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and interaction effects of these scalar covariates should be included in the model. This means to
consider initially seven scalar covariates: denoting by Z1 and Z2 the protein and moisture contents,
respectively, they considered as linear covariates Z2j−1 = Zj
1 and Z2j = Zj
2 (for j = 1, 2, 3), and
Z7 = Z1Z2. They fitted both an SFPLM and an SFPLSIM to the data, using PeLS estimation with a
SCAD penalty combined with a kernel-based technique. In addition, they used the BIC criterion for | [
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] |
SCAD penalty combined with a kernel-based technique. In addition, they used the BIC criterion for
selecting the tuning parameters. In this context, we can illustrate and compare functions in Table 5
by extending that application, thereby demonstrating the wide range of capabilities of the fsemipar
package.
As an example, the following code shows the implementation of the SFPLSIM using the func-
tion sfplsim.kernel.fit(). In addition to the data, we have provided some specific input argu- | [
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tion sfplsim.kernel.fit(). In addition to the data, we have provided some specific input argu-
ments to improve the fit: we have reduced the interval in which the bandwidth is selected (setting
max.q.h=0.35) discarding the largest values (given that by default num.h=10, reducing these interval
22
will lead to more precision in the selection of �h without increasing the computational cost), we have
also increased the minimum value of the sequence for selecting λ (lambda.min=0.01) and the number | [
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] |
also increased the minimum value of the sequence for selecting λ (lambda.min=0.01) and the number
of iterations (max.iter=5000) trying to avoid convergence problems in this context of high dependence
between predictors; we also chose the BIC as the criterion for selecting the tuning parameters, which
leads to slightly faster computation than GCV. As in Section 2.3.2, we set nknot.theta=4, nknot=20
and range.grid=c(850,1050).
>z1<-Tecator$protein
>z2<-Tecator$moisture | [
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] |
and range.grid=c(850,1050).
>z1<-Tecator$protein
>z2<-Tecator$moisture
>z.com<-cbind(z1,z2,z1^2,z2^2,z1^3,z2^3,z1*z2)
>fit2<-sfplsim.kernel.fit(x=x[train,], z=z.com[train,], y=y[train],
max.q.h=0.35,lambda.min.l=0.01,max.iter=5000, nknot.theta=4,criterion="BIC",
nknot=20,range.grid=c(850,1050))
The fitted object resulting from the functions listed in Table 5 contains valuable information about the
performed estimation. This includes the fitted values for the response (fitted.values), the residuals | [
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] |
(residuals), and the estimates for the linear coefficients (beta.est). It also provides the selected
values for the tuning parameters, such as ˆh (h.opt) or ˆk (k.opt), and ˆλ (lambda.opt). Additionally,
the object includes the optimal value of the criterion used to select ˆh/ˆk and ˆλ (IC), as well as the
optimal value of the objective function of the PeLS procedure (see 16) (Q). In the case of SFPLSIM,
it also contains the coefficients of �θ in the B-spline basis (theta.est), among other details. | [
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] |
it also contains the coefficients of �θ in the B-spline basis (theta.est), among other details.
In the case of sfplsim.kernel.fit() (class sfplsim.kernel), the fitted object can be used in
the functions print.sfplsim.kernel() and summary.sfplsim.kernel() to display summaries of the
fitted model. It can also be used in predict.sfplsim.kernel() to obtain predictions for new values
of the covariates provided by the arguments newdata.x (curves) and newdata.z (scalar predictors). | [
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] |
of the covariates provided by the arguments newdata.x (curves) and newdata.z (scalar predictors).
This function implements two possibilities for prediction:
• If option=1, we maintain all the estimations (h.opt, beta.est and theta.est) to predict the
functional single-index component of the model.
• If option=2, we maintain beta.est and theta.est, while the tuning parameter (h.opt) is
selected again to predict the functional single-index component of the model. This selection is | [
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] |
selected again to predict the functional single-index component of the model. This selection is
performed using the LOOCV criterion in the associated FSIM.
For each option, the MSEP is also obtained when the user provides the actual responses via the y.test
argument. The remaining functions of Table 5 offer analogous possibilities for prediction.
> summary(fit2)
*** SFPLSIM fitted using penalized least-squares combined with kernel estimation with
Nadaraya-Watson weights *** | [
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Nadaraya-Watson weights ***
-Call: sfplsim.kernel.fit(x = x[train, ], z = z.com[train, ], y = y[train],
nknot.theta = 4, max.q.h = 0.35, range.grid = c(850, 1050), nknot = 20,
lambda.min= 0.01, criterion = "BIC", max.iter = 5000)
23
-Bandwidth (h): 27.11844
-Theta coefficients in the B-spline basis: 0.1656316 -0.1656316 0 0.1656316 -0.1656316
0 0.1656316
-Linear coefficients (beta): 0.6852112 -1.98299 0 0.01086126 -0.001282229 0 0
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-Number of non-zero linear coefficients: 4
-Indexes non-zero beta-coefficients: 1 2 4 5
-Lambda: 0.01244154
-IC: 523.0534
-Q: 102.0231
-Penalty: grSCAD
-Criterion: BIC
-vn: 7
> predict(fit2,newdata.x=x[161:215,],newdata.z=z.com[161:215,],y.test=y[161:215],
option=2)$MSEP
[1] 1.262723
Regarding the prediction, Table 7 displays the MSEP and selected scalar covariates corresponding to
each primary fitting function when using the respective BIC selectors for nr (only in the SFPSLIM), | [
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] |
each primary fitting function when using the respective BIC selectors for nr (only in the SFPSLIM),
λ, and h, or k. In the functions for the SFPLM, the results were obtained using the semimetric based
on the derivatives (semimetric="deriv", the option by default) and as x contains already the second
derivative of the curves, we used the option q=0 (also, the option by default).
sfpl.kNN<-sfpl.kNN.fit(x=x[train,],y=y[train],z=z.com[train,], step=1, min.knn=10, | [
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] |
sfpl.kNN<-sfpl.kNN.fit(x=x[train,],y=y[train],z=z.com[train,], step=1, min.knn=10,
max.knn=15,criterion="BIC",range.grid=c(850,1050),lambda.min=0.01,nknot=20,max.iter=
5000)
As can be observed from Table 7, under the conditions described, the best result in MSEP is obtained
with the SFPLM estimated using PeLS with kNN estimation, which retains only 3 scalar covariates
in the model. Finally, the output of the S3 method plot() for the functions in Table 5 represents two
MSEP
Function
Selected variables
�nr
1 | [
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] |
MSEP
Function
Selected variables
�nr
1
2
sfpl.kernel.fit()
Z1, Z2
-
1.317
1.317
sfpl.kNN.fit()
Z2, Z3, Z4
-
0.797
0.797
sfplsim.kernel.fit()
Z1, Z2, Z4, Z5
4
1.260
1.263
sfplsim.kNN.fit()
Z1, Z2, Z3, Z4
4
1.349
1.349
Table 7: MSEP obtained for each function using the respective BIC selectors for tuning parameters
(h/k, λ; also nr in the SFPLSIM fitting functions).
diagnostic plots in addition to the estimated functional index for the SFPLSIM fitting functions. In | [
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] |
Figures 4 and 5, we provide the outputs of the 4 main functions, obtained under the same conditions
as Table 7.
