Zarrachalar filtri: Versiyalar orasidagi farq

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„Particle filter“ sahifasi tarjima qilib yaratildi
 
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Qator 2: Qator 2:
'''Zarrachalar filtrlari''' yoki '''ketma-ket Monte-Karlo''' usullari - signalni qayta ishlash va Bayes statistik xulosasi kabi chiziqli bo'lmagan holat-kosmik tizimlar uchun filtrlash muammolari uchun taxminiy yechimlarni topish uchun ishlatiladigan [[Montekarlo usuli|Monte-Karlo]] algoritmlari to'plami hisoblanadi. <ref name="Wills">{{Cite journal|last=Wills|first1=Adrian G.|last2=Schön|first2=Thomas B.|title=Sequential Monte Carlo: A Unified Review|journal=Annual Review of Control, Robotics, and Autonomous Systems|date=3 May 2023|volume=6|issue=1|pages=159–182|doi=10.1146/annurev-control-042920-015119|url=https://www.annualreviews.org/doi/full/10.1146/annurev-control-042920-015119|language=en|issn=2573-5144}}</ref> Filtrlash muammosi qisman kuzatishlar olib borilganda va sensorlarda, shuningdek dinamik tizimda tasodifiy buzilishlar mavjud bo'ladigan holatlar dinamik tizimlardagi ichki holatlarni baholashdan iborat. Maqsad shovqinli va qisman kuzatuvlarni hisobga olgan holda Markov jarayoni holatining posterior taqsimotini hisoblashdir. "Zarracha filtrlari" atamasi birinchi marta 1996 yilda Per Del Moral tomonidan kiritilgan bo'lib,1960-yillarning boshidan suyuqlik mexanikasida qo'llaniladigan o'rtacha maydon o'zaro ta'sir qiluvchi zarrachalar usullari haqida kiritilgan. <ref name="dm962">{{Cite journal|last=Del Moral|first1=Pierre|title=Non Linear Filtering: Interacting Particle Solution.|journal=Markov Processes and Related Fields|date=1996|volume=2|issue=4|pages=555–580|url=http://people.bordeaux.inria.fr/pierre.delmoral/delmoral96nonlinear.pdf}}</ref>Keyinchalik "Sequential Monte Carlo" atamasi 1998 yilda Jun S. Liu va Rong Chen tomonidan kiritilgan <ref>{{Cite journal|last=Liu|first1=Jun S.|last2=Chen|first2=Rong|date=1998-09-01|title=Sequential Monte Carlo Methods for Dynamic Systems|journal=Journal of the American Statistical Association|volume=93|issue=443|pages=1032–1044|doi=10.1080/01621459.1998.10473765|issn=0162-1459}}</ref>
'''Zarrachalar filtrlari''' yoki '''ketma-ket Monte-Karlo''' usullari - signalni qayta ishlash va Bayes statistik xulosasi kabi chiziqli bo'lmagan holat-kosmik tizimlar uchun filtrlash muammolari uchun taxminiy yechimlarni topish uchun ishlatiladigan [[Montekarlo usuli|Monte-Karlo]] algoritmlari to'plami hisoblanadi. <ref name="Wills">{{Cite journal|last=Wills|first1=Adrian G.|last2=Schön|first2=Thomas B.|title=Sequential Monte Carlo: A Unified Review|journal=Annual Review of Control, Robotics, and Autonomous Systems|date=3 May 2023|volume=6|issue=1|pages=159–182|doi=10.1146/annurev-control-042920-015119|url=https://www.annualreviews.org/doi/full/10.1146/annurev-control-042920-015119|language=en|issn=2573-5144}}</ref> Filtrlash muammosi qisman kuzatishlar olib borilganda va sensorlarda, shuningdek dinamik tizimda tasodifiy buzilishlar mavjud bo'ladigan holatlar dinamik tizimlardagi ichki holatlarni baholashdan iborat. Maqsad shovqinli va qisman kuzatuvlarni hisobga olgan holda Markov jarayoni holatining posterior taqsimotini hisoblashdir. "Zarracha filtrlari" atamasi birinchi marta 1996 yilda Per Del Moral tomonidan kiritilgan bo'lib,1960-yillarning boshidan suyuqlik mexanikasida qo'llaniladigan o'rtacha maydon o'zaro ta'sir qiluvchi zarrachalar usullari haqida kiritilgan. <ref name="dm962">{{Cite journal|last=Del Moral|first1=Pierre|title=Non Linear Filtering: Interacting Particle Solution.|journal=Markov Processes and Related Fields|date=1996|volume=2|issue=4|pages=555–580|url=http://people.bordeaux.inria.fr/pierre.delmoral/delmoral96nonlinear.pdf}}</ref>Keyinchalik "Sequential Monte Carlo" atamasi 1998 yilda Jun S. Liu va Rong Chen tomonidan kiritilgan <ref>{{Cite journal|last=Liu|first1=Jun S.|last2=Chen|first2=Rong|date=1998-09-01|title=Sequential Monte Carlo Methods for Dynamic Systems|journal=Journal of the American Statistical Association|volume=93|issue=443|pages=1032–1044|doi=10.1080/01621459.1998.10473765|issn=0162-1459}}</ref>


Zarrachalarni filtrlash shovqinli va/yoki qisman kuzatuvlarni hisobga olgan holda stokastik jarayonning posterior taqsimotini ifodalash uchun zarralar to'plamidan (namunalar deb ham ataladi) foydalanadi. Holat-kosmos modeli chiziqli bo'lmagan bo'lishi mumkin va boshlang'ich holat va shovqin taqsimoti talab qilinadigan har qanday shaklni olishi mumkin. Zarrachalarni filtrlash texnikasi davlat-kosmik modeli yoki holat taqsimoti haqida taxminlarni talab qilmasdan kerakli taqsimotdan namunalar yaratish uchun yaxshi o'rnatilgan metodologiyani ta'minlaydi <ref name="dm962">{{Cite journal|last=Del Moral|first1=Pierre|title=Non Linear Filtering: Interacting Particle Solution.|journal=Markov Processes and Related Fields|date=1996|volume=2|issue=4|pages=555–580|url=http://people.bordeaux.inria.fr/pierre.delmoral/delmoral96nonlinear.pdf}}<cite class="citation journal cs1" data-ve-ignore="true" id="CITEREFDel_Moral1996">Del Moral, Pierre (1996). [http://people.bordeaux.inria.fr/pierre.delmoral/delmoral96nonlinear.pdf "Non Linear Filtering: Interacting Particle Solution"] <span class="cs1-format">(PDF)</span>. ''Markov Processes and Related Fields''. '''2''' (4): 555–580.</cite></ref> <ref name=":22">{{Cite journal|last=Del Moral|first1=Pierre|title=Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems|journal=Annals of Applied Probability|date=1998|edition=Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996)|volume=8|issue=2|pages=438–495|url=http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535|doi=10.1214/aoap/1028903535}}</ref> <ref name=":1">{{Kitob manbasi|title=Feynman-Kac formulae. Genealogical and interacting particle approximations.|last=Del Moral|first=Pierre|publisher=Springer. Series: Probability and Applications|year=2004|isbn=978-0-387-20268-6|url=https://www.springer.com/gp/book/9780387202686|pages=556}}</ref> .
Zarrachalarni filtrlash shovqinli va/yoki qisman kuzatuvlarni hisobga olgan holda stokastik jarayonning posterior taqsimotini ifodalash uchun zarralar to'plamidan (namunalar deb ham ataladi) foydalanadi. Holat-kosmos modeli chiziqli bo'lmagan bo'lishi mumkin va boshlang'ich holat va shovqin taqsimoti talab qilinadigan har qanday shaklni olishi mumkin. Zarrachalarni filtrlash texnikasi davlat-kosmik modeli yoki holat taqsimoti haqida taxminlarni talab qilmasdan kerakli taqsimotdan namunalar yaratish uchun yaxshi o'rnatilgan metodologiyani ta'minlaydi <ref name="dm962">{{Cite journal|last=Del Moral|first1=Pierre|title=Non Linear Filtering: Interacting Particle Solution.|journal=Markov Processes and Related Fields|date=1996|volume=2|issue=4|pages=555–580|url=http://people.bordeaux.inria.fr/pierre.delmoral/delmoral96nonlinear.pdf}}</ref> <ref name=":22">{{Cite journal|last=Del Moral|first1=Pierre|title=Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems|journal=Annals of Applied Probability|date=1998|edition=Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996)|volume=8|issue=2|pages=438–495|url=http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535|doi=10.1214/aoap/1028903535}}</ref> <ref name=":1">{{Kitob manbasi|title=Feynman-Kac formulae. Genealogical and interacting particle approximations.|last=Del Moral|first=Pierre|publisher=Springer. Series: Probability and Applications|year=2004|isbn=978-0-387-20268-6|url=https://www.springer.com/gp/book/9780387202686|pages=556}}</ref> .


