INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 17 |NUMBER: 3 |2019 |SEPTEMBER
Adap i e Robus E icien Me hods o Pe iodic
Signal P ocessing Obse ed wi h Colou s Noises
E geny PCHELINTSEV 1,2, Se guei PERGAMENSHCHIKOV 3,4, Ma iana MARCOKOVA5,6
1Depa men o Ma hema ical Analysis and Theo y o Func ions, Facul y o Mechanics and Ma hema ics,
Tomsk S a e Uni e si y, Lenin A enue 36, 634050 Tomsk, Russia
2In e na ional Labo a o y o S a is ics o S ochas ic P ocesses and Quan i a i e Finance,
Tomsk S a e Uni e si y, Lenin A enue 36, 634050 Tomsk, Russia
3Labo a o y o Ma hema ics Raphael Salem, Uni e si y o Rouen No mandie,
A enue de l’Uni e si e 12, 76801 Sain E ienne du Rou ay, F ance
4In e na ional Labo a o y o S a is ics o S ochas ic P ocesses and Quan i a i e Finance,
Tomsk S a e Uni e si y, Lenin A enue 36, 634050 Tomsk, Russia
5Depa men o S uc u al Mechanics and Applied Ma hema ics, Facul y o Ci il Enginee ing,
Uni e si y o Zilina, Uni e zi na 8215/1, 010 26 Zilina, Slo ak Republic
6Depa men o Elec onics and Nanoelec onics,
Na ional Resea ch Uni e si y "Moscow Powe Enginee ing Ins i u e",
K asnokaza mennaya 14, 111250 Moscow, Russia
e gen-pch@yandex. u, se ge.pe gamench chik[email p o ec ed], ma iana.ma coko a@ s a .uniza.sk
DOI: 10.15598/aeee. 17i3.3132
Abs ac . In his pape , we conside he p oblem
o obus adap i e e icien es ima ing a pe iodic sig-
nal obse ed in he ansmission channel wi h he
dependen noise de ined by non-Gaussian O ns ein-
Uhlenbeck p ocesses wi h unknown co ela ion p ope -
ies. Adap i e model selec ion p ocedu es, based on
he sh inkage weigh ed leas squa es es ima es, a e p o-
posed. The compa ison be ween sh inkage and leas
squa es me hods is s udied and he ad an ages o he
sh inkage me hods a e analyzed. Es ima ion p ope ies
o p oposed s a is ical algo i hms a e s udied on he
basis o he obus mean squa e accu acy de ined as he
maximum mean squa e es ima ion e o o e all pos-
sible alues o unknown noise pa ame e s. Sha p o a-
cle inequali ies o he obus isks ha e been ob ained.
The obus e iciency o he model selec ion p ocedu e
has been es ablished.
Keywo ds
Asymp o ic e iciency, model selec ion, non-
pa ame ic eg ession, O ns ein-Uhlenbeck p o-
cess, pe iodic signals, obus quad a ic isk,
sha p o acle inequali y, sh inkage es ima ion,
weigh ed leas squa es es ima es.
1. In oduc ion
In his pape , we conside he es ima ion p oblem o
he 1-pe iodic signal S( )on he basis o obse a ions
(y )0≤ ≤ngi en by he s ochas ic di e en ial equa ion:
dy =S( )d +dξ ,0≤ ≤n, (1)
whe e nis he du a ion o obse a ion and (ξ )0≤ ≤n
is unobse ed colou noise. No e ha i (ξ )0≤ ≤nis
B ownian mo ion, hen we ob ain he well-known "sig-
nal + whi e noise" model which is e y popula in s a-
is ical adio-physics (see, o example, [1], [2] and [3]).
