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bounded and analytic in the open right (resp. left) half-plane. The non-tangential limit of h
exists almost everywhere on the imaginary axis, and by the Maximum Modulus Principle, we
must have
∥h∥∞ := ess supRe(s)=0∥h(s)∥2 =
sup
Re(s)>0
∥h(s)∥2,
h ∈ H∞
+
and
∥h∥∞ := ess supRe(s)=0∥h(s)∥2 =
sup
Re(s)<0
∥h(s)∥2,
h ∈ H∞
− .
The Adamyan–Arov–Krein (AAK) Theory [1] says the following:
σk+1 = inf
Rk,F ∥G − Rk − F∥∞,
0 ≤ k ≤ n − 1,
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σk+1 = inf
Rk,F ∥G − Rk − F∥∞,
0 ≤ k ≤ n − 1,
(12)
where Rk ranges over all rational functions with at most k poles, all contained in the open left
half-plane, and F ranges over H∞
− . In fact, as discussed in section 2, Rk is the transfer function
of a reduced system ˜Γ = ( ˜A, ˜B, ˜C, ˜D) with ˜A ∈ Ck×k. Hence, eq. (12) tells us that if the
transfer function of Γ can be well-approximated by rational functions, then it has fast-decaying | [
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] |
transfer function of Γ can be well-approximated by rational functions, then it has fast-decaying
Hankel singular values. The other direction is not true, due to the existence of F. That is, if
the Hankel singular values decay fast, then it does not necessarily mean that G can be well-
approximated by rational functions. However, section 2 shows that this is true if the sum of
the tails of the Hankel singular values decay rapidly.
C
Proof of Theorem 1 | [
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the tails of the Hankel singular values decay rapidly.
C
Proof of Theorem 1
Let G be the transfer function of a random system Γ2, i.e.,
G(z) = C(zI − A)−1B + D.
By the AAK theory, the Hankel singular values of Γ can be studied via the rational approxi-
mation of G. Since the matrix D does not affect the Hankel singular values of a system, we
assume, without loss of generality, that D = 0. We let σ1, . . . , σn be the random variables that | [
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] |
assume, without loss of generality, that D = 0. We let σ1, . . . , σn be the random variables that
are equal to the singular values of the random LTI system Γ2. We approach Theorem 1 in three
steps:
1. We separate out the poles of the transfer function G of (A, B, C, 0) that are close to the
boundary of the unit disk. This breaks G into two low-rank systems G1 and G2, where
G1 is low-rank because it has few poles and G2 is low-rank because its poles are far away
from the boundary. | [
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0.638948380947113,
0.12334708124399185,
-0.18349580466747284,
-0.16154374182224274,
0.2520529329776764
] |
from the boundary.
2. We then estimate the decay of the singular values of G2 using the information of the
maximum moduli of its poles.
3. Finally, we control σ1 in probability. This gives a control on the relative singular values.
We will make these three steps into three lemmas and use them to derive the result at the end.
Lemma 1. Given γ > 0, with probability at least 1 − δ, there are at most nβ poles of G(z)
outside the disk D(0, 1 − n−γ), where
β = 1 + logn
�
n−γα +
�
ln(1/δ)
2n
�
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outside the disk D(0, 1 − n−γ), where
β = 1 + logn
�
n−γα +
�
ln(1/δ)
2n
�
.
Proof. Let Z be the number of poles inside D(0, 1 − n−γ). Then, Z has a binomial distribution
Z ∼ B(n, 1 − n−γα).
17
From Hoeffding’s inequality, we have
P(Z ≤ n − nβ) ≤ exp
�
−2n
�
1 − n−γα − n − nβ
n
�2�
= exp
�
−2n
�
−n−γα + nβ−1�2�
Set
β = 1 + logn
�
n−γα +
�
ln(1/δ)
2n
�
= 1 − γα + logn
�
1 +
�
ln(1/δ)
2n1−2γα
�
.
Then, we have
nβ−1 = n−γα +
�
ln(1/δ)
2n
so that
P(Z ≤ n − nβ) ≤ exp
�
−2nln(1/δ)
2n
�
= δ. | [
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0.8780071139335632,
0.8704891204833984,
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0.027120238170027733,
0.3013457655906677,
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0.2152644395828247,
0.2381865680217743,
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0.5409378409385681,
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0.038146570324897766,
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0.0019093109294772148,
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0.43289411067962646,
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0.23640765249729156,
0.23386189341545105,
0.33420485258102417,
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0.21544532477855682,
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1.0150738954544067,
1.1645681858062744,
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0.5090258717536926,
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0.33129817247390747,
0.9733411073684692,
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0.19046206772327423,
-0.34372860193252563,
0.2425379455089569,
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0.13669778406620026,
0.5965659022331238,
0.0808885395526886,
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-0.07012378424406052,
0.3112950921058655,
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0.6961710453033447,
0.014909482561051846,
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0.20390906929969788,
0.7332159280776978,
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0.4083985388278961,
0.15743422508239746,
0.3286522328853607,
0.6075748205184937,
0.0882776603102684,
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0.672664225101471,
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0.24547621607780457,
0.8045720458030701,
0.7942322492599487,
0.3823554217815399,
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0.5658798217773438,
0.34273138642311096,
0.6653136610984802,
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-0.4162499010562897,
-0.7871291041374207,
0.700266420841217,
0.0883093997836113,
0.14644593000411987,
-0.06639239937067032,
-0.16713213920593262,
-0.2823512852191925,
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-0.13055101037025452,
0.06336662173271179,
0.00791146606206894,
0.011022389866411686
] |
�
.
Then, we have
nβ−1 = n−γα +
�
ln(1/δ)
2n
so that
P(Z ≤ n − nβ) ≤ exp
�
−2nln(1/δ)
2n
�
= δ.
This finishes the proof.
Lemma 2. For any γ > 0, let the random variable k be the number of poles of G(z) inside
D(0, 1 − n−γ). Let κ > γ be given. Then, with conditional probability (given k) at least 1 − δ,
we have
σ(n−k)+nκ+2 ≤ O
��
nγ+1(ln(1/δ)+n)
�
e−(n(κ−γ)),
where the constant in O is universal.
Proof. Let z1, . . . , zk be the poles of G(z) inside D(0, 1 − n−γ). Assume G(z) can be written as | [
-0.13430474698543549,
-0.4114075303077698,
0.14038802683353424,
-0.32076898217201233,
-0.20068512856960297,
0.13894058763980865,
0.039490971714258194,
-0.574231743812561,
-0.13966189324855804,
-0.5946131944656372,
-0.29364514350891113,
0.15603852272033691,
-0.5080857276916504,
0.45882448554039,
0.0707288384437561,
0.5709880590438843,
0.6846514344215393,
0.28968310356140137,
-0.6489038467407227,
-0.9964244961738586,
0.5074799656867981,
-0.03437824174761772,
0.16481609642505646,
0.2990240752696991,
-0.1835799664258957,
0.1316160261631012,
0.5304344296455383,
-0.2548867464065552,
0.6815653443336487,
0.37893185019493103,
-0.245533287525177,
0.548651933670044,
0.04003460332751274,
0.23711885511875153,
0.18334199488162994,
0.09168744087219238,
0.7050296068191528,
0.371871680021286,
0.444780558347702,
0.007281027268618345,
0.18044358491897583,
-0.2005082070827484,
0.1622450351715088,
0.6018947958946228,
-0.2607133090496063,
-0.8689016103744507,
-0.9913968443870544,
0.003679135348647833,
0.2565792202949524,
-0.5526059865951538,
-0.08717389404773712,
-0.05439547449350357,
0.15788359940052032,
0.5562930703163147,
-0.48156678676605225,
0.2912808954715729,
-0.3298599421977997,
-0.10953240096569061,
0.5542507171630859,
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0.09781288355588913,
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0.34029000997543335,
0.3745444416999817,
0.10011584311723709,
1.0715702772140503,
0.21238499879837036,
0.2791547477245331,
-0.3593718111515045,
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0.21809913218021393,
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0.26264968514442444,
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0.4876459538936615,
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0.07170144468545914,
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0.08995579928159714,
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0.11983585357666016,
0.9727886915206909,
0.28715962171554565,
-0.35956475138664246,
0.6705031991004944,
0.1156812310218811,
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0.3463432788848877,
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0.7140363454818726,
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0.06486958265304565,
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0.19291460514068604,
0.35264110565185547,
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0.8047395348548889,
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0.825846254825592,
0.13682155311107635,
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0.5028326511383057,
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0.04426300525665283,
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0.7265111207962036,
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0.4953448176383972,
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0.21801452338695526,
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0.0022128368727862835,
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0.21724581718444824,
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1.4436979293823242,
0.2990332841873169,
-0.28540703654289246,
0.47421228885650635,
-0.18850606679916382,
0.1394800841808319,
0.05188468471169472,
0.045710038393735886,
0.3633531928062439,
0.39341336488723755,
-0.10035483539104462,
0.6812131404876709,
-0.008310595527291298,
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0.919200599193573,
0.19997243583202362,
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0.22954867780208588,
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0.30368396639823914,
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0.3241827189922333,
0.15655475854873657,
0.7025310397148132,
1.0597012042999268,
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] |
Proof. Let z1, . . . , zk be the poles of G(z) inside D(0, 1 − n−γ). Assume G(z) can be written as
G(z) = G1(z) +
k
�
i=1
ci
z − zi
,
where G1(z) is a degree-(n − k) rational function with poles inside the annulus of inner radius
1 − n−γ and outer radius 1 and ci’s are i.i.d. random variables with distribution N(0, 1). We
can further write
k
�
i=1
ci
z − zi
=
k
�
i=1
ci
∞
�
j=0
zj
i z−j−1 =
∞
�
j=1
z−1−j
� k
�
i=1
cizj
i
�
.
