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

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):

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)

Evaluating Convex Polytopes

Description

Given the system of inequalities AAMat%*%ZVec <= BBVec describing a convex polytope/contour with an interior point IPVec in the Euclidean space of dimension two to six, this function identifies all nonredundant constraints and computes some characteristics of the resulting convex polytope such as its vertices, facets, volume and surface area.

Usage

evalContour(AAMat, BBVec = NULL, IPVec = NULL)

Arguments

Details

This function is included to be used for evaluating (regression) quantile contours or their cuts.

In fact, the function analyzes not the polytope itself, but its regularized intersection with the zero-centered hypercube of the edge length 2 000 that is employed as an artificial bounding box to avoid the problems with unbounded contours. The regularization consists of rounding the vertices (i.e., all of their coordinates) of such an intersection to the seventh decimal digit and of considering only the polytope determined by all the distinct rounded vertices for the final analysis.

Value

evalContour returns a list with the following components describing the resulting convex polytope:

Examples

##- a simple example using a tilted zero-centered square 
AAMat <- rbind(c(-1,-1), c(1,-1), c(1,1), c(-1,1))
BBVec <- c(1, 1, 1, 1)
IPVec <- c(0, 0)
CST <- evalContour(AAMat, BBVec, IPVec)
print(CST)

##- computing and evaluating the 0.15-quantile contour of 199 
##random points uniformly distributed in the unit square 
##centered at zero
Tau  <- 0.15
YMat <- matrix(runif(2*199, -0.5, 0.5), 199, 2)
C <- compContourM1u(Tau, YMat)
CST <- evalContour(-C$CharST$HypMat[,1:2], -C$CharST$HypMat[,3])
print(CST)

##See also the examples ExampleA to ExampleE for some 
##more elaborate ways of computing, evaluating and 
##plotting the (regression) quantile contours.

Getting the List of Options CTechST for compContourM1/2u

Description

The functions getCTechSTM1u and getCTechSTM2u set the default list of options CTechST for computing all the directional (regression) quantiles by means of compContourM1u and compContourM2u, respectively.

Usage

getCTechSTM1u()
getCTechSTM2u()

Arguments

none

Details

Fortunately, the default list of options usually leads to a satisfactory performance in all but very large problems.

Value

Both getCTechSTM1u and getCTechSTM2u produce a list with a few components whose default values are stated below after the equality sign.

The components OutFilePrefS and getCharST are initialized in a method-specific way.

The components CubRegWiseI, ArchAllFI, and SkipRedI are relevant only if D2SpecI is zero or if the dimension of directions/responses is higher than two, i.e., if the breadth-first search algorithm is used.

Most of the components are generated by both functions. Nevertheless, the component SkipRedI is only generated by getCTechSTM2u and used by compContourM2u.

The output components are as follows:

Examples

##- a typical use of getCTechSTM1u: 
##computing all directional 0.01-quantiles of 49 random points 
##(uniformly distributed in the unit cube centered at zero) 
##after changing the default settings 
Tau  <- 0.01
XMat <- matrix(1, 49, 1)
YMat <- matrix(runif(3*49, -0.5, 0.5), 49, 3)
CTechST <- getCTechSTM1u()
CTechST$ReportI <- 1
COutST <- compContourM1u(Tau, YMat, XMat, CTechST)

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

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:

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.

Computing Some Overall Characteristics in compContourM2u

Description

The function computes some overall characteristics of directional regression quantiles in the output of compContourM2u, 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

getCharSTM2u(Tau, N, M, P, BriefDQMat, CharST, IsFirst)

Arguments

Details

This function is called inside compContourM2u. 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+P*M+M of the form c(ConeID, Nu, UVec, vec(ACOMat), MuBRow) 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 negative residuals corresponding to the interior directions of the cone.

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).

ACOMat

is the matrix describing AVec, AVec = ACOMat%*%UVec.

MuBRow

is the constant vector of the Lagrange multipliers corresponding to BVec. Its inner product with UVec is equal to the optimal value Psi of the objective function for that direction.

Recall that c(BVec, AVec) stands for the coefficients of the regression quantile hyperplane associated with UVec and always BVec = UVec.

If CTechST$BriefOutputI = 0, then the rows of the potential output file(s) are longer (of length 1+1+P*M+M+P+P) because they contain two more vectors appended at the end. The rows are of the form c(ConeID, Nu, UVec, vec(ACOMat), MuBRow, MuR0Row, IZ) where

MuRORow

is the constant vector of the Lagrange multipliers corresponding to zero residuals associated with the interior of the cone. That is to say that all directions from the interior of the cone result in the regression Tau-quantile hyperplanes containing the same P observations.

IZ

is the vector containing original indices of the P observations with zero residuals for all directions from the interior of the cone.

Value

getCharSTM2u returns a list with the following components:

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(getCharSTM2u) to examine the default setting.