In his pape , we assume ha he use ul signal Sis
dis o ed by he impulse low desc ibed by he non-
Gaussian O ns ein-Uhlenbeck p ocesses, which allows
s udying he signal es ima ion p oblems wi h depen-
den pulse noises, i.e. we assume ha he noise p ocess
(ξ )0≤ ≤nobeys he equa ion:
dξ =aξ d +du ,(2)
u = ´n1w + ´n2z and z =x∗(µ−˜µ) ,
whe e a,´n1and ´n2a e some unknown cons an s,
(w ) ≥0is he s anda d B ownian mo ion, µ(ds dx)
is he jump measu e wi h de e minis ic compensa o
˜µ(ds dx) = dsΠ (dx),Π(·)is he Le y measu e, i.e.
some posi i e measu e on R∗=R {0}, such ha
Πx2= 1 and Πx6<∞. He e we use he no-
a ion Π (|x|m) = RR∗|y|mΠ (dy). No e ha he Le y
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measu e Π (R∗)could be equal o +∞. We use ∗ o he
s ochas ic in eg als wi h espec o andom measu es
(see [4], Chap e s 2 and 3), i.e.:
x∗(µ−˜µ) =
Z
0Z
R∗
y(µ−˜µ) (ds, dy).(3)
I should be no ed ha i a= 0, hen we ob ain he
Le y eg ession model conside ed in [5]. In he case
when Π (·) = 0 we ob ain he well-known Gaussian
O ns ein-Uhlenbeck eg ession model in oduced in [6]
and [7]. The model in he Eq. (1) and Eq. (2) in which
he jump p ocess (z ) ≥0is de ined by he compound
Poisson p ocess was s udied in [8] and [9]. Howe e ,
he compound Poisson p ocesses can desc ibe only he
la ge noise impulses o small ixed equency, bu he
elecommunica ion and loca ion sys ems may ha e he
impulse noises wi h any equency wi hou any condi-
ion. We no e ha in he pape s [8] and [9] he p o-
posed s a is ical p ocedu es a e based on he classical
weigh ed leas squa es es ima o s.
The main goal o his pape is o de elop a new
imp o ed adap i e obus e icien signal es ima ion
me hods o he non-Gaussian O ns ein-Uhlenbeck
noise (ξ )0≤ ≤nbased on he gene al Le y p ocesses
wi h unknown dis ibu ion Q. We assume ha his
dis ibu ion belongs o he class Q∗
nde ined as a am-
ily o all hese dis ibu ions o which he pa ame e s
−a∗≤a < 0,´n1≥ξ∗and ´n2
1+ ´n2
2≤ξ∗, whe e a∗,
ξ∗and ξ∗a e some ixed posi i e bounds. The quali y
o an es ima e ˆ
Sno he unknown signal S, i.e. some
unc ion o (y )0≤ ≤n, will be measu ed wi h he obus
quad a ic isk:
R∗ˆ
Sn, S= sup
Q∈Q∗
n
RQˆ
Sn, S,(4)
whe e
RQˆ
Sn, S:=EQ,S 


ˆ
Sn−S


2
and (5)
kSk2=
1
Z
0
S2( )d .
He e EQ,S is he expec a ion wi h espec o he dis i-
bu ion PQ,S o he p ocess in he Eq. (1) wi h a ixed
dis ibu ion Qo he noise (ξ )0≤ ≤nand a gi en unc-
ion S.
2. Sh inkage Es ima ion
Me hods
Le (φj)j≥1be a igonome ic basis in L2[0,1]. We
ex end hese unc ions by he pe iodic way on Ri.e.
φj( ) = φj( + 1) o any ∈R. Fo es ima ing he
unknown unc ion Sin he Eq. (1), we conside i s
Fou ie expansion:
S( ) = ∞
X
j=1
θjφj( )
and (6)
θj= (S, φj) =
1
Z
0
S( )φj( )d .
The Fou ie coe icien s θjcan be es ima ed as:
ˆ
θj,n =1
n
n
Z
0
φj( )dy .(7)
We de ine a class o weigh ed leas squa es es ima es
o S( )as:
ˆ
Sλ=
n
X
j=1
λ(j)ˆ
θj,nφj,(8)
whe e he weigh s λ∈Rnbelong o some ini e se Λ
om [0,1]n.
Now, o he i s d≤nFou ie coe icien s in
Eq. (6), we use he imp o ed es ima ion me hod p o-
posed o pa ame ic models in [10] and [11]. To his
end we se ˜
θn=ˆ
θj,n1≤j≤d. In he sequel, we will use
he no m |x|2
d=
d
P
j=1
x2
j o any ec o x= (xj)1≤j≤d
om Rn. Now we de ine he sh inkage es ima o s as:
θ∗
j,n = (1 −g(j)) ˆ
θj,n,(9)
whe e g(j) = cn
|˜
θn|d
1{1≤j≤d},1Ais he indica o o
he se Aand cnis some known pa ame e such ha
cn≈d
nas n→ ∞. Now we in oduce a class o sh ink-
age weigh ed leas squa es es ima es o Sas:
S∗
λ=
n
X
j=1
λ(j)θ∗
j,nφj.(10)
We deno e he di e ence o quad a ic isks o he es i-
ma es in Eq. (10) and Eq. (8) as ∆Q(S):=RQ(S∗
λ, S)−
RQˆ
Sλ, S. Now o his de ia ion, we ob ain he ol-
lowing esul .