24
−0.1
0.0
0.1
850
900
950
1000
1050
range.grid X
θ0
G
G
G
G
G
GG
G
G
G
G
G GGG
GGGG
G
GGG
G
GG
G G
G
G
GG
G GG
GG
G
GG
G
GGG
G
G
G
GGG
G
GG
GGG
GGGG
GG
GGGG
GG
GGGG
GG
GGG
G
GGG
GGGGG
GGG
GGGGG
GGGG
GGGGGGGGG G
GG
G
GG
GGGG
GGG
G
GGG
GGG
G
G
G
G
GGG
G
G
G
G
G
G
G
G
G
GGG
GG
GGGG G
G
GG GG
0
10
20
30
40
50
0
10
20
30
40
50
Fitted values
y
Response vs Fitted values
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G | [
-0.4119992256164551,
0.030033918097615242,
0.9464821815490723,
0.058962736278772354,
-0.5594223141670227,
0.7661035060882568,
0.43765994906425476,
-0.5004286766052246,
-0.5570571422576904,
-0.19516988098621368,
-0.1771269291639328,
0.5134049654006958,
-0.09535520523786545,
0.16773360967636108,
-0.11390727013349533,
0.6103711128234863,
0.8555548191070557,
0.24125678837299347,
-0.5013660788536072,
-0.6702399849891663,
0.43440374732017517,
-0.23457175493240356,
0.8534182906150818,
0.0798632800579071,
-0.04063275828957558,
-0.09254626929759979,
0.31058773398399353,
-0.016560345888137817,
0.4095921516418457,
0.46235981583595276,
-0.23804467916488647,
0.9143392443656921,
0.06346849352121353,
0.0033022193238139153,
-0.024042988196015358,
-0.14632098376750946,
0.2689460217952728,
-0.398366779088974,
0.16179834306240082,
0.010688891634345055,
0.13130849599838257,
-0.5240570902824402,
0.2020776867866516,
0.19917751848697662,
0.006947667337954044,
-0.4326751232147217,
-1.0621360540390015,
-0.09111836552619934,
0.06910251080989838,
0.07539371401071548,
-0.826858639717102,
0.6299338340759277,
-0.032782480120658875,
0.6954599618911743,
-0.0379444882273674,
0.08980749547481537,
-0.3664674460887909,
0.018124425783753395,
0.6452255845069885,
-0.5993524193763733,
-0.04543958231806755,
-0.3825959265232086,
0.4607812464237213,
0.08165979385375977,
-0.4708613157272339,
0.3772595524787903,
-0.18560515344142914,
0.29713472723960876,
0.00824514776468277,
-0.14868995547294617,
0.29374268651008606,
0.0023866111878305674,
0.13924100995063782,
-0.31985223293304443,
-0.08707691729068756,
0.12353856861591339,
-0.25053834915161133,
-0.4827598035335541,
0.39374032616615295,
0.1794450730085373,
0.18532662093639374,
-0.14285612106323242,
0.6649707555770874,
-0.268236368894577,
0.15746484696865082,
0.22502698004245758,
0.07710558921098709,
0.11522023379802704,
-0.29407867789268494,
0.5973220467567444,
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] |
30
40
50
0
10
20
30
40
50
Fitted values
y
Response vs Fitted values
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
GG
G
G
G
G
G
G
G
G
G
G
G
GG
GG
G
G
G
G
G
G
GG
G
G
G
G
G
G
G
GG
G
G
GG
GGG
GGG
G
G
G
G
GG
G
G
G
G
GG
G
G
G
G
G
G
G
GG
G
G
G
G
GG
GG
G
GGG
G
G
G
GG
G
G
G
G
GG
G
G
G
G
G
G
G
G
GG
G
G
GG
G
G
G
G
G
GG
G
G
G
G
G
G
G G
G
G
G
G
G
G
G
G
G
G
GG
G
G
GG
GG
G
G
G
G
GG
GG
−2
0
2
0
10
20
30
40
50
Fitted values
Residuals
Residuals vs Fitted Values
(a) sfplsim.kernel.fit() (�nr = 4)
−0.04
0.00
0.04
0.08
0.12
850
900
950
1000 | [
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(a) sfplsim.kernel.fit() (�nr = 4)
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Response vs Fitted values
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Fitted values
Residuals
Residuals vs Fitted Values
(b) sfplsim.kNN.fit() (�n∗
r = 4)
Figure 4: Outputs of the S3 method plot() applied to the fitted objects of functions for the SFPLSIM, | [
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] |
using the corresponding BIC selectors for the tuning parameters and nr.
4
Multiple scalar predictors with functional origin and a functional
covariate
In Section 4, we focus on SoF regression including as predictors multiple scalar variables with linear
effect coming from the discretisation of a curve. Section 4.1 introduces the concept of points of impact
and some modelling alternatives based on linear or partial linear approaches when more than one | [
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] |
and some modelling alternatives based on linear or partial linear approaches when more than one
functional variable is available. In Section 4.2, we discuss the techniques implemented in fsemipar for
estimation and selection of points of impact in such models. In Section 4.3 we present the associated
functions of the package and illustrate their use through two real chemometric datasets.
