of given K grows up with help.degree=q-2. Avoiding open B-Splines at the boundary, we simulate 6 extra knots at both ends of the support, saved in knots.val\\$all. Therefore, we get normal B-Splines at the given knots 'knots.val\\$val'. For these knots, we construct the B-Spline base of order 'q' and for order 'q+1' (using for calculation the distribution). Additionally, we save q-1 knots at both ends of the support of 'knots.val\\$val'. After construction, we get a base of dimension K=K+help.degree. So, we define our value K and cut our B-Spline base at both ends to get the adequate base due to the order 'q' and the number of knots 'K'. For the base of order 'q+1', we need to get an additional base, due to the construction of the B-Splines. Due to the fact, that we use equidistant knots, we can integrate our B-Splines very simple to the value of 1. The integration is done by the well-known factor q/(knots.val\\$help[i+q]-knots.val\\$help[i]). This results the standardization coefficients 'stand.num' for each B-spline (which are identically for equidistant knots). Moreover, one can draw the integrated base and, if one calls this function with the argument 'plot.bsp=TRUE'.

## Value

 `base.den` The integrated B-Spline base of order q `base.den2` The integrated B-Spline base of order q+1 `stand.num` The coefficients for standardization of the ordinary B-Spline base `knots.val` This return is a list. It consider of the used knots 'knots.val\\$val', the help knots 'knots.val\\$help' and the additional knots 'knots.val\\$all', used for the construction of the base and the calculation of the distribution function of each B-Spline. `K` The transformed value of K, due to used order 'q' and the input of 'K' `help.degree` Due to the recursive construction of the B-Spline, for all orders greater than 2, the dimension of the B-Spline base of given K grows up with 'help.degree=q-2'. This value is returned for later use.

## Note

This functions uses the fda-package for the construction of the B-Spline Base.

## Author(s)

Christian Schellhase <cschellhase@wiwi.uni-bielefeld.de>

pendensity documentation built on May 2, 2019, 3:58 a.m.