Theo em 1 Assume ha o any ec o λ∈Λ he e
exis s some ixed in ege d=d(λ)such ha hei i s
dcomponen s equal o one. Then o any n≥1and
> 0:
sup
Q∈Qn
sup
kSk≤
∆Q(S)<−c2
n.(11)
The inequali y in Eq. (11) means ha non-
asymp o ically, i.e. o any n≥1 he es ima e in he
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Eq. (10) ou pe o ms in mean squa e accu acy he es-
ima e in he Eq. (8). Mo eo e , as we will see below,
ncn→ ∞ as n→ ∞. This means ha he imp o e-
men e ec in he nonpa ame ic case is mo e signi i-
can han o pa ame ic eg ession [11].
3. Model Selec ion P ocedu e
This Sec ion gi es he cons uc ion o a model selec ion
p ocedu e o es ima ing a unc ion Sin he Eq. (1) on
he basis o imp o ed weigh ed leas squa es es ima es
and s a es he sha p o acle inequali y o he obus
isk o he p oposed p ocedu e.
The model selec ion p ocedu e o he unknown unc-
ion Sin he Eq. (1) will be cons uc ed on he basis o
a amily o es ima es (S∗
λ)λ∈Λ. The pe o mance o any
es ima e S∗
λwill be measu ed by he empi ical squa ed
e o :
E n(λ) = kS∗
λ−Sk2.(12)
In o de o ob ain a good es ima e, we ha e o w i e
a ule o choose a weigh ec o λ∈Λin he Eq. (6).
I is ob ious ha he bes app oach is o minimize he
empi ical squa ed e o wi h espec o λ. Making use
he es ima e de ini ion in he Eq. (6) and he Fou ie
ans o ma ion o Simplies:
E n(λ) =
n
X
j=1
λ2(j)θ∗
j,n2−2
n
X
j=1
λ(j)θ∗
j,nθj+
n
X
j=1
θ2
j.
(13)
Since he Fou ie coe icien s (θj)j≥1a e unknown, he
weigh coe icien s (λj)j≥1canno be ound by mini-
mizing his quan i y. To ci cum en his di icul y one
needs o eplace he e ms θ∗
j,nθjby hei es ima o s
˜
θj,n. We se :
˜
θj,n =θ∗
j,n ˆ
θj,n −ˆσn
n,(14)
whe e ˆσnis he es ima e o he noise a iance o
σQ=EQξ2
j,n which we choose in he ollowing o m:
ˆσn=
n
X
j=[√n]+1
ˆ
2
j,n and ˆ
j,n =1
n
n
Z
0
φj( )dy .(15)
Fo his change in he empi ical squa ed e o , one has
o pay some penal y. Thus, one comes o he cos unc-
ion o he o m:
Jn(λ) =
n
X
j=1
λ2(j)θ∗
j,n2−2
n
X
j=1
λ(j)˜
θj,n +δˆ
Pn(λ),
(16)
whe e δis some posi i e cons an and ˆ
Pn(λ)is he
penal y e m de ined as:
ˆ
Pn(λ) = ˆσn|λ|2
n
n.(17)
Subs i u ing he weigh coe icien s, minimizing he
cos unc ion:
λ∗= a gmin
λ∈Λ
Jn(λ)(18)
in he Eq. (10) leads o he imp o ed model selec ion
p ocedu e:
S∗=S∗
λ∗.(19)
I will be no ed ha λ∗exis s because Λis a ini e se .
I he minimizing sequence in he Eq. (18) λ∗is no
unique, one can ake any minimize . In he case, when
he alue o σQis known, one can ake ˆσn=σQand
Pn(λ) = σQ|λ|2
nn−1.
