start.valgrid: Calculating the start values 'b' for the first iteration of the
quadratic program.
Description
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.
Usage
start.valgrid(penden.env)
Arguments
penden.env
Containing all information, environment of pencopula()
Value
X.knots.gIf 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.allSet 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.
Details
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.
References
Flexible Copula Density Estimation with Penalized
Hierarchical B-Splines, Kauermann G., Schellhase C. and Ruppert, D. (2011), to appear.