IASSMR.kNN.fit {fsemipar} | R Documentation |
Impact point selection with IASSMR and kNN estimation
Description
This function implements the Improved Algorithm for Sparse Semiparametric Multi-functional Regression (IASSMR) with kNN estimation. This algorithm is specifically designed for estimating multi-functional partial linear single-index models, which incorporate multiple scalar variables and a functional covariate as predictors. These scalar variables are derived from the discretisation of a curve and have linear effects while the functional covariate exhibits a single-index effect.
IASSMR is a two-stage procedure that selects the impact points of the discretised curve and estimates the model. The algorithm employs a penalised least-squares regularisation procedure, integrated with kNN estimation using Nadaraya-Watson weights. It uses B-spline expansions to represent curves and eligible functional indexes. Additionally, it utilises an objective criterion (criterion
) to determine the initial number of covariates in the reduced model (w.opt
), the number of neighbours (k.opt
), and the penalisation parameter (lambda.opt
).
Usage
IASSMR.kNN.fit(x, z, y, train.1 = NULL, train.2 = NULL,
seed.coeff = c(-1, 0, 1), order.Bspline = 3, nknot.theta = 3, knearest = NULL,
min.knn = 2, max.knn = NULL, step = NULL, range.grid = NULL,
kind.of.kernel = "quad", nknot = NULL, lambda.min = NULL, lambda.min.h = NULL,
lambda.min.l = NULL, factor.pn = 1, nlambda = 100, vn = ncol(z), nfolds = 10,
seed = 123, wn = c(10, 15, 20), criterion = "GCV", penalty = "grSCAD",
max.iter = 1000, n.core = NULL)
Arguments
x |
Matrix containing the observations of the functional covariate collected by row (functional single-index component). |
z |
Matrix containing the observations of the functional covariate that is discretised collected by row (linear component). |
y |
Vector containing the scalar response. |
train.1 |
Positions of the data that are used as the training sample in the 1st step. The default setting is |
train.2 |
Positions of the data that are used as the training sample in the 2nd step. The default setting is |
seed.coeff |
Vector of initial values used to build the set |
order.Bspline |
Positive integer giving the order of the B-spline basis functions. This is the number of coefficients in each piecewise polynomial segment. The default is 3. |
nknot.theta |
Positive integer indicating the number of regularly spaced interior knots in the B-spline expansion of |
knearest |
Vector of positive integers containing the sequence in which the number of nearest neighbours |
min.knn |
A positive integer that represents the minimum value in the sequence for selecting the number of nearest neighbours |
max.knn |
A positive integer that represents the maximum value in the sequence for selecting number of nearest neighbours |
step |
A positive integer used to construct the sequence of k-nearest neighbours as follows: |
range.grid |
Vector of length 2 containing the endpoints of the grid at which the observations of the functional covariate |
kind.of.kernel |
The type of kernel function used. Currently, only Epanechnikov kernel ( |
nknot |
Positive integer indicating the number of interior knots for the B-spline expansion of the functional covariate. The default value is |
lambda.min |
The smallest value for lambda (i. e., the lower endpoint of the sequence in which |
lambda.min.h |
The lower endpoint of the sequence in which |
lambda.min.l |
The lower endpoint of the sequence in which |
factor.pn |
Positive integer used to set |
nlambda |
Positive integer indicating the number of values in the sequence from which |
vn |
Positive integer or vector of positive integers indicating the number of groups of consecutive variables to be penalised together. The default value is |
nfolds |
Number of cross-validation folds (used when |
seed |
You may set the seed for the random number generator to ensure reproducible results (applicable when |
wn |
A vector of positive integers indicating the eligible number of covariates in the reduced model. For more information, refer to the section |
criterion |
The criterion used to select the tuning and regularisation parameters: |
penalty |
The penalty function applied in the penalised least-squares procedure. Currently, only "grLasso" and "grSCAD" are implemented. The default is "grSCAD". |
max.iter |
Maximum number of iterations allowed across the entire path. The default value is 1000. |
n.core |
Number of CPU cores designated for parallel execution. The default is |
Details
The multi-functional partial linear single-index model (MFPLSIM) is given by the expression
Y_i=\sum_{j=1}^{p_n}\beta_{0j}\zeta_i(t_j)+r\left(\left<\theta_0,X_i\right>\right)+\varepsilon_i,\ \ \ (i=1,\dots,n),
where:
-
Y_i
represents a real random response andX_i
denotes a random element belonging to some separable Hilbert space\mathcal{H}
with inner product denoted by\left\langle\cdot,\cdot\right\rangle
. The second functional predictor\zeta_i
is assumed to be a curve defined on the interval[a,b]
, observed at the pointsa\leq t_1<\dots<t_{p_n}\leq b
. -
\mathbf{\beta}_0=(\beta_{01},\dots,\beta_{0p_n})^{\top}
is a vector of unknown real coefficients, andr(\cdot)
denotes a smooth unknown link function. In addition,\theta_0
is an unknown functional direction in\mathcal{H}
. -
\varepsilon_i
denotes the random error.