By the AAK theory, for any K > 0, we have
σ(n−k)+K ≤ sup
|z|=1
������
∞
�
j=K−2 | [
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] |
�
i=1
cizj
i
�
.
By the AAK theory, for any K > 0, we have
σ(n−k)+K ≤ sup
|z|=1
������
∞
�
j=K−2
z−1−j
� k
�
i=1
cizj
i
�������
≤
∞
�
j=K−2
∥c zj∥1 ≤ ∥c∥2
∞
�
j=K−2
∥zj∥2,
where c = [c1, . . . , ck]⊤ and zj = [zj
1, . . . , zj
k]⊤, and the last step follows from H¨older’s inequality.
Since ∥c∥2
2 follows the χ2
k distribution, for M > n, we have that
P(∥c∥2
2 > M) ≤ exp
�
−n
2
�M
n − 1 − ln
�M
n
���
.
Hence, there exists a universal constant C > 0 such that when M ≥ C(ln(1/δ) + n)1, we have
P(∥c∥2 | [
-0.12731988728046417,
0.012545231729745865,
0.44202500581741333,
-0.40648406744003296,
-0.06037169322371483,
0.2290438711643219,
0.23260757327079773,
-0.5854731202125549,
-0.2672256529331207,
-0.3363499641418457,
-0.028167415410280228,
0.13814112544059753,
-0.22870226204395294,
0.4343429207801819,
0.27449551224708557,
0.8024638891220093,
0.5952537655830383,
0.5470640659332275,
-1.0170356035232544,
-0.7745198011398315,
0.5439488291740417,
-0.12856516242027283,
0.2683965861797333,
0.17064103484153748,
-0.14613556861877441,
0.05972316861152649,
0.12958967685699463,
0.0013140043010935187,
0.49784186482429504,
0.06418165564537048,
-0.08598361909389496,
-0.06943286210298538,
0.44144535064697266,
-0.17001010477542877,
0.1846429854631424,
-0.018413467332720757,
0.32757568359375,
0.2696899175643921,
0.2701773941516876,
-0.2642863690853119,
0.18425095081329346,
-0.12446729838848114,
0.15331140160560608,
0.5490325093269348,
-0.4213554263114929,
-0.6649543642997742,
-1.688173532485962,
0.023759812116622925,
0.16991424560546875,
-0.11855452507734299,
0.06811759620904922,
0.0010244195582345128,
0.22393350303173065,
-0.20406201481819153,
-0.18367473781108856,
0.208415687084198,
-0.4786309599876404,
-0.24164463579654694,
0.4440029263496399,
-0.34636297821998596,
0.20522959530353546,
-0.28108033537864685,
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] |
.
Hence, there exists a universal constant C > 0 such that when M ≥ C(ln(1/δ) + n)1, we have
P(∥c∥2
2 > M) ≤ exp
�
−n
2
M
n
�
≤ δ.
1Here, n is to guarantee that M > n
18
Moreover, since
���zj
i
��� ≤ (1 − n−γ)j for all 1 ≤ i ≤ k, we have
∥zj∥2 ≤ √n(1 − n−γ)j.
That is, with probability at least 1 − δ, we have
σ(n−k)+K ≤ O
��
ln(1/δ)+n
�√n
∞
�
j=K−2
(1 − n−γ)j = O
��
nγ+1(ln(1/δ)+n)
�
(1−n−γ)K−2.
Suppose K ≥ nκ + 2. Then, we have
(1−n−γ)K−2 ≤ (1−n−γ)(nκ) =
�
(1−n−γ)(nγ)�(nκ−γ)
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] |
�
(1−n−γ)K−2.
Suppose K ≥ nκ + 2. Then, we have
(1−n−γ)K−2 ≤ (1−n−γ)(nκ) =
�
(1−n−γ)(nγ)�(nκ−γ)
.
Since (1−n−γ)nγ → e−1 as n → ∞, if κ > γ, then (1−n−γ)K−2 decays faster than any negative
power of n. Hence, we have
σ(n−k)+K ≤ O
��
nγ+1(ln(1/δ)+n)
�
e−(n(κ−γ)),
as desired.
Lemma 3. With probability at least 1 − δ, the leading Hankel singular value σ1 satisfies
σ1 ≥ O(√nδ),
where the constant in O is universal.
Proof. The leading Hankel singular value σ1 is equivalent to the spectral norm of the Hankel | [
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] |
Proof. The leading Hankel singular value σ1 is equivalent to the spectral norm of the Hankel
matrix
CB
CAB
CA
2B
· · ·
CAB
CA
2B
· · ·
· · ·
CA
2B
...
...
...
...
...
· · ·
...
.
Hence, we have
σ1 ≥
��CB
�� ,
where CB/√n ∼ N(0, 1). Hence, if we set M = C√nδ for some universal constant C > 0, we
have
P(σ1 ≤ M) ≤ P
� ��CB
�� ≤ M
�
= P
� ��CB
�� /√n ≤ M/√n
�
≤ δ.
This completes the proof.
Now, let’s assemble our ultimate statement. | [
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] |
� ��CB
�� /√n ≤ M/√n
�
≤ δ.
This completes the proof.
Now, let’s assemble our ultimate statement.
Proof of Theorem 1. By Lemma 2 and Lemma 3, with probability at least 1−δ/2, we have that
σ(n−k+nκ+2)
σ1
≤ O
��
nγ(ln(1/δ)+n)δ−1�
e−(n(κ−γ)).
Hence, if we set κ so that
nκ ≥ nγ ln
�
Cδ−1ϵ−1�
nγ(log(1/δ) + n)
�
= nγ ln
�
O
�
δ−3/2ϵ−1n(γ+1)/2��
19
for a sufficiently large universal constant C > 0, then we guarantee that
σ(n−k+nκ+2)
σ1
≤ ϵ.
By Lemma 1, we have that with probability at least 1 − δ/2,
n − k ≤ nβ, | [
0.06665156781673431,
0.4496968686580658,
0.39201390743255615,
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0.07722261548042297,
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] |
σ(n−k+nκ+2)
σ1
≤ ϵ.
By Lemma 1, we have that with probability at least 1 − δ/2,
n − k ≤ nβ,
β = 1 + logn
�
n−γα +
�
ln(2/δ)
2n
�
.
Set γ = 1/(1 + α). Then, we have
β = 1 + logn
�
n−γα
�
1 +
�
ln(2/δ)
2n1−2γα
��
= 1 − γα + logn
�
1 +
�
ln(2/δ)
2n1−2γα
�
≤
1
1 + α + logn(1 +
�
ln(2/δ)/2) <
1
1 + α + ln(2 +
�
ln(1/δ)/2)
ln(n)
.