4.1
Linear and partial linear modelling approaches | [
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] |
4.1
Linear and partial linear modelling approaches
In several studies, the interpretability of the outcomes prompted researchers to revisit the concept of
discretising functional objects. They discovered that the discretised values of a random curve ζ(t),
ζ(t1), ..., ζ(tpn), might hold insights that are not accessible via the continuous curve ζ(t), and vice
versa (see, for instance, McKeague and Sen 2010). So, in certain applications we want to sift through | [
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] |
the discretised observations of a curve ζ(t) and identify the points of the domain at which the curve
most influence a scalar variable of interest Y . In some papers (for example, Kneip et al. 2016 or Novo
25
G
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G
G
GG
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GG G
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G
G
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G
G G
GG
GGGG
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GG
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GG
GG
GGGG
GG
GGG
GGGG
GGGG
GG
GG
GGGGG
GGG
G
GGGGGGG
GGG
GG
G
GGGGGG
GGG
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0
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Fitted values
y | [
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] |
G
G
G
G
GGG
GG
G
G
G
G
G
G
G
G
GG
GGGGGG G
G
GGGG
0
10
20
30
40
50
10
20
30
40
50
Fitted values
y
Response vs Fitted values
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
GGG
G
G
G
G
G
G
G
G
G
G
G
G
GG GG
G
G
G
G G
G
G
GG
G
G
G
G
G
G
G
GG
GGGG
GGGG
G
G
G
G
G
G
G
G
G
GGG
G
G
G
G
G
GGGG
GG
G
G
G
G
GG
GG
GG
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
GG
G
GGG
G
G
G
G
G
GG
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
GGGG
G
G
G
G
GGG
G
−4
−2
0
2
10
20
30
40
50
Fitted values
Residuals
Residuals vs Fitted Values
(a) sfplm.kernel.fit()
G | [
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] |
−2
0
2
10
20
30
40
50
Fitted values
Residuals
Residuals vs Fitted Values
(a) sfplm.kernel.fit()
G
G
G
G
G
GG
G
GG
G
GGGGGGGG
G
GGGGGG
GG
GGG
G
G GG
GGG
GG
G
GG
G
G
G
G
G
GGG
GGG
GGG
GG
GG
G
GG
GG
GG
GG
GG
GG
GGG
G
GGGG
GGG
GGGG
GGGGG
GGGG
GGGGGGGGGG
GG
G
G
GGGGG
GGG
GGGG
GGG
G
G
G
G
GGG GG
G
G
G
G
G
G
G
G
GG
GGG
GGGG
G
GGG G
0
10
20
30
40
50
0
10
20
30
40
50
Fitted values
y
Response vs Fitted values
G
G
G
G
G
G
G
G
G
G
G
GG
G
G
GGG
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G GG
G
G
G
G
G
G
GG
G
G
G
GGG
G
GG
G | [
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Fitted values
Residuals
Residuals vs Fitted Values
(b) sfplm.kNN.fit()
Figure 5: Outputs of the S3 method plot() applied to the fitted objects of functions for the SFPLM, | [
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] |
Figure 5: Outputs of the S3 method plot() applied to the fitted objects of functions for the SFPLM,
using the corresponding BIC selectors for the tuning parameters.
et al. 2021b, among others), these points of the domain are called points of impact, so we will follow
such terminology along the paper. In the following lines, we summarise the modelling alternatives
involving the discretised values of a curve (scalar variables with functional origin) implemented in the
package fsemipar: | [
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] |
package fsemipar:
• The multiple linear model (MLM) studied in Aneiros and Vieu (2014), which is given by the
expression:
Yi = β00 +
pn
�
j=1
β0jζi(tj) + εi,
i = 1, . . . , n,
(19)
where ζi is a random curve defined on some interval [a, b] and is observed in the points a ≤ t1 <
· · · < tpn ≤ b. In addition, (β00, β01, . . . , β0pn)⊤ is a vector of unknown real coefficients.
• The multi-functional partial linear model (MFPLM) examined in Aneiros and Vieu (2015). This
model follows the expression
Yi =
pn
� | [
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] |
model follows the expression
Yi =
pn
�
j=1
β0jζi(tj) + m(Xi) + εi,
i = 1, . . . , n,
(20)
where Xi denotes another functional variable valued in some semi-metric space, which influences
the response nonparametrically.
• The multi-functional partial linear single-index model (MFPLSIM) studied in Novo et al. (2021b),
defined by the following relationship:
Yi =
pn
�
j=1
β0jζi(tj) + r (⟨θ0, Xi⟩) + εi
i = 1, . . . , n.
(21) | [
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] |
Yi =
pn
�
j=1
β0jζi(tj) + r (⟨θ0, Xi⟩) + εi
i = 1, . . . , n.
(21)
Here, Xi is valued in a separable Hilbert space and its effect on the response is given through a
functional single-index component.
In models (19)-(21), it is assumed that pn → ∞ as n → ∞, and only a few of the discretised points
ζ(tj), j = 1, . . . , pn, influence the response Y . We could consider applying to these models the same
26
techniques described in Section 3.2 to estimate and select relevant variables in the linear component. | [
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] |
However, these methods are often inadequate for two main reasons.
First, there exists a strong
dependence between variables, a consequence of their functional origin. Second, the sheer number
of observations (n << pn) makes it difficult to obtain results in a reasonable time frame. Then, it
is necessary to develop specialised methods tailored to these particular scenarios. Therefore, in the
following section, we provide a short review of the techniques implemented in fsemipar for estimating
these models. | [
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these models.
4.2
Selection of points of impact and model estimation
Aneiros and Vieu (2014) proposed an algorithm called Partitioning Variable Selection (PVS) for the
MLM (19). This two-step procedure is based on the idea that observations ζi(tj) and ζi(tk), where
tj and tk are closely spaced points in the domain of the curve, contain similar information about the
response. The method proceeds as follows:
1. In the first step, it considers a reduced linear model incorporating only a few covariates, wn, | [
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1. In the first step, it considers a reduced linear model incorporating only a few covariates, wn,
covering the entire discretisation interval of ζi(t). For this purpose:
• We assume, without loss of generality, that pn = qnwn.
• The wn variables taken into account are ζi(t(2k−1)qn/2) with k ∈ {1, . . . , wn}.
• The remaining pn − wn covariates from the original set are directly discarded.
A PeLS procedure (see Section 3.2) is applied to the reduced model.
In this manner, the | [
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] |
A PeLS procedure (see Section 3.2) is applied to the reduced model.
In this manner, the
dependence among covariates is mitigated before the variable selection is performed.
2. In the second step, a linear model is considered that includes the variables selected in the first
step and those in their neighbourhood. Specifically:
• Let’s denote �S1n = {k = 1, . . . , wn, �β(2k−1)qn/2 ̸= 0}.
• The set of covariates considered in the second step is
R = ∪k∈�S1{ζi(t(k−1)qn+1), ζi(t(k−1)qn+2) . . . , ζi(tkqn)}. | [
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] |
R = ∪k∈�S1{ζi(t(k−1)qn+1), ζi(t(k−1)qn+2) . . . , ζi(tkqn)}.
A PeLS procedure is applied again to the resultant model. By doing so, relevant information
that may have been overlooked in the first step is reconsidered.
The outputs of the second step of the PVS algorithm are the estimates for the MLM (19). The selected
points of impact are those belonging to the set �S2n = {j = 1, . . . , pn, such that ζi(tj) ∈ R and �β0j ̸= | [
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] |
0 in the second step}, and �β0j is the estimate of the second step if ζi(tj) ∈ R and 0 otherwise. The
PVS algorithm requires dividing the sample to execute the two steps. The natural choice involves
using half of the sample in the first step and the other half in the second step, that is n1 = n2 = n/2,
although in some applications this may not be the optimal option. The consistency of this procedure
was showed in Aneiros and Vieu (2014), who also provided the rates of convergence of the estimators. | [
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] |
Note that the PVS algorithm has an extra tuning parameter: the number of covariates in the
reduced model, wn, in the first step of the method. This parameter can be selected using the same
objective criterion to select the parameter λ in the PeLS procedure involved.
