compContour: Directional Regression Quantile Computation
In modQR: Multiple-Output Directional Quantile Regression
compContourM1/2u R Documentation
Directional Regression Quantile Computation
Description
The functions compContourM1u and compContourM2u
may be used to obtain not only directional regression quantiles
for all directions, but also some related overall
statistics. Their output may also be used for the evaluation
of the corresponding regression quantile regions by means of
evalContour. The functions use different
methods and algorithms, namely compContourM1u is based
on [01] and [06] and compContourM2u results from [03]
and [07]. The corresponding regression quantile regions are
nevertheless virtually the same. See all the references below
for further details and possible applications.
Usage
compContourM1u(Tau = 0.2, YMat = NULL, XMat = NULL, CTechST = NULL)
compContourM2u(Tau = 0.2, YMat = NULL, XMat = NULL, CTechST = NULL)
Arguments
Tau
the quantile level in (0, 0.5).
YMat
the N x M response matrix with two to six
columns, N > M+P-1.
Each row corresponds to one observation.
XMat
the N x P design matrix including the (first)
intercept column. The default NULL value
corresponds to the unit vector of the right
length.
Each row corresponds to one observation.
CTechST
the (optional) list with some parameters
influencing the computation and its output.
Its default value can be generated by
method-dependent getCTechSTM1/2u
and then modified by the user before its use
in compContourM1/2u.
Details
Generally, the performance of the functions deteriorates with
increasing Tau, N, M, and P as for their reliability and
time requirements. Nevertheless, they should work fine
at least for two-dimensional problems
up to N = 10000 and P = 10,
for three-dimensional problems
up to N = 500 and P = 5,
and for four-dimensional problems
up to N = 150 and P = 3.
Furthemore, common problems related to the computation
can fortunately be prevented or overcome easily.
Bad data - the computation may fail if the processed
data points are in a bad configuration (i.e., if they are not
in general position or if they would lead to a quantile
hyperplane with at least one zero coefficient), which mostly
happens when discrete-valued/rounded/repeated observations,
dummy variables or bad random number generators are employed.
Such problems can often be prevented if one perturbs the data
with a random noise of a reasonably small magnitude before the
computation, splits the model into separate or independent
submodels, cleverly uses affine equivariance, or replaces
a few identical observations with a copy of them weighted
by the total number of their occurrences.
Bad Tau - the computation may fail for a finite number of
problematic quantile levels, e.g., if Tau is an integer multiple
of 1/N in the location case with unit weights (when the sample
quantiles are not uniquely defined). Such a situation may occur
easily for Tau's with only a few decimal digits or in a
fractional form, especially when the number of observations
changes automatically during the computation. The problem
can be fixed easily by perturbing Tau with a sufficiently small
number in the right direction, which should not affect the
resulting regression quantile contours although it may slightly
change the other output. The strategy is also adopted
by compContourM1/2u, but only in the location case and
with a warning output message explaining it.
Bad scale - the computation may fail easily for badly
scaled data. That is to say that the functionality has been
heavily tested only for the observations coming from
a centered unit hypercube. Nevertheless, you can always
change the units of measurements or employ full affine
equivariance to avoid all the troubles. Similar problems
may also arise when properly scaled data are used with highly
non-uniform weights, which frequently happens in local(ly)
polynomial regression. Then the weights can be rescaled
in a suitable way and the observations with virtually zero
weights can be excluded from the computation.
Bad expectations - the computation and its output need
not meet false expectations. Every user should be aware of
the facts that the computation may take a long time or fail
even for moderately sized three-dimensional data sets, that the
HypMat component is not always present in the list
COutST$CharST by default, and that the sample regression
quantile contours can be not only empty, but also unbounded and
crossing one another in the general regression case.
Bad interpretation - the output results may be easily
interpreted misleadingly or erroneously. That is to say that
the quantile level Tau is not linked to the probability
content of the sample (regression) Tau-quantile region in
any straightforward way. Furthermore, any meaningful
parametric quantile regression model should include
as regressors not only the variables influencing the
trend, but also all those affecting the dispersion
of the multivariate responses. Even then the cuts
of the resulting regression quantile contours parallel
to the response space cannot be safely interpreted
as conditional multivariate quantiles except for
some very special cases. Nevertheless, such a
conclusion could somehow be warranted in case of
nonparametric multiple-output quantile regression; see [09].