Since
nκ + 2 = O
�
nγ ln
�
δ−3/2ϵ−1n
��
≤ O
�
nβ ln
�
δ−3/2ϵ−1n
��
.
The claim is proved.
D
Proof of Theorem 2
In this section, we prove Theorem 2. | [
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] |
�
δ−3/2ϵ−1n
��
.
The claim is proved.
D
Proof of Theorem 2
In this section, we prove Theorem 2.
Our proof focuses on the worst-case perturbation by
construction. As a consequence, it simultaneously proves the sharpness of the result. Intuitively,
consider a rational function
s �→
bc
s − a,
since we only care about its values on the imaginary axis, the closer the pole a is to the imaginary
axis, the less stable it is. On the other hand, it is obvious that the size of bc also controls the | [
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0.45239177346229553,
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] |
axis, the less stable it is. On the other hand, it is obvious that the size of bc also controls the
(absolute) conditioning of the rational function. We state the rigorous proof below.
Proof of Theorem 2. Without loss of generality, we assume that B = ˜B =
�
1
1
· · ·
1
�⊤. 2
The transfer functions of Γ and ˜Γ are
G(s) =
n
�
j=1
cj
s − sj
and
˜G(s) =
n
�
j=1
˜cj
s − ˜sj
,
respectively. Then, for any s on the imaginary axis, we have
����
cj
s − sj
−
˜cj
s − ˜sj
���� =
����
cj(s − ˜sj) − ˜cj(s − sj) | [
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0.7452864646911621,
0.19513438642024994,
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0.5333638191223145,
0.21975119411945343,
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0.1164473295211792,
0.2724566161632538,
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] |
����
cj
s − sj
−
˜cj
s − ˜sj
���� =
����
cj(s − ˜sj) − ˜cj(s − sj)
(s − sj)(s − ˜sj)
���� =
����
cjs − cj˜sj − ˜cjs + ˜cjsj + ˜cj˜sj − ˜cj˜sj
(s − sj)(s − ˜sj)
����
≤ |cj − ˜cj| |s − ˜sj| + |˜cj| |sj − ˜sj|
|s − sj| |s − ˜sj|
= |cj − ˜cj|
|s − sj| +
|˜cj| |sj − ˜sj|
|s − sj| |s − ˜sj|
≤
∆B
|Re(sj)| +
2 |cj| ∆A
|Re(sj)|2/2.
2Otherwise, we can redefine B = ˜B =
�
1
1
· · ·
1
�⊤, C = B⊤ ◦ C, and ˜C = ˜B⊤ ◦ ˜C, and the redefined
transfer functions are unchanged.
20
Hence, we have
∥G − ˜G∥∞ ≤
n
�
j=1
�
∆B | [
-0.2610430121421814,
0.10779551416635513,
0.5819446444511414,
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0.07942827045917511,
0.32145431637763977,
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0.13349947333335876,
-0.43478158116340637,
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0.18743260204792023,
-0.17920374870300293,
0.4880068898200989,
0.4859106242656708,
0.8939389586448669,
0.29903003573417664,
0.3523752987384796,
-0.26517462730407715,
-0.860959529876709,
0.113684743642807,
0.26586923003196716,
0.5418871641159058,
0.06914136558771133,
0.010018904693424702,
0.22563202679157257,
0.36310216784477234,
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0.0624554418027401,
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0.2872176766395569,
0.20075975358486176,
0.463007390499115,
0.0005439628730528057,
0.29887932538986206,
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0.44228023290634155,
0.14740677177906036,
0.48396244645118713,
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0.48487231135368347,
0.8239635229110718,
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0.1328340321779251,
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0.3521207273006439,
0.2625719904899597,
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0.059996142983436584,
0.6617597341537476,
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0.6379050612449646,
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0.23081053793430328,
-0.68227618932724,
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0.6909599900245667,
0.10947740077972412,
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0.21212127804756165,
0.24578751623630524,
-0.6752389073371887,
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0.2781520485877991,
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0.5597041845321655,
0.2525440454483032,
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0.6224256753921509,
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0.7612552046775818,
-0.35465192794799805,
-0.2439505159854889,
0.10629679262638092,
0.09747931361198425,
0.13653236627578735,
0.5274232029914856,
0.01687987893819809,
-0.5550240278244019,
0.6708576679229736,
0.19190074503421783,
-0.23746991157531738,
-0.3997972309589386,
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0.1538001298904419,
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0.022869180887937546,
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0.028896864503622055,
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0.04528101161122322,
0.8759439587593079,
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0.48714450001716614,
-0.14963315427303314,
0.29701757431030273,
0.5208714008331299,
-0.23876745998859406,
0.37869638204574585,
0.8492669463157654,
-0.7626438140869141,
-0.0016993898898363113,
0.5941107273101807,
0.6918816566467285,
0.2598143219947815,
0.3606138825416565,
-0.6491615176200867,
-0.770995020866394,
-0.08466719090938568,
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-0.7943638563156128,
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0.12480180710554123,
-0.3305065631866455,
0.0454322025179863,
0.4899330735206604,
0.006175613962113857,
-0.44614899158477783,
0.3187732696533203,
-0.08541405200958252,
0.0038723796606063843,
-0.10460720956325531,
0.2638266384601593,
-0.5249696969985962,
0.1736619621515274,
-0.501995861530304,
0.07349091023206711,
-0.20714613795280457,
0.10168083757162094,
-0.41103672981262207,
0.00801402423530817,
0.20781834423542023,
0.4499172568321228,
-0.05855555087327957,
-0.11546260863542557,
0.6961621642112732,
-0.24186073243618011,
0.1901838332414627,
-0.2142900824546814,
0.03428765758872032,
-0.07939641922712326,
-0.45594921708106995,
0.09751997888088226,
0.03371517360210419,
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-0.413117915391922,
0.09189434349536896,
0.3768811821937561,
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0.028861291706562042,
0.32621243596076965,
0.4425799548625946,
0.26808324456214905,
0.4451132118701935,
0.50210040807724,
-0.840307354927063,
-0.12999895215034485,
0.2914344072341919,
-1.2200008630752563,
1.2929648160934448,
-0.1873542219400406,
-0.3891366720199585,
0.32352757453918457,
-0.5153436064720154,
-0.14108265936374664,
-0.3272795081138611,
0.557153046131134,
0.27834051847457886,
-0.12602752447128296,
-0.2604866027832031,
0.7936883568763733,
0.23585772514343262,
-0.056033842265605927,
-0.592261016368866,
0.13917817175388336,
0.10682177543640137,
-0.18750928342342377,
0.10849053412675858,
-0.3988966941833496,
0.23249825835227966,
-0.33900225162506104,
0.005569320637732744,
-0.3669899106025696,
0.31532877683639526,
0.162059485912323,
0.9110427498817444,
-0.14758259057998657,
0.447403222322464,
0.8586671948432922,
-0.09884995967149734,
0.5034322738647461,
0.8366214036941528,
-0.010716165415942669,
0.5335038900375366,
-0.4075888991355896,
0.12436607480049133,
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0.13172049820423126,
-0.051260851323604584,
-0.19235444068908691,
0.6326709389686584,
-0.021144479513168335,
0.1323479413986206,
0.2149796187877655,
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0.2471514344215393,
0.45624038577079773,
-0.5496582984924316,
-0.006136061158031225,
0.011542947962880135,
0.08225221931934357,
-0.42670634388923645,
0.16577273607254028,
-0.21929217875003815,
-0.17823894321918488,
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] |
transfer functions are unchanged.
20
Hence, we have
∥G − ˜G∥∞ ≤
n
�
j=1
�
∆B
|Re(sj)| + 4 |cj| ∆A
|Re(sj)|2
�
≤ n∆B max
j
1
|Re(sj)| + 4n∆A max
j
|cj|
|Re(sj)|2 .
This proves the upper bound. To prove the lower bound, let j1 be an index that maximizes
1/|Re(sj)| and j2 be an index that maximizes |cj|/|Re(sj)|2. Define ˜ΓB by perturbing cj1 to
cj1 + ∆B. Then, we have
∥G − ˜GB∥∞ =
����
∆B
s − sj1
����
∞
= ∆B max
j
1
|Re(sj)|.