Zarrachalar filtrlari o'zlarining bashoratlarini taxminiy (statistik) tarzda yangilaydi. Tarqatishdan olingan namunalar zarrachalar to'plami bilan ifodalanadi; har bir zarrachaga tayinlangan ehtimollik og'irligi mavjud bo'lib, u zarrachaning [[Ehtimolik zichligi funksiyasi|ehtimollik zichligi funktsiyasidan]] namuna olish [[Ehtimollik|ehtimolini]] ifodalaydi. Og'irlikning pasayishiga olib keladigan vazn nomutanosibligi ushbu filtrlash algoritmlarida tez-tez uchraydigan muammodir. Biroq, og'irliklar notekis bo'lishidan oldin, qayta namuna olish bosqichini kiritish orqali uni yumshatish mumkin. Bir nechta moslashtirilgan qayta namuna olish mezonlaridan foydalanish mumkin, shu jumladan og'irliklarning o'zgarishi va bir xil taqsimotga tegishli nisbiy [[entropiya]] . <ref name=":0">{{Cite journal|last=Del Moral|first1=Pierre|last2=Doucet|first2=Arnaud|last3=Jasra|first3=Ajay|title=On Adaptive Resampling Procedures for Sequential Monte Carlo Methods|journal=Bernoulli|date=2012|volume=18|issue=1|pages=252–278|url=http://hal.inria.fr/docs/00/33/25/83/PDF/RR-6700.pdf|doi=10.3150/10-bej335}}</ref>
Zarrachalar filtrlari o'zlarining bashoratlarini taxminiy (statistik) tarzda yangilaydi. Tarqatishdan olingan namunalar zarrachalar to'plami bilan ifodalanadi; har bir zarrachaga tayinlangan ehtimollik og'irligi mavjud bo'lib, u zarrachaning [[Ehtimolik zichligi funksiyasi|ehtimollik zichligi funktsiyasidan]] namuna olish [[Ehtimollik|ehtimolini]] ifodalaydi. Og'irlikning pasayishiga olib keladigan vazn nomutanosibligi ushbu filtrlash algoritmlarida tez-tez uchraydigan muammodir. Biroq, og'irliklar notekis bo'lishidan oldin, qayta namuna olish bosqichini kiritish orqali uni yumshatish mumkin. Bir nechta moslashtirilgan qayta namuna olish mezonlaridan foydalanish mumkin, shu jumladan og'irliklarning o'zgarishi va bir xil taqsimotga tegishli nisbiy [[entropiya]] . <ref name=":0">{{Cite journal|last=Del Moral|first1=Pierre|last2=Doucet|first2=Arnaud|last3=Jasra|first3=Ajay|title=On Adaptive Resampling Procedures for Sequential Monte Carlo Methods|journal=Bernoulli|date=2012|volume=18|issue=1|pages=252–278|url=http://hal.inria.fr/docs/00/33/25/83/PDF/RR-6700.pdf|doi=10.3150/10-bej335}}</ref>
Qator 18: Qator 18:




Biologiya va [[Genetika|genetikada]] avstraliyalik genetik Aleks Freyzer 1957 yilda organizmlarning [[Chatishtirish|sun'iy tanlanishining]] genetik turini simulyatsiya qilish bo'yicha bir qator maqolalarni nashr etdi. <ref>{{Cite journal|last=Fraser|first1=Alex|author-link=Alex Fraser (scientist)|year=1957|title=Simulation of genetic systems by automatic digital computers. I. Introduction|journal=Aust. J. Biol. Sci.|volume=10|issue=4|pages=484–491|doi=10.1071/BI9570484}}</ref> Biologlar tomonidan evolyutsiyani kompyuter simulyatsiyasi 1960-yillarning boshlarida keng tarqalgan va usullar Freyzer va Burnel (1970) <ref>{{Kitob manbasi|last=Fraser|first=Alex|authorlink=Alex Fraser (scientist)|year=1970|title=Computer Models in Genetics|publisher=McGraw-Hill|location=New York|isbn=978-0-07-021904-5}}</ref> va Krosbi (1973) kitoblarida tasvirlangan. <ref>{{Kitob manbasi|last=Crosby|first=Jack L.|year=1973|title=Computer Simulation in Genetics|publisher=John Wiley & Sons|location=London|isbn=978-0-471-18880-3}}</ref> Freyzerning simulyatsiyalari zamonaviy mutatsiya-tanlash genetik zarrachalar algoritmlarining barcha muhim elementlarini o'z ichiga olgan.Matematik nuqtai nazardan, ba'zi bir qisman va shovqinli kuzatuvlar berilgan signalning tasodifiy holatlarining shartli taqsimlanishi ehtimollik potentsial funktsiyalari ketma-ketligi bilan og'irlikdagi signalning tasodifiy traektoriyalari bo'yicha Feynman-Kac ehtimolligi bilan tavsiflanadi. <ref name="dp042">{{Kitob manbasi|last=Del Moral|first=Pierre|title=Feynman-Kac formulae. Genealogical and interacting particle approximations|year=2004|publisher=Springer|quote=Series: Probability and Applications|url=https://www.springer.com/mathematics/probability/book/978-0-387-20268-6|pages=575|isbn=9780387202686|series=Probability and its Applications}}<cite class="citation book cs1" data-ve-ignore="true">Del Moral, Pierre (2004). [https://www.springer.com/mathematics/probability/book/978-0-387-20268-6 ''Feynman-Kac formulae. Genealogical and interacting particle approximations'']. Probability and its Applications. Springer. p.&nbsp;575. [[ISBN]]&nbsp;[[Special:BookSources/9780387202686|<bdi>9780387202686</bdi>]]. <q>Series: Probability and Applications</q></cite></ref> <ref name="dmm002">{{Kitob manbasi|last=Del Moral|first=Pierre|chapter=Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering|title=Séminaire de Probabilités XXXIV|editor=Jacques Azéma|series=Lecture Notes in Mathematics|date=2000|pages=1–145|url=http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf|doi=10.1007/bfb0103798|isbn=978-3-540-67314-9}}<cite class="citation book cs1" data-ve-ignore="true" id="CITEREFDel_MoralMiclo2000">Del Moral, Pierre; Miclo, Laurent (2000). "Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering". In Jacques Azéma; Michel Ledoux; Michel Émery; Marc Yor (eds.). [http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf ''Séminaire de Probabilités XXXIV''] <span class="cs1-format">(PDF)</span>. Lecture Notes in Mathematics. Vol.&nbsp;1729. pp.&nbsp;1–145. [[Raqamli obyekt identifikatori(DOI)|doi]]:[[doi:10.1007/bfb0103798|10.1007/bfb0103798]]. [[ISBN]]&nbsp;[[Maxsus:Kitob manbalari/978-3-540-67314-9|<bdi>978-3-540-67314-9</bdi>]].</cite></ref> Kvant Monte-Karlo, va aniqrog'i, Diffuziya Monte-Karlo usullari, shuningdek, Feynman-Kac yo'l integrallarining o'rtacha maydon genetik turi zarrachalar yaqinlashishi sifatida talqin qilinishi mumkin. <ref name="dp042" /> <ref name="dmm002" /> <ref name="dmm00m2">{{Cite journal|last=Del Moral|first1=Pierre|last2=Miclo|first2=Laurent|title=A Moran particle system approximation of Feynman-Kac formulae.|journal=Stochastic Processes and Their Applications|date=2000|volume=86|issue=2|pages=193–216|doi=10.1016/S0304-4149(99)00094-0}}<cite class="citation journal cs1" data-ve-ignore="true">Del Moral, Pierre; Miclo, Laurent (2000). [[doi:10.1016/S0304-4149(99)00094-0|"A Moran particle system approximation of Feynman-Kac formulae"]]. ''Stochastic Processes and Their Applications''. '''86''' (2): 193–216. [[Raqamli obyekt identifikatori(DOI)|doi]]:<span class="cs1-lock-free" title="Freely accessible">[[doi:10.1016/S0304-4149(99)00094-0|10.1016/S0304-4149(99)00094-0]]</span>.</cite></ref> <ref name="h84">{{Cite journal|last=Hetherington|first1=Jack, H.|title=Observations on the statistical iteration of matrices|journal=Phys. Rev. A|date=1984|volume=30|issue=2713|doi=10.1103/PhysRevA.30.2713|pages=2713–2719|bibcode=1984PhRvA..30.2713H}}<cite class="citation journal cs1" data-ve-ignore="true" id="CITEREFHetherington1984">Hetherington, Jack, H. (1984). "Observations on the statistical iteration of matrices". ''Phys. Rev. A''. '''30''' (2713): 2713–2719. [[Bibkod (identifikator)|Bibcode]]:[https://ui.adsabs.harvard.edu/abs/1984PhRvA..30.2713H 1984PhRvA..30.2713H]. [[Raqamli obyekt identifikatori(DOI)|doi]]:[[doi:10.1103/PhysRevA.30.2713|10.1103/PhysRevA.30.2713]].</cite></ref> <ref name="dm-esaim032">{{Cite journal|last=Del Moral|first1=Pierre|title=Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups|journal=ESAIM Probability & Statistics|date=2003|volume=7|pages=171–208|url=http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a|doi=10.1051/ps:2003001}}<cite class="citation journal cs1" data-ve-ignore="true" id="CITEREFDel_Moral2003">Del Moral, Pierre (2003). [http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a "Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups"]. ''ESAIM Probability & Statistics''. '''7''': 171–208. [[Raqamli obyekt identifikatori(DOI)|doi]]:<span class="cs1-lock-free" title="Freely accessible">[[doi:10.1051/ps:2003001|10.1051/ps:2003001]]</span>.</cite></ref> <ref name="caffarel1">{{Cite journal|last=Assaraf|first1=Roland|last2=Caffarel|first2=Michel|last3=Khelif|first3=Anatole|title=Diffusion Monte Carlo Methods with a fixed number of walkers|journal=Phys. Rev. E|url=http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|date=2000|volume=61|issue=4|pages=4566–4575|doi=10.1103/physreve.61.4566|pmid=11088257|bibcode=2000PhRvE..61.4566A|archiveurl=https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|archivedate=2014-11-07}}</ref> <ref name="caffarel2">{{Cite journal|last=Caffarel|first1=Michel|last2=Ceperley|first2=David|last3=Kalos|first3=Malvin|title=Comment on Feynman-Kac Path-Integral Calculation of the Ground-State Energies of Atoms|journal=Phys. Rev. Lett.|date=1993|volume=71|issue=13|doi=10.1103/physrevlett.71.2159|bibcode=1993PhRvL..71.2159C|pages=2159|pmid=10054598}}</ref> Kvant Monte-Karlo usullarining kelib chiqishi ko'pincha Enriko Fermi va Robert Richtmyer bilan bog'liq bo'lib, ular 1948 yilda neytron zanjiri reaktsiyalarining o'rtacha maydon zarrachalari talqinini <ref>{{Cite journal|last=Fermi|first1=Enrique|last2=Richtmyer|first2=Robert, D.|title=Note on census-taking in Monte Carlo calculations|journal=LAM|date=1948|volume=805|issue=A|url=http://scienze-como.uninsubria.it/bressanini/montecarlo-history/fermi-1948.pdf|quote=Declassified report Los Alamos Archive}}</ref> ishlab chiqdilar, lekin birinchi evristik va genetik turdagi zarrachalar algoritmi (aka) Kvant tizimlarining asosiy holat energiyalarini (kamaytirilgan matritsali modellarda) baholash uchun qayta namunalangan yoki qayta konfiguratsiya qilingan Monte-Karlo usullari) 1984 yilda Jek X. Xeterington tomonidan amalga oshirilgan <ref name="h84" /> Shuningdek, Teodor E. Xarris va Herman Kanning zarrachalar fizikasi bo'yicha 1951 yilda nashr etilgan, zarracha uzatish energiyasini baholash uchun o'rtacha maydon, lekin evristik genetik usullardan foydalangan holda oldingi muhim ishlaridan iqtibos keltirish mumkin. <ref>{{Cite journal|last=Herman|first1=Kahn|last2=Harris|first2=Theodore, E.|title=Estimation of particle transmission by random sampling|journal=Natl. Bur. Stand. Appl. Math. Ser.|date=1951|volume=12|pages=27–30|url=https://dornsifecms.usc.edu/assets/sites/520/docs/kahnharris.pdf}}</ref> Molekulyar kimyoda genetik evristik o'xshash zarrachalar metodologiyalaridan foydalanish (aka kesish va boyitish strategiyalari) 1955 yilda Marshallning asosiy ishi bilan kuzatilishi mumkin. N. Rosenbluth va Arianna. V. Rozenblut. <ref name=":5">{{Cite journal|last=Rosenbluth|first1=Marshall, N.|last2=Rosenbluth|first2=Arianna, W.|title=Monte-Carlo calculations of the average extension of macromolecular chains|journal=J. Chem. Phys.|date=1955|volume=23|issue=2|pages=356–359|doi=10.1063/1.1741967|bibcode=1955JChPh..23..356R|url=https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7}}<cite class="citation journal cs1" data-ve-ignore="true" id="CITEREFRosenbluthRosenbluth1955">Rosenbluth, Marshall, N.; Rosenbluth, Arianna, W. (1955). [https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7 "Monte-Carlo calculations of the average extension of macromolecular chains"]. ''J. Chem. Phys''. '''23''' (2): 356–359. [[Bibkod (identifikator)|Bibcode]]:[https://ui.adsabs.harvard.edu/abs/1955JChPh..23..356R 1955JChPh..23..356R]. [[Raqamli obyekt identifikatori(DOI)|doi]]:[[doi:10.1063/1.1741967|10.1063/1.1741967]]. [[S2CID (identifikator)|S2CID]]&nbsp;[https://api.semanticscholar.org/CorpusID:89611599 89611599].</cite></ref> Ilg'or signallarni qayta ishlash va Bayes xulosasida genetik zarrachalar algoritmlaridan foydalanish yaqinroqdir. 1993 yil yanvar oyida Genshiro Kitagava "Monte-Karlo filtrini" ishlab chiqdi, <ref name="Kitagawa1993">{{Cite journal|last=Kitagawa|first1=G.|date=January 1993|title=A Monte Carlo Filtering and Smoothing Method for Non-Gaussian Nonlinear State Space Models|journal=Proceedings of the 2nd U.S.-Japan Joint Seminar on Statistical Time Series Analysis|pages=110–131|url=https://www.ism.ac.jp/~kitagawa/1993_US-Japan.pdf}}</ref> ushbu maqolaning biroz o'zgartirilgan versiyasi 1996 yilda paydo bo'ldi <ref>{{Cite journal|last=Kitagawa|first1=G.|year=1996|title=Monte carlo filter and smoother for non-Gaussian nonlinear state space models|volume=5|issue=1|journal=Journal of Computational and Graphical Statistics|pages=1–25|doi=10.2307/1390750|jstor=1390750}}