27
In addition, the first step of the PVS algorithm relies on the assumption pn = wnqn.
When
this assumption fails, we can use the solution proposed by Aneiros and Vieu (2015), which is based | [
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this assumption fails, we can use the solution proposed by Aneiros and Vieu (2015), which is based
on consider non-fixed qn = qn,k values (for k = 1, . . . , wn) when pn/wn is not an integer number.
Specifically:
qn,k =
�
[pn/wn] + 1
k ∈ {1, . . . , pn − wn[pn/wn]},
[pn/wn]
k ∈ {pn − wn[pn/wn] + 1, . . . , wn},
where [z] denotes the integer part of z ∈ R.
Aneiros and Vieu (2015) extended the PVS algorithm to estimate the MFPLM (20). The idea | [
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] |
Aneiros and Vieu (2015) extended the PVS algorithm to estimate the MFPLM (20). The idea
was to transform the MFPLM (20) into a linear model using the same strategy described in Section
3.2. Once an approximate linear model was obtained, they applied the PVS algorithm to that model.
They proved the consistency of the method and obtained the rates of convergence for the estimators.
Novo et al. (2021b) addressed the MFPLSIM (21). In this case, there were two additional challenges | [
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] |
Novo et al. (2021b) addressed the MFPLSIM (21). In this case, there were two additional challenges
when extending the PVS: first, estimating the functional index θ0 was computationally intensive,
and second, it required a relatively large sample size (see Aneiros et al. 2022). They then explored
two variants of the PVS algorithm: the improved algorithm for sparse multi-functional regression
(IASSMR), which essentially adapts the PVS to the MFPLSIM, and the fast algorithm for sparse | [
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(IASSMR), which essentially adapts the PVS to the MFPLSIM, and the fast algorithm for sparse
multi-functional regression (FASSMR), which employs only the first step of the PVS. The motivation
for the latter variant was to reduce the computational cost for large discretisation sizes (pn) and
improve performance in situations with small sample sizes, as the sample no longer needs to be divided.
Both the IASSMR and the FASSMR require the transformation of the expression (21) into a linear | [
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] |
Both the IASSMR and the FASSMR require the transformation of the expression (21) into a linear
model, using the strategy described in Section 3.2. In Novo et al. (2021b), the authors demonstrated
the consistency of both methods and derived the convergence rates for the estimators of the former.
4.3
Fitting the MLM, MFPLM and MFPLSIM with the fsemipar package
The fsemipar package implements the algorithms described in Section 4.2 to estimate models (19)- | [
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] |
The fsemipar package implements the algorithms described in Section 4.2 to estimate models (19)-
(21). In the case of models with partial linear structure, it provides the flexibility to perform both
kernel- and kNN-based fitting. The primary related functions are summarised in the Table 8 along
with the corresponding S3 classes.
Model
Function
S3 class
Description
MLM
PVS.fit
‘PVS’
PVS algorithm.
MFPLM
PVS.kernel.fit
‘PVS.kernel’
PVS algorithm with kernel estimation.
PVS.kNN.fit
‘PVS.kNN’ | [
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] |
MFPLM
PVS.kernel.fit
‘PVS.kernel’
PVS algorithm with kernel estimation.
PVS.kNN.fit
‘PVS.kNN’
PVS algorithm with kNN estimation.
MFPLSIM
IASSMR.kernel.fit
‘IASSMR.kernel’
IASSMR with kernel estimation.
IASSMR.kNN.fit
‘IASSMR.kNN’
IASSMR with kNN estimation.
FASSMR.kernel.fit
‘FASSMR.kernel’
FASSMR with kernel estimation.
FASSMR.kNN.fit
‘FASSMR.kNN’
FASSMR with kNN estimation.
Table 8: Summary of main functions for MLM, MFPLM and MFPLSIM together with the associated
S3 classes.
28
4.3.1 | [
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] |
S3 classes.
28
4.3.1
Main fitting functions
The main fitting functions for MLM, MFPLM and MFPLSIM (Table 8) share the following required
arguments:
• The functional predictor z (pointwise effect), a matrix of dimension (n, pn) containing n curves
discretised in pn points.
• The scalar response variable y, a vector of dimension n.
PVS.fit(z,y,...)
In the case of the MFPLM and MFPLSIM, the functions also require: | [
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PVS.fit(z,y,...)
In the case of the MFPLM and MFPLSIM, the functions also require:
• The functional predictor x (continuous effect), a matrix of dimension (n, p) containing n curves
discretised in p points.
PVS.kernel.fit(x,z,y,...)
PVS.kNN.fit(x,z,y,...)
IASSMR.kernel.fit(x,z,y,...)
IASSMR.kNN.fit(x,z,y,...)
FASSMR.kernel.fit(x,z,y,...)
FASSMR.kNN.fit(x,z,y,...)
These functions additionally accept numerous optional parameters. Many of them have been already
introduced in Section 3.3.1: | [
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] |
introduced in Section 3.3.1:
• Optional arguments for the two steps of the algorithm. In functions tailored to PVS and IASSMR
(which are two-step procedures), users can control the indices of the data used for the first
and second steps with the arguments train.1 (default 1:ceiling(n/2)) and train.2 (default
(ceiling(n/2)+1):n), respectively. Additionally, for the seven functions listed in Table 8, the
argument wn (default c(10,15,20)) allows users to provide a sequence of possible values for wn | [
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] |
argument wn (default c(10,15,20)) allows users to provide a sequence of possible values for wn
in the reduced model of step 1.
• Optional arguments for the PeLS procedure. These arguments are listed in Table 6 and com-
mented in Section 3.3.1. In this case, the criterion chosen by the user to select the tuning
parameters of the estimation/variable selection procedure is also used to select �wn from the
values provided in wn. | [
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] |
values provided in wn.
In addition, functions related to the MFPLM and MFPLSIM admit the following optional arguments:
• Optional arguments for the nonparametric fit.
This item refers to the arguments collected
in Table 2.
Kernel-based functions accept the same arguments as fsim.kernel.fit() and
fsim.kernel.fit.optim(), and kNN-based ones accept arguments consistent with fsim.kNN.fit()
and fsim.kNN.fit.optim(). Additionally, the functions for the MFPLM admit the arguments | [
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] |
and fsim.kNN.fit.optim(). Additionally, the functions for the MFPLM admit the arguments
semimetric and q (see Optional arguments for the nonparametric fit in Section 3.3.1).
