Central quantile subspace

cqs estimates the directions of the central quantile subspace.

cqs(x, y, tau = 0.5, dtau = NULL)

Arguments

x

A design matrix (n x p). The rows represent observations and the columns represent predictor variables.

y

A vector of the response variable.

tau

A quantile level, a number strictly between 0 and 1.

dtau

An optional dimension of the central quantile subspace. If specified, it should be an integer between 1 and p, the number of columns of the design matrix x. In the context of the algorithm, if dtau is known to be one, i.e., the assumed model is a single-index model, then the algorithm stops after estimating the initial vector and saves computational time. However, if dtau is greater than one or (more realistically) unknown, then the algorithm continues on creating more vectors.

Value

cqs computes the directions of the central quantile subspace and returns:

qvectors:

The estimated directions of the \(\tau\)th central quantile subspace.

qvalues:

The eigenvalues resulting from the eigenvalue decomposition of the matrix with column vectors that span the central quantile subspace. If dtau is one, the qvalues output is not produced.

dtau:

Suggested dimension of the central quantile subspace. If dtau is specified by the user then the algorithm outputs the user-defined value. If dtau is not specified by the user then the algorithm outputs a suggested dimension using the modified-BIC type criterion of Zhu et al. (2010). Note that this is one suggested method to estimate the structural dimension and is not necessarily a perfect one. The user has the option to use the eigenvalues qvalues on other criteria, like cross-validation, and determine the estimated dimension of the subspace.

Details

The function computes the directions that span the \(\tau\)th central quantile subspace, i.e., the directions that define linear combinations of the predictor x that contain all the information available on the conditional quantile function.

The function starts by estimating the initial vector, which is defined as the least-squares estimator from regressing the conditional quantile on the predictor variable x. Then, if the dimension of the central quantile subspace is one, the algorithm stops and reports that vector as the basis of the central quantile subspace. Otherwise, the algorithm continues by creating more vectors and applying an eigenvalue decomposition to extract linearly independent vectors.

References

Zhu, L.-P., Zhu, L.-X., Feng, Z.-H. (2010) Dimension reduction in regression through cumulative slicing estimation. Journal of the American Statistical Association, 105, 1455-1466.

Examples

# estimate the directions of a single-index model
set.seed(1234)
n <- 100
p <- 10
x <- matrix(rnorm(n * p), n, p)
error <- rnorm(n)
y <- 3 * x[, 1] + x[, 2] + error
tau <- 0.5
out <- cqs(x, y, tau, dtau = 1)
out
#> $qvectors
#>               [,1]
#>  [1,]  0.952948139
#>  [2,]  0.287974471
#>  [3,]  0.042667367
#>  [4,]  0.038590479
#>  [5,] -0.003885815
#>  [6,] -0.042769158
#>  [7,]  0.028417264
#>  [8,]  0.018789922
#>  [9,]  0.023914784
#> [10,]  0.045541197
#> 
#> $dtau
#> [1] 1
#> 
# without specifying dtau
out <- cqs(x, y, tau)
out
#> $qvectors
#>              [,1]        [,2]         [,3]        [,4]        [,5]        [,6]
#>  [1,]  0.94285965 -0.04373861  0.323724575  0.10473035  0.06290808 -0.04667851
#>  [2,]  0.30671396  0.29208998 -0.401758607 -0.16287563  0.15202916 -0.21945183
#>  [3,]  0.03101309 -0.28351089 -0.199986929 -0.59193015 -0.45455105 -0.19721619
#>  [4,]  0.03394904 -0.16427022 -0.478614025  0.35884778 -0.13384503  0.13125004
#>  [5,] -0.00658252 -0.09730212 -0.205810821 -0.08802898  0.75194191 -0.04145494
#>  [6,] -0.02433910  0.31968626  0.006386568 -0.65344860  0.37885008 -0.06677676
#>  [7,]  0.02707587  0.03398101  0.354491863 -0.21839222  0.12549555  0.77574347
#>  [8,]  0.01569753 -0.12261129 -0.358513162  0.23379874  0.04091144  0.40756218
#>  [9,]  0.06317845  0.79095189  0.144685863  0.19507455 -0.18329861 -0.20917231
#> [10,]  0.07962450  0.54997345 -0.443034398 -0.38301190 -0.35618445  0.33809519
#>              [,7]        [,8]         [,9]       [,10]
#>  [1,] -0.07376381 -0.16872174  0.008474194  0.14020956
#>  [2,]  0.00725682  0.78950190  0.134953357  0.14996842
#>  [3,]  0.06903646 -0.25806832  0.566790028 -0.28810369
#>  [4,] -0.31360476 -0.09073977 -0.387039672 -0.57510228
#>  [5,] -0.39371036 -0.29970274  0.123874984 -0.05495289
#>  [6,]  0.47061321 -0.12265678 -0.366212884 -0.45406911
#>  [7,] -0.07727430  0.30837138  0.410192154 -0.40082730
#>  [8,]  0.74653472 -0.32831788  0.339507633  0.04297342
#>  [9,]  0.22776133 -0.25554310  0.624061136 -0.30765592
#> [10,] -0.10565655 -0.04530856 -0.094568566  0.39571912
#> 
#> $qvalues
#>  [1] 9.943967e+01 9.879354e-02 3.285674e-03 4.633843e-04 1.760913e-04
#>  [6] 3.195217e-05 1.161632e-05 4.207074e-06 1.322628e-06 1.268059e-10
#> 
#> $dtau
#> [1] 1
#> 
out$qvectors[, 1:out$dtau]
#>  [1]  0.94285965  0.30671396  0.03101309  0.03394904 -0.00658252 -0.02433910
#>  [7]  0.02707587  0.01569753  0.06317845  0.07962450