Value
Both compContourM1u and compContourM2u may display some
auxiliary information regarding the computation on the screen
(if CTechST$ReportI = 1) or store their in-depth
output (determined by CTechST$BriefOutputI) in the output
files (if CTechST$OutSaveI = 1) with
the filenames beginning with the string contained in
CTechST$OutFilePrefS, followed by the file number
padded with zeros to form six digits
and by the extension ‘.dqo’, respectively. The first
output file produced by compContourM1u would
thus be named ‘DQOutputM1_000001.dqo’.
Both compContourM1u and compContourM2u always return
a list with the same components. Their interpretation is
also the same (except for CharST that itself contains some
components that are method-specific):
CharST
the list with some default or
user-defined output.
The default one is provided
by function getCharSTM1u
for compContourM1u and by
function getCharSTM2u
for compContourM2u.
A user-defined function generating
its own output can be employed instead
by changing CTechST$getCharST.
CTechSTMsgS
the (possibly empty) string that informs
about the problems with input CTechST.
ProbSizeMsgS
the (possibly empty) string that warns
if the input problem is very large.
TauMsgS
the (possibly empty) string that announces
an internal perturbation of Tau.
CompErrMsgS
the (possibly empty) string that decribes
the error interrupting the computation.
NDQFiles
the counter of (possible) output files,
i.e., as if CTechST$OutSaveI = 1.
NumB
the counter of (not necessarily distinct) optimal
bases considered.
PosVec
the vector of length N that desribes
the position of individual (regression)
observations with respect to the
exact (regression) Tau-quantile
contour.
The identification is reliable only after a
successful computation.
PosVec[i] = 0/1/2 if the i-th
observation is in/on/out of the contour.
If compContourM2u is used with
CTechST$SkipRedI = 1, then PosVec
correctly detects only all the outer observations.
MaxLWidth
the maximum width of one layer of the
internal algorithm.
NIniNone
the number of trials when the initial
solution could not be found at all.
NIniBad
the number of trials when the found
initial solution did not have
the right number of clearly nonzero
coordinates.
NSkipCone
the number of skipped cones (where
an interior point could not be found).
If CTechST.CubRegWiseI = 1, then the last four
components are calculated over all the individual
orthants.
References
[01] Hallin, M., Paindaveine, D. and Šiman, M. (2010)
Multivariate quantiles and multiple-output regression quantiles:
from L1 optimization to halfspace depth.
Annals of Statistics 38, 635–669.
[02] Hallin, M., Paindaveine, D. and Šiman, M. (2010)
Rejoinder (to [01]).
Annals of Statistics 38, 694–703.
[03] Paindaveine, D. and Šiman, M. (2011)
On directional multiple-output quantile regression.
Journal of Multivariate Analysis 102, 193–212.
[04] Šiman, M. (2011)
On exact computation of some statistics based on projection
pursuit in a general regression context.
Communications in Statistics - Simulation and Computation 40, 948–956.
[05] McKeague, I. W., López-Pintado, S., Hallin, M. and Šiman, M. (2011)
Analyzing growth trajectories.
Journal of Developmental Origins of Health and Disease 2, 322–329.
[06] Paindaveine, D. and Šiman, M. (2012)
Computing multiple-output regression quantile regions.
Computational Statistics & Data Analysis 56, 840–853.
[07] Paindaveine, D. and Šiman, M. (2012)
Computing multiple-output regression quantile regions
from projection quantiles.
Computational Statistics 27, 29–49.
[08] Šiman, M. (2014)
Precision index in the multivariate context.
Communications in Statistics - Theory and Methods 43, 377–387.
[09] Hallin, M., Lu, Z., Paindaveine, D. and Šiman, M. (2015)
Local bilinear multiple-output quantile/depth regression.
Bernoulli 21, 1435–1466.