Define ˜ΓA by perturbing sj2 to sj2 + ∆A. Then, we have
∥G − ˜GA∥∞ = |cj2|
����
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-0.22945092618465424,
0.241104394197464,
-0.12363585829734802,
0.7430375814437866,
-0.5347041487693787,
-0.7638797760009766,
-0.35835298895835876,
-0.08734951168298721
] |
j
1
|Re(sj)|.
Define ˜ΓA by perturbing sj2 to sj2 + ∆A. Then, we have
∥G − ˜GA∥∞ = |cj2|
����
1
s − sj2
−
1
s − sj2 + ∆A
����
∞
= |cj2| ∆A
����
1
(s − sj2)(s − sj2 + ∆A)
����
∞
≥ |cj2| ∆A
1
|Re(sj2)|2 = ∆A max
j
|cj|
|Re(sj)|2 .
This proves the sharpness of the theorem.
E
Proof of Theorem 3 and Theorem 4
The proof of Theorem 3 is a straightforward assembly of two results in random matrix theory.
The first result, due to [24], controls the Hankel norm σ1(Hn) of a random Hankel matrix, | [
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] |
The first result, due to [24], controls the Hankel norm σ1(Hn) of a random Hankel matrix,
whereas the second result by [8] studies the distribution of all absolute singular values σj(Hn)
of a random Hankel matrix. Our study of the relative Hankel singular values is achieved by
taking the quotient of the subjects of the two prior works.
Proof of Theorem 3. By [24], with probability 1, we have that
σ1(Hn) = ∥Hn∥ = O(
√
n ln n). | [
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] |
Proof of Theorem 3. By [24], with probability 1, we have that
σ1(Hn) = ∥Hn∥ = O(
√
n ln n).
Define Kn = Hn(1:⌈n/2⌉, 1:⌈n/2⌉). Then, by [8], with probability 1, we have that µ(Kn/√n)
converges in distribution to a fixed probability measure, where
µ(Kn/√n) =
1
⌈n/2⌉
⌈n/2⌉
�
j=1
δλj(Kn/√n)
is the spectral measure of Kn/√n. Since Kn is symmetric, the singular values of Kn are the
moduli of the eigenvalues of Kn. Hence, fix some ϵ > 0, we have that
���{j | σj(Kn)/σ1(Hn) > ϵ/
�
ln(n)}
��� = Ω(n). | [
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] |
���{j | σj(Kn)/σ1(Hn) > ϵ/
�
ln(n)}
��� = Ω(n).
Since Kn is a submatrix of Hn, we have σj(Hn) ≥ σj(Kn) for all 1 ≤ j ≤ ⌈n/2⌉. Hence, its
(ϵ/
�
ln(n))-rank can be controlled as
���{j | σj(Hn)/σ1(Hn) > ϵ/
�
ln(n)}
��� ≥
���{j | σj(Kn)/σ1(Hn) > ϵ/
�
ln(n)}
��� = Ω(n).
This finishes the proof.
21
We remark that in the statement of Theorem 3, we study the ϵ/
�
ln(n)-rank of the Hankel
matrix instead of the ϵ-rank. The reason is that, unlike the singular values of a random matrix, | [
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] |
matrix instead of the ϵ-rank. The reason is that, unlike the singular values of a random matrix,
the spectral measure of a normalized random Hankel matrix Hn/√n has unbounded support.
In order to pull the 1/
�
ln(n) factor out and make it into the Ω(n) bound, we need to study the
distribution of the spectral measure of Hn/√n. As pointed out by [7], however, this seems to be
a hard problem. Nevertheless, we can empirically test the statement by numerical experiments. | [
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0.19243882596492767,
0.5830415487289429,
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] |
a hard problem. Nevertheless, we can empirically test the statement by numerical experiments.
Next, we provide the proof of Theorem 4, which is almost immediate from H¨older’s inequality.
Proof of Theorem 4. By H¨older’s inequality, we have
∥G − ˜G∥∞ = ∥G − ˜G∥∞ ≤ max
|z|=1
n−1
�
j=0
|hj − ˜hj||z|−j−1 ≤ ∥h − ˜h∥2
√n.
F
Some numerical experiments on Hankel matrices
F.1
Numerical ranks of random LTI systems and random Hankel matrices
in practice | [
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] |
F.1
Numerical ranks of random LTI systems and random Hankel matrices
in practice
While the theoretical part of our paper focuses on the ϵ-rank of an LTI system, in the main
text, we showed the distribution of all Hankel singular values of different LTI systems. The
main reason for showing the histograms instead of a single number (i.e., the ϵ-rank) is that the
histogram gives us more information while the ϵ-rank is merely a cutoff. In this section, we | [
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] |
histogram gives us more information while the ϵ-rank is merely a cutoff. In this section, we
empirically compute the ϵ-rank to verify the two theorems (i.e., Theorem 1 and Theorem 3).
In this experiment, we always set ϵ = 0.01. For every n in our experiment, we first randomly
initialize a random n × n Hankel matrix
Hn =
h0
h1
h2
· · ·
hn−1
h1
h2
· · ·
hn−1
0
h2
· · ·
hn−1
0
0
...
...
...
...
...
hn−1
0
· · ·
0
0
, | [
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] |
h2
· · ·
hn−1
h1
h2
· · ·
hn−1
0
h2
· · ·
hn−1
0
0
...
...
...
...
...
hn−1
0
· · ·
0
0
,
where hj are i.i.d. random Gaussian variables with variance of 1. We compute its ϵ-rank and
we repeat the experiment for 1000 trials. Similarly, for every n, we randomly initialize an LTI
system with
A = diag(a1, . . . , an),
aj ∼ Uniform(D),
where D is the open unit disk in the complex plane and the elements of B ◦ C⊤ are sampled
i.i.d.
from N(0, 1). | [
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] |
i.i.d.
from N(0, 1).
We compute its ϵ-rank and also repeat the experiment for 1000 trials.
From Figure 7, we see that a random LTI system has a low rank, whereas a random Hankel
matrix has a high rank in the sense that it is about proportional to n. This observation aligns
with our theory in Theorem 1 and Theorem 3,
F.2
Hankel matrices are stable to perturbation in practice
Theorem 4 predicts that the Hankel matrices are very stable when being perturbed. In this | [
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] |
Theorem 4 predicts that the Hankel matrices are very stable when being perturbed. In this
section, we run experiments in parallel with those in Figure 4, where we perturb a Hankel
matrix H. As in Figure 4, we also set the size of the random perturbation to be 1% and 0.1%,
respectively, of the original system. Theorem 4 is corroborated by Figure 8, where we see that
a small perturbation has a minimal effect on the Hankel singular values.
22
101
102
103
101
102
103
Random Hankel Matrix
Random LTI System | [
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] |
22
101
102
103
101
102
103
Random Hankel Matrix
Random LTI System
Ref. Line (slope = 1)
Ref. Line (slope = 1/2)
n
rankϵ(Γ)
Figure 7: The ϵ-rank of a random LTI system of size n, where ϵ = 0.01. The random systems
are parameterized by a Hankel matrix or by the matrices A, B, and C. The lines are the average
rank and the shaded regions indicate the 10%-90% range over 1000 trials.
10
20
30
40
50
60
10-6
10-4
10-2
100
A = 0
A
A = 1%
j
σj/σ1 | [
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] |
10
20
30
40
50
60
10-6
10-4
10-2
100
A = 0
A
A = 1%
j
σj/σ1
Figure 8: A random perturbation is added to H. The magnitude of the perturbation is set to
0.1% and 1% of the original matrix. We show the relative Hankel singular values σj/σ1 of the
original and perturbed systems. In this case, the three curves are almost overlapping.
G
Adjusting the discretization step via NUFFT
Assume we have a discrete LTI system whose transfer function is G. As explained in section 4, | [
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] |
Assume we have a discrete LTI system whose transfer function is G. As explained in section 4,
if we assume ∆t = 1, then the output of the system given an input sequence can be computed
as
y = iFFT(FFT(u) ◦ G(ω(L))),
G(ω(L)) =
�
G(ω(L)
0
)
· · ·
G(ω(L)
L−1)
�⊤
.
