Description Usage Arguments Details Value References See Also Examples
Performs a Monte Carlo simulation to estimate the distribution of the null thresholding statistic required for computation of the quantile universal threshold, and computes its upper alpha-quantile if alpha is provided.
1 2 3 4 |
X |
input matrix of dimension |
q |
size of noise dictionaries. A noise dictionary consists in a
Gaussian matrix G of size |
M |
number of noise dictionaries used. |
alpha |
level of the quantile universal threshold. By default
|
sigma |
standard deviation of the noise. If |
intercept |
if |
standardizeX |
whether the columns of |
standardizeG |
either a positive numerical value indicating the desired
Euclidean norm of all columns of the noise dictionaries, or a logical value
indicating whether the columns of the noise dictionaries should be
standardized to have unit standard deviation. If |
MCrep |
number of Monte Carlo replications. Default is |
GEVapprox |
whether to approximate the distribution of the null
thresholding statistic by a GEV distribution. If |
parallel |
if |
var.subset |
subset of variables for which QUT is computed (i.e. we will compute the parameter value for which P(betahat[var.subset]) = 0) = 1-alpha when beta = 0. |
If the noise level sigma
is known, the statistic of interest is simply
the sup-norm of the Lasso-Zero coefficients obtained under the null
hypothesis (i.e. when all coefficients all zero) when the threshold
tau
is set to 0, and its upper alpha-quantile is the quantile
universal threshold. If sigma = NULL
(sigma unknown) a pivotized
statistic is used, which is obtained by dividing the statistic described
above by the MAD of all nonzero noise coefficients obtained by Lasso-Zero.
An object of class "qut.MC"
, which is a list with the
following components:
allMC |
all |
GEVpar |
MLE estimates of the GEV distribution parameters
( |
GEVfit |
set to NULL is GEVapprox is FALSE. If GEVapprox is TRUE, GEVfit is either the result of the gev.fit function, or the character string "error" if gev.fit produced an error. |
upperQuant |
upper alpha-quantile of the null thresholding statistis (either the empirical quantile, or the quantile of the fitted GEV distribution). |
call |
matched call. |
lass0settings |
a list
containing the chosen settings for the computation of lasso-zero: |
Descloux, P., & Sardy, S. (2018). Model selection with lasso-zero: adding straw to the haystack to better find needles. arXiv preprint arXiv:1805.05133. https://arxiv.org/abs/1805.05133
Giacobino, C., Sardy, S., Diaz-Rodriguez, J., & Hengartner, N. (2017). Quantile universal threshold. Electronic Journal of Statistics, 11(2), 4701-4722.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | ### Fast toy example with 5x10 input matrix and a small number (MCrep = 50)
### of Monte Carlo replications.
### Illustrates how to tune Lasso-Zero with QUT for the same input matrix but
### different responses and/or different alpha values, without calling
### qut.MC several times:
### (for faster computation when X and MCrep are larger: register a parallel
### backend and choose parallel = TRUE when calling lass0 and qut.MC functions.)
set.seed(3)
## input matrix:
n <- 5
p <- 10
X <- matrix(rnorm(n*p), n, p)
## two sparse vectors and corresponding responses:
S1 <- 1:2 # first support
beta1 <- numeric(p)
beta1[S1] <- 5
y1 <- X[, S1] %*% beta1[S1] + rnorm(n)
S2 <- 3:4 # second support
beta2 <- numeric(p)
beta2[S2] <- 5
y2 <- X[, S2] %*% beta2[S2] + rnorm(n)
## Monte Carlo simulation giving empirical distribution for the statistic P (see paper below):
qut.MC.output <- qut.MC(X, parallel = FALSE, MCrep = 50)
## lasso-zero estimates:
## for y1 with alpha = 0.1:
lass01 <- lass0(X, y1, alpha = 0.1, qut.MC.output = qut.MC.output, parallel = FALSE)
plot(lass01)
## for y2 with alpha = 0.05:
lass02 <- lass0(X, y2, alpha = 0.05, qut.MC.output = qut.MC.output, parallel = FALSE)
plot(lass02)
|
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