Biologiya va [[Genetika|genetikada]] avstraliyalik genetik Aleks Freyzer 1957 yilda organizmlarning [[Chatishtirish|sun'iy tanlanishining]] genetik turini simulyatsiya qilish bo'yicha bir qator maqolalarni nashr etdi. <ref>{{Cite journal|last=Fraser|first1=Alex|author-link=Alex Fraser (scientist)|year=1957|title=Simulation of genetic systems by automatic digital computers. I. Introduction|journal=Aust. J. Biol. Sci.|volume=10|issue=4|pages=484–491|doi=10.1071/BI9570484}}</ref> Biologlar tomonidan evolyutsiyani kompyuter simulyatsiyasi 1960-yillarning boshlarida keng tarqalgan va usullar Freyzer va Burnel (1970) <ref>{{Kitob manbasi|last=Fraser|first=Alex|authorlink=Alex Fraser (scientist)|year=1970|title=Computer Models in Genetics|publisher=McGraw-Hill|location=New York|isbn=978-0-07-021904-5}}</ref> va Krosbi (1973) kitoblarida tasvirlangan. <ref>{{Kitob manbasi|last=Crosby|first=Jack L.|year=1973|title=Computer Simulation in Genetics|publisher=John Wiley & Sons|location=London|isbn=978-0-471-18880-3}}</ref> Freyzerning simulyatsiyalari zamonaviy mutatsiya-tanlash genetik zarrachalar algoritmlarining barcha muhim elementlarini o'z ichiga olgan.Matematik nuqtai nazardan, ba'zi bir qisman va shovqinli kuzatuvlar berilgan signalning tasodifiy holatlarining shartli taqsimlanishi ehtimollik potentsial funktsiyalari ketma-ketligi bilan og'irlikdagi signalning tasodifiy traektoriyalari bo'yicha Feynman-Kac ehtimolligi bilan tavsiflanadi. <ref name="dp042">{{Kitob manbasi|last=Del Moral|first=Pierre|title=Feynman-Kac formulae. Genealogical and interacting particle approximations|year=2004|publisher=Springer|quote=Series: Probability and Applications|url=https://www.springer.com/mathematics/probability/book/978-0-387-20268-6|pages=575|isbn=9780387202686|series=Probability and its Applications}}</ref> <ref name="dmm002">{{Kitob manbasi|last=Del Moral|first=Pierre|chapter=Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering|title=Séminaire de Probabilités XXXIV|editor=Jacques Azéma|series=Lecture Notes in Mathematics|date=2000|pages=1–145|url=http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf|doi=10.1007/bfb0103798|isbn=978-3-540-67314-9}}</ref> Kvant Monte-Karlo, va aniqrog'i, Diffuziya Monte-Karlo usullari, shuningdek, Feynman-Kac yo'l integrallarining o'rtacha maydon genetik turi zarrachalar yaqinlashishi sifatida talqin qilinishi mumkin. <ref name="dp042" /> <ref name="dmm002" /> <ref name="dmm00m2">{{Cite journal|last=Del Moral|first1=Pierre|last2=Miclo|first2=Laurent|title=A Moran particle system approximation of Feynman-Kac formulae.|journal=Stochastic Processes and Their Applications|date=2000|volume=86|issue=2|pages=193–216|doi=10.1016/S0304-4149(99)00094-0}}</ref> <ref name="h84">{{Cite journal|last=Hetherington|first1=Jack, H.|title=Observations on the statistical iteration of matrices|journal=Phys. Rev. A|date=1984|volume=30|issue=2713|doi=10.1103/PhysRevA.30.2713|pages=2713–2719|bibcode=1984PhRvA..30.2713H}}</ref> <ref name="dm-esaim032">{{Cite journal|last=Del Moral|first1=Pierre|title=Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups|journal=ESAIM Probability & Statistics|date=2003|volume=7|pages=171–208|url=http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a|doi=10.1051/ps:2003001}}</ref> <ref name="caffarel1">{{Cite journal|last=Assaraf|first1=Roland|last2=Caffarel|first2=Michel|last3=Khelif|first3=Anatole|title=Diffusion Monte Carlo Methods with a fixed number of walkers|journal=Phys. Rev. E|url=http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|date=2000|volume=61|issue=4|pages=4566–4575|doi=10.1103/physreve.61.4566|pmid=11088257|bibcode=2000PhRvE..61.4566A|archiveurl=https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|archivedate=2014-11-07}}</ref> <ref name="caffarel2">{{Cite journal|last=Caffarel|first1=Michel|last2=Ceperley|first2=David|last3=Kalos|first3=Malvin|title=Comment on Feynman-Kac Path-Integral Calculation of the Ground-State Energies of Atoms|journal=Phys. Rev. Lett.|date=1993|volume=71|issue=13|doi=10.1103/physrevlett.71.2159|bibcode=1993PhRvL..71.2159C|pages=2159|pmid=10054598}}</ref> Kvant Monte-Karlo usullarining kelib chiqishi ko'pincha Enriko Fermi va Robert Richtmyer bilan bog'liq bo'lib, ular 1948 yilda neytron zanjiri reaktsiyalarining o'rtacha maydon zarrachalari talqinini <ref>{{Cite journal|last=Fermi|first1=Enrique|last2=Richtmyer|first2=Robert, D.|title=Note on census-taking in Monte Carlo calculations|journal=LAM|date=1948|volume=805|issue=A|url=http://scienze-como.uninsubria.it/bressanini/montecarlo-history/fermi-1948.pdf|quote=Declassified report Los Alamos Archive}}</ref> ishlab chiqdilar, lekin birinchi evristik va genetik turdagi zarrachalar algoritmi (aka) Kvant tizimlarining asosiy holat energiyalarini (kamaytirilgan matritsali modellarda) baholash uchun qayta namunalangan yoki qayta konfiguratsiya qilingan Monte-Karlo usullari) 1984 yilda Jek X. Xeterington tomonidan amalga oshirilgan <ref name="h84" /> Shuningdek, Teodor E. Xarris va Herman Kanning zarrachalar fizikasi bo'yicha 1951 yilda nashr etilgan, zarracha uzatish energiyasini baholash uchun o'rtacha maydon, lekin evristik genetik usullardan foydalangan holda oldingi muhim ishlaridan iqtibos keltirish mumkin. <ref>{{Cite journal|last=Herman|first1=Kahn|last2=Harris|first2=Theodore, E.|title=Estimation of particle transmission by random sampling|journal=Natl. Bur. Stand. Appl. Math. Ser.|date=1951|volume=12|pages=27–30|url=https://dornsifecms.usc.edu/assets/sites/520/docs/kahnharris.pdf}}</ref> Molekulyar kimyoda genetik evristik o'xshash zarrachalar metodologiyalaridan foydalanish (aka kesish va boyitish strategiyalari) 1955 yilda Marshallning asosiy ishi bilan kuzatilishi mumkin. N. Rosenbluth va Arianna. V. Rozenblut. <ref name=":5">{{Cite journal|last=Rosenbluth|first1=Marshall, N.|last2=Rosenbluth|first2=Arianna, W.|title=Monte-Carlo calculations of the average extension of macromolecular chains|journal=J. Chem. Phys.|date=1955|volume=23|issue=2|pages=356–359|doi=10.1063/1.1741967|bibcode=1955JChPh..23..356R|url=https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7}}</ref> Ilg'or signallarni qayta ishlash va Bayes xulosasida genetik zarrachalar algoritmlaridan foydalanish yaqinroqdir. 1993 yil yanvar oyida Genshiro Kitagava "Monte-Karlo filtrini" ishlab chiqdi, <ref name="Kitagawa1993">{{Cite journal|last=Kitagawa|first1=G.|date=January 1993|title=A Monte Carlo Filtering and Smoothing Method for Non-Gaussian Nonlinear State Space Models|journal=Proceedings of the 2nd U.S.-Japan Joint Seminar on Statistical Time Series Analysis|pages=110–131|url=https://www.ism.ac.jp/~kitagawa/1993_US-Japan.pdf}}</ref> ushbu maqolaning biroz o'zgartirilgan versiyasi 1996 yilda paydo bo'ldi <ref>{{Cite journal|last=Kitagawa|first1=G.|year=1996|title=Monte carlo filter and smoother for non-Gaussian nonlinear state space models|volume=5|issue=1|journal=Journal of Computational and Graphical Statistics|pages=1–25|doi=10.2307/1390750|jstor=1390750}}
</ref> 1993 yil aprel oyida Gordon va boshqalar o'zlarining asosiy ishlarida <ref name="Gordon1993">{{Cite journal|title=Novel approach to nonlinear/non-Gaussian Bayesian state estimation|journal=IEE Proceedings F - Radar and Signal Processing|date=April 1993|issn=0956-375X|pages=107–113|volume=140|issue=2|first1=N.J.|last=Gordon|first2=D.J.|last2=Salmond|first3=A.F.M.|last3=Smith|doi=10.1049/ip-f-2.1993.0015}}</ref> Bayes statistik xulosasida genetik turdagi algoritmni qo'llashni nashr etdilar. Mualliflar o'zlarining algoritmlarini "bootstrap filter" deb nomladilar va boshqa filtrlash usullari bilan solishtirganda ularning yuklash algoritmi ushbu holat maydoni yoki tizim shovqini haqida hech qanday taxminni talab qilmasligini ko'rsatdi. Mustaqil ravishda, Per Del Moral <ref name="dm962" /> va Ximilkon Karvalyo, Per Del Moral, Andre Monin va Jerar Salut <ref>{{Cite journal|last=Carvalho|first1=Himilcon|last2=Del Moral|first2=Pierre|last3=Monin|first3=André|last4=Salut|first4=Gérard|title=Optimal Non-linear Filtering in GPS/INS Integration.|journal=IEEE Transactions on Aerospace and Electronic Systems|date=July 1997|volume=33|issue=3|pages=835|url=http://homepages.laas.fr/monin/Version_anglaise/Publications_files/GPS.pdf|bibcode=1997ITAES..33..835C|doi=10.1109/7.599254}}</ref> tomonidan 1990-yillarning oʻrtalarida nashr etilgan zarracha filtrlari boʻyicha. Zarrachalar filtrlari 1989-1992 yillar boshida signalni qayta ishlashda P. Del Moral, JC Noyer, G. Rigal va G. Salut tomonidan LAAS-CNRSda STCAN (Service Technique des) bilan cheklangan va tasniflangan bir qator tadqiqot hisobotlarida ishlab chiqilgan. Constructions et Armes Navales), IT kompaniyasi DIGILOG va [https://www.laas.fr/public/en LAAS-CNRS] (Tizimlar tahlili va arxitekturasi laboratoriyasi) RADAR/SONAR va GPS signallarini qayta ishlash muammolari. <ref>P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : An unified framework for particle solutions <br />
</ref> 1993 yil aprel oyida Gordon va boshqalar o'zlarining asosiy ishlarida <ref name="Gordon1993">{{Cite journal|title=Novel approach to nonlinear/non-Gaussian Bayesian state estimation|journal=IEE Proceedings F - Radar and Signal Processing|date=April 1993|issn=0956-375X|pages=107–113|volume=140|issue=2|first1=N.J.|last=Gordon|first2=D.J.|last2=Salmond|first3=A.F.M.|last3=Smith|doi=10.1049/ip-f-2.1993.0015}}</ref> Bayes statistik xulosasida genetik turdagi algoritmni qo'llashni nashr etdilar. Mualliflar o'zlarining algoritmlarini "bootstrap filter" deb nomladilar va boshqa filtrlash usullari bilan solishtirganda ularning yuklash algoritmi ushbu holat maydoni yoki tizim shovqini haqida hech qanday taxminni talab qilmasligini ko'rsatdi. Mustaqil ravishda, Per Del Moral <ref name="dm962">{{Cite journal|last=Del Moral|first1=Pierre|title=Non Linear Filtering: Interacting Particle Solution.|journal=Markov Processes and Related Fields|date=1996|volume=2|issue=4|pages=555–580|url=http://people.bordeaux.inria.fr/pierre.delmoral/delmoral96nonlinear.pdf}}</ref> va Ximilkon Karvalyo, Per Del Moral, Andre Monin va Jerar Salut <ref>{{Cite journal|last=Carvalho|first1=Himilcon|last2=Del Moral|first2=Pierre|last3=Monin|first3=André|last4=Salut|first4=Gérard|title=Optimal Non-linear Filtering in GPS/INS Integration.|journal=IEEE Transactions on Aerospace and Electronic Systems|date=July 1997|volume=33|issue=3|pages=835|url=http://homepages.laas.fr/monin/Version_anglaise/Publications_files/GPS.pdf|bibcode=1997ITAES..33..835C|doi=10.1109/7.599254}}</ref> tomonidan 1990-yillarning oʻrtalarida nashr etilgan zarracha filtrlari boʻyicha. Zarrachalar filtrlari 1989-1992 yillar boshida signalni qayta ishlashda P. Del Moral, JC Noyer, G. Rigal va G. Salut tomonidan LAAS-CNRSda STCAN (Service Technique des) bilan cheklangan va tasniflangan bir qator tadqiqot hisobotlarida ishlab chiqilgan. Constructions et Armes Navales), IT kompaniyasi DIGILOG va [https://www.laas.fr/public/en LAAS-CNRS] (Tizimlar tahlili va arxitekturasi laboratoriyasi) RADAR/SONAR va GPS signallarini qayta ishlash muammolari. <ref>P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : An unified framework for particle solutions <br />
LAAS-CNRS, Toulouse, Research Report no. 91137, DRET-DIGILOG- LAAS/CNRS contract, April (1991).</ref> <ref>P. Del Moral, G. Rigal, and G. Salut. Nonlinear and non-Gaussian particle filters applied to inertial platform repositioning.<br />
LAAS-CNRS, Toulouse, Research Report no. 91137, DRET-DIGILOG- LAAS/CNRS contract, April (1991).</ref> <ref>P. Del Moral, G. Rigal, and G. Salut. Nonlinear and non-Gaussian particle filters applied to inertial platform repositioning.<br />
LAAS-CNRS, Toulouse, Research Report no. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN no. A.91.77.013, (94p.) September (1991).</ref> <ref>P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. Experimental results.<br />
LAAS-CNRS, Toulouse, Research Report no. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN no. A.91.77.013, (94p.) September (1991).</ref> <ref>P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. Experimental results.<br />
Qator 26: Qator 26:
LAAS-CNRS, Toulouse, Research report no. 92495, December (1992).</ref> <ref>P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. <br />
LAAS-CNRS, Toulouse, Research report no. 92495, December (1992).</ref> <ref>P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. <br />
Studies on: Filtering, optimal control, and maximum likelihood estimation. Convention DRET no. 89.34.553.00.470.75.01. Research report no.4 (210p.), January (1993).</ref>
Studies on: Filtering, optimal control, and maximum likelihood estimation. Convention DRET no. 89.34.553.00.470.75.01. Research report no.4 (210p.), January (1993).</ref>

