Calculating the start values 'b' for the first iteration of the quadratic program. Moreover, the grid of values for the side condition c>=0 of the quadratic program are calculated. If "adapt.grid", the number of grid points is reduced to speed up the quadratic program.

1 | ```
start.valgrid(penden.env)
``` |

`penden.env` |
Containing all information, environment of pencopula() |

The grid of values for the side conditions of the quadratic program c>=0 is constructed as the tensor product of all knots. If $p$ and $d$ increase, the number of conditions and computational time of the quadratic programm increase enormously, e.g. a full tensor product $u$ for $p=4$ and $d=4$ contains 83521 entries. If the data $u$ is not high correlated, i.e. the data is not from a extreme value copula like a Clayton copula, one can reduce the full tensor product. In 'pencopula' one can choose the option 'adapt.grid' which effects the following and may reduce the calculating time without any loss of accuracy. One can omit points in $u$ in sections of $[0,1]^p$ which are in the neighbourhood of many observations in $u$, because the data itself induces a positive density in these areas by construction. Therefore, we calculate the minimal $p$-dimensional euclidean distance $e_i$ of each $u_i, i=1,...,(2^d+1)^p$ to the data $u$ and omit the points corresponding to the first quartile of minimal euclidean distance $e_i$ in $u$, we call this new set of points $u_min$. This amout of points is used in the first iteration step to estimate weights $b$ corresponding to a copula density.

`X.knots.g` |
If adapt.grid=TRUE, set of reduced grid values, in which the side condition of the quadratic program c(u,b)>=0 will be postulated. |

`X.knots.g.all` |
Set of all grid values, in which the side condition of the quadratic program c(u,b)>=0 will be postulated. |

The values are saved in the environment.

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

Flexible Copula Density Estimation with Penalized Hierarchical B-Splines, Kauermann G., Schellhase C. and Ruppert, D. (2013), Scandinavian Journal of Statistics 40(4), 685-705.

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