To understand why this relationship holds, assume there exists a continuous function u on the
unit circle ∂D where the discrete inputs u are sampled from. Then, FFT allows us to write u
into the Fourier expansion:
u(z) =
L−1
�
j=0
[FFT(u)]j exp
� | [
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] |
into the Fourier expansion:
u(z) =
L−1
�
j=0
[FFT(u)]j exp
�
−2πi j
Lz
�
.
23
By the property of the transfer function eq. (3), we know that the output function y is equal to
y(z) =
L−1
�
j=0
�
[FFT(u)]j G(ω(L)
j
)
�
�
��
�
ˆyj
exp
�
−2πi j
Lz
�
.
To compute the discrete output y, one samples y at z = ω(L)
0
, . . . , ω(L)
L−1, which is equivalent to
an inverse FFT on FFT(u) ◦ G(ω(L)).
Now, if we want to change ∆t, one way to think of it is as if our LTI system is unchanged, but | [
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] |
Now, if we want to change ∆t, one way to think of it is as if our LTI system is unchanged, but
the time domain of u(z) is scaled by a factor of ∆t. That is, we now have3
u(∆t)(z) =
L−1
�
j=0
[FFT(u)]j exp
�
−2πi j
Lz(∆t)
�
,
z(∆t) = 1 + s/∆t
1 − s/∆t
,
s = z − 1
z + 1.
The output function y(∆t) is now equal to
y(∆t)(z) =
L−1
�
j=0
�
[FFT(u)]j G(ω(L,∆t)
j
)
�
exp
�
−2πi j
Lz(∆t)
�
.
The only real difficulty is that when we sample y(∆t) at z = ω(L)
0
, . . . , ω(L)
L−1 to obtain y(∆t), | [
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0
, . . . , ω(L)
L−1 to obtain y(∆t),
we note that z(∆t) = ω(L,∆t)
0
, . . . , ω(L,∆t)
L−1
are not uniform on the unit circle. Hence, it cannot
be achieved via inverse FFT. However, this sampling can be done via the so-called nonuniform
FFT (NUFFT), which also takes O(L log L). The stability of the NUFFT procedure is studied
in [6, 33, 4], and fast algorithms can be found in [10, 6, 31]. This is how we obtained
y(∆t) = NUFFT(FFT(u) ◦ G(ω(L,∆t))).
Now, consider the following function | [
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y(∆t) = NUFFT(FFT(u) ◦ G(ω(L,∆t))).
Now, consider the following function
˜y(∆t)(z) = y(∆t)(z(1/∆t)) =
L−1
�
j=0
�
[FFT(u)]j G(ω(L,∆t)
j
)
�
exp
�
−2πi j
Lz
�
.
This output function is a scaled version of y(∆t), where we scale the time domain by a factor of
1/∆t. One can sample ˜y(∆t) at z = ω(L)
0
, . . . , ω(L)
L−1 using iFFT:
˜y(∆t) = iFFT(FFT(u) ◦ G(ω(L,∆t))).
This leads us to eq. (7).
H
Experimental details
In this section, we provide the details of the three experiments presented in the main text. | [
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] |
In this section, we provide the details of the three experiments presented in the main text.
3Note that we could alternatively scale the angular domain instead of the time domain, i.e., z(∆t) = z/∆t.
The difference is on the level of discretization. However, we find that discretizing the time domain gives us a
better performance in general.
24
Figure 9: A visualization of appendix G.
H.1
Analyzing Hankel singular values using the sCIFAR-10 task | [
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] |
H.1
Analyzing Hankel singular values using the sCIFAR-10 task
As mentioned in section 3, in the experiments presented in Figure 2 and Figure 5, we always
train an SSM with 4 layers and 128 channels. Each channel in a layer is modeled by an LTI
system with n = 64 states. When we parameterize the LTI systems using A, B, C, and D, we
assign a learning rate of 0.001 to A and of 0.01 to the rest. When we freeze the system matrices, | [
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assign a learning rate of 0.001 to A and of 0.01 to the rest. When we freeze the system matrices,
then we set the learning rate of A, B and C to 0 while keeping that of D to be 0.01. Note
that the matrix D does not affect the Hankel singular values. All other model parameters are
trained with a learning rate of 0.01. For an LTI system parameterized by the Hankel matrix
H, we adopt the same setting, except that H is trained with a non-reduced learning rate of | [
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] |
H, we adopt the same setting, except that H is trained with a non-reduced learning rate of
0.01. To compute the Hankel singular values of an LTI system (A, B, C, D), we use its balanced
realization (see Appendix B). To compute the Hankel singular values of a system parameterized
by H, we apply an SVD to the matrix H.
H.2
Testing HOPE-SSMs long memory using noisy-sCIFAR
In this experiment (see Figure 6), we modify the sequential CIFAR-10 dataset by padding | [
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] |
In this experiment (see Figure 6), we modify the sequential CIFAR-10 dataset by padding
random sequences to the right. For each sequence of length 1024 from the original dataset, we
pad another sequence of length 1024 to the end of it. The entries are sampled independently
from the Gaussian distribution on the same magnitude as the entries in the original sequences.
We adopt the same model architectures and learning rates as described in Appendix H.1 but | [
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] |
We adopt the same model architectures and learning rates as described in Appendix H.1 but
make two exceptions. First, we fix the discretization size to be ∆t = 0.1, and therefore, it does
not need a learning rate. In addition, a canonical SSM decodes the output sequence by first
doing a pooling. Here, instead of pooling over all 2048 output vectors, to make the problem
more challenging and require longer memory, we only pool over the last 1024 output vectors. | [
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] |
more challenging and require longer memory, we only pool over the last 1024 output vectors.
These correspond to the output vectors when the noises are fed. We also test the models using
different discretization sizes ∆t. We see that when ∆t = 1, the S4D model fails to converge
while our HOPE-SSM performs relatively well; on the other hand, when ∆t = 0.01, both models
tend to have a relatively good performance. These observations align with our theory because | [
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] |
tend to have a relatively good performance. These observations align with our theory because
a larger ∆t means that we put the discrete data on a continuous time domain with a longer
span; hence, longer memory capacity is needed.
25
0
20
40
60
80
100
0
20
40
60
80
100
HOPE-SSM
S4D
Accuracy
Epochs
∆t = 1
0
20
40
60
80
100
0
20
40
60
80
100
HOPE-SSM
S4D
Accuracy
Epochs
∆t = 0.1
0
20
40
60
80
100
0
20
40
60
80
100
HOPE-SSM
S4D
Accuracy
Epochs
∆t = 0.01 | [
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] |
Accuracy
Epochs
∆t = 0.1
0
20
40
60
80
100
0
20
40
60
80
100
HOPE-SSM
S4D
Accuracy
Epochs
∆t = 0.01
Figure 10: Performance of the HOPE-SSM and the S4D model on the noise-padded sCIFAR-
10 task using different values of ∆t.
H.3
Hyperparameters of HOPE-SSMs in the Long-Range Arena
In this section, we present the table of hyperparameters used to train our HOPE-SSM on the
LRA tasks [28] (Apache License, Version 2.0). (See Table 2.) Our codes are adapted from the | [
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] |
LRA tasks [28] (Apache License, Version 2.0). (See Table 2.) Our codes are adapted from the
code associated with the original S4 and S4D papers [14, 13] (Apache License, Version 2.0).
Note that compared to the hyperparameters used to train S4 and S4D, we use the same model
hyperparameters and only slightly tune the training hyperparameters. All experiments are done
on a NVIDIA A30 Tensor Core GPU with 24 GB of memory. The time efficiency of our model
is roughly the same as that of the S4D model.
Task
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is roughly the same as that of the S4D model.
Task
Depth
#Features
Norm
Prenorm
DO
LR
BS
Epochs
WD
∆ Range
ListOps
8
256
BN
False
0.
0.002
20
500
0.03
(0.0001,1)
Text
6
256
BN
True
0.
0.01
16
150
0.005
(0.001,0.1)
Retrieval
6
128
BN
True
0.
0.008
32
50
0.03
(0.001,0.1)
Image
6
128
LN
False
0.1
0.004
32
1500
0.01
(0.001,10)
Pathfinder
6
128
BN
True
0.
0.001
8
250
0.01
(0.0001,0.1)
Path-X
6
128
BN
True
0.