=== Matematik asoslar ===
1950 yildan 1996 yilgacha zarrachalar filtrlari va genetik algoritmlarga oid barcha nashrlar, shu jumladan hisoblash fizikasi va molekulyar kimyoda joriy qilingan Monte-Karlo usullarini kesish va qayta namunalash, ularning izchilligining yagona isbotisiz turli vaziyatlarga qo'llaniladigan tabiiy va evristik algoritmlarni taqdim etadi., shuningdek, hisob-kitoblarning noto'g'riligi va genealogik va ajdodlar daraxtiga asoslangan algoritmlar haqida munozara ham yo'q. Ushbu zarracha algoritmlarining matematik asoslari va birinchi qat'iy tahlili 1996 yilda Per Del Moral <ref name="dm962" /> <ref name=":22" /> tomonidan amalga oshirilgan. Maqolada <ref name="dm962" />, shuningdek, zarrachaning ehtimollik funktsiyalarining yaqinlashuvi va normallashtirilmagan shartli ehtimollik o'lchovlarining xolis xususiyatlarining isboti mavjud. Ushbu maqolada keltirilgan ehtimollik funktsiyalarining xolis zarracha baholovchisi bugungi kunda Bayes statistik xulosasida qo'llaniladi.


Den Crisan, Jessica Gaines va Terri Lyons <ref name=":42">{{Cite journal|last=Crisan|first1=Dan|last2=Gaines|first2=Jessica|last3=Lyons|first3=Terry|date=1998|title=Convergence of a branching particle method to the solution of the Zakai|url=https://semanticscholar.org/paper/99e8759a243cd0568b0f32cbace2ad0525b16bb6|journal=SIAM Journal on Applied Mathematics|volume=58|issue=5|pages=1568–1590|doi=10.1137/s0036139996307371}}</ref> <ref>{{Cite journal|last=Crisan|first1=Dan|last2=Lyons|first2=Terry|date=1997|title=Nonlinear filtering and measure-valued processes|journal=Probability Theory and Related Fields|volume=109|issue=2|pages=217–244|doi=10.1007/s004400050131}}</ref> <ref>{{Cite journal|last=Crisan|first1=Dan|last2=Lyons|first2=Terry|date=1999|title=A particle approximation of the solution of the Kushner–Stratonovitch equation|journal=Probability Theory and Related Fields|volume=115|issue=4|pages=549–578|doi=10.1007/s004400050249}}</ref>, shuningdek, Den Crisan, Per Del Moral va Terri Lyons <ref name=":52">{{Cite journal|last=Crisan|first1=Dan|last2=Del Moral|first2=Pierre|last3=Lyons|first3=Terry|date=1999|title=Discrete filtering using branching and interacting particle systems|url=http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf|journal=Markov Processes and Related Fields|volume=5|issue=3|pages=293–318}}</ref> 2001-yilning oxirlarida har xil populyatsiya kattaligiga ega boʻlgan shoxlangan zarrachalar texnikasini yaratdilar. 1990-yillar. P. Del Moral, A. Guionnet va L. Miklo <ref name="dmm002">{{Kitob manbasi|last=Del Moral|first=Pierre|chapter=Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering|title=Séminaire de Probabilités XXXIV|editor=Jacques Azéma|series=Lecture Notes in Mathematics|date=2000|pages=1–145|url=http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf|doi=10.1007/bfb0103798|isbn=978-3-540-67314-9}}<cite class="citation book cs1" data-ve-ignore="true" id="CITEREFDel_MoralMiclo2000">Del Moral, Pierre; Miclo, Laurent (2000). "Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering". In Jacques Azéma; Michel Ledoux; Michel Émery; Marc Yor (eds.). [http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf ''Séminaire de Probabilités XXXIV''] <span class="cs1-format">(PDF)</span>. Lecture Notes in Mathematics. Vol.&nbsp;1729. pp.&nbsp;1–145. [[Raqamli obyekt identifikatori(DOI)|doi]]:[[doi:10.1007/bfb0103798|10.1007/bfb0103798]]. [[ISBN]]&nbsp;[[Maxsus:Kitob manbalari/978-3-540-67314-9|<bdi>978-3-540-67314-9</bdi>]].</cite></ref> <ref name="dg99" /> <ref name="dg01" /> 2000 yilda bu borada koʻproq yutuqlarga erishdilar. Birinchi markaziy chegara teoremalarini Per Del Moral va Elis Guionnet <ref name=":2">{{Cite journal|last=Del Moral|first1=P.|last2=Guionnet|first2=A.|date=1999|title=Central limit theorem for nonlinear filtering and interacting particle systems|journal=The Annals of Applied Probability|volume=9|issue=2|pages=275–297|doi=10.1214/aoap/1029962742|issn=1050-5164}}</ref> 1999 yilda, Per Del Moral va Loran Miklo <ref name="dmm002" /> 2000 yilda isbotladilar. Zarrachalar filtrlari uchun vaqt parametri bo'yicha birinchi yagona yaqinlashuv natijalari 1990-yillarning oxirida Per Del Moral va Elis Guionnet tomonidan ishlab chiqilgan. <ref name="dg99">{{Cite journal|last=Del Moral|first1=Pierre|last2=Guionnet|first2=Alice|title=On the stability of Measure Valued Processes with Applications to filtering|journal=C. R. Acad. Sci. Paris|date=1999|volume=39|issue=1|pages=429–434}}</ref> <ref name="dg01">{{Cite journal|last=Del Moral|first1=Pierre|last2=Guionnet|first2=Alice|date=2001|title=On the stability of interacting processes with applications to filtering and genetic algorithms|journal=Annales de l'Institut Henri Poincaré|volume=37|issue=2|pages=155–194|bibcode=2001AIHPB..37..155D|doi=10.1016/s0246-0203(00)01064-5|url=http://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps|archiveurl=https://web.archive.org/web/20141107004539/https://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps|archivedate=2014-11-07}}</ref> Genealogik daraxtga asoslangan zarracha filtri silliqlash vositalarining birinchi qat'iy tahlili 2001 yilda P. Del Moral va L. Miklo tomonidan amalga oshirilgan