29
• Optional arguments for B-spline expansions. Functions for the MFPLSIM allow all the argu-
ments listed in Table 3; functions for the MFPLM, 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 MFPLSIM allow for | [
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] |
• Optional arguments for the parallel computation. The functions for the MFPLSIM allow for
parallel computation, so they admit the argument n.core (see the item Optional arguments for
the parallel computation in Section 2.3.1).
4.3.2
Case study I, part III: Tecator dataset
The purpose of this section is twofold: to illustrate the implementation of the models (19)–(21) using
the functions in Table 8 and to show the predictive advantage of including impact points in the
modelling of the Tecator dataset. | [
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] |
modelling of the Tecator dataset.
Although MFPLM (20) (so as to the MFPLSIM (21)) is tailored to address situations with more
than one functional covariate, in Aneiros and Vieu (2015), the authors applied it in the case ζ(t) =
X(t). They model the Tecator dataset (one functional covariate) using the model:
Yi =
100
�
j=1
β0jXi(tj) + m(Xi) + εi,
i = 1, . . . , 160,
where Xi is the second derivative of the absorbance spectrum of each piece of meat, which was recorded | [
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] |
in 100 equally spaced wavelengths in the interval 850-1050. To illustrate and compare the functions
in Table 8, we extended that application, fitting also the models:
Yi =
100
�
j=1
β0jXi(tj) + r(⟨Xi, θ0⟩) + εi,
i = 1, . . . , 160.
Yi = β00 +
100
�
j=1
β0jXi(tj) + εi,
i = 1, . . . , 160.
As an example, the following code shows the implementation of the MFPLSIM using the functions
FASSMR.kernel.fit() and IASSMR.kernel.fit(). In addition to the data, we have provided some
specific input arguments. | [
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] |
specific input arguments.
Following the comments made in Section 3.3.2, we set max.q.h=0.35,
criterion="BIC", nknot.theta=4, nknot=20, and range.grid=c(850,1050).
We increased the
smallest value in the sequence to select �λ (lambda.min=0.03), trying to obtain a simpler model.
Note that in this case, if we use half of the sample in each stage of the two-step procedures (that is,
n1 = n2 = 80 by setting train.1=1:80 and train.2=81:160), then n1 > wn and n2 is likely to exceed | [
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] |
n1 = n2 = 80 by setting train.1=1:80 and train.2=81:160), then n1 > wn and n2 is likely to exceed
the cardinality of R. Consequently, it may not be worthwhile to specify lambda.min.h, lambda.min.l,
and factor.pn (refer to Optional Arguments for the PeLS Procedure in Section 3.3.1) to adjust the
level of penalization in each step of the IASSMR. Typically, we assign a value to factor.pn that is
≥ 1, aiming to impose greater penalization in situations where the sample size is smaller than the | [
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] |
≥ 1, aiming to impose greater penalization in situations where the sample size is smaller than the
number of covariates in the model.
fit3 <- FASSMR.kernel.fit(x=x[train,],z=x[train,], y=y[train], nknot.theta=4,
max.q.h=0.35, range.grid=c(850,1050), nknot=20,criterion="BIC", lambda.min=0.03)
30
fit4<- IASSMR.kernel.fit(x=x[train,],z=x[train,], y=y[train],train.1=1:80,train.2=81:160,
nknot.theta=4,max.q.h=0.35, range.grid=c(850,1050), nknot=20,criterion="BIC",
lambda.min=0.03) | [
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nknot.theta=4,max.q.h=0.35, range.grid=c(850,1050), nknot=20,criterion="BIC",
lambda.min=0.03)
The objects fit3 and fit4 belong to an S3 class (see Table 8), which implements S3 methods.
In addition to the information listed in Section 3.3.2, these objects include the selected value for
�wn (w.opt).
As in the Sections 2.3.2 and 3.3.2, we can display a summary of the fitted models
using summary.FASSMR.kernel() (or print.FASSMR.kernel()) and summary.IASSMR.kernel() (or | [
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] |
using summary.FASSMR.kernel() (or print.FASSMR.kernel()) and summary.IASSMR.kernel() (or
print.IASSMR.kernel()), respectively. To make predictions for the fitted object fit3, we can use
predict.FASSMR.kernel(). This function implements the two possibilities for prediction outlined in
Section 3.3.2. To make predictions for fit4, we can use predict.IASSMR.kernel(), which offers the
following possibilities:
• If option=1, we retain all the estimates (h.opt, theta.est, and beta.est) to predict the | [
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• If option=1, we retain all the estimates (h.opt, theta.est, and beta.est) to predict the
functional single-index component of the model. Since we use the estimates from the second
step of the algorithm, only train.2 was used as the training sample for prediction. Therefore,
h.opt might not be suitable for predicting the functional single-index component of the model.
• If option=2, we keep theta.est and beta.est, while the tuning parameter �h is selected anew | [
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] |
• If option=2, we keep theta.est and beta.est, while the tuning parameter �h is selected anew
for predicting the functional single-index component of the model. This selection is performed
using the LOOCV criterion in the associated FSIM and the complete training sample (i.e.,
train=c(train.1,train.2)).
• If option=3, we maintain only the indices of the relevant variables selected by the IASSMR. We
re-estimate the linear coefficients and the functional index using sfplsim.kernel.fit, without | [
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] |
re-estimate the linear coefficients and the functional index using sfplsim.kernel.fit, without
penalization (by setting lambda.seq=0) and using the entire training sample (train=c(train.1,
train.2)). This method provides two predictions (and MSEPs):
a) The prediction associated with option=1 for the sfplsim.kernel class.
b) The prediction associated with option=2 for the sfplsim.kernel class.
Note that the objects resulting from the functions IASSMR.kNN.fit, PVS.kernel.fit and PVS.kNN.fit | [
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-0.09735820442438126
] |
Note that the objects resulting from the functions IASSMR.kNN.fit, PVS.kernel.fit and PVS.kNN.fit
offer analogous possibilities for prediction. For more details, see Section 4.3.3 or refer to the manual
of the fsemipar package (Aneiros and Novo 2024).
Table 9 displays the MSEP and the selected points of impact for each primary fitting function,
using the respective BIC selectors for λ, wn, and h/k in the MFPLM and MFPLSIM models. The | [
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] |
using the respective BIC selectors for λ, wn, and h/k in the MFPLM and MFPLSIM models. The
points of impact, selected from the original set of 100, are represented in red on a timeline for enhanced
visualization and comparison. The column �sn indicates the number of points of impact selected by
each function.
For the MFPLM functions, the results were obtained using semimetric="deriv".
For comparative purposes, the same values for common input arguments were used across the seven | [
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] |
For comparative purposes, the same values for common input arguments were used across the seven
functions in Table 8. Additionally, h and k were selected using the same grid for each pair of functions
employing kernel/kNN estimation, setting min.knn=2, max.knn=10, and step=1 in the kNN case.
