R: Computing Some Overall Characteristics in 'compContourM1u'
getCharSTM1u
R Documentation
Computing Some Overall Characteristics in compContourM1u
Description
The function computes some overall characteristics
of directional regression quantiles in the output
of compContourM1u, namely the list
COutST$CharST. It makes possible
to obtain some useful information without saving
any file on the disk, and it can be easily modified
by the users according to their wishes.
Usage
getCharSTM1u(Tau, N, M, P, BriefDQMat, CharST, IsFirst)
Arguments
Tau
the quantile level in (0, 0.5).
N
the number of observations.
M
the dimension of responses.
P
the dimension of regressors including the intercept.
BriefDQMat
the method-specific matrix containing
the rows of a potential individual output
file corresponding to
CTechST$BriefOutputI = 1. See
the details below.
CharST
the output list, updated with each run
of the function.
IsFirst
the indicator equal to one in the
first run of getCharSTM1u
(when CharST is initialized)
and equal to zero otherwise.
Details
This function is called inside compContourM1u.
First, it is called with
BriefDQMat = NULL,
CharST = NULL and
IsFirst = 1 to initialize
the output list CharST, and then
it is called with IsFirst = 0
successively for the content of each potential
output file corresponding to
CTechST$BriefOutputI = 1, i.e., even if
the output file(s) are not stored on the disk owing to
CTechST$OutSaveI = 0.
It still remains to describe in detail the content of possible
output files, describing the optimal conic segmentation of
the directional space that lies behind the optimization
problem involved.
If CTechST$BriefOutputI = 1, then the rows of such
files are vectors of length 1+1+M+M+P+1 of the form
c(ConeID, Nu, UVec, BDVec, ADVec, LambdaD) where
ConeID
is the number/order of the cone related to the
line. If M > 2, then a cone can appear in
the output repeatedly (under different numbers).
Nu
is the number of corresponding negative residuals.
UVec
is a normalized vector of the cone.
It is usually its vertex direction but it may
also be its interior vector pointing to a vertex
of the artificial intersection of the cone with
the bounding box [-1,1]^M. The
max normalization is used if the breadth-first
search algorithm is employed and the L2
normalization is used in the other case (when
M = 2 and CTechST$D2SpecI = 1).
BDVec
is the vector c(b_1,...,b_M), i.e.,
the constant vector denominator of BVec,
where
BVec = BDVec/(t(BDVec)%*%UVec).
ADVec
is the vector c(a_1,...,a_P), i.e.,
the constant vector denominator of AVec,
where
AVec = ADVec/(t(BDVec)%*%UVec).
LambdaD
is the constant scalar denominator of
Lambda = LambdaD/(t(BDVec)%*%UVec).
Recall that c(BVec, AVec) stands for the coefficients
of the regression quantile hyperplane associated with
UVec and that Lambda denotes the Lagrange
multiplier equal to the optimal value Psi of the objective
function for that direction.
If CTechST$BriefOutputI = 0, then the rows of the
potential output file(s) are longer
(of length 1+1+M+M+P+1+(P+M-1)*M+(P+M-1))
because they contain two more vectors appended at the end.
The rows are of the form
c(ConeID, Nu, UVec, BDVec, ADVec, LambdaD, vec(VUMat), IZ)
where
VUMat
is the matrix for computing the multiplier vector
MuR0Vec associated with zero residuals,
MuR0Vec
= (VUMat%*%UVec)/(t(BDVec)%*%UVec).
That is to say that all directions from the
interior of the cone result in the regression
Tau-quantile hyperplanes containing
the same P+M-1 observations because all
such hyperplanes are the same up to a scaling
factor multiplying their coefficients.
IZ
is the vector containing original indices of the
M+P-1 observations with zero residuals
for all directions from the interior of the cone.
Value
getCharSTM1u returns a list with the following
components:
NUESkip
the number of (skipped) directions (and
corresponding hyperplanes) artificially
induced by intersecting the cones with
the [-1,1]^M bounding box.
NAZSkip
the number of (skipped) hyperplanes (and
corresponding directions) not counted in NUESkip
and with at least one coordinate of
AVec zero.
NBZSkip
the number of (skipped) hyperplanes (and
corresponding directions) not counted in NUESkip
and with at least one coordinate of
BVec zero.
HypMat
(for M > 4) the component is missing
(for M <= 4) the matrix with M + P
columns containing (in rows) all the distinct
regression Tau-quantile hyperplane
coefficients c(BVec, AVec) normalized
with |BVec|, rounded to the eighth
decimal digit, and sorted lexicographically.
This matrix can be used for the computation
of the regression Tau-quantile contour.
CharMaxMat
the matrix with the (slightly rounded) maxima
of certain directional regression Tau-quantile
characteristics over all remaining vertex
directions.
If P = 1, then CharMaxMat has
only three rows:
c(UVec, max(|BVec|)),
c(UVec, max(Lambda)), and
c(UVec, max(Lambda/|BVec|)),
respectively.
If P > 1, then the rows of
CharMaxMat are as follows:
c(UVec, max(|BVec|)),
c(UVec, max(Lambda)),
c(UVec, max(Lambda/|BVec|)),
c(UVec, max(|c(a_2,...,a_P)|)),
c(UVec, max(|c(a_2,...,a_P)|/|BVec|)),
c(UVec, max(|a_2|)),
c(UVec, max(|a_2|/|BVec|)),
...,
c(UVec, max(|a_P|)), and
c(UVec, max(|a_P|/|BVec|)),
respectively. If P = 2,
then the last two rows are missing for not
being included twice.
CharMinMat
the matrix with the (slightly rounded) minima
of certain directional regression Tau-quantile
characteristics over all remaining vertex
directions.
If P = 1, then CharMinMat has
only three rows:
c(UVec, min(|BVec|)),
c(UVec, min(Lambda)), and
c(UVec, min(Lambda/|BVec|)),
respectively.
If P > 1, then CharMinMat
has five rows:
c(UVec, min(|BVec|)),
c(UVec, min(Lambda)),
c(UVec, min(Lambda/|BVec|)),
c(UVec, min(|c(a_2,...,a_P)|)), and
c(UVec, min(|c(a_2,...,a_P)|/|BVec|)),
respectively.
Note that || symbolizes the Euclidean norm,
and that the vertices (UVec) in the rows of
CharMaxMat and CharMinMat are generally
different and denote (one of) the direction(s) where
the row maximum or minimum is attained.
Examples
##Run print(getCharSTM1u) to examine the default setting.