30-May 2023, 17:29 dagi koʻrinishi

Zarrachalar filtrlari yoki ketma-ket Monte-Karlo usullari - signalni qayta ishlash va Bayes statistik xulosasi kabi chiziqli bo'lmagan holat-kosmik tizimlar uchun filtrlash muammolari uchun taxminiy yechimlarni topish uchun ishlatiladigan Monte-Karlo algoritmlari to'plami hisoblanadi. [1] Filtrlash muammosi qisman kuzatishlar olib borilganda va sensorlarda, shuningdek dinamik tizimda tasodifiy buzilishlar mavjud bo'ladigan holatlar dinamik tizimlardagi ichki holatlarni baholashdan iborat. Maqsad shovqinli va qisman kuzatuvlarni hisobga olgan holda Markov jarayoni holatining posterior taqsimotini hisoblashdir. "Zarracha filtrlari" atamasi birinchi marta 1996 yilda Per Del Moral tomonidan kiritilgan bo'lib,1960-yillarning boshidan suyuqlik mexanikasida qo'llaniladigan o'rtacha maydon o'zaro ta'sir qiluvchi zarrachalar usullari haqida kiritilgan. [2]Keyinchalik "Sequential Monte Carlo" atamasi 1998 yilda Jun S. Liu va Rong Chen tomonidan kiritilgan [3]

Zarrachalarni filtrlash shovqinli va/yoki qisman kuzatuvlarni hisobga olgan holda stokastik jarayonning posterior taqsimotini ifodalash uchun zarralar to'plamidan (namunalar deb ham ataladi) foydalanadi. Holat-kosmos modeli chiziqli bo'lmagan bo'lishi mumkin va boshlang'ich holat va shovqin taqsimoti talab qilinadigan har qanday shaklni olishi mumkin. Zarrachalarni filtrlash texnikasi davlat-kosmik modeli yoki holat taqsimoti haqida taxminlarni talab qilmasdan kerakli taqsimotdan namunalar yaratish uchun yaxshi o'rnatilgan metodologiyani ta'minlaydi [2] [4] [5] .

Zarrachalar filtrlari o'zlarining bashoratlarini taxminiy (statistik) tarzda yangilaydi. Tarqatishdan olingan namunalar zarrachalar to'plami bilan ifodalanadi; har bir zarrachaga tayinlangan ehtimollik og'irligi mavjud bo'lib, u zarrachaning ehtimollik zichligi funktsiyasidan namuna olish ehtimolini ifodalaydi. Og'irlikning pasayishiga olib keladigan vazn nomutanosibligi ushbu filtrlash algoritmlarida tez-tez uchraydigan muammodir. Biroq, og'irliklar notekis bo'lishidan oldin, qayta namuna olish bosqichini kiritish orqali uni yumshatish mumkin. Bir nechta moslashtirilgan qayta namuna olish mezonlaridan foydalanish mumkin, shu jumladan og'irliklarning o'zgarishi va bir xil taqsimotga tegishli nisbiy entropiya . [6]

Statistik va ehtimollik nuqtai nazaridan, zarracha filtrlari Feynman-Kac ehtimollik o'lchovlarining o'rtacha maydon zarracha talqini sifatida talqin qilinishi ham mumkin. [7] [8] [9] [10] [11] Ushbu zarrachalarni integratsiyalash usullari molekulyar kimyo va elementar zarrachalar hisoblash fizikasida 1951 yilda ikki fizik Teodor E. Xarris va Herman Kan, 1955 yilda Marshall N. Rozenblyut va Arianna V. Rozenblyut, [12] va yaqinda 1984 yilda Jek X. Xeterington tomonidan ishlab chiqilgan [13] Hisoblash fizikasida ushbu Feynman-Kac tipidagi zarrachalarni birlashtirish usullari Kvant Monte-Karloda, aniqrog'i Diffuziya Monte-Karlo usullarida ham qo'llaniladi. [14] [15] [16] Feynman-Kac o'zaro ta'sir qiluvchi zarrachalar usullari, shuningdek, murakkab optimallashtirish muammolarini hal qilish uchun evolyutsion hisoblashda hozirda qo'llaniladigan mutatsiya-tanlash genetik algoritmlari bilan kuchli bog'liq bo'ladi.

Zarrachalarni filtrlash metodologiyasi Yashirin Markov Modeli (HMM) va chiziqli bo'lmagan filtrlash muammolarini hal qilish uchun ishlatiladi. Chiziqli-Gauss signal-kuzatish modellari ( Kalman filtri ) yoki modellarning kengroq sinflari (Benes filtri [17] )ni isbotlashdan tashqari, Mirey Chaleyat-Maurel va Dominik Mishel 1984 yilda tasodifiy holatlarning posterior taqsimoti ketma-ketligini isbotladilar. Berilgan kuzatishlarda ya'ni optimal filtr berilgan signalda chekli rekursiya yo'q. [18] Ruxsat etilgan tarmoqni taxmin qilish, Markov zanjiri Monte-Karlo texnikasi, an'anaviy chiziqlilashtirish, kengaytirilgan Kalman filtrlari yoki eng yaxshi chiziqli tizimni aniqlash (kutilayotgan xarajat-xato ma'nosida) keng ko'lamli tizimlar, beqaror jarayonlar bilan kurashishga qodir emas., yoki etarli darajada silliq bo'lmagan chiziqli.

Zarrachalar filtrlari va Feynman-Kac zarrachalar metodologiyasi signal va tasvirni qayta ishlash, Bayes xulosasi, mashinani o'rganish, xavf tahlili va noyob hodisalardan namuna olish, muhandislik va robototexnika, sun'iy intellekt, bioinformatika, [19] filogenetika, hisoblash fanlari, iqtisod va matematika moliyasida ham keng qo'llaniladi., bundan tashqari molekulyar kimyo, hisoblash fizikasi, farmakokinetika va boshqa sohalarda ham.