It is remarkable that the MLM does not offer good predictive power in this case. This could be
expected since the relationship between the response and the predictor is non-linear, as pointed out | [
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] |
in Novo et al. (2019), among others. Another important observation is that the MFPLSIM appears
31
to offer the lowest MSEP under our conditions, although the MFPLM, estimated using the PVS with
kNN smoothing, is also competitive in terms of MSEP. Among the algorithms for estimating the
MFPLSIM, it is important to note that even the 1-step procedure offers a very competitive result,
especially when combined with kernel estimation. In the two-step procedures, estimating again the | [
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0.5142868161201477,
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] |
especially when combined with kernel estimation. In the two-step procedures, estimating again the
tuning parameters (option=3) offers the higher predictive power.
MSEP
Model
Function
Points of impact
�sn
1
2
3a
3b
MLM
PVS.fit()
850
1050
3
19.46
-
-
-
MFPLM
PVS.kernel.fit()
850
1050
7
6.94
5.14
4.66
4.66
PVS.kNN.fit()
850
1050
5
2.65
1.84
1.29
1.29
MFPLSIM
FASSMR.kernel.fit()
850
1050
9
1.60
1.79
-
-
FASSMR.kNN.fit()
850
1050
7
2.25
2.25
-
-
IASSMR.kernel.fit()
850
1050
6
1.66
1.69
1.10
1.15 | [
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] |
-
-
FASSMR.kNN.fit()
850
1050
7
2.25
2.25
-
-
IASSMR.kernel.fit()
850
1050
6
1.66
1.69
1.10
1.15
IASSMR.kNN.fit()
850
1050
7
1.50
1.66
1.04
1.04
Table 9: MSEP obtained for each function using the respective BIC selectors for the tuning parameters
(λ and wn; also h or k in models with functional nonparametric/single-index component).
Focusing on the selection of impact points, we observe four distinct clusters of impact points. | [
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] |
Focusing on the selection of impact points, we observe four distinct clusters of impact points.
It seems that all algorithms consistently select observations in the second quartile of the timeline
(wavelengths between 900 and 950 nm). Additionally, some algorithms select impact points at the
beginning of the first quartile, while others choose them at the end of the fourth quartile. Another
subset of algorithms selects points towards the end of the third quartile. If we combine this infor- | [
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] |
mation with the MSEP, the best choice appears to be made by the functions IASSMR.kNN.fit() and
IASSMR.kernel.fit().
A comparison of Tables 4, 7, and 9 reveals that the lowest MSEP value is obtained when the protein
and moisture contents are included in the model (using the SFPLM estimated using sfpl.kNN.fit()).
However, acquiring these measurements necessitates chemical analyses of the meat samples, which we
aim to circumvent by utilizing spectrometric information. A comparison between Table 4 and Table | [
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] |
aim to circumvent by utilizing spectrometric information. A comparison between Table 4 and Table
9 clearly demonstrates that incorporating both continuous and pointwise effects of the functional
predictor not only enhances predictive power but also preserves interpretability: each variable is
associated with an interpretable parameter in the model.
Finally, the output of the S3 method plot() for the functions in Table 8 represents the curves | [
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] |
Finally, the output of the S3 method plot() for the functions in Table 8 represents the curves
with their impact points (in dashed vertical lines), the same two diagnostic plots as the functions of
Section 3.3, and the estimated functional index for the MFPLSIM fitting functions. In Figure 6, we
provide the outputs of the 7 main functions (indicating option=1 to get just the first plot), obtained
under the same conditions as Table 9.
32
−100
−50
0
50
850
900
950
1000
1050
t
ζi(t)
(a) PVS.fit()
−100
−50
0 | [
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32
−100
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t
ζi(t)
(a) PVS.fit()
−100
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ζi(t)
(b) PVS.kernel.fit()
−100
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(c) PVS.kNN.fit()
−100
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(d) FASSMR.kernel.fit()
−100
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(e) FASSMR.kNN.fit()
−100
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(f) IASSMR.kernel.fit()
−100
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ζi(t)
(g) IASSMR.kNN.fit() | [
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t
ζi(t)
(f) IASSMR.kernel.fit()
−100
−50
0
50
850
900
950
1000
1050
t
ζi(t)
(g) IASSMR.kNN.fit()
Figure 6: Outputs of the S3 method plot() applied to the fitted objects of functions for the MLM,
MFPLM and MFPLSIM, using the corresponding BIC selectors for the tuning parameters.
4.3.3
Case study II: Sugar dataset
The purpose of this section is to demonstrate the implementation of the models (20) and (21) when | [
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] |
The purpose of this section is to demonstrate the implementation of the models (20) and (21) when
dealing with two functional covariates. Additionally, it addresses scenarios where the discretisation
size may exceed the sample size in the second step of the algorithms.
The Sugar dataset contains information about 268 samples obtained at a sugar plant in Scandi-
navia (see https://ucphchemometrics.com/sugar-process-data/ for details). For each sample, the | [
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] |
navia (see https://ucphchemometrics.com/sugar-process-data/ for details). For each sample, the
absorbance spectra from 275 nm to 560 nm were measured in 0.5 nm intervals (pn = 571). But in this
case, unlike Tecator dataset, measurements at several different excitation wavelengths were recorded,
generating various functional variables. In this application, we are going to consider two, those taken
33
into account in Novo et al. (2021b): the absorbance spectra at excitation wavelengths 240 nm (the | [
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] |
into account in Novo et al. (2021b): the absorbance spectra at excitation wavelengths 240 nm (the
first functional variable, which will be denoted as ζ) and at excitation wavelengths 290 nm (the sec-
ond functional variable, which will be named X). Figure 7 shows a sample of 20 curves from each
functional variable.
0
50
100
150
300
400
500
Wavelength
ζi(t)
0
200
400
600
300
400
500
Wavelength
χi(t)
Figure 7: Sample of 20 spectrometric curves measured at excitation wavelengths 240 nm (left panel) | [
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] |
Figure 7: Sample of 20 spectrometric curves measured at excitation wavelengths 240 nm (left panel)
and 290 nm (right panel).
In addition, Sugar dataset also contains the ash content of sugar. The target of the statistical
analysis is to predict the ash content of sugar having the measurements of the absorbance spectra
at those two excitation wavelengths. For that, we used models (20) and (21). In the bi-functional | [
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] |
at those two excitation wavelengths. For that, we used models (20) and (21). In the bi-functional
models, to select the most accurate effect of each functional predictor, continuous or pointwise-linear,
we followed the study performed in Novo et al. (2021b). They concluded that the absorbance spectra
at excitation wavelengths 240 nm (ζ(t)) should be included with the pointwise-linear effect. Then, we
fitted models
Yi =
571
�
j=1
β0jζi(tj) + m(Xi) + εi,
i = 1, . . . , 216,
and
Yi =
571
�
j=1 | [
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] |
fitted models
Yi =
571
�
j=1
β0jζi(tj) + m(Xi) + εi,
i = 1, . . . , 216,
and
Yi =
571
�
j=1
β0jζi(tj) + r(⟨Xi, θ0⟩) + εi,
i = 1, . . . , 216.