0.001
16
80
0.03
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] |
6
128
BN
True
0.
0.001
8
250
0.01
(0.0001,0.1)
Path-X
6
128
BN
True
0.
0.001
16
80
0.03
(0.001,1)
Table 2: Configurations of the HOPE-SSM model, where DO, LR, BS, and WD stand for
dropout rate, learning rate, batch size, and weight decay, respectively.
26 | [
0.14874310791492462,
0.11437102407217026,
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] |
fsemipar: an R package for SoF semiparametric
regression
Silvia Novo*
Germ´an Aneiros**
*Statistics Department, Universidad Carlos III de Madrid
**Mathematics Department, CITIC, Universidade da Coru˜na
Abstract
Functional data analysis has become a tool of interest in applied areas such as economics,
medicine, and chemistry. Among the techniques developed in recent literature, functional semipara-
metric regression stands out for its balance between flexible modelling and output interpretation. | [
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] |
metric regression stands out for its balance between flexible modelling and output interpretation.
Despite the large variety of research papers dealing with scalar-on-function (SoF) semiparametric
models, there is a notable gap in software tools for their implementation. This article introduces the
R package fsemipar, tailored for these models. fsemipar not only estimates functional single-index
models using kernel smoothing techniques but also estimates and selects relevant scalar variables | [
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] |
models using kernel smoothing techniques but also estimates and selects relevant scalar variables
in semi-functional models with multivariate linear components. A standout feature is its ability
to identify impact points of a curve on the response, even in models with multiple functional co-
variates, and to integrate both continuous and pointwise effects of functional predictors within a
single model. In addition, it allows the use of location-adaptive estimators based on the k-nearest- | [
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neighbours approach for all the semiparametric models included. Its flexible interface empowers
users to customise a wide range of input parameters and includes the standard S3 methods for
prediction, statistical analysis, and estimate visualization (predict, summary, print, and plot),
enhancing clear result interpretation. Throughout the article, we illustrate the functionalities and
the practicality of fsemipar using two chemometric datasets.
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] |
the practicality of fsemipar using two chemometric datasets.
Keywords:
R package; functional data analysis; semiparametric regression; functional single-index
model; semi-functional models; variable selection; impact point selection; k-nearest neighbours
1
Introduction
The technological advances of recent decades have enabled the recording of vast amounts of data,
leading to a revolution in data analysis.
Applied disciplines now require tools to handle complex | [
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leading to a revolution in data analysis.
Applied disciplines now require tools to handle complex
datasets, posing ongoing challenges for statisticians in both methodological and computational aspects.
For example, real-world applications often involve an extensive number of random variables. Data
may even be infinite-dimensional; that is, the observations of these variables are measured either
continuously or in a finely-spaced grid over a time interval, over a surface or even more intricate | [
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] |
continuously or in a finely-spaced grid over a time interval, over a surface or even more intricate
structures. For instance, in economics, intraday stock price can be considered as a curve (see, e.g.,
Rice et al. 2020); in chemometrics, the measurement of a spectrometer results in a curve (see, e.g.,
1
arXiv:2405.14048v1 [stat.ME] 22 May 2024
Dai and Genton 2018); in medicine, magnetic resonance imaging outputs are images (see, e.g. Li et al. | [
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2019). In such cases, data “atoms” are random functions, leading to the categorization of this data
type as functional data.
The term Functional Data Analysis (FDA) encompasses those statistical methods designed to
handle functional data.
Currently, functional regression has garnered significant attention within
the statistical community. In particular, scalar-on-function (SoF) regression, i.e. the case of scalar | [
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] |
response and functional covariates, has emerged as a prolific area of research (see Greven and Scheipl
2017 and Reiss et al. 2017 for recent reviews). Several SoF regression models have been formulated,
including parametric, nonparametric, and semiparametric approaches (if we follow the terminology of
conventional finite-dimensional regression). Parametric models, like the functional linear model (see, e.
g., chapter 12 of Ramsay and Silverman 2005 for a comprehensive description and Febrero-Bande et al. | [
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] |
2017 for a discussion on estimation techniques), provide a straightforward practical interpretation but
are limited in terms of flexibility. In contrast, nonparametric models (see, e. g., Ferraty and Vieu 2006
for a detailed development and Ling and Vieu 2018 for a review) provide a high level of flexibility, but
the outputs are challenging to interpret. As a middle point, semiparametric regression (see Ling and | [
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] |
Vieu 2021 for a recent survey) provides a balance between parametric and nonparametric approaches
allowing for flexible models while yielding interpretable results.
The practical application of functional linear and functional nonparametric regression has been
facilitated by computational tools designed for their implementation, typically developed within the R
programming environment (R Core Team 2022). Among these, the R packages fda (see Ramsay et al. | [
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] |
programming environment (R Core Team 2022). Among these, the R packages fda (see Ramsay et al.
2022) and fda.usc (see Febrero-Bande and Oviedo de la Fuente 2012) stand out as key references in
the FDA field. In the context of SoF regression, the former allows for the estimation of the functional
linear model using the techniques described in Ramsay and Silverman 2005.
The latter includes
several parametric and nonparametric functional regression models. It can estimate the functional | [
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several parametric and nonparametric functional regression models. It can estimate the functional
linear model using several approaches (including those described in Cardot et al. 1999, Ramsay and
Silverman 2005 and Preda and Saporta 2005), the functional nonparametric model (described in
Ferraty and Vieu 2006) and various generalised additive models for functional data (described in
Febrero-Bande and Gonz´alez-Manteiga 2013).
There are also two other major packages designed | [
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] |
Febrero-Bande and Gonz´alez-Manteiga 2013).
There are also two other major packages designed
specifically for regression problems involving functional variables: FDBoost (see Brockhaus et al. 2020)
and refund (see Goldsmith et al. 2023). In the SoF context, the package FDBoost fits additive models
using a component-wise gradient boosting algorithm. It allows for functional and scalar covariates, but
the effects of functional predictors currently implemented are functional linear and smooth interactions. | [
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] |
Regarding the refund package, it implements penalised functional regression, including generalised
additive models for functional data using spline-based methods (developed in McLean et al. 2014).
While there are other packages tailored to specific models and applications (see Scheipl et al. 2022
for an overview of R packages for functional data analysis), to the best of our knowledge there was no
R package dealing with the estimation of functional single-index models prior to the introduction of | [
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] |
fsemipar. It is worth noting that despite the practical advantages of functional semiparametric regres-
sion over parametric and nonparametric approaches, implementation is usually complex and requires
advanced knowledge of mathematical programming. To address this gap, we created the R package
fsemipar (see Aneiros and Novo 2024) aiming to simplify the use of SoF semiparametric models. It
can estimate functional single-index models (see Amato et al. 2006) and can also simultaneously esti-
2 | [
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2
mate and select relevant scalar variables in models composed of a multivariate linear component and
either a functional single-index part (see Novo et al. 2021a) or a functional nonparametric part (see
Aneiros et al. 2015). For that, it allows both kernel-based and k-nearest-neighbours-based estimators,
allowing the user to choose between a global smooth or a location-adaptive one. A remarkable feature | [
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] |
of fsemipar it is the selection of impact points of a curve on the scalar response variable (see Aneiros
and Vieu 2014). Essentially, this means the package can incorporate discretised observations from
a curve into the regression model. From these observations, it then identifies and selects the most
predictive points of the discretisation interval, termed “impact points” or “points of impact”. As a
consequence, it allows for the inclusion of both pointwise and continuous effects of functional variables | [
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in the response (see Aneiros and Vieu 2015 and Novo et al. 2021b), leading to interpretability, model
simplification and good predictive performance. An important feature of its functions is their ability
to accommodate a wide range of customisable input parameters. This flexibility enables expert users
to achieve a more precise fit. At the same time, the functions require only the data as a mandatory
argument, thereby simplifying usage for non-experts. | [
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] |
argument, thereby simplifying usage for non-experts.