Tarix

Evristik algoritmlar

Statistik va ehtimollik nuqtai nazaridan, zarracha filtrlari tarmoqlanuvchi / genetik turdagi algoritmlar va o'rtacha maydon tipidagi o'zaro ta'sir qiluvchi zarrachalar metodologiyalari sinfiga kiradi. Ushbu zarracha usullarini talqin qilish ilmiy intizomga bog'liqdir. Evolyutsion hisoblashda o'rtacha maydon genetik tipidagi zarrachalar metodologiyalari ko'pincha evristik va tabiiy qidiruv algoritmlari sifatida ishlatiladi Metaevristik ). Hisoblash fizikasi va molekulyar kimyoda evristik algaritmlar Feynman-Kac yo'lini integratsiyalash muammolarini hal qilish yoki Boltsmann-Gibbs o'lchovlari, eng yuqori xos qiymatlari va Shredinger operatorlarining asosiy holatlarini hisoblash uchun ishlatiladi. Biologiya va genetikada ular ba'zi bir muhitda individlar yoki genlar populyatsiyasining evolyutsiyasini ham ifodalashi mumkin.


Biologiya va genetikada avstraliyalik genetik Aleks Freyzer 1957 yilda organizmlarning sun'iy tanlanishining genetik turini simulyatsiya qilish bo'yicha bir qator maqolalarni nashr etdi. [20] Biologlar tomonidan evolyutsiyani kompyuter simulyatsiyasi 1960-yillarning boshlarida keng tarqalgan va usullar Freyzer va Burnel (1970) [21] va Krosbi (1973) kitoblarida tasvirlangan. [22] Freyzerning simulyatsiyalari zamonaviy mutatsiya-tanlash genetik zarrachalar algoritmlarining barcha muhim elementlarini o'z ichiga olgan.Matematik nuqtai nazardan, ba'zi bir qisman va shovqinli kuzatuvlar berilgan signalning tasodifiy holatlarining shartli taqsimlanishi ehtimollik potentsial funktsiyalari ketma-ketligi bilan og'irlikdagi signalning tasodifiy traektoriyalari bo'yicha Feynman-Kac ehtimolligi bilan tavsiflanadi. [7] [8] Kvant Monte-Karlo, va aniqrog'i, Diffuziya Monte-Karlo usullari, shuningdek, Feynman-Kac yo'l integrallarining o'rtacha maydon genetik turi zarrachalar yaqinlashishi sifatida talqin qilinishi mumkin. [7] [8] [9] [13] [14] [23] [24] Kvant Monte-Karlo usullarining kelib chiqishi ko'pincha Enriko Fermi va Robert Richtmyer bilan bog'liq bo'lib, ular 1948 yilda neytron zanjiri reaktsiyalarining o'rtacha maydon zarrachalari talqinini [25] ishlab chiqdilar, lekin birinchi evristik va genetik turdagi zarrachalar algoritmi (aka) Kvant tizimlarining asosiy holat energiyalarini (kamaytirilgan matritsali modellarda) baholash uchun qayta namunalangan yoki qayta konfiguratsiya qilingan Monte-Karlo usullari) 1984 yilda Jek X. Xeterington tomonidan amalga oshirilgan [13] Shuningdek, Teodor E. Xarris va Herman Kanning zarrachalar fizikasi bo'yicha 1951 yilda nashr etilgan, zarracha uzatish energiyasini baholash uchun o'rtacha maydon, lekin evristik genetik usullardan foydalangan holda oldingi muhim ishlaridan iqtibos keltirish mumkin. [26] Molekulyar kimyoda genetik evristik o'xshash zarrachalar metodologiyalaridan foydalanish (aka kesish va boyitish strategiyalari) 1955 yilda Marshallning asosiy ishi bilan kuzatilishi mumkin. N. Rosenbluth va Arianna. V. Rozenblut. [12] Ilg'or signallarni qayta ishlash va Bayes xulosasida genetik zarrachalar algoritmlaridan foydalanish yaqinroqdir. 1993 yil yanvar oyida Genshiro Kitagava "Monte-Karlo filtrini" ishlab chiqdi, [27] ushbu maqolaning biroz o'zgartirilgan versiyasi 1996 yilda paydo bo'ldi [28] 1993 yil aprel oyida Gordon va boshqalar o'zlarining asosiy ishlarida [29] Bayes statistik xulosasida genetik turdagi algoritmni qo'llashni nashr etdilar. Mualliflar o'zlarining algoritmlarini "bootstrap filter" deb nomladilar va boshqa filtrlash usullari bilan solishtirganda ularning yuklash algoritmi ushbu holat maydoni yoki tizim shovqini haqida hech qanday taxminni talab qilmasligini ko'rsatdi. Mustaqil ravishda, Per Del Moral [2] va Ximilkon Karvalyo, Per Del Moral, Andre Monin va Jerar Salut [30] tomonidan 1990-yillarning oʻrtalarida nashr etilgan zarracha filtrlari boʻyicha. Zarrachalar filtrlari 1989-1992 yillar boshida signalni qayta ishlashda P. Del Moral, JC Noyer, G. Rigal va G. Salut tomonidan LAAS-CNRSda STCAN (Service Technique des) bilan cheklangan va tasniflangan bir qator tadqiqot hisobotlarida ishlab chiqilgan. Constructions et Armes Navales), IT kompaniyasi DIGILOG va LAAS-CNRS (Tizimlar tahlili va arxitekturasi laboratoriyasi) RADAR/SONAR va GPS signallarini qayta ishlash muammolari. [31] [32] [33] [34] [35] [36]

Matematik asoslar

1950 yildan 1996 yilgacha zarrachalar filtrlari va genetik algoritmlarga oid barcha nashrlar, shu jumladan hisoblash fizikasi va molekulyar kimyoda joriy qilingan Monte-Karlo usullarini kesish va qayta namunalash, ularning izchilligining yagona isbotisiz turli vaziyatlarga qo'llaniladigan tabiiy va evristik algoritmlarni taqdim etadi., shuningdek, hisob-kitoblarning noto'g'riligi va genealogik va ajdodlar daraxtiga asoslangan algoritmlar haqida munozara ham yo'q. Ushbu zarracha algoritmlarining matematik asoslari va birinchi qat'iy tahlili 1996 yilda Per Del Moral [2] [4] tomonidan amalga oshirilgan. Maqolada [2], shuningdek, zarrachaning ehtimollik funktsiyalarining yaqinlashuvi va normallashtirilmagan shartli ehtimollik o'lchovlarining xolis xususiyatlarining isboti mavjud. Ushbu maqolada keltirilgan ehtimollik funktsiyalarining xolis zarracha baholovchisi bugungi kunda Bayes statistik xulosasida qo'llaniladi.


Den Crisan, Jessica Gaines va Terri Lyons [37] [38] [39], shuningdek, Den Crisan, Per Del Moral va Terri Lyons [40] 2001-yilning oxirlarida har xil populyatsiya kattaligiga ega boʻlgan shoxlangan zarrachalar texnikasini yaratdilar. 1990-yillar. P. Del Moral, A. Guionnet va L. Miklo [8] [41] [42] 2000 yilda bu borada koʻproq yutuqlarga erishdilar. Birinchi markaziy chegara teoremalarini Per Del Moral va Elis Guionnet [43] 1999 yilda, Per Del Moral va Loran Miklo [8] 2000 yilda isbotladilar. Zarrachalar filtrlari uchun vaqt parametri bo'yicha birinchi yagona yaqinlashuv natijalari 1990-yillarning oxirida Per Del Moral va Elis Guionnet tomonidan ishlab chiqilgan. [41] [42] Genealogik daraxtga asoslangan zarracha filtri silliqlash vositalarining birinchi qat'iy tahlili 2001 yilda P. Del Moral va L. Miklo tomonidan amalga oshirilgan