From the original set of 268 samples, 2 were discarded as extreme outliers. As a consequence, the
available sample size is 266. To compare procedures we split the sample into two subsamples: a training
sample composed of the first n = 216 observations and a testing sample containing the ntest = 50 last.
>data(Sugar)
>y<-Sugar$ash
>x<-Sugar$wave.290
>z<-Sugar$wave.240 | [
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] |
>data(Sugar)
>y<-Sugar$ash
>x<-Sugar$wave.290
>z<-Sugar$wave.240
>index.y.25 <- y > 25
34
>index.atip <- index.y.25
>(1:268)[index.atip]
[1]
71 129
>x.sug <- x[!index.atip,]
>z.sug<- z[!index.atip,]
>y.sug <- y[!index.atip]
>train<-1:216
The following code demonstrates the implementation of the MFPLM using the functions PVS.kernel.
fit() and PVS.kNN.fit(). We have chosen a semimetric based on derivatives, specified as semimetric= | [
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fit() and PVS.kNN.fit(). We have chosen a semimetric based on derivatives, specified as semimetric=
"deriv" with q=0 (default options). In each stage of the two-step procedures, we use half of the sam-
ple, setting n1 = n2 = 108 by defining train.1=1:108 and train.2=109:216. Consequently, n1 > wn,
but n2 may be smaller than the cardinality of R (denoted as ♯R). In such cases, it is advisable to
specify lambda.min.h, lambda.min.l, and factor.pn to adjust the level of penalisation in each step | [
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specify lambda.min.h, lambda.min.l, and factor.pn to adjust the level of penalisation in each step
of the PVS. Therefore, we set lambda.min.l=0.05, lambda.min.h=0.08, and factor.pn=1. This
ensures that if n2 < ♯R, then lambda.min=0.08; otherwise, lambda.min=0.05.
>fit5 <- PVS.kNN.fit(x=x.sug[train,],z=z.sug[train,], y=y.sug[train],train.1=1:108,
train.2=109:216, lambda.min.l=0.05,lambda.min.h=0.08, max.knn=10,nknot=20,
criterion="BIC") | [
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] |
train.2=109:216, lambda.min.l=0.05,lambda.min.h=0.08, max.knn=10,nknot=20,
criterion="BIC")
>fit6 <- PVS.kernel.fit(x=x.sug[train,],z=z.sug[train,], y=y.sug[train], train.1=1:108,
train.2=109:216, lambda.min.l=0.05,lambda.min.h=0.08, num.h = 10, max.q.h=0.35,
nknot=20,criterion="BIC")
The objects fit5 and fit6 belong to an S3 class (see Table 8), which implements S3 methods. The
functionalities of these methods were detailed throughout the case studies in Sections 2.3.2, 3.3.2, | [
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] |
and 4.3.2. They include displaying a summary of the fitted models using summary.PVS.kNN() and
summary.PVS.kernel(), respectively (alternatively, print.PVS.kNN() and print.PVS.kernel() can
be used); obtaining graphical representations of the outputs through plot.PVS.kNN() and plot.PVS.
kernel(); and making predictions for the fitted objects using predict.PVS.kNN() and predict.PVS.
kernel(), respectively. These latter functions implement the three prediction possibilities outlined | [
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] |
in Section 4.3.2. In addition, predict.PVS.kNN() implements option=4, which involves reselecting k
for predicting the functional nonparametric component of the model, but this time in a local manner.
This selection is performed using the LOOCV criterion in the functional nonparametric model, applied
to the complete training sample (i.e., train=c(train.1,train.2)).
Table 10 shows the MSEP, setting option=2, and �sn for each primary fitting function listed in | [
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] |
Table 10 shows the MSEP, setting option=2, and �sn for each primary fitting function listed in
Table 8 allowing for 2 curves in the model (i.e., the MFPLM and MFPLSIM fitting functions). The
results were obtained using the respective BIC selectors for λ, wn, and h/k in two different scenar-
ios: (i) lambda.min=0.05 (i.e., lambda.min.h=lambda.min.l=0.05) and (ii) lambda.min=0.05 and
lambda.min=0.08, with factor.pn=1. It should be noted that in the 1-step procedures (FASSMR.kernel | [
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] |
lambda.min=0.08, with factor.pn=1. It should be noted that in the 1-step procedures (FASSMR.kernel
.fit() and FASSMR.kNN.fit()), the same results will be obtained in both scenarios (i) and (ii) as
long as n >factor.pn*wn. For comparative purposes, the same values for common input arguments
were used across all six functions. Additionally, h and k were selected using the same input options:
35
setting max.knn=10 for kNN-based estimation functions and max.q.h=0.35 for kernel-based ones. The | [
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] |
setting max.knn=10 for kNN-based estimation functions and max.q.h=0.35 for kernel-based ones. The
remaining arguments were set to their default options.
It is noteworthy that both the MFPLM and MFPLSIM fitting functions improve their predictive
ability by setting lambda.min.h=0.08. Additionally, under this setting, they tend to select an equal
or lower number of impact points, leading to a simpler model. The functions fitting the MFPLSIM | [
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] |
or lower number of impact points, leading to a simpler model. The functions fitting the MFPLSIM
show a more significant improvement in MSEP and a greater reduction in the number of impact
points selected. However, the best result in terms of MSEP is provided by the MFPLM fitted using
PVS.kNN.fit().
Model
Function
lambda.min=0.05
lambda.min.l=0.05
lambda.min.h=0.08
MSEP
ˆsn
MSEP
ˆsn
MFPLM
PVS.kernel.fit()
2.00
10
1.92
9
PVS.kNN.fit()
1.20
7
1.19
7
MFPLSIM
FASSMR.kernel.fit()
4.70
6
4.70
6
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] |
10
1.92
9
PVS.kNN.fit()
1.20
7
1.19
7
MFPLSIM
FASSMR.kernel.fit()
4.70
6
4.70
6
FASSMR.kNN.fit()
3.60
8
3.60
8
IASSMR.kernel.fit()
3.28
11
2.44
5
IASSMR.kNN.fit()
3.29
8
1.85
4
Table 10: MSEP obtained for each function using the respective BIC selectors for the tuning param-
eters (λ, wn and h or k).
