FLAME: Global FLAME method: from the definition of the kernel to the...

In ardeeshany/FLAME: FLAME for high-dimensional Function-on-Scalar regression problems

Description

It computes FLAME for the Function-on-Scalar regression problem. From a set of functions -stored in a matrix or in an fd object- and a set of predictors, FLAME identifies the set of meaningful predictors and their smooth representation in the kernel space.

Usage

FLAME(Y, X, type_kernel = "sobolev", param_kernel = 8, thres_eigen = 0.99,
  period_kernel = NULL, NoI = 10, thres_CD = 0.01,
  number_non_zeros = NULL, ratio_lambda = 0.01, number_lambda = 100,
  proportion_training_set = 0.75, verbose = FALSE)

Arguments

Y	fd object or list. N functional responses of the Function-on-Scalar regression probelm. It can be an fd object or a list with 2 elements: data and time_domain. time_domain is the m-length vector of the absicissa grid of the functions and data is the N \times m matrix of the point-wise evaluation of the response functions.
X	matrix. N \times I design matrix. It has standardized columns.
type_kernel	string. Four possible choices are implemented. gaussian, exponential, sobolev or periodic. For other kernels, define manually the eigenfunctions and eigenvectors and then use the estimation_beta function. Defualt is sobolev.
param_kernel	scalar. Value of the characteristic smoothing parameter of the kernel. It is the σ parameter of the Gaussian and the Exponential kernel, as introduced in rbfdot and laplacedot functions; the σ parameter of the Sobolev kernel as in the sobolev_kernel_generation function or the σ paramter of the periodic kernel of the generation_kernel_periodic function. Default is 8.
thres_eigen	scalar. Threshold to identify the significant eigenvalues of the kernel. The number of significant eigennvalues J is the minimum J s.t. ∑_{j = 1}^{J} θ_j ≥q \textrm{thres\_eigen} ∑_{j = 1}^{∞} θ_j. Default is 0.99.
period_kernel	scalar. Period of the kernel. In case of type_kernel = "periodic", it is a mandatory parameter with no default. If other types of kernel are chosen, it is ignored.
NoI	scalar. integer, maximum number of iterations in the Coordinate-Descent loop. Default is 10.
thres_CD	scalar. tolerance in the increment of the K-norm of the estimation to stop the Coordinate-Descent loop. Default is 0.01
number_non_zeros	scalar. integer, threshold on the number of non zeros parameters to be detected. It is the kill switch parameter. See the Vignette for further details. Default is NULL meaning that no kill switch paramter is imposed.
ratio_lambda	scalar. ratio to compute the minimum value of lambda. The maximum λ (λ_{\textrm{max}}) is computed as the minimum value which makes all the coefficients equal to zero. And the minimum is the product ratio_lambda\times λ_{\max}. Default is 0.01.
number_lambda	scalar. integer, length of the grid for the λ parameter. Default is 100.
proportion_training_set	scalar. value in (0,1), the proportion for the training set for the Cross Validation. Defualt is 0.75.
verbose	bool. If TRUE the progression of the algorithm in the adaptive and non adaptive step is shown. If FALSE no output is shown. Default is FALSE.

Value

list containing:

• beta fd object or list. I functional coefficients estimated by FLAME. If Y is an fd object. then also beta is an fd object, and beta is the projection of the estimations on a 10 elements cubic Bspline basis. If Y is a list, then also beta is a list with 2 elements: data and time_domain. time_domain is the m-length domain grid, while data is I \times m matrix of the point-wise evaluation of the estimated coefficients.

• predictors vector of the indices of the non-zero estimated predictors.

Examples

## Not run: 
data(simulation)
data(SobolevKernel)
time <- proc.time()
FLAME_estimation <- FLAME()
duration <- proc.time()-time
duration
names(FLAME_estimation)

## End(Not run)

ardeeshany/FLAME documentation built on May 14, 2019, 8:41 a.m.