In the MFPLSIM, it is assumed that only a few scalar variables from the set \{\zeta(t_1),\dots,\zeta(t_{p_n})\}
are part of the model. Therefore, the relevant variables in the linear component (the impact points of the curve \zeta
on the response) must be selected, and the model estimated.
In this function, the MFPLSIM is fitted using the IASSMR. The IASSMR is a two-step procedure. For this, we divide the sample into two independent subsamples, each asymptotically half the size of the original (n_1\sim n_2\sim n/2
). One subsample is used in the first stage of the method, and the other in the second stage.The subsamples are defined as follows:
\mathcal{E}^{\mathbf{1}}=\{(\zeta_i,\mathcal{X}_i,Y_i),\quad i=1,\dots,n_1\},
\mathcal{E}^{\mathbf{2}}=\{(\zeta_i,\mathcal{X}_i,Y_i),\quad i=n_1+1,\dots,n_1+n_2=n\}.
Note that these two subsamples are specified in the program through the arguments train.1
and train.2
. The superscript \mathbf{s}
, where \mathbf{s}=\mathbf{1},\mathbf{2}
, indicates the stage of the method in which the sample, function, variable, or parameter is involved.
To explain the algorithm, we assume that the number p_n
of linear covariates can be expressed as follows: p_n=q_nw_n
, with q_n
and w_n
being integers.
-
First step. The FASSMR (see
FASSMR.kNN.fit
) combined with kNN estimation is applied using only the subsample\mathcal{E}^{\mathbf{1}}
. Specifically:Consider a subset of the initial
p_n
linear covariates, which contains onlyw_n
equally spaced discretized observations of\zeta
covering the entire interval[a,b]
. This subset is the following:\mathcal{R}_n^{\mathbf{1}}=\left\{\zeta\left(t_k^{\mathbf{1}}\right),\ \ k=1,\dots,w_n\right\},
where
t_k^{\mathbf{1}}=t_{\left[(2k-1)q_n/2\right]}
and\left[z\right]
denotes the smallest integer not less than the real numberz
.The size (cardinality) of this subset is provided to the program in the argumentwn
(which contains a sequence of eligible sizes).Consider the following reduced model, which involves only the
w_n
linear covariates belonging to\mathcal{R}_n^{\mathbf{1}}
:Y_i=\sum_{k=1}^{w_n}\beta_{0k}^{\mathbf{1}}\zeta_i(t_k^{\mathbf{1}})+r^{\mathbf{1}}\left(\left<\theta_0^{\mathbf{1}},X_i\right>\right)+\varepsilon_i^{\mathbf{1}}.
The penalised least-squares variable selection procedure, with kNN estimation, is applied to the reduced model. This is done using the function
sfplsim.kNN.fit
, which requires the remaining arguments (seesfplsim.kNN.fit
). The estimates obtained after that are the outputs of the first step of the algorithm.
-
Second step. The variables selected in the first step, along with those in their neighborhood, are included. The penalised least-squares procedure, combined with kNN estimation, is carried out again considering only the subsample
\mathcal{E}^{\mathbf{2}}
. Specifically:Consider a new set of variables:
\mathcal{R}_n^{\mathbf{2}}=\bigcup_{\left\{k,\widehat{\beta}_{0k}^{\mathbf{1}}\not=0\right\}}\left\{\zeta(t_{(k-1)q_n+1}),\dots,\zeta(t_{kq_n})\right\}.