Model
Function
lambda.min=0.05
lambda.min.l=0.05
lambda.min.h=0.08
Points of impact
Points of impact
MFPLM
PVS.kernel.fit()
275
560 275
560
PVS.kNN.fit()
275
560 275
560
MFPLSIM
FASSMR.kernel.fit() 275
560 | [
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0.4608931243419647,
-0.4422176778316498,
0.040301427245140076,
0.5122500658035278,
0.22812695801258087,
0.09248940646648407,
-0.001141839660704136,
0.174408957362175,
-0.18234778940677643,
0.1919509917497635,
-0.5280349850654602,
0.17728550732135773,
-0.486500084400177,
-0.4232819676399231,
-0.6159854531288147,
0.5893691182136536,
-0.18451480567455292,
0.46364808082580566,
-0.14523568749427795,
0.2201933115720749,
-0.16514231264591217,
0.5136563181877136,
0.19893476366996765,
-0.029226182028651237,
-0.06588016450405121,
-0.28957951068878174,
0.8333648443222046,
-0.6886635422706604,
0.5465303659439087,
0.1864287555217743,
0.8188175559043884,
0.16286227107048035,
-0.4680618345737457,
-0.5804263949394226,
-0.404478520154953,
-0.39551475644111633,
0.13271288573741913,
0.10156131535768509,
0.3409656882286072,
-0.6268191337585449,
-0.6848537921905518,
-0.8676173686981201,
0.6476825475692749,
0.3097454905509949,
0.1315101832151413,
-0.11286593973636627,
-0.5636313557624817,
-0.16480636596679688,
-0.5652187466621399,
-0.32717785239219666,
-0.14231520891189575,
-0.2236187905073166,
-0.08866249769926071
] |
PVS.kernel.fit()
275
560 275
560
PVS.kNN.fit()
275
560 275
560
MFPLSIM
FASSMR.kernel.fit() 275
560
FASSMR.kNN.fit()
275
560
IASSMR.kernel.fit() 275
560 275
560
IASSMR.kNN.fit()
275
560 275
560
Table 11: Impact points obtained for each function using the respective BIC selectors for the tuning
parameters (λ, wn and h or k).
To gain a deeper understanding, Table 11 displays (as explained in Section 4.3.2) the impact points | [
-0.10314462333917618,
0.2696310877799988,
0.4976988434791565,
0.008542663417756557,
-0.2608427405357361,
0.6582861542701721,
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-0.12744933366775513,
0.6427077054977417,
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0.19843193888664246,
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0.6886183619499207,
0.5876815915107727,
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0.49465247988700867,
-0.13488531112670898,
0.6044439673423767,
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0.3267078995704651,
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0.197938472032547,
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0.4045497477054596,
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0.08206279575824738,
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0.05791844427585602,
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0.061415188014507294,
0.15197062492370605,
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0.21104270219802856,
0.25355246663093567,
0.07782302051782608,
-0.7075639963150024,
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0.365933358669281,
-0.2077319324016571,
0.668044924736023,
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] |
To gain a deeper understanding, Table 11 displays (as explained in Section 4.3.2) the impact points
selected by the six main functions in scenarios (i) and (ii). In the case of 1-step functions, the selection
36
remains the same under both scenarios, as previously detailed. Comparing the results of each method
in the two settings, it is noteworthy that the PVS functions obtain almost the same impact point
selection: exactly the same using the PVS.kNN.fit() function, and with just one additional impact | [
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] |
selection: exactly the same using the PVS.kNN.fit() function, and with just one additional impact
point using PVS.kernel.fit(). However, for the IASSMR functions, there is a significant change in
the impact point selection, both in the number and position of the impact points. This fact explains
the substantial change in MSEP in the two scenarios.
Analysing the results in scenario (ii) for the two-step functions, we observe four distinct clusters of | [
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] |
impact points. It appears that all algorithms consistently select a set of observations at the beginning
of the first quartile of the timeline. Additionally, both IASSMR algorithms select an impact point in
the middle of the third quartile, while the PVS algorithms choose it at the end of the third quartile.
Futhermore, PVS procedures select another point at the end of the second quartile and PVS.kNN.fit()
also selects an impact point at the beginning of the second quartile. | [
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] |
also selects an impact point at the beginning of the second quartile.
Combining the information from Tables 10 and 11, the best choice of points appears to be made
by the function PVS.kNN.fit(). Furthermore, in this real data analysis, it seems that the second step
plays a crucial role in the IASSMR algorithms, as the results provided in MSEP and impact point
selection by the FASSMR are substantially worse.
5
Discussion | [
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] |
selection by the FASSMR are substantially worse.
5
Discussion
This article introduces a new R package, fsemipar (see Aneiros and Novo 2024), tailored for SoF semi-
parametric regression. The package implements various models with at least one functional predictor.
The effects included for functional covariates are functional single-index, functional nonparametric,
and pointwise linear. The link function of functional single-index components and the functional op- | [
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erator of the functional nonparametric ones are estimated using kernel smoothing techniques. This
includes not only kernel-based estimation but also kNN-based estimation, allowing for both global and
local smoothing. In the case of the pointwise linear effect of curves, the package includes algorithms
to handle the high dependence between the discretized observations, allowing the selection of points
of impact. This feature has proven to be highly beneficial from a predictive perspective in the real | [
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] |
data applications included in the article.
The package fsemipar also allows the inclusion of scalar variables as predictors with a linear ef-
fect, enabling simultaneous estimation and variable selection using penalised least-squares procedures.
Through the real data applications, the article demonstrates the capabilities of the package, provid-
ing a comprehensive review of its features. Despite the large number of optional arguments for each | [
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ing a comprehensive review of its features. Despite the large number of optional arguments for each
function, all functions require only the data as a mandatory argument, facilitating easy use.
Despite the wide range of possibilities it currently offers, the package will continue to be updated,
striving to enhance its functionalities. This includes optimizing the computational cost of functions
that estimate functional single-index components, exploring implementations in C++ using the Rcpp | [
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that estimate functional single-index components, exploring implementations in C++ using the Rcpp
or RcppArmadillo packages. It also involves incorporating goodness-of-fit tests for functional semi-
parametric regression (see, e.g., Chan et al. 2023), and increasing the flexibility of the included models.
Enhancements may involve increasing the flexibility in the effect of scalar variables. For instance, this | [
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] |
could include accounting for measurement errors (see, e.g., Zhu et al. 2020) and varying coefficients
(see, e.g., Wu et al. 2010), or handling responses that are missed at random (see, e.g., Ling et al.
37
2019).
References
Ait-Sa¨ıdi, A., Ferraty, F., Kassa, R., and Vieu, P. (2008). Cross-validated estimations in the single-
functional index model. Statistics, 42(6):475–494.
Amato, U., Antoniadis, A., and De Feis, I. (2006). Dimension reduction in functional regression with | [
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Subsets and Splits