In this tutorial, we present the R package fsemipar, available from the Comprehensive R Archive
Network (CRAN) at https://CRAN.R-project.org/package=fsemipar. The remainder of the paper
is structured as follows. Section 2 addresses SoF semiparametric regression with a single functional
covariate. Section 3 deals with SoF semiparametric models incorporating multiple scalar predictors | [
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covariate. Section 3 deals with SoF semiparametric models incorporating multiple scalar predictors
and a functional covariate. Section 4 discusses SoF semiparametric regression with multiple scalar
predictors derived from the discretization of a curve and a functional covariate. The three sections
share the same structure (I=2,3,4): Section I.1 introduces the modeling approaches implemented in
fsemipar, Section I.2 summarizes the techniques for estimating such models, and Section I.3 presents | [
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] |
the software infrastructure, introducing the main functions and classes of the package fsemipar tai-
lored for each model. To highlight the capabilities of the package, Section I.3 also contains practical
applications using two real datasets from the chemometric field. Finally, Section 5 includes a discussion
about the functions included and some directions for future development. Due to space constraints, | [
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] |
about the functions included and some directions for future development. Due to space constraints,
we have not delved into every facet of the package. For a detailed description of each function, readers
are directed to the package manual.
2
Single functional covariate
In Section 2, we focus on SoF semiparametric models with single functional predictor. Section 2.1
introduces the functional single-index model as a competitor of the linear model and the nonparametric | [
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model. In Section 2.2, we discuss the estimation techniques implemented in fsemipar. Finally, Section
2.3 presents the associated functions of the package and illustrates their use with a real chemometric
dataset.
2.1
The functional single-index model
Let Y be a scalar random variable, X (also denoted as X(t) with t ∈ I and I ⊂ R) a functional
covariate valued in some infinite-dimensional space H, and ε, a random error. We assume that we | [
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covariate valued in some infinite-dimensional space H, and ε, a random error. We assume that we
observe data pairs {(Yi, Xi), i = 1, . . . , n} independent and identically distributed (i.i.d.) to (Y, X).
3
We are interested in predict Y using X.
Literature in FDA focused on three different regression strategies to address this situation:
• The linear (parametric) approach. The functional linear model (FLM) is formulated under the | [
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] |
• The linear (parametric) approach. The functional linear model (FLM) is formulated under the
assumption of a linear relationship between the functional covariate and the response, repre-
sented as: Yi =
�
I θ0(t)Xi(t)dt + εi i = 1, . . . , n.
Here, H = L2(I) and θ0 is an unknown
parameter-function defined on I. One of the strengths of this model is its ability to provide an
interpretable output, θ0, and its relative ease of estimation compared to alternative models. The | [
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] |
interpretable output, θ0, and its relative ease of estimation compared to alternative models. The
main disadvantage is that it may not be reliable in some real applications due to its dependence
on the linearity assumption.
• The nonparametric approach. To address the inflexibility of the FLM, a natural step is to move
away from the strict linear hypothesis and instead adopt a basic smooth condition. This lead
us to the functional nonparametric model (FNM) given by the expression: Yi = m(Xi) + εi | [
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] |
us to the functional nonparametric model (FNM) given by the expression: Yi = m(Xi) + εi
i = 1, . . . , n. In this context, H is a semimetric space and m(·) an unknown, smooth, non-linear
functional operator. The FNM is more reliable than the FLM in practice, but m(·) is harder to
represent and interpret, and also more difficult to estimate.
• The semiparametric approach. To combine the strengths of both parametric and nonparamet- | [
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] |
• The semiparametric approach. To combine the strengths of both parametric and nonparamet-
ric approaches, an appealing technique is to assume that the functional predictor acts on the
response only through its projection onto some one-dimensional subspace. This leads to the
functional single-index model (FSIM), represented as:
Yi = r (⟨θ0, Xi⟩) + εi,
i = 1, . . . , n.
(1)
In this case, H is a separable Hilbert space and ⟨·, ·⟩ denotes its the inner product; θ0 ∈ H is | [
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] |
In this case, H is a separable Hilbert space and ⟨·, ·⟩ denotes its the inner product; θ0 ∈ H is
the functional index: a function-parameter that allows to summarize the information carried in
Xi to predict Yi, and r is the unknown, non-linear, smooth link function. The FSIM can be
viewed as an extension of the FLM, offering enhanced flexibility due to the smooth link function
r. Conversely, it can be considered a specific instance of the FNM, but with the advantage of | [
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] |
r. Conversely, it can be considered a specific instance of the FNM, but with the advantage of
dimensionality reduction, given that r operates on a scalar rather than a function like m(·). As
a consequence, the FSIM offers interpretable and representable outputs (θ0 and r).
We can implement the FLM using several R functions: fRegress (estimation based on basis rep-
resentation) of the fda R package, as well as the functions fregre.basis (estimation based on ba- | [
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] |
resentation) of the fda R package, as well as the functions fregre.basis (estimation based on ba-
sis representation), fregre.pc (functional principal component estimation) and fregre.pls (partial
least-squares estimation) of the fda.usc package. For the implementation of the FNM, we can use
the function fregre.np (nonparametric kernel estimation) of the fda.usc package. We can also refer
to the codes created by F. Ferraty, available on his website: https://www.math.univ-toulouse.fr/ | [
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] |
to the codes created by F. Ferraty, available on his website: https://www.math.univ-toulouse.fr/
~ferraty/SOFTWARES/NPFDA/index.html. However, to the best of our knowledge, there is no func-
tion available for the direct implementation of the FSIM. In the subsequent section, we will introduce
the techniques for estimating the FSIM that are included in the fsemipar package.
4
2.2
Link function and functional index estimation | [
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] |
4
2.2
Link function and functional index estimation
Since the FSIM can be seen as a particular case of the FNM, we can estimate the former by adapting the
estimation techniques used for the latter. In this section, we will briefly discuss how this is achieved.
It is important to note that the package can also estimate models with functional nonparametric
components. This capability is precisely why we have explored functional nonparametric estimation
in depth throughout this section. | [
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] |
in depth throughout this section.
Ferraty and Vieu (2006) proposed the functional extension of the classical Nadaraya-Watson kernel
estimator (see Nadaraya, 1964 and Watson, 1964) for the FNM. The following expression provides the
kernel estimator for m(·):
�mh(χ) =
�n
i=1 YiK
�
h−1d(Xi, χ)
�
�n
i=1 K (h−1d(Xi, χ)) , ∀χ ∈ H.
(2)
Here, h ∈ R+ represents the bandwidth, d is a general semi-metric, which allows the computation of | [
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] |
Here, h ∈ R+ represents the bandwidth, d is a general semi-metric, which allows the computation of
the proximity between functional data, and K is a real-valued asymmetrical kernel.
Alternatively, Burba et al. (2009) studied the functional version of the kNN estimator (see Collomb,
1979 and Devroye et al., 1994, among others) for m(·). For each element χ within H, it calculates
the regression solely based on the k sample observations that exhibit the closest proximity to said | [
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] |
the regression solely based on the k sample observations that exhibit the closest proximity to said
element. Specifically, the kNN estimator for m(·) is defined as follows
�m∗
k(χ) =
�n
i=1 YiK
�
H−1
k,χd(Xi, χ)
�
�n
i=1 K
�
H−1
k,χd(Xi, χ)
� , ∀χ ∈ H.
(3)
Here, k ∈ Z+ is the tuning parameter and Hk,χ = min
�
h ∈ R+ such that �n
i=1 1B(χ,h)(Xi) = k
�
,
where B(χ, h) = {z ∈ H : d (χ, z) ≤ h}, the derived local bandwidth which depends on k.
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] |
�
,
where B(χ, h) = {z ∈ H : d (χ, z) ≤ h}, the derived local bandwidth which depends on k.
Unlike
the kernel case, in the kNN estimator (3) the smoothing parameter Hk,χ depends on χ (and k),
which provides the desired location-adaptive property. Furthermore, this local smoothing estimator
exclusively relies on a discrete parameter k that takes values from a finite set {1, . . . , n}. This attribute
presents an additional advantage over the kernel estimator, where selecting the tuning parameter h | [
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] |
presents an additional advantage over the kernel estimator, where selecting the tuning parameter h
needs consideration of a continuous interval.
