Adaptive Robust Efficient Methods for Periodic Signal Processing Observed with Colours Noises

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 c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 270 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 17 |NUMBER: 3 |2019 |SEPTEMBER 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 c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 271 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 17 |NUMBER: 3 |2019 |SEPTEMBER 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) c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 272 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 17 |NUMBER: 3 |2019 |SEPTEMBER 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. Re e ences [1] IBRAGIMOV, I. A. and R. Z. KHASMINSKII. S a is ical Es ima ion: Asymp o ic Theo y. 1s ed. New Yo k: Sp inge , 1981. ISBN 978-1-4899-0027- 2. [2] KASSAM, S. A. Signal De ec ion in Non- Gaussian Noise. 1s ed. New Yo k: Sp inge , 1988. ISBN 978-1-4612-3834-8. [3] CHERNOYAROV, O. V., M. VACULIK, A. SHIRIKYAN and A. V. SALNIKOVA. S a is- ical Analysis o Fas Fluc ua ing Random Signals wi h A bi a y - Func ion En elope and Unknown Pa ame e s. Communica ions - Scien i ic Le e s o he Uni e si y o Zilina. 2015, ol. 17, no. 1a, pp. 35–43. ISSN 1335-4205. c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 273 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 17 |NUMBER: 3 |2019 |SEPTEMBER [4] CONT, R. and P. TANKOV. Financial Modelling wi h Jump P ocesses. 1s ed. Boca Ra on: Chap- man & Hall, 2003. ISBN 1-58488-413-4. [5] PCHELINTSEV, E., V. PCHELINTSEV and S. PERGAMENSHCHIKOV. Non asymp o ic sha p o acle inequali ies o he imp o ed model selec ion p ocedu es o he adap i e nonpa ame - ic signal es ima ion p oblem. Communica ions - Scien i ic Le e s o he Uni e si y o Zilina. 2018, ol. 20, no. 1, pp. 72–76. ISSN 1335-4205. [6] HOPFNER, R. and Y. A. KUTOYANTS. On LAN o pa ame ized con inuous pe iodic sig- nals in a ime-inhomogeneous di usion. S a is i- cal Decisions. 2009, ol. 27, iss. 4, pp. 309–326. ISSN 0721-2631. DOI: 10.1524/s nd.2009.1064. [7] HOPFNER, R. and Y. A. KUTOYANTS. Es i- ma ing discon inuous pe iodic signals in a ime- inhomogeneous di usion. S a is ical In e ence o S ochas ic P ocesses. 2010, ol. 13, iss. 3, pp. 193– 230. ISSN 1387-0874. DOI: 10.1007/s11203-010- 9046-7. [8] KONEV, V. V. and S. PERGAMENSHCHIKOV. E icien obus nonpa ame ic es ima ion in a semima ingale eg ession model. Annales de l’Ins i u Hen i Poinca e (B) P obabili y and S a is ics. 2012, ol. 48, no. 4, pp. 1217–1244. ISSN 0246-0203. DOI: 10.1214/12-AIHP488. [9] KONEV, V. 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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. c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 274