Theo em 2 Fo any n≥2and 0< δ < 1
2, he obus
isks de ined in he Eq. (4) o es ima e in he Eq. (19)
o con inuously di e en iable unc ion Ssa is ies he
o acle inequali y:
R∗(S∗
λ∗, S)≤1+5δ
1−δmin
λ∈ΛR∗(S∗
λ, S) + B∗
n
nδ ,(20)
whe e he e m B∗
nis independen o Sand such ha
B∗
nn−→0as n→ ∞ o any  > 0.
The inequali y in Eq. (20) allows us o es ablish ha
he p ocedu e in he Eq. (19) is op imal in he o acle
inequali ies sense. This p ope y enables o p o ide
asymp o ic e iciency in he adap i e se ing, i.e. when
in o ma ion abou he signal egula i y is unknown.
4. Asymp o ic E iciency
In o de o s udy he asymp o ic e iciency, we de ine
he ollowing unc ional Sobole ball:
Wk, =( ∈Ck
p[0,1] :
k
X
i=0 

 (i)


2≤ ),(21)
whe e > 0and k≥1a e some unknown pa ame e s,
Ck
p[0,1] is he space o k imes di e en iable 1-pe iodic
unc ions such ha o any 0≤i≤k−1 : (i)(0) =
(i)(1). In o de o o mula e ou asymp o ic esul s
we se :
n=n
ξ∗, lk( ) = ((2k+ 1) )1
(2k+1) k
π(k+ 1)2k
(2k+1)
(22)
and we deno e by Σno all es ima es ˆ
Sno Smea-
su able wi h espec o he σ-algeb a gene a ed by he
p ocess in he Eq. (1).
Theo em 3 The obus isk de ined in he Eq. (4) ad-
mi s he ollowing asymp o ic lowe bound:
lim in
n→∞ in
ˆ
Sn∈Σn
2k/(2k+1)
nsup
S∈Wk,
R∗ˆ
Sn, S≥lk( ).
(23)
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This lowe bound is sha p in he ollowing sense.
Theo em 4 The obus isk de ined in he Eq. (4) o
he es ima ing p ocedu e in he Eq. (19) has he ollow-
ing asymp o ic uppe bound:
lim sup
n→∞
2k/(2k+1)
nsup
S∈Wk,
R∗(S∗, S)≤lk( ).(24)
Theo em 3 and Thm. 4 imply ha he model selec ion
p ocedu e S∗is e icien and he pa ame e lk( )de-
ined in he Eq. (22) is he Pinske cons an in his case
[3].
5. Mon e Ca lo Simula ions
In his sec ion, we epo he esul s o a Mon e Ca lo
expe imen o assess he pe o mance o he p oposed
model selec ion p ocedu e in he Eq. (19). In he
Eq. (1) we choose 1-pe iodic unc ion Swhich is de ined
as S( ) = sin(2π ) + 2(1 − ) cos(2π ), o 0≤ ≤1.
We simula e he Eq. (1) wi h he noise p ocess de ined
as:
dξ =−ξ d + 0.5dw + 0.5dz ,(25)
whe e z =
N
P
j=1
Yj,N is a Poisson p ocess wi h he
in ensi y λ= 1 and (Yj)j≥1is i.i.d. Gaussian (0,1).
We use he model selec ion p ocedu e de ined in he
Eq. (19) wi h he weigh s p oposed in [8]: k∗=
100 + √ln n,=1
ln nand m=1
2. We used he
cos unc ion wi h δ= (3 + ln n)−2. We de ine he em-
pi ical isk as ¯
R˜
S, S=1
p
p
P
j=1
ˆ
E˜
Sn( j)−S( j)2
and ˆ
E˜
Sn(·)−S(·)2=1
N
N
P
l=1 ˜
Sl
n(·)−S(·)2wi h
he equency o obse a ions p= 100001 and numbe s
o eplica ions N= 10000.
Table 1 gi es he alues o he sample isks o di -
e en numbe s o obse a ion pe iod n.
Tab. 1: Empi ical isks.
n¯
R˜
S, S¯
R(S∗, S)¯
R˜
S, S/¯
R(S∗, S)
100 0.0457 0.0289 1.6
200 0.0216 0.0089 2.4
500 0.0133 0.0021 6.3
1000 0.098 0.0011 8.9
6. Conclusion
In his pape , we conside ed he p oblem o nonpa a-
me ic signal p ocessing on he basis o he obse a-
ions wi h he dependen non-Gaussian impulse noises.
