Fit two-step estimator for inference in sparse high-dimensional additive models.
This is an R package which accompanies the paper:
Karl Gregory. Enno Mammen. Martin Wahl. “Statistical inference in sparse high-dimensional additive models.” Ann. Statist. 49 (3) 1514 - 1536, June 2021. https://doi.org/10.1214/20-AOS2011
Install with the R commands:
install.packages("devtools")
devtools::install_github("gregorkb/spaddinf")
Y = f1(X1) + … + fq(Xq) + ε, where we assume 𝔼Y = 0 and 𝔼fj(Xj) = 0 for all j = 1, …, q.
The estimator of fj is constructed in two steps:
- First, an undersmoothed presmoothing estimator f̂j(pre) of fj is obtained.
- Second, a less-wiggly smoother is applied to the evaluations of f̂j(pre) at the design points.
The two functions spadd.presmth.Bspl and spadd.presmth.Legr are for
obtaining the presmoothing estimators; they construct the presmoothing
estimators from cubic B-spline and Legendre polynomial bases,
respectively.
It is computationally expensive to compute the presmoothing estimator
f̂j(pre) for all j = 1, …, q when q is
large. Therefore, both of these functions take an argument n.foi with
which the user may specify the number of functions of interest for
which to obtain the presmoothing estimator. For example, n.foi = 5
will cause the presmoothing estimators for the first 5 components
(corresponding to the first 5 columns of the design matrix) to be
returned.
The following code uses the function data_gen to generate data
according to the simulation design in Gregory et al. (2021) and then
obtains the presmoothing estimators for the first 6 components. The
argument d.pre is the number of basis functions to be used for fitting
each component. The arguments lambda and eta are for the tuning
parameters λ and η from the paper.
The output is plotted with the function plot_presmth_Bspl.
library(spaddinf)
## Loading required package: splines
## Loading required package: grplasso
# generate some data as in Gregory et al. (2020)
data <- data_gen(n = 200, q = 10, r = .5)
# obtain presmoothing estimators for first 6 components
spadd.presmth.Bspl.out <- spadd.presmth.Bspl(X = data$X,
Y = data$Y,
d.pre = 20,
lambda = 1,
eta = 3,
n.foi = 6)
# make plots
plot_presmth_Bspl(spadd.presmth.Bspl.out,
true.functions = list( f = data$f,
X = data$X))
The following code does the same as the previous code, but constructs
the presmoothing estimators from Legendre polynomial bases. Note the
additional argument K for choosing the order of the Legendre
polynomials; the default is K=1, which results in piecewise linear
functions.
# obtain presmoothing estimators for first 6 components
spadd.presmth.Legr.out <- spadd.presmth.Legr(X = data$X,
Y = data$Y,
d.pre = 20,
lambda = 1,
eta = 3,
n.foi = 6,
K = 1)
# make plots
plot_presmth_Legr(spadd.presmth.Legr.out,
true.functions = list( f = data$f,
X = data$X))
The functions spadd.presmth.Bspl.cv and spadd.presmth.Legr.cv make
choices of the tuning parameters λ and η from grids of values, while
constructing the presmoothing estimators from cubic B-spline and
Legendre polynomial bases, respectively.
Rather than searching over a 2-dimensional grid for the best (λ,η)
pair, these functions first select λ in order to minimize
crossvalidation prediction error for
f̂1L + … + f̂qL (see
section 3.1 of the paper), and then select η by minimizing the average
crossvalidation prediction error for
Π̂−1Lbj1, …, Π̂−1Lbj**d,
where d is d.pre, the number of basis functions (see Section 3.3 of
the paper). To reduce computation time, η is only chosen for j = 1
(this could be easily changed in the code).
The user specifies by the arguments n.lambda and n.eta the number of
λ and η values to consider. Grids of values are constructed on a
logarithmic scale (see code).
The following code generates data according to the simulation design in
Gregory et al. (2020) and uses the function spadd.presmth.Bspl.cv to
choose λ and η using crossvalidation for the cubic B-spline
presmoother. The crossvalidation code takes a couple of minutes to run.
The values chosen in crossvalidation are then given to the function
spadd.presmth.Bspl so that the presmoothing estimator can be obtained.
Some plots of the output are generated.
# generate some data as in Gregory et al. (2020)
data <- data_gen(n = 150,q = 30,r = .9)
# get CV choices of lambda and eta: takes a couple of minutes to run
spadd.presmth.Bspl.cv.out <- spadd.presmth.Bspl.cv(X = data$X,
Y = data$Y,
d.pre = 15,
n.lambda = 25,
n.eta = 25,
n.folds = 5,
plot = TRUE)
## [1] "lambda fold: 1"
## [1] "lambda fold: 2"
## [1] "lambda fold: 3"
## [1] "lambda fold: 4"
## [1] "lambda fold: 5"
## [1] "eta fold: 1"
## [1] "eta fold: 2"
## [1] "eta fold: 3"
## [1] "eta fold: 4"
## [1] "eta fold: 5"
# obtain presmoothing estimators for first 6 components
spadd.presmth.Bspl.out <- spadd.presmth.Bspl(X = data$X,
Y = data$Y,
d.pre = 20,
lambda = spadd.presmth.Bspl.cv.out$cv.lambda,
eta = spadd.presmth.Bspl.cv.out$cv.eta,
n.foi = 6)
# make plots
plot_presmth_Bspl(spadd.presmth.Bspl.out)
The functions spadd.presmth.Bspl.Bspl and spadd.presmth.Legr.Bspl
compute the two-step estimators when the presmoothers are constructed
with cubic B-spline and Legendre polynomial bases, respectively, with
the resmoothers constructed from cubic B-spline bases.
The user provides d.pre, which is as before, and, optionally, d.re,
which is the number of functions in the bases used for resmoothing. If
d.re = NULL, then this is chosen using a crossvalidation procedure.
Values of λ and η for constructing the presmoothers must be
provided. Setting plot = TRUE produces plots of each resmoother with
(1−α)100% pointwise confidence intervals of the form in equation (14)
of the paper, where α is specified by the argument alpha. The red
circles in the plots are the values of the presmoothing estimators at
the design points.
The code below demonstrates this.
# generate some data as in Gregory et al. (2020)
data <- data_gen(n = 200, q = 50, r = .9)
# get two-step estimator with B-spline presmoother and B-spline resmoother
preresmth.Bspl.Bspl.out <- preresmth.Bspl.Bspl(Y = data$Y,
X = data$X,
d.pre = 20,
d.re = 10,
lambda = 1,
eta = 3,
n.foi = 2,
plot = TRUE,
alpha = 0.05)
# get two-step estimator with Legendre polynomial presmoother and B-spline resmoother
preresmth.Legr.Bspl.out <- preresmth.Legr.Bspl(Y = data$Y,
X = data$X,
d.pre = 20,
d.re = 10,
lambda = 1,
eta = 3,
n.foi = 2,
plot = TRUE,
alpha = 0.05)