Regarding the tuning parameters, in FDA the typical approach for addressing the selection of h
and k is cross-validation (CV), specifically, leave-one-out CV (LOOCV), where the objective functions
are defined as
CV (h) = n−1
n
�
j=1
�
Yj − �m(−j)
h
(Xj)
�2
and CV ∗(k) = n−1
n
�
j=1
�
Yj − �m∗(−j)
k
(Xj)
�2
,
respectively. Here, �m(−j)
h
(·) and �m∗(−j)
k | [
0.426159143447876,
0.1695919781923294,
0.5812005996704102,
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0.24499988555908203,
0.08841549605131149,
-0.14190474152565002,
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-0.03708812966942787,
0.1524546891450882,
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1.2155250310897827,
0.08107324689626694,
0.051068879663944244,
-0.3584541976451874,
0.7007302641868591,
1.142549991607666,
0.17768219113349915,
0.3240647315979004,
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0.30125007033348083,
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0.2839130461215973,
0.4255034625530243,
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0.11031597852706909,
0.04388895630836487,
0.5972753763198853,
0.11280860751867294,
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0.12140689790248871,
0.5102609992027283,
0.5312755107879639,
0.08602697402238846,
0.06107967719435692,
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0.3825611174106598,
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0.1856151819229126,
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0.2422199249267578,
0.6717934012413025,
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0.6484129428863525,
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1.0712032318115234,
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0.28813186287879944,
0.38230279088020325,
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] |
and CV ∗(k) = n−1
n
�
j=1
�
Yj − �m∗(−j)
k
(Xj)
�2
,
respectively. Here, �m(−j)
h
(·) and �m∗(−j)
k
(·) are the leave-one-out versions of �mh(·) and �m∗
k(·), respec-
tively. Then, we select �h = arg minh∈[a,b] CV (h) and �k = arg mink∈{k1,...,k2} CV ∗ (k). The theoretical
validation of both the data-driven selectors was showed in Kara-Zaitri et al. (2017b,a). In practise, it is
usual to minimize over an interval [a, b] ⊂ R+ (respectively, over a set {k1, . . . , k2} ⊂ {1, . . . , n}) that | [
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] |
encompasses a range of reasonable values for h (k, respectively). The issue of automatically selecting
the interval [a, b] remains unresolved in one-dimensional nonparametric statistics. However, it is often
considered of lesser importance due to the tendency of the CV function to exhibit relative flatness
5
around its minimum, see Novo et al. (2019). In Section 2.3.1 we delve into the options included in the
fsemipar package. | [
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] |
fsemipar package.
In the functional context, we must pay attention to the selection of the semi-metric d: since H repre-
sents an infinite-dimensional space, the equivalence between norms fails (in contrast to what happens
in the finite-dimensional Euclidean space). Ferraty and Vieu (2006) provided some alternatives of
semi-metric, like the based on the functional principal component expansion, on the Fourier expan- | [
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] |
semi-metric, like the based on the functional principal component expansion, on the Fourier expan-
sion, and on the B-spline expansion of the derivatives. We can compute these semi-metrics through
the functions semimetric.pca, semimetric.fourier, and semimetric.deriv, respectively, included
in the fda.usc package.
An interesting special case arises when the semi-metric is contingent upon a parameter. This cir-
cumstance is observed in the FSIM (1), where we consider the operator rθ0(·) : H −→ R defined | [
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] |
cumstance is observed in the FSIM (1), where we consider the operator rθ0(·) : H −→ R defined
as rθ0(χ) = r (⟨θ0, χ⟩) , ∀χ ∈ H. In this special case, if we want to quantify the proximity between
elements of the functional space to obtain a nonparametric estimator for rθ0(·), the semi-metric adopts
the following form:
dθ0 (χ1, χ2) = |⟨θ0, χ1⟩ − ⟨θ0, χ2⟩| = |⟨θ0, χ1 − χ2⟩| ,
for χ1, χ2 ∈ H,
(4)
thus relying on the function-parameter θ0. When θ0 is unknown, the expressions (2) and (3) become | [
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] |
thus relying on the function-parameter θ0. When θ0 is unknown, the expressions (2) and (3) become
statistics (and not estimators) since they also depend on a parameter and can’t be directly employed
in practice. That is, for each θ ∈ H, they turn into
�rh,θ(χ) =
�n
i=1 YiK
�
h−1dθ(Xi, χ)
�
�n
i=1 K (h−1dθ(Xi, χ)) ,
�r∗
k,θ(χ) =
�n
i=1 YiK
�
H−1
k,χ,θdθ(Xi, χ)
�
�n
i=1 K
�
H−1
k,χ,θdθ(Xi, χ)
� ,
∀χ ∈ H.
(5)
Here, Hk,χ,θ = min
�
h ∈ R+ such that �n
i=1 1Bθ(χ,h)(Xi) = k
� | [
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] |
H−1
k,χ,θdθ(Xi, χ)
� ,
∀χ ∈ H.
(5)
Here, Hk,χ,θ = min
�
h ∈ R+ such that �n
i=1 1Bθ(χ,h)(Xi) = k
�
with Bθ(χ, h) = {z ∈ H : dθ (χ, z) ≤ h}
In this scenario, the semi-metric selection is direct; however, as a trade-off, we are required to esti-
mate θ0. To address the estimation of the regression in the FSIM (selection of tuning parameter and
estimation of θ0), we can use LOOCV. That is, considering the following objective functions
CV (h, θ) = n−1
n
�
j=1
�
Yj − �r(−j)
h,θ (Xj)
�2
and CV ∗(k, θ) = n−1
n
� | [
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] |
CV (h, θ) = n−1
n
�
j=1
�
Yj − �r(−j)
h,θ (Xj)
�2
and CV ∗(k, θ) = n−1
n
�
j=1
�
Yj − �r∗(−j)
k,θ
(Xj)
�2
,
(6)
respectively, and minimising them respect to both parameters. Consequently, the kernel-based and
kNN-based estimators of θ0 are
�θh = arg min
θ∈Θ CV (h, θ),
�θ∗
k = arg min
θ∈Θ CV ∗(k, θ),
(7)
where Θ ⊂ H, while the selectors of h and k are, respectively, �h = arg minh∈[a,b] CV
�
h, �θh
�
, and �k =
arg mink1≤k≤k2 CV ∗ �
k, �θk
� | [
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] |
�
h, �θh
�
, and �k =
arg mink1≤k≤k2 CV ∗ �
k, �θk
�
. By substituting these estimators into (5), we obtain �r�h,�θ�h(χ) and �r∗
�k,�θ∗
�k
(χ),
respectively. The theoretical validation of both the data-driven selectors for the tuning parameters (h
or k) and the estimators for θ0 was established by Novo et al. (2019).
The focal aspect in practical application is the implementation of the minimization problems described | [
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] |
in equation (7) to estimate θ0. Note that the minimisation is performed on a subset of the functional
space Θ ⊂ H, so the question of how to build this subset seems challenging. Ait-Sa¨ıdi et al. (2008)
6
or Ferraty et al. (2013) proposed to solve this issue using B-Spline approximation of the functional
directions and reduce the infinite-dimensional optimization problem to a multivariate one. Specifically,
each direction θ ∈ Θ is obtained from a d-dimensional space generated by B-spline basis functions, | [
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] |
each direction θ ∈ Θ is obtained from a d-dimensional space generated by B-spline basis functions,
denoted as {e1(·), . . . , ed(·)}. Consequently, our attention is focused on directions expressed as
θ(·) =
d
�
j=1
αjej(·) where (α1, . . . , αd) ∈ V.
(8)
and the problem in expression (7) turns into:
�θh = arg
min
(α1,...,αd)∈V CV (h, θ),
�θ∗
k = arg
min
(α1,...,αd)∈V CV ∗(k, θ).
(9)
To construct the set of coefficients V in (8), it is necessary to consider the model identifiability | [
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] |
conditions (see Ferraty et al., 2003). Such restrictions hold true when θ ∈ Θ satisfies the constraints
⟨θ, θ⟩ = 1 and θ(t0) > 0 for some arbitrary t0 in the domain of θ0 (see Ait-Sa¨ıdi et al., 2008).
Taking into account the outlined considerations, the fsemipar package includes two procedures to
carry out the minimisation of the CV functions defined in (6): (i) a direct method based on Ait-Sa¨ıdi
et al. (2008); (ii) an iterative algorithm based on Ferraty et al. (2013).
2.2.1 | [
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Subsets and Splits