Examples
##computing all directional 0.15-quantiles of 199 random points
##uniformly distributed in the unit square centered at zero
##- preparing the input
Tau <- 0.15
XMat <- matrix(1, 199, 1)
YMat <- matrix(runif(2*199, -0.5, 0.5), 199, 2)
##- Method 1:
COutST <- compContourM1u(Tau, YMat, XMat)
##- Method 2:
COutST <- compContourM2u(Tau, YMat, XMat)
modQR documentation built on May 11, 2022, 5:18 p.m.
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| compContourM1/2u | R Documentation |
Directional Regression Quantile Computation
Description
The functions compContourM1u and compContourM2u
may be used to obtain not only directional regression quantiles
for all directions, but also some related overall
statistics. Their output may also be used for the evaluation
of the corresponding regression quantile regions by means of
evalContour. The functions use different
methods and algorithms, namely compContourM1u is based
on [01] and [06] and compContourM2u results from [03]
and [07]. The corresponding regression quantile regions are
nevertheless virtually the same. See all the references below
for further details and possible applications.
Usage
compContourM1u(Tau = 0.2, YMat = NULL, XMat = NULL, CTechST = NULL) compContourM2u(Tau = 0.2, YMat = NULL, XMat = NULL, CTechST = NULL)
Arguments
Tau |
the quantile level in (0, 0.5). |
YMat |
the N x M response matrix with two to six
columns, |
XMat |
the N x P design matrix including the (first) intercept column. The default NULL value corresponds to the unit vector of the right length. Each row corresponds to one observation. |
CTechST |
the (optional) list with some parameters
influencing the computation and its output.
Its default value can be generated by
method-dependent |
Details
Generally, the performance of the functions deteriorates with increasing Tau, N, M, and P as for their reliability and time requirements. Nevertheless, they should work fine at least for two-dimensional problems up to N = 10000 and P = 10, for three-dimensional problems up to N = 500 and P = 5, and for four-dimensional problems up to N = 150 and P = 3.
Furthemore, common problems related to the computation can fortunately be prevented or overcome easily.
Bad data - the computation may fail if the processed data points are in a bad configuration (i.e., if they are not in general position or if they would lead to a quantile hyperplane with at least one zero coefficient), which mostly happens when discrete-valued/rounded/repeated observations, dummy variables or bad random number generators are employed. Such problems can often be prevented if one perturbs the data with a random noise of a reasonably small magnitude before the computation, splits the model into separate or independent submodels, cleverly uses affine equivariance, or replaces a few identical observations with a copy of them weighted by the total number of their occurrences.
Bad Tau - the computation may fail for a finite number of
problematic quantile levels, e.g., if Tau is an integer multiple
of 1/N in the location case with unit weights (when the sample
quantiles are not uniquely defined). Such a situation may occur
easily for Tau's with only a few decimal digits or in a
fractional form, especially when the number of observations
changes automatically during the computation. The problem
can be fixed easily by perturbing Tau with a sufficiently small
number in the right direction, which should not affect the
resulting regression quantile contours although it may slightly
change the other output. The strategy is also adopted
by compContourM1/2u, but only in the location case and
with a warning output message explaining it.
Bad scale - the computation may fail easily for badly scaled data. That is to say that the functionality has been heavily tested only for the observations coming from a centered unit hypercube. Nevertheless, you can always change the units of measurements or employ full affine equivariance to avoid all the troubles. Similar problems may also arise when properly scaled data are used with highly non-uniform weights, which frequently happens in local(ly) polynomial regression. Then the weights can be rescaled in a suitable way and the observations with virtually zero weights can be excluded from the computation.
Bad expectations - the computation and its output need
not meet false expectations. Every user should be aware of
the facts that the computation may take a long time or fail
even for moderately sized three-dimensional data sets, that the
HypMat component is not always present in the list
COutST$CharST by default, and that the sample regression
quantile contours can be not only empty, but also unbounded and
crossing one another in the general regression case.