We de eloped adap i e e icien s a is ical model selec-
ion p ocedu es based on he sh inkage me hods and we
ha e shown ha he sh inkage es ima ion me hods con-
side ably imp o e he non-asymp o ic es ima ion accu-
acy. The ob ained heo e ical esul s a e con i med by
he nume ical simula ion. I u ns ou ha nume ically
he imp o emen e ec may inc ease 10 imes. Nex ,
o he de eloped s a is ical me hods we ob ained he
adap i e e iciency p ope y, which means ha we p o-
ide he bes mean squa es accu acy wi hou using he
smoo hness in o ma ion abou he o m o unknown
signal. Mo eo e , in his pape , we s udied he accu-
acy p ope ies o he p oposed me hods on he basis
o he obus app oach, i.e. uni o mly o e all possible
unknown noise dis ibu ions. This allows us o syn-
hesize he s a is ical algo i hms possessing he high
noise immuni y p ope ies. The esul s ( hei sa is ac-
o y conco dance wi h he co esponding expe imen al
da a) can be used o he es ima ion o he signals.
Such p oblems a e o a g ea impo ance in he ields
o adio-and-hyd oacous ic communica ions and posi-
ioning, adio-and-hyd oloca ion, e c. (see [12] and e -
e ences he ein).
Acknowledgmen
The esul s o his wo k a e suppo ed by he Min-
is y o Science and Highe Educa ion o he Russian
Fede a ion in he amewo k o he esea ch p ojec
no. 2.3208.2017/4.6. The second au ho is pa ially
suppo ed by he Russian Fede al P o esso P og am,
p ojec no. 1.472.2016/1.4 (Minis y o Science and
Highe Educa ion o he Russian Fede a ion) and by
he p ojec X e M-Fede , Uni e si y o Rouen. The
esul s o Sec. 4. and Sec. 5. a e suppo ed by he
RSF g an numbe 17-11-01049.
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s a e and obse a ion equa ions. Communica ions
- Scien i ic Le e s o he Uni e si y o Zilina.
2018, ol. 20, no. 1, pp. 66–71. ISSN 1335-4205.
Abou Au ho s
E geny PCHELINTSEV was bo n in Alma y,
Kazakhs an. He ecei ed his M.Sc. in Ma hema ics
om Tomsk S a e Uni e si y in 2009, Ph.D. in
Applied Ma hema ics and S a is ics om Tomsk
S a e Uni e si y and Rouen Uni e si y in 2012. His
esea ch in e es s include s a is ical modelling and
iden i ica ion o s ochas ic dynamic sys ems, so
compu ing, and simula ion.
Se guei PERGAMENSHCHIKOV was bo n
in Tomsk, Russia. He ecei ed his M.Sc. om Tomsk
S a e Uni e si y in 1980, Ph.D. in P obabili y Theo y
and S a is ics om Tomsk S a e Uni e si y in 1986,
Doc o o Sciences in Ma hema ics om Cen al
Economics and Ma hema ics Ins i u e o RAS in 1994.
His esea ch in e es s include s a is ical modelling
and iden i ica ion o s ochas ic dynamic sys ems,
s ochas ic di e en ial equa ions, op imiza ion, and
con ol.
Ma iana MARCOKOVA was bo n in T en-
cianska Tepla, Slo ak Republic. She ecei ed he
Mas e deg ee in eache aining o ma hema ics
and physics a Comenius Uni e si y in B a isla a,
Slo ak Republic in 1970. Since 1970 she wo ked as
a eache o ma hema ics a se e al ma hema ical
depa men s a Uni e si y o Zilina, Slo ak Republic
( o me he Uni e si y o T anspo in Zilina), in he
las 25 yea s as an associa e p o esso o ma hema ics
and in he las 5 yea s also as he associa e p o esso
a Moscow Powe Enginee ing Ins i u e, Russia. A e
doc o al s udies in ma hema ical analysis a Palacky
Uni e si y in Olomouc, Czech Republic, she ob ained
he deg ees: RND . and CSc. He esea ch in e es s
include special unc ions, o hogonal polynomials, and
applied ma hema ics.
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