Denoting by
r_n=\sharp(\mathcal{R}_n^{\mathbf{2}})
, the variables in\mathcal{R}_n^{\mathbf{2}}
can be renamed as follows:\mathcal{R}_n^{\mathbf{2}}=\left\{\zeta(t_1^{\mathbf{2}}),\dots,\zeta(t_{r_n}^{\mathbf{2}})\right\},
Consider the following model, which involves only the linear covariates belonging to
\mathcal{R}_n^{\mathbf{2}}
Y_i=\sum_{k=1}^{r_n}\beta_{0k}^{\mathbf{2}}\zeta_i(t_k^{\mathbf{2}})+r^{\mathbf{2}}\left(\left<\theta_0^{\mathbf{2}},X_i\right>\right)+\varepsilon_i^{\mathbf{2}}.
The penalised least-squares variable selection procedure, with kNN estimation, is applied to this model using the function
sfplsim.kNN.fit
.
The outputs of the second step are the estimates of the MFPLSIM. For further details on this algorithm, see Novo et al. (2021).
Remark: If the condition p_n=w_n q_n
is not met (then p_n/w_n
is not an integer number), the function considers variable q_n=q_{n,k}
values k=1,\dots,w_n
. Specifically:
q_{n,k}= \left\{\begin{array}{ll}
[p_n/w_n]+1 & k\in\{1,\dots,p_n-w_n[p_n/w_n]\},\\
{[p_n/w_n]} & k\in\{p_n-w_n[p_n/w_n]+1,\dots,w_n\},
\end{array}
\right.
where [z]
denotes the integer part of the real number z
.
The function supports parallel computation. To avoid it, we can set n.core=1
.
Value
call |
The matched call. |
fitted.values |
Estimated scalar response. |
residuals |
Differences between |
beta.est |
|
theta.est |
Coefficients of |
indexes.beta.nonnull |
Indexes of the non-zero |
k.opt |
Selected number of nearest neighbours (when |
w.opt |
Selected initial number of covariates in the reduced model. |
lambda.opt |
Selected value of the penalisation parameter |
IC |
Value of the criterion function considered to select |
vn.opt |
Selected value of |
beta2 |
Estimate of |
theta2 |
Estimate of |
indexes.beta.nonnull2 |
Indexes of the non-zero linear coefficients after the step 2 of the method for each value of the sequence |
knn2 |
Selected number of neighbours in the second step of the algorithm for each value of the sequence |
IC2 |
Optimal value of the criterion function in the second step for each value of the sequence |
lambda2 |
Selected value of penalisation parameter in the second step for each value of the sequence |
index02 |
Indexes of the covariates (in the entire set of |
beta1 |
Estimate of |
theta1 |
Estimate of |
knn1 |
Selected number of neighbours in the first step of the algorithm for each value of the sequence |
IC1 |
Optimal value of the criterion function in the first step for each value of the sequence |
lambda1 |
Selected value of penalisation parameter in the first step for each value of the sequence |
index01 |
Indexes of the covariates (in the whole set of |
index1 |
Indexes of the non-zero linear coefficients after the step 1 of the method for each value of the sequence |
... |
Author(s)
German Aneiros Perez german.aneiros@udc.es
Silvia Novo Diaz snovo@est-econ.uc3m.es
References
Novo, S., Vieu, P., and Aneiros, G., (2021) Fast and efficient algorithms for sparse semiparametric bi-functional regression. Australian and New Zealand Journal of Statistics, 63, 606–638, doi:10.1111/anzs.12355.
See Also
See also sfplsim.kNN.fit, predict.IASSMR.kNN
, plot.IASSMR.kNN
and FASSMR.kNN.fit
.
Alternative method IASSMR.kernel.fit
Examples
data(Sugar)
y<-Sugar$ash
x<-Sugar$wave.290
z<-Sugar$wave.240
#Outliers
index.y.25 <- y > 25
index.atip <- index.y.25
(1:268)[index.atip]
#Dataset to model
x.sug <- x[!index.atip,]
z.sug<- z[!index.atip,]
y.sug <- y[!index.atip]
train<-1:216
ptm=proc.time()
fit<- IASSMR.kNN.fit(x=x.sug[train,],z=z.sug[train,], y=y.sug[train],
train.1=1:108,train.2=109:216,nknot.theta=2,lambda.min.h=0.07,
lambda.min.l=0.07, max.knn=20, nknot=20,criterion="BIC", max.iter=5000)
proc.time()-ptm
fit
names(fit)