Bad interpretation - the output results may be easily interpreted misleadingly or erroneously. That is to say that the quantile level Tau is not linked to the probability content of the sample (regression) Tau-quantile region in any straightforward way. Furthermore, any meaningful parametric quantile regression model should include as regressors not only the variables influencing the trend, but also all those affecting the dispersion of the multivariate responses. Even then the cuts of the resulting regression quantile contours parallel to the response space cannot be safely interpreted as conditional multivariate quantiles except for some very special cases. Nevertheless, such a conclusion could somehow be warranted in case of nonparametric multiple-output quantile regression; see [09].
Value
Both compContourM1u and compContourM2u may display some
auxiliary information regarding the computation on the screen
(if CTechST$ReportI = 1) or store their in-depth
output (determined by CTechST$BriefOutputI) in the output
files (if CTechST$OutSaveI = 1) with
the filenames beginning with the string contained in
CTechST$OutFilePrefS, followed by the file number
padded with zeros to form six digits
and by the extension ‘.dqo’, respectively. The first
output file produced by compContourM1u would
thus be named ‘DQOutputM1_000001.dqo’.
Both compContourM1u and compContourM2u always return a list with the same components. Their interpretation is also the same (except for CharST that itself contains some components that are method-specific):
CharST |
the list with some default or
user-defined output.
The default one is provided
by function |
CTechSTMsgS |
the (possibly empty) string that informs
about the problems with input |
ProbSizeMsgS |
the (possibly empty) string that warns if the input problem is very large. |
TauMsgS |
the (possibly empty) string that announces
an internal perturbation of |
CompErrMsgS |
the (possibly empty) string that decribes the error interrupting the computation. |
NDQFiles |
the counter of (possible) output files,
i.e., as if |
NumB |
the counter of (not necessarily distinct) optimal bases considered. |
PosVec |
the vector of length N that desribes
the position of individual (regression)
observations with respect to the
exact (regression) Tau-quantile
contour.
The identification is reliable only after a
successful computation.
|
MaxLWidth |
the maximum width of one layer of the internal algorithm. |
NIniNone |
the number of trials when the initial solution could not be found at all. |
NIniBad |
the number of trials when the found initial solution did not have the right number of clearly nonzero coordinates. |
NSkipCone |
the number of skipped cones (where an interior point could not be found). |
If CTechST.CubRegWiseI = 1, then the last four
components are calculated over all the individual
orthants.
References
[01] Hallin, M., Paindaveine, D. and Šiman, M. (2010) Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth. Annals of Statistics 38, 635–669.
[02] Hallin, M., Paindaveine, D. and Šiman, M. (2010) Rejoinder (to [01]). Annals of Statistics 38, 694–703.
[03] Paindaveine, D. and Šiman, M. (2011) On directional multiple-output quantile regression. Journal of Multivariate Analysis 102, 193–212.
[04] Šiman, M. (2011) On exact computation of some statistics based on projection pursuit in a general regression context. Communications in Statistics - Simulation and Computation 40, 948–956.
[05] McKeague, I. W., López-Pintado, S., Hallin, M. and Šiman, M. (2011) Analyzing growth trajectories. Journal of Developmental Origins of Health and Disease 2, 322–329.
[06] Paindaveine, D. and Šiman, M. (2012) Computing multiple-output regression quantile regions. Computational Statistics & Data Analysis 56, 840–853.
[07] Paindaveine, D. and Šiman, M. (2012) Computing multiple-output regression quantile regions from projection quantiles. Computational Statistics 27, 29–49.
[08] Šiman, M. (2014) Precision index in the multivariate context. Communications in Statistics - Theory and Methods 43, 377–387.
[09] Hallin, M., Lu, Z., Paindaveine, D. and Šiman, M. (2015) Local bilinear multiple-output quantile/depth regression. Bernoulli 21, 1435–1466.
Examples
##computing all directional 0.15-quantiles of 199 random points ##uniformly distributed in the unit square centered at zero ##- preparing the input Tau <- 0.15 XMat <- matrix(1, 199, 1) YMat <- matrix(runif(2*199, -0.5, 0.5), 199, 2) ##- Method 1: COutST <- compContourM1u(Tau, YMat, XMat) ##- Method 2: COutST <- compContourM2u(Tau, YMat, XMat)
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