  1. ↑ Wills, Adrian G.; Schön, Thomas B. (3 May 2023). "Sequential Monte Carlo: A Unified Review" (en). Annual Review of Control, Robotics, and Autonomous Systems 6 (1): 159–182. doi:10.1146/annurev-control-042920-015119. ISSN 2573-5144. https://www.annualreviews.org/doi/full/10.1146/annurev-control-042920-015119. 
  2. ↑ 2,0 2,1 2,2 2,3 2,4 Del Moral, Pierre (1996). "Non Linear Filtering: Interacting Particle Solution.". Markov Processes and Related Fields 2 (4): 555–580. http://people.bordeaux.inria.fr/pierre.delmoral/delmoral96nonlinear.pdf. 
  3. ↑ Liu, Jun S.; Chen, Rong (1998-09-01). "Sequential Monte Carlo Methods for Dynamic Systems". Journal of the American Statistical Association 93 (443): 1032–1044. doi:10.1080/01621459.1998.10473765. ISSN 0162-1459. 
  4. ↑ 4,0 4,1 Del Moral, Pierre (1998). "Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems". Annals of Applied Probability 8 (2): 438–495. doi:10.1214/aoap/1028903535. http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535. 
  5. ↑ Del Moral, Pierre. Feynman-Kac formulae. Genealogical and interacting particle approximations.. Springer. Series: Probability and Applications, 2004 — 556 bet. ISBN 978-0-387-20268-6. 
  6. ↑ Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay (2012). "On Adaptive Resampling Procedures for Sequential Monte Carlo Methods". Bernoulli 18 (1): 252–278. doi:10.3150/10-bej335. http://hal.inria.fr/docs/00/33/25/83/PDF/RR-6700.pdf. 
  7. ↑ 7,0 7,1 7,2 Del Moral, Pierre. Feynman-Kac formulae. Genealogical and interacting particle approximations, Probability and its Applications. Springer, 2004 — 575 bet. ISBN 9780387202686. „Series: Probability and Applications“ 
  8. ↑ 8,0 8,1 8,2 8,3 8,4 Del Moral, Pierre „Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering“,. Séminaire de Probabilités XXXIV, Lecture Notes in Mathematics Jacques Azéma: , 2000 — 1–145 bet. DOI:10.1007/bfb0103798. ISBN 978-3-540-67314-9.  Manba xatosi: Invalid <ref> tag; name "dmm002" defined multiple times with different content
  9. ↑ 9,0 9,1 Del Moral, Pierre; Miclo, Laurent (2000). "A Moran particle system approximation of Feynman-Kac formulae.". Stochastic Processes and Their Applications 86 (2): 193–216. doi:10.1016/S0304-4149(99)00094-0. 
  10. ↑ Del Moral, Pierre. Mean field simulation for Monte Carlo integration. Chapman & Hall/CRC Press, 2013 — 626 bet. „Monographs on Statistics & Applied Probability“ 
  11. ↑ Moral, Piere Del; Doucet, Arnaud (2014). "Particle methods: An introduction with applications". ESAIM: Proc. 44: 1–46. doi:10.1051/proc/201444001. 
  12. ↑ 12,0 12,1 Rosenbluth, Marshall, N.; Rosenbluth, Arianna, W. (1955). "Monte-Carlo calculations of the average extension of macromolecular chains". J. Chem. Phys. 23 (2): 356–359. doi:10.1063/1.1741967. https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7. 
  13. ↑ 13,0 13,1 13,2 Hetherington, Jack, H. (1984). "Observations on the statistical iteration of matrices". Phys. Rev. A 30 (2713): 2713–2719. doi:10.1103/PhysRevA.30.2713. 
  14. ↑ 14,0 14,1 Del Moral, Pierre (2003). "Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups". ESAIM Probability & Statistics 7: 171–208. doi:10.1051/ps:2003001. http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a. 
  15. ↑ Assaraf, Roland; Caffarel, Michel; Khelif, Anatole (2000). "Diffusion Monte Carlo Methods with a fixed number of walkers". Phys. Rev. E 61 (4): 4566–4575. doi:10.1103/physreve.61.4566. PMID 11088257. Archived from the original on 2014-11-07. https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf. 
  16. ↑ Caffarel, Michel; Ceperley, David; Kalos, Malvin (1993). "Comment on Feynman-Kac Path-Integral Calculation of the Ground-State Energies of Atoms". Phys. Rev. Lett. 71 (13): 2159. doi:10.1103/physrevlett.71.2159. PMID 10054598. 
  17. ↑ Ocone, D. L. (January 1, 1999). "Asymptotic stability of beneš filters". Stochastic Analysis and Applications 17 (6): 1053–1074. doi:10.1080/07362999908809648. ISSN 0736-2994. 
  18. ↑ Maurel, Mireille Chaleyat; Michel, Dominique (January 1, 1984). "Des resultats de non existence de filtre de dimension finie". Stochastics 13 (1–2): 83–102. doi:10.1080/17442508408833312. ISSN 0090-9491. 
  19. ↑ Hajiramezanali, Ehsan; Imani, Mahdi; Braga-Neto, Ulisses; Qian, Xiaoning; Dougherty, Edward R. (2019). "Scalable optimal Bayesian classification of single-cell trajectories under regulatory model uncertainty". BMC Genomics 20 (Suppl 6): 435. doi:10.1186/s12864-019-5720-3. PMID 31189480. PMC 6561847. //www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=6561847. 
  20. ↑ Fraser, Alex (1957). "Simulation of genetic systems by automatic digital computers. I. Introduction". Aust. J. Biol. Sci. 10 (4): 484–491. doi:10.1071/BI9570484. 
  21. ↑ Fraser, Alex. Computer Models in Genetics. New York: McGraw-Hill, 1970. ISBN 978-0-07-021904-5. 
  22. ↑ Crosby, Jack L.. Computer Simulation in Genetics. London: John Wiley & Sons, 1973. ISBN 978-0-471-18880-3. 
  23. ↑ Assaraf, Roland; Caffarel, Michel; Khelif, Anatole (2000). "Diffusion Monte Carlo Methods with a fixed number of walkers". Phys. Rev. E 61 (4): 4566–4575. doi:10.1103/physreve.61.4566. PMID 11088257. Archived from the original on 2014-11-07. https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf. 
  24. ↑ Caffarel, Michel; Ceperley, David; Kalos, Malvin (1993). "Comment on Feynman-Kac Path-Integral Calculation of the Ground-State Energies of Atoms". Phys. Rev. Lett. 71 (13): 2159. doi:10.1103/physrevlett.71.2159. PMID 10054598. 
  25. ↑ Fermi, Enrique; Richtmyer, Robert, D. (1948). "Note on census-taking in Monte Carlo calculations". LAM 805 (A). http://scienze-como.uninsubria.it/bressanini/montecarlo-history/fermi-1948.pdf. "Declassified report Los Alamos Archive" 
  26. ↑ Herman, Kahn; Harris, Theodore, E. (1951). "Estimation of particle transmission by random sampling". Natl. Bur. Stand. Appl. Math. Ser. 12: 27–30. https://dornsifecms.usc.edu/assets/sites/520/docs/kahnharris.pdf. 
  27. ↑ Kitagawa, G. (January 1993). "A Monte Carlo Filtering and Smoothing Method for Non-Gaussian Nonlinear State Space Models". Proceedings of the 2nd U.S.-Japan Joint Seminar on Statistical Time Series Analysis: 110–131. https://www.ism.ac.jp/~kitagawa/1993_US-Japan.pdf. 
  28. ↑ Kitagawa, G. (1996). "Monte carlo filter and smoother for non-Gaussian nonlinear state space models". Journal of Computational and Graphical Statistics 5 (1): 1–25. doi:10.2307/1390750. 
  29. ↑ Gordon, N.J.; Salmond, D.J.; Smith, A.F.M. (April 1993). "Novel approach to nonlinear/non-Gaussian Bayesian state estimation". IEE Proceedings F - Radar and Signal Processing 140 (2): 107–113. doi:10.1049/ip-f-2.1993.0015. ISSN 0956-375X. 
  30. ↑ Carvalho, Himilcon; Del Moral, Pierre; Monin, André; Salut, Gérard (July 1997). "Optimal Non-linear Filtering in GPS/INS Integration.". IEEE Transactions on Aerospace and Electronic Systems 33 (3): 835. doi:10.1109/7.599254. http://homepages.laas.fr/monin/Version_anglaise/Publications_files/GPS.pdf. 
  31. ↑ P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : An unified framework for particle solutions
    LAAS-CNRS, Toulouse, Research Report no. 91137, DRET-DIGILOG- LAAS/CNRS contract, April (1991).
  32. ↑ P. Del Moral, G. Rigal, and G. Salut. Nonlinear and non-Gaussian particle filters applied to inertial platform repositioning.
    LAAS-CNRS, Toulouse, Research Report no. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN no. A.91.77.013, (94p.) