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et al. (2008); (ii) an iterative algorithm based on Ferraty et al. (2013).
2.2.1
Procedure in Ait-Sa¨ıdi et al. (2008)
Ait-Sa¨ıdi et al. (2008) proposed the joint minimisation in the tuning parameter and the functional
direction in (6). As a consequence, their method requires intensive computation. Thus, it is necessary
to strike a balance between the size of Θ and the performance of the estimators. The following lines
describe how to construct the set of coefficients V in (8):
|
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0.1471264809370041,
0.24842891097068787
] |
describe how to construct the set of coefficients V in (8):
1. Choose the dimension of the B-spline basis d. We can decompose d as d = ℓ + nr, where ℓ is the
order of the B-spline functions (degree ℓ − 1) and nr is the number of regularly spaced interior
knots (see De Boor 1978).
2. For each (β1, . . . , βd) ∈ Cd, where C = {c1, . . . , cJ} ⊂ RJ denotes a set of J “seed-coefficients”,
build the initial functional direction
θinit(·) =
d
�
j=1
βjej(·).
(10)
|
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] |
build the initial functional direction
θinit(·) =
d
�
j=1
βjej(·).
(10)
3. For each θinit in Step 1 that verifies the condition θinit(t0) > 0, where t0 denotes a fixed value
in the domain of θinit(·), compute ⟨θinit, θinit⟩ and construct
(α1, . . . , αd) =
(β1, . . . , βd)
⟨θinit, θinit⟩1/2 .
4. Construct V as the set of vectors (α1, . . . , αd) obtained in Step 2.
Therefore, the final set of eligible functional directions is given by
Θ =
θ(·) =
d
�
j=1
αjej(·); (α1, . . . , αd) ∈ V
.
7
|
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] |
Θ =
θ(·) =
d
�
j=1
αjej(·); (α1, . . . , αd) ∈ V
.
7
Subsequently, the goal is to compute the value of the CV functions in (6) for each pair (θ, h) (respec-
tively (θ, k)) where θ ∈ Θ, and select pair that minimises (6).
Remark 2.1. To achieve a balance between computation time and accurate estimation, Ait-Sa¨ıdi
et al. 2008 recommended using the seed-coefficients C = {1, 0, −1}. This set has been shown to offer
|
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] |
good practical performance (see Novo et al., 2019). Regarding the dimension of the B-spline basis,
d, the usual procedure is to set the order of the B-spline functions (ℓ) and try different number of
internal knots, nr.
Then, we can choose the value of nr that minimises the CV function: �nr =
arg minnr∈Z+ CV (�θ�h,�h) (respectively, �n∗
r = arg minnr∈Z+ CV ∗(�θ�k, �k)).
2.2.2
Procedure in Ferraty et al. (2013)
|
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0.06401396542787552,
-0.05962246283888817,
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-0.2862256169319153,
-0.538836658000946,
-0.19889335334300995,
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] |
r = arg minnr∈Z+ CV ∗(�θ�k, �k)).
2.2.2
Procedure in Ferraty et al. (2013)
Instead of the direct joint minimisation, Ferraty et al. (2013) and Chan et al. (2023) proposed an
iterative algorithm. For a given d:
1. Choose an initial set of coefficients γγγ = (γ1, . . . , γd) ∈ Rd to give rise to a functional direction
θ(0,m) as in (10). Calibrate them to verify the identifiability conditions of the model. As a result
|
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] |
you obtain another set of coefficients ααα = (α1, . . . , αd) corresponding to a functional direction
θ(m).
2. Select the optimal tuning parameter setting θ(m) in (6): �h(m) = arg minh∈[am,bm] CV (h, θ(m))
(respectively, �k(m) = arg minkm1<k<km2 CV ∗(k, θ(m))).
3. Estimate the direction setting �h(m) (respectively, �k(m)) in (6): �θ(0,m+1) = arg minγγγ∈Rd CV (�h(m), θ),
(�θ∗(0,m+1) = arg minγγγ∈Rd CV ∗(�k(m), θ)).
|
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] |
(�θ∗(0,m+1) = arg minγγγ∈Rd CV ∗(�k(m), θ)).
4. Update θ(0,m) = �θ(0,m+1) (θ(0,m) = �θ∗(0,m+1)). Continue the process until convergence is achieved:
the algorithm halts when the change in CV from one iteration to the next (scaled by the variance
of the response) is positive and falls below a predetermined threshold.
Remark 2.2. In step 3 of the method, the minimisation can be carried out using general-purpose
|
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] |
Remark 2.2. In step 3 of the method, the minimisation can be carried out using general-purpose
optimisation algorithms for multidimensional spaces, such as Nelder and Mead (1965), included in the
R function optim.
2.3
Fitting the FSIM with the fsemipar package
The fsemipar package contains several functions for estimating the FSIM. It provides the flexibility
to perform both kernel and kNN fitting, with implementation options based on either Ait-Sa¨ıdi et al.
|
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] |
(2008) or Ferraty et al. (2013). The primary related functions are summarised in the Table 1. Each
primary function produces an object that belongs to an S3 class. These S3 classes are designed to
include essential S3 methods, such as print(), summary(), plot() and predict() to facilitate the
posterior analysis. Each function offers a wide range of customisable parameters for the user, while
also providing a straightforward option to use default settings for the input parameters.
8
Function
S3 class
|
[
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-0.1810072362422943,
-0.06821652501821518,
-0.39883798360824585,
0.35006996989250183,
0.6854225397109985,
-0.30490967631340027,
-0.44506779313087463,
0.21977616846561432
] |
8
Function
S3 class
Description
fsim.kernel.fit
‘fsim.kernel’
kernel estimation based on Ait-Sa¨ıdi et al. (2008)
fsim.kNN.fit
‘fsim.kNN’
kNN estimation based on Ait-Sa¨ıdi et al. (2008)
fsim.kernel.fit.optim
‘fsim.kernel’
kernel estimation based on Ferraty et al. (2013)
fsim.kNN.fit.optim
‘fsim.kNN’
kNN estimation based on Ferraty et al. (2013)
Table 1: Summary of main functions for the FSIM and associated S3 classes.
2.3.1
Main fitting functions
|
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] |
2.3.1
Main fitting functions
The main fitting functions for the FSIM (Table 1) implement either the procedure detailed in Ait-Sa¨ıdi
et al. (2008) (fsim.kernel.fit() and fsim.kNN.fit()) or the procedure outlined in Ferraty et al.
(2013) (fsim.kernel.fit.optim() and fsim.kNN.fit.optim()). In each case, these procedures are
combined with either kernel or kNN estimation.
Regarding the functional predictor, all the functions of the package fsemipar require the values
|
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] |
Regarding the functional predictor, all the functions of the package fsemipar require the values
of the curves evaluated on a grid of equally-spaced discretisation points. The functions internally
represent the discretised curves in a B-spline basis of dimension that can be customised by the user.
Functions in Table 1 share two required arguments: the functional predictor x, a matrix of dimension
(n, p) containing n curves discretised in p points, and the scalar response variable y, a vector of
|
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] |
(n, p) containing n curves discretised in p points, and the scalar response variable y, a vector of
dimension n.
fsim.kernel.fit(x,y,...)
fsim.kNN.fit(x,y,...)
fsim.kernel.fit.optim(x,y,...)
fsim.kNN.fit.optim(x,y,...)
However, these functions also take several optional arguments for finer control:
• Optional arguments for the nonparametric fit.
These arguments are summarised in Table 2
and refer to the type of kernel used and the choice of the tuning parameters: h (kernel), or
|
[
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] |
and refer to the type of kernel used and the choice of the tuning parameters: h (kernel), or
k (kNN). In the kernel-based functions fsim.kernel.fit() and fsim.kernel.fit.optim(),
there is a default option for constructing the sequence of bandwidths from which �h will be
selected. This default option takes into account the proximity between sample curves using the
semimetric (in the FSIM, the projection semimetric). Specifically, the endpoints of the sequence
|
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] |
semimetric (in the FSIM, the projection semimetric). Specifically, the endpoints of the sequence
are determined by calculating quantiles of a certain order, controlled by the user through the
arguments min.q.h (default 0.05) and max.q.h (default 0.5), of the distances between curves.
Additionally, the number of bandwidths in the sequence can also be customised using num.h
(default 10). Alternatively, we can directly specify the sequence of bandwidths with the argument
h.seq.
|
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] |
h.seq.
When h.seq is provided, the arguments min.q.h, max.q.h and num.h are ignored.
Regarding kNN-based functions fsim.kNN.fit() and fsim.kNN.fit.optim(), the user has the
flexibility to choose the endpoints of the sequence for k (number of neighbours) using min.knn
(default 2) and max.knn (default integer division n/5). In addition, the user can specify the step
between two consecutive elements of the sequence using the argument step (default smallest
9
|
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] |
between two consecutive elements of the sequence using the argument step (default smallest
9
integer greater than or equal to n/100). Alternatively, we can directly provide the sequence of
number of neighbours using knearest. It is important to note that if we specify the knearest
argument, min.knn, max.knn and step are ignored.
The four primary functions share the argument kind.of.kernel that allows user to choose the
type of kernel function. Currently, only Epanechnikov kernel (“quad”) is available.
|
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] |
type of kernel function. Currently, only Epanechnikov kernel (“quad”) is available.
Function
Argument
Description
fsim.kernel.fit
fsim.kernel.fit.optim
min.q.h
Minimum quantile order for the distances.
max.q.h
Maximum quantile order for the distances.
num.h
Number of h values in the sequence.
h.seq
Sequence of h values.
fsim.kNN.fit
fsim.kNN.fit.optim
min.knn
Minimum k for kNN fitting.
max.knn
Maximum k for kNN fitting.
step
Step between elements of the sequence.
knearest
Sequence of k values.
All
|
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] |
step
Step between elements of the sequence.
knearest
Sequence of k values.
All
kind.of.kernel
Type of kernel used.
Table 2: Optional input arguments related to nonparametric estimation (kernel or kNN).
• Optional arguments for B-spline expansions. These arguments are summarised in Table 3 and
refer to the B-spline expansion for the functional directions and predictor. The four functions
allow to customize the order of the B-spline functions, ℓ, through the argument order.Bspline
|
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] |
allow to customize the order of the B-spline functions, ℓ, through the argument order.Bspline
(default 3), the number of interior knots, nr, for directions in Θ using nknot.theta (default 3),
and the number of interior knots for the sample curves with nknot (default floor (p − ℓ − 1)/2),
and their definition interval with range.grid.
In addition, functions fsim.kernel.fit() and fsim.kNN.fit() allow to choose the set of seed
|
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] |
In addition, functions fsim.kernel.fit() and fsim.kNN.fit() allow to choose the set of seed
coefficients C through the argument seed.coeff (see Section 2.2.1). Furthermore, the func-
tions fsim.kernel.fit() and fsim.kNN.fit.optim() enable the choice of the set of initial
coefficients γ ∈ Rd through the argument gamma.
• Optional arguments for the parallel computation. For functions fsim.kernel.fit() and fsim.kN
N.fit() the implementation is computationally expensive. For that, they both allow the parallel
|
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] |
N.fit() the implementation is computationally expensive. For that, they both allow the parallel
computation (internally, they use the functions makeCluster(), registerDoParallel(), and
foreach() of the R packages parallel, doParallel and foreach, respectively). Users can ex-
plicitly specify the number of CPU cores to use for parallel execution with the argument n.core.
By default, the both functions take n.core as the number of available cores minus 1 if more
|
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] |
By default, the both functions take n.core as the number of available cores minus 1 if more
than 1 is available, otherwise 1 (this is done internally using the function availableCores() of
the R package parallely). For sequential execution, it is sufficient to set n.core=1.
• Optional arguments for controlling the iterative algorithm. The functions fsim.kernel.fit.opt
im() and fsim.kNN.fit.optim() are built upon an iterative procedure (see Section 2.2.2).
10
|
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] |
im() and fsim.kNN.fit.optim() are built upon an iterative procedure (see Section 2.2.2).
10
Users have the flexibility to customize the convergence threshold using the argument threshold
(default: 5e-3).
Function
Arguments
Description
fsim.kernel.fit
fsim.kNN.fit
fsim.kernel.fit.optim
fsim.kNN.fit.optim
order.Bspline
Order of the B-spline basis functions.
nknot.theta
Number of interior knots for the functional
directions in Θ.
nknot
Number of interior knots for the curves.
range.grid
|
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] |
directions in Θ.
nknot
Number of interior knots for the curves.
range.grid
Endpoints of the definition interval for the
curves.
fsim.kernel.fit
fsim.kNN.fit
seed.coeff
Set of seed coefficients C.
fsim.kernel.fit.optim
fsim.kNN.fit.optim
gamma
Set of initial coefficients γ.
Table 3: Optional input arguments related to B-spline expansions.
2.3.2
Case study I, part I: Tecator dataset
The Tecator dataset is a well-known benchmark database in FDA, as evidenced by the numerous
|
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] |
The Tecator dataset is a well-known benchmark database in FDA, as evidenced by the numerous
research works in the field that have used it to illustrate their techniques (see, e.g., Amato et al.
2006, Ferraty and Vieu 2006, Galeano et al. 2015, among others). This dataset contains spectral
measurements of 215 pork samples collected using a Tecator Infratec Food and Feed Analyser. The
spectrum includes absorbances at 100 wavelengths in the range 850 to 1050 nm, which are commonly
|
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] |
spectrum includes absorbances at 100 wavelengths in the range 850 to 1050 nm, which are commonly
considered to be functional data. In addition, the dataset records the fat, protein and moisture content
of each meat sample.
The Tecator dataset is included in the fsemipar package. In addition to the discretised original
spectrometric curves, the first and second derivatives are included.
Figure 1 includes a graphical
|
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] |
spectrometric curves, the first and second derivatives are included.
Figure 1 includes a graphical
representation of the first 20 curves (left panel) and its second derivative (right panel). We illustrate
the primary FSIM fitting functions of the package by predicting the fat percentage using the available
spectrometric data, specifically the second derivative of the curves (usual choice in the literature, see,
|
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] |
e. g., Aneiros and Vieu 2015, Novo et al. 2019). Note that spectrometric data is more readily obtain-
able, as analysing the chemical composition typically involves cost-intensive and lengthier chemical
experiments. To evaluate the performance of the procedures we split the sample into two subsamples:
the training sample is composed of the first 160 entries in the dataset, whereas the test sample includes
the remaining 55 observations.
> data(Tecator)
> str(Tecator)
List of 6
$ fat
|
[
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] |
the remaining 55 observations.
> data(Tecator)
> str(Tecator)
List of 6
$ fat
: num [1:215] 22.5 40.1 8.4 5.9 25.5 42.7 42.7 10.6 19.9 19.9 ...
11
2.5
3.0
3.5
4.0
4.5
850
900
950
1000
1050
Wavelength
χ(t)
−100
−50
0
50
850
900
950
1000
1050
Wavelength
χ(2)(t)
Figure 1: Sample of 20 spectrometric curves (left panel) and their second derivatives (right panel).
$ protein
: num [1:215] 16.7 13.5 20.5 20.7 15.5 13.7 13.7 19.3 17.7 17.7 ...
$ moisture
: num [1:215] 60.5 46 71 72.8 58.3 44 44 69.3 61.4 61.4 ...
|
[
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] |
$ moisture
: num [1:215] 60.5 46 71 72.8 58.3 44 44 69.3 61.4 61.4 ...
$ absor.spectra : num [1:215, 1:100] 2.62 2.83 2.58 2.82 2.79 ...
$ absor.spectra1: num [1:215, 1:100] 0.0352 0.4171 0.1772 0.1776 0.168 ...
$ absor.spectra2: num [1:215, 1:100] 1.118 -4.412 -4.764 -6.778 -0.601 ...
As an example, the following code shows the implementation of the FSIM using the fsim.kNN.fit()
function. In addition to the data, we have provided some specific input arguments to improve the fit:
|
[
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] |
we have narrowed the sequence in which the number of neighbours is selected (setting max.knn=15),
discarding the largest values which in this case lead to oversmoothing and an increment of the com-
putational time. We also set step=1 to try all the values in this sequence. Additionally, we have
specified the number of interior knots for the B-spline representation of the unknown functional di-
rection (nknot.theta=4) and for the curves (nknot=20), also providing the range in which the curves
|
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] |
rection (nknot.theta=4) and for the curves (nknot=20), also providing the range in which the curves
were observed (range.grid=c(850,1050)). It is important to note that it is reasonable to seek a very
precise representation of the curves on the B-spline basis, which leads to the use of a B-spline basis of
greater dimension than in the case of the functional directions. In fact, large values of nknot.theta
will tend to undersmoothing and capture the noise of the training sample. Furthermore, the com-
|
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] |
will tend to undersmoothing and capture the noise of the training sample. Furthermore, the com-
putational cost of estimating the functional direction is strongly related to nknot.theta (greater
nknot.theta more intensive computation), as we can deduce from Figure 2. For these reasons, it is
usually recommended to take nknot.theta between 2 and 8.
> y<-Tecator$fat
> x<-Tecator$absor.spectra2
> train=1:160
> test=161:215
> fit<-fsim.kNN.fit(y[train],x=x[train,],max.knn=15,nknot.theta=4,nknot=20,
|
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] |
> test=161:215
> fit<-fsim.kNN.fit(y[train],x=x[train,],max.knn=15,nknot.theta=4,nknot=20,
range.grid=c(850,1050),step=1)
The fitted object of the functions listed in the Table 1 contains useful information about the esti-
mation performed: the fitted values for the response (fitted.values), the residuals (residuals),
12
the coefficients of �θ in the B-spline basis (theta.est), the selected value of the tuning parameter, ˆh
|
[
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] |
(h.opt) or ˆk (k.opt), the coefficient of determination (r.squared), the residual variance (var.res),
the degrees of freedom (df) or the minimum value of the CV function (CV.opt). The following code
shows the information contained in the output of the function fsim.kNN.fit().
> names(fit)
[1] "fitted.values"
"residuals"
"theta.est"
"k.opt"
[5] "r.squared"
"var.res"
"df"
"yhat.cv"
[9] "CV.opt"
"CV.values"
"H"
"m.opt"
[13] "theta.seq.norm" "k.seq"
"call"
"y"
[17] "x"
"n"
"kind.of.kernel" "range.grid"
|
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] |
"H"
"m.opt"
[13] "theta.seq.norm" "k.seq"
"call"
"y"
[17] "x"
"n"
"kind.of.kernel" "range.grid"
[21] "nknot"
"order.Bspline"
"nknot.theta"
> class(fit)
[1] "fsim.kNN"
Each fitted object belongs to an S3 class (see Table 1), which implements S3 methods. In the par-
ticular case of fsim.kNN.fit() (class fsim.kNN), the returned object can be used in the functions
print.fsim.kNN() and summary.fsim.kNN() to display summaries of the fitted model. It can also be
|
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] |
print.fsim.kNN() and summary.fsim.kNN() to display summaries of the fitted model. It can also be
employed in predict.fsim.kNN() to obtain predictions for new curves (which can be provided using
the newdata argument), and the mean squared error of prediction (MSEP) if the actual responses are
provided via the y.test argument. The MSEP is calculated as follows:
MSEP =
1
ntest
n+ntest
�
i=n+1
�
Yi − �Yi
�2
,
(11)
|
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] |
MSEP =
1
ntest
n+ntest
�
i=n+1
�
Yi − �Yi
�2
,
(11)
where ntest is the size of the test sample and �Yi, the predicted values. The code below illustrates these
functionalities:
> summary(fit)
*** FSIM fitted using kNN estimation with Nadaraya-Watson weights ***
-Call: fsim.kNN.fit(x = x[1:160, ], y = y[1:160], nknot.theta = 4,
max.knn = 15, step = 1, range.grid = c(850, 1050), nknot = 20)
-Number of neighbours (k): 9
-Theta coefficients in the B-spline basis: 0.1656316 -0.1656316 0
|
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] |
-Number of neighbours (k): 9
-Theta coefficients in the B-spline basis: 0.1656316 -0.1656316 0
0.1656316 -0.1656316 0 0.1656316
-CV: 3.921117
-R squared: 0.9824381
-Residual variance: 3.323535 on 132.9207 degrees of freedom
> predict(fit,newdata=x[test,],y.test=y[test])$MSEP
[1] 2.68585
13
Regarding the prediction, Table 4 displays the MSEP corresponding to each primary fitting function
when using the respective cross-validation (CV) selectors for nr, h, or k and θ0. In this example, the
|
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] |
kNN-based functions yield lower MSEP values than the kernel-based ones. Furthermore, the procedure
described in Ait-Sa¨ıdi et al. (2008) results in a lower MSEP than the one presented in Ferraty et al.
(2013). However, for the functions based on the former procedure, when we increase nr (nknot.theta),
the computation time also increases, while for the functions based on the latter, it remains constant.
This fact can be observed in Figure 2, where we show the execution times for the four functions over
|
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0.11090222746133804,
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] |
nr. Note that fsim.kernel.fit() and fsim.kNN.fit() use parallel computation and were executed
with n.core=19. For each function we set all the remaining optional input parameters (in particular,
order.Bspline=3) while we vary nknot.theta.
Function
�nk
MSEP
fsim.kernel.fit()
4
3.49
fsim.kNN.fit()
4
2.69
fsim.kernel.fit.optim()
2
3.64
fsim.kNN.fit.optim()
4
2.81
Table 4: MSEP obtained for each function using the respective CV selectors for tuning parameters,
functional index, and nr.
G
G
G
G
G
G
G
G
G
G
G
G
|
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] |
functional index, and nr.
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
0
30
60
90
2
3
4
5
6
7
nknot.theta
Time (s)
G
G
G
G
fsim.kernel.fit()
fsim.kernel.fit.optim()
fsim.kNN.fit()
fsim.kNN.fit.optim()
Execution time
Figure 2: Execution time for functions in Table 1 over nr obtained in a computer with the following
features: Intel Core i9-10850K, 64GB, 1TB SSD and 2TB HDD, RTX 3090 24GB.
The S3 classes listed in Table 1 also implement the S3 method plot(). The routines implementing
|
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] |
The S3 classes listed in Table 1 also implement the S3 method plot(). The routines implementing
this S3 method use internally the R package ggplot2, trying to produce elegant and high quality
charts. The output of this method represents the regression fit and the estimated functional index.
In Figure 3, we provide the outputs of the 4 main functions when using the CV selectors for tuning
parameters, functional index, and nr.
14
−0.1
0.0
0.1
850
900
950
1000
1050
range.grid X
θ0
G
G
G
G
G
G
G
G
GG
G
G
G
|
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850
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1050
range.grid X
θ0
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0
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50
−300
−200
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0
100
<θ0,X>
r(<θ0,X>)
(d) fsim.knn.fit.optim() (�n∗
r = 4)
Figure 3: Outputs of the S3 method plot() applied to the fitted objects of functions in the Table 1,
|
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] |
using the corresponding CV estimator for θ0 and CV selectors for the tuning parameters and nr.
3
Multiple scalar predictors and a functional covariate
In Section 3, we focus on SoF semiparametric regression including as predictors multiple scalar vari-
ables and a functional one. Section 3.1 introduces the modelling alternatives based on the partial
linear structure. In Section 3.2, we discuss the techniques implemented in fsemipar for simultaneous
|
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] |
estimation and variable selection in the linear component. Finally, Section 3.3 presents the associated
functions of the package and illustrates their use with the Tecator dataset (see Section 2.3.2).
3.1
Semi-functional partial linear regression
In practical applications, it is common to encounter more than one covariate. Particularly, situations
arise where, in addition to a functional predictor, several scalar variables are linked to the response.
|
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] |
Take, for example, the Tecator dataset (see Section 2.3.2), which includes measurements of not only
fat content but also the percentage of protein and moisture for each meat sample. These additional
variables can be valuable in predicting fat content.
Statistically, this implies that, in addition to Y and X, there exists a vector ZZZ = (Z1, . . . , Zpn)⊤ of
scalar variables potentially related to the response. We observe tuples {(Yi,ZZZi, Xi), i = 1, . . . , n} i.i.d.
|
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] |
to (Y,ZZZ, X) and our goal is to predict Y using the information carried by the remaining variables. In
these complex scenarios, combining the partial linear approach with functional nonparametric or func-
15
tional single-index modelling typically yields interpretable results with good predictive performance.
Consequently, literature in FDA focused on two regression strategies:
• The semi-functional partial linear model (SFPLM) (see Aneiros-P´erez and Vieu 2006, Aneiros
|
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] |
• The semi-functional partial linear model (SFPLM) (see Aneiros-P´erez and Vieu 2006, Aneiros
et al. 2015, Boente and Vahnovan 2017, among others), which is given by the expression
Yi = Zi1β01 + · · · + Zipnβ0pn + m (Xi) + εi
i = 1, . . . , n.
(12)
Here, βββ0 = (β01, . . . , β0pn)⊤ is a vector of unknown real coefficients and m(·) is an unknown
functional operator with the same features as in the FNM.
• The semi-functional partial linear single-index model (SFPLSIM) (Wang et al. 2016, Novo et al.
|
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] |
• The semi-functional partial linear single-index model (SFPLSIM) (Wang et al. 2016, Novo et al.
2021a), which is given by the relationship
Yi = Zi1β01 + · · · + Zipnβ0pn + r (⟨θ0, Xi⟩) + εi
i = 1, . . . , n.
(13)
Here, the vector βββ0 = (β01, . . . , β0pn)⊤ ∈ Rp is assumed to be unknown, as is the functional
index θ0 ∈ H and the real-valued link function r(·), which both have the same characteristics as
in the FSIM.
|
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] |
in the FSIM.
An interesting situation arises when only some of the pn scalar covariates have a real effect on the
response. Then estimation techniques must be combined with a variable selection tool.
The SFPLM, along with the kernel-based estimation procedure introduced in Aneiros-P´erez and
Vieu 2006, can be implemented in R using the function fregre.plm from the fda.usc package. How-
ever, it is worth noting that this function does not support variable selection in the linear component or
|
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] |
kNN-based estimation. Furthermore, as far as we know, there is currently no function available for es-
timating the SFPLSIM and performing variable selection excluding those belonging to the fsemipar
package. Consequently, in the subsequent section we introduce the techniques implemented in the
fsemipar related to the SFPLM and SFPLSIM.
3.2
Estimation and variable selection in the linear component
Aneiros et al. (2015) introduced a procedure based on penalised least-squares (PeLS) for simultaneous
|
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] |
estimation and variable selection in the linear component for the SFPLM (12). This procedure entails
transforming the model into a linear model using nonparametric regression. Specifically:
1. The first step is to extract the effect of the functional covariate from the response and predictors:
Yi − E (Yi|Xi) = (ZZZi − E (ZZZi|Xi))⊤ βββ0 + εi,
i = 1, . . . , n,
(14)
where ZZZi = (Zi1, . . . , Zipn). Given that the expectations in the expression (14) are unknown, it
|
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] |
becomes necessary to derive estimations for them. To accomplish this, we can employ functional
nonparametric regression, utilizing either the kernel (2) or the kNN (3) estimators; so we need
to choose an appropriate semimetric (see Section 2.2). After that, an approximate linear model
is obtained:
�YYY ≈ �ZZZβββ0 + εεε,
(15)
where �YYY = (�
Y1, . . . , �
Yn)⊤ and �ZZZ = (�
ZZZ1, . . . , �
ZZZn)⊤ are the residuals of the regressions of the re-
|
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] |
Yn)⊤ and �ZZZ = (�
ZZZ1, . . . , �
ZZZn)⊤ are the residuals of the regressions of the re-
sponse variable and the predictors, respectively, over the functional covariate, and εεε = (ε1, . . . , εn)⊤.
16
2. The second step entails the application of the PeLS approach to model (15), minimising the
profile least-squares function with respect to βββ = (β1, . . . , βpn)⊤:
Q (βββ) = 1
2
�
�YYY − �ZZZβββ
�⊤ �
�YYY − �ZZZβββ
�
+ n
pn
�
j=1
Pλj (βj) .
(16)
|
[
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] |
Q (βββ) = 1
2
�
�YYY − �ZZZβββ
�⊤ �
�YYY − �ZZZβββ
�
+ n
pn
�
j=1
Pλj (βj) .
(16)
Here Pλj (·) represents a penalty function and λj > 0 is a tuning parameter that controls the
amount of penalisation. In fact, under suitable conditions on Pλ (see, e.g., Fan and Li 2001),
the PeLS estimators produce sparse solutions, that is, many estimated coefficients are zero (see
Aneiros et al. 2022 for a review on variable selection in the functional setting using shrinkage
|
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] |
Aneiros et al. 2022 for a review on variable selection in the functional setting using shrinkage
methods). Therefore, we perform simultaneous estimation and variable selection in the linear
component.
3. The third step is the estimation of the functional operator m(·). After obtaining �βββ0, a natural
way to perform this estimation is to smooth the partial residuals, Yi − ZZZ⊤
i �βββ0. For that it is
enough to replace Yi by Yi − ZZZ⊤
i �βββ0 in (2) for kernel estimation, or in (3) for kNN techniques.
|
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] |
i �βββ0 in (2) for kernel estimation, or in (3) for kNN techniques.
Regarding the choice of the penalty function Pλ (·) in (16), there is a large list of possibilities in the
literature for linear models. The LASSO (Tibshirani 1996), defined as Pλ(βj) = λ |βj|, is the most
famous one, despite not satisfying oracle properties (see Fan and Li 2001). A prominent alternative
to norm-based penalties like the LASSO is the SCAD (Fan 1997), defined, for a > 2, as
Pλ(βj) =
λ |βj|
|βj| < λ,
|
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] |
Pλ(βj) =
λ |βj|
|βj| < λ,
(a2 − 1)λ2 − (|βj| − aλ)2
2(a − 1)
λ ≤ |βj| < aλ,
(a + 1)λ2
2
|βj| ≥ aλ.
(17)
(Fan and Li 2001 suggested taking a = 3.7) The SCAD satisfies the oracle properties and, in general,
improves the features of the LASSO (Fan and Li 2001).
Aneiros et al. (2015) demonstrated the
consistency of variable selection achieved using the PeLS procedure combined with the SCAD penalty.
|
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] |
consistency of variable selection achieved using the PeLS procedure combined with the SCAD penalty.
They also derived the convergence rates for the estimators and established the oracle property in the
context of the SFPLM.
Both penalty functions, LASSO and SCAD, are included in the fsemipar package. In fact, the
package features a generalization of both penalties. Specifically, it facilitates dealing with grouped
|
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] |
partial linear models. In this scenario, the scalar regressors are divided into M groups, denoted as
ZZZim = (Zim1, . . . , Zimvm)⊤ for m = 1, . . . , M, where �M
m=1 vm = pn. In such situations, the focus is
not on selecting variables individually, but rather on choosing important factors, each corresponding
to a group of covariates. Yuan and Lin (2005) proposed the group LASSO penalty defined as Pλ(βββ) =
λ �M
m=1
�
βββ⊤
|
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] |
λ �M
m=1
�
βββ⊤
mKmβββm, where Km are positive definite matrices for m ∈ 1, . . . , M. They used Km =
vmIvm, with m ∈ 1, . . . , M, where Ivm is the identity matrix of dimension vm. Wang et al. (2007)
proposed the group SCAD penalty, defined as Pλ(βββ) = λ �M
m=1 Pλ
��vm
r=1 β2
mr
�
, where Pλ(·) was
defined in (17). Both group LASSO and group SCAD penalties are included in the fsemipar package.
It is important to note that when M = 1, individual selection is performed.
|
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] |
It is important to note that when M = 1, individual selection is performed.
Another important aspect is the selection of the parameter λj (j = 1, . . . , pn) on which the penalty
functions depend (see expression (16)). To reduce the number of parameters to select, we consider
17
λj = λ�σβ0,j,OLS, where β0,j,OLS denotes the ordinary least-squares (OLS) estimate of β0,j and �σβ0,j,OLS
is the estimated standard deviation.
|
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] |
is the estimated standard deviation.
Remark 3.1. Note that in step 1 we must perform nonparametric regression, which involves a tuning
parameter, h (kernel) or k (kNN). Therefore we compute the nonparametric fit for a grid of values
for the tuning parameter, with each value leading to a different linear model in expression (15). In the
package fsemipar, the PeLS objective function (16) is minimised using the function grpreg of the
|
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] |
package fsemipar, the PeLS objective function (16) is minimised using the function grpreg of the
package grpreg (Breheny and Huang 2015). Specifically, for each linear model (15) (setting the value
of h or k), the function grpreg fits the regularization path over a grid of values for λ (obtains the value
of β that minimises (16) for each λ in the grid). This optimisation is carried out using the coordinate
descent approach in Breheny and Huang (2011). Then, the function select (grpreg package) is used
|
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] |
descent approach in Breheny and Huang (2011). Then, the function select (grpreg package) is used
to determine the optimal value of λ for each linear model using a predefined objective criterion, such
as Generalised CV (GCV), k-folds CV or Bayesian Information Criterion (BIC). Finally, the same
criterion is used to select the optimal value for �h or �k. Therefore, once selected the linear model, �λ is
also determined and, as a consequence, we have obtained �β0. The estimated values �h or �k and �β0 are
|
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] |
then used in step 3 to fit the nonparametric component of the model.
Next, we provide the pseudocode for kernel-based SFPLM estimation; kNN-based estimation
would be analogous: Novo et al. (2021a) adapted the described procedure to fit the SFPLSIM (13).
Algorithm 1 SFPLM estimation
1: for each h in the grid do
2:
Perform functional nonparametric regression to estimate E(Yi|Xi) and E (ZZZi|Xi).
3:
Denote by �myi
h and �mzzzi
h the respective estimates. Obtain �Yi = Yi − �myi
h and �ZZZi = ZZZi − �mzzzi
h .
|
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] |
h and �mzzzi
h the respective estimates. Obtain �Yi = Yi − �myi
h and �ZZZi = ZZZi − �mzzzi
h .
4:
Construct a linear model as in (15).
5:
for each λ = λh in the grid do
6:
Use grpreg to obtain the βββh,λ that minimises (16). Denote it by �βββh,λ.
7:
end for
8:
Use select to determine the optimal λh using with a pre-specified criterion. Denote it by �λh.
9: end for
10: Use the same criterion to select the optimal h. Denote it by �h. Then, �λ = �λ�h and �βββ0 = �βββ�h,�λ.
|
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] |
11: Perform functional nonparametric regression using �h to fit the residuals Yi − ZZZi�βββ0 to obtain �m�h.
In this case, conditional expectations in the expression (14) become E (Yi|Xi) = E (Yi| ⟨θ0, Xi⟩) and
E (ZZZi|Xi) = E (ZZZi| ⟨θ0, Xi⟩), which must be estimated through functional single-index regression. For
that we can use either kernel- or kNN-based procedures in expression (5). In the second step, we apply
|
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] |
the PeLS procedure, see expression (16), but now the profile least-squares function also depends on θ.
Q (βββ, θ) = 1
2
�
�YYY θ − �ZZZθβββ
�⊤ �
�YYY θ − �ZZZθβββ
�
+ n
pn
�
j=1
Pλj (βj) .
(18)
Therefore, the minimisation must be performed over both βββ and θ. Once �βββ0 and �θ0 are obtained, we
estimate the real-valued link function r in the same way as m(·) in the SFPLM, but now using one of
the expressions in (5). Novo et al. (2021a) demonstrated the consistency of the PeLS procedure with
18
|
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] |
18
the SCAD penalty in the context of the SFPLSIM. They also obtained the convergence rates for the
estimators and established the oracle property in this setting.
Remark 3.2. In the case of the SFPLSIM, step 1 involves using functional single-index regression.
We follow the procedure in Section 2.2.1 to obtain the set of functional directions Θ. Therefore we
compute the nonparametric fit for a grid of values for the tuning parameter (h or k) and for each
|
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] |
compute the nonparametric fit for a grid of values for the tuning parameter (h or k) and for each
θ ∈ Θ. As a result, each pair (θ, h) or (θ, k) leads to a different linear model in expression (15). For
each linear model (15) (setting the values of h or k and θ), the minimisation of expression (16) is
carried out using the function grpreg over a grid of values for λ. Then, we use the function select
to determine the optimal value for λ using a predefined objective criterion for each linear model. The
|
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] |
same criterion is used to select the optimal value for h or k, therefore we obtain the optimal β for
each θ ∈ Θ. After that, the �θ0 is the θ ∈ Θ that offers a lower value of the objective function Q(·, ·)
and �h/�k and �β0 the associated optimal values for the remaining parameters. The estimated values �h
or �k, �θ0 and �β0 are then used in step 3 to fit the semiparametric component of the model. As can
be observed we perform a joint selection of the tuning parameters of the model, so we must take into
|
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] |
account the considerations in Section 2.2.1.
Next, we provide the pseudocode for kernel-based SFPLSIM estimation; kNN-based estimation
would be analogous:
Algorithm 2 SFPLSIM estimation
1: for each θ ∈ Θ do
2:
for each h in the grid do
3:
Perform functional single-index regression to estimate E (Yi| ⟨θ, Xi⟩) and E (ZZZi| ⟨θ, Xi⟩).
4:
Denote by ˆryi
θ,h and ˆrzzzi
θ,h the respective estimates. Obtain �Yi = Yi − ˆryi
θ,h and �ZZZi = ZZZi − ˆrzzzi
θ,h.
5:
Construct a linear model as in (15).
6:
|
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] |
θ,h and �ZZZi = ZZZi − ˆrzzzi
θ,h.
5:
Construct a linear model as in (15).
6:
for each λ = λh,θ in the grid do
7:
Use grpreg to obtain the βββh,λ,θ that minimises (18). Denote it by �βββh,λ,θ.
8:
end for
9:
Use select to determine the optimal λh,θ using with a prespecified criterion. Denote it by
�λh,θ.
10:
end for
11:
Use the same criterion to select the optimal h = hθ. Denote it by �hθ.
12: end for
|
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] |
10:
end for
11:
Use the same criterion to select the optimal h = hθ. Denote it by �hθ.
12: end for
13: Select the θ that minimises (18) when βββ = �βββ�hθ,�λ�hθ,θ,θ is considered. Denote it by �θ0. Then, �h = �h�θ0,
�λ = �λ�h,�θ0 and �βββ0 = �βββ�h,�λ,�θ0.
14: Perform functional single-index regression using �h and �θ0 to fit the residuals Yi − ZZZi�βββ0 to obtain
�r�h,�θ0.
3.3
Fitting SFPLM and SFPLSIM with the fsemipar package
|
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] |
�r�h,�θ0.
3.3
Fitting SFPLM and SFPLSIM with the fsemipar package
The fsemipar package contains functions to estimate the SFPLM and SFPLSIM using the procedures
described in Section 3.2. As in the FSIM case, it provides the flexibility to perform both kernel- and
19
kNN-based fitting.
The primary related functions are summarised in the Table 5 along with the
corresponding S3 classes designed to include essential S3 methods (print(), summary(), plot() and
predict()).
Model
Function
S3 class
Description
|
[
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] |
predict()).
Model
Function
S3 class
Description
SFPLM
sfpl.kernel.fit
‘sfpl.kernel’
PeLS with kernel estimation.
sfpl.kNN.fit
‘sfpl.kNN’
PeLS with kNN estimation.
SFPLSIM
sfplsim.kernel.fit
‘sfplsim.kernel’
PeLS with kernel estimation.
sfplsim.kNN.fit
‘sfplsim.kNN’
PeLS with kNN estimation.
Table 5: Summary of main functions for the SFPLM and SFPLSIM, together with the associated S3
classes.
3.3.1
Main fitting functions
|
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] |
classes.
3.3.1
Main fitting functions
The main fitting functions for the SFPLM and the SFPLSIM (Table 5) share three required arguments:
• The functional predictor x, a matrix of dimension (n, p) containing n curves discretised in p
points
• The scalar covariates z, a matrix of dimension (n, pn).
• The scalar response variable y, a vector of dimension n.
sfpl.kernel.fit(x,z,y,...)
sfpl.kNN.fit(x,z,y,...)
sfplsim.kernel.fit(x,z,y,...)
sfplsim.kNN.fit(x,z,y,...)
|
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] |
sfpl.kNN.fit(x,z,y,...)
sfplsim.kernel.fit(x,z,y,...)
sfplsim.kNN.fit(x,z,y,...)
However, these functions additionally accept numerous optional parameters to provide more precise
customisation; some of them have been already introduced in the main fitting functions for the FSIM:
• Optional arguments for the PeLS procedure. In this item, we include the arguments related to
the penalisation function and the selection of the tuning parameter λ. These arguments are
|
[
-0.09461792558431625,
-0.10205886512994766,
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] |
the penalisation function and the selection of the tuning parameter λ. These arguments are
listed in Table 6. The PeLS objective function is minimised using the function grpreg of the
package grpreg (Breheny and Huang 2015). The four functions listed in Table 5 share a default
option for constructing the sequence of λ values from which �λ will be chosen. This sequence is
generated by starting from a minimum value (expressed as a fraction of the maximum value)
|
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] |
generated by starting from a minimum value (expressed as a fraction of the maximum value)
controlled by the user, and it can contain a customisable number of values. More specifically, the
user has the option to specify a value for the minimum λ to face two different situations: when
n is smaller than factor.pn (default 1) times pn, this value can be set using the lambda.min.h
argument (default 0.05); when n is greater, the lambda.min.l argument (default 0.00001) allows
|
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] |
argument (default 0.05); when n is greater, the lambda.min.l argument (default 0.00001) allows
the user to set a different value. The utility of these options will be showed in Sections 4.3.2 and
4.3.3. Alternatively, we can directly specify a unique value for the minimum lambda using the
20
lambda.min argument. When lambda.min is provided, lambda.min.h and lambda.min.l are
disregarded. Furthermore, the number of λ values in the sequence can be customized using the
|
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] |
disregarded. Furthermore, the number of λ values in the sequence can be customized using the
nlambda argument (default 100). Additionally, the user has the option to provide the sequence
of λ values directly through the lambda.seq argument. When lambda.seq is provided, all of
the previously mentioned arguments are ignored. Note that setting lambda.seq to 0 results in
non-penalized fitting, which corresponds to the ordinary least-squares estimates.
|
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] |
non-penalized fitting, which corresponds to the ordinary least-squares estimates.
In addition, we can choose the criterion to select λ and the tuning parameter h (kernel-based
functions) or k (kNN-based functions).
This can be done through the argument criterion
(default "GCV") and the available options are "GCV", "AIC", "BIC" and "k-fold-CV". If we
choose k-fold CV, we can specify the number or CV folds using nfolds (default 10) and also set
|
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] |
choose k-fold CV, we can specify the number or CV folds using nfolds (default 10) and also set
the seed to obtain reproducible results using the argument seed (default 123).
The functions listed in Table 5 also allow for grouped variable selection. Using the argument vn
(default vn=ncol(z), which leads to individual penalisation), the user can directly specify the
number of groups or to provide a vector of possibilities so that the routines select the optimal
|
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] |
number of groups or to provide a vector of possibilities so that the routines select the optimal
number of groups according to the provided criterion.
For instance, vn = pn means that
each variable is a group (ungrouped case); whereas vn=c(pn,2) means that routines will choose
between the ungrouped case and the case with two groups of consecutive variables (both of size
n/2, if n is even).
Finally, the user can also choose between the penalty function SCAD/group SCAD (grSCAD)
|
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] |
Finally, the user can also choose between the penalty function SCAD/group SCAD (grSCAD)
and LASSO/group LASSO (grLASSO) using the argument penalty (default grSCAD), and set
the maximum number of iterations over each regularisation path though the argument max.iter
(default 1000).
• Optional arguments for the nonparametric fit.
This item refers to the arguments that are
collected in Table 2.
Functions that perform the kernel-based procedure accept the same
|
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] |
collected in Table 2.
Functions that perform the kernel-based procedure accept the same
arguments as fsim.kernel.fit() and fsim.kernel.fit.optim().
Similarly, functions im-
plementing the kNN-based estimation accept arguments consistent with fsim.kNN.fit() and
fsim.kNN.fit.optim(). The four functions in Table 5 accept the argument kind.of.kernel.
Additionally, the functions for the SFPLM allow the choice of the semimetric using the argument
|
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] |
Additionally, the functions for the SFPLM allow the choice of the semimetric using the argument
semimetric (default "deriv"). Currently, only the semimetric based on the B-spline represen-
tation of derivatives, semimetric="deriv", and the based on FPCA, semimetric="pca", are
implemented. The code of these functions is based on the Frederic Ferraty routines included
in the web page https://www.math.univ-toulouse.fr/~ferraty/SOFTWARES/NPFDA/index.
html.
|
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0.2516961097717285,
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0.22226005792617798,
-0.28559279441833496,
-0.6597527861595154,
-0.353740930557251,
0.3585529625415802
] |
in the web page https://www.math.univ-toulouse.fr/~ferraty/SOFTWARES/NPFDA/index.
html.
If semimetric="deriv", the argument q allows the user to control the order of the
derivative (default q=0). If semimetric="pca", the argument q controls the number of principal
components retained (default q=2).
• Optional arguments for B-spline expansions. Functions for the SFPLSIM allow all the arguments
listed in Table 3; functions for the SFPLM, only those related to the B-spline expansion of the
|
[
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1.0819339752197266,
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0.05985075607895851,
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0.43008822202682495,
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0.42478883266448975,
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-0.4275406301021576,
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0.08894835412502289,
0.42353782057762146,
0.10178320854902267,
0.037929534912109375,
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-0.12263225018978119,
0.19303540885448456,
-0.4891170561313629,
-0.4399511516094208,
-0.4238891899585724,
0.027809392660856247
] |
listed in Table 3; functions for the SFPLM, only those related to the B-spline expansion of the
curves (order.Bspline, nknot and range.grid).
• Optional arguments for the parallel computation. The functions for the SFPLSIM, which are
time consuming in contrast to functions for the SFPLM, allow for parallel computation. Similar
21
Function
Argument
Description
sfpl.kernel.fit
sfpl.kNN.fit
sfplsim.kernel.fit
sfplsim.kNN.fit
lambda.min.h
Smallest value for λ if n<factor.pn*pn.
lambda.min.l
|
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] |
sfplsim.kNN.fit
lambda.min.h
Smallest value for λ if n<factor.pn*pn.
lambda.min.l
Smallest value for λ if n>factor.pn*pn.
lambda.min
Smallest value for λ.
factor.pn
Positive integer used to set the smallest value for λ.
nlambda
Number of λ values in the sequence.
lambda.seq
Sequence of λ values.
vn
Integer/vector
of
integers
indicating
the
num-
ber/numbers of groups of variables to be penalised
together.
criterion
Criterion for selecting ˆλ and ˆh/ˆk: "GCV", "BIC",
"AIC", "k-fold-CV".
nfolds
|
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] |
together.
criterion
Criterion for selecting ˆλ and ˆh/ˆk: "GCV", "BIC",
"AIC", "k-fold-CV".
nfolds
If criterion= "k-fold-CV": number of CV folds.
seed
If criterion= "k-fold-CV": Seed to obtain repro-
ducible results.
penalty
Penalty function: "grLASSO", "grSCAD".
max.iter
Maximum number of iterations (across the entire
path).
Table 6: Optional input arguments related to the PeLS procedure.
to the FSIM functions, the user can specify the number of CPU cores to use for parallel execution
|
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] |
to the FSIM functions, the user can specify the number of CPU cores to use for parallel execution
by setting the argument n.core (see the item Optional arguments for the parallel computation
in Section 2.3.1).
3.3.2
Case study I, part II: Tecator dataset
In Novo et al. (2021a), the authors studied the inclusion of protein and moisture content as covariates
to predict the fat content in each piece of meat. They also investigated whether the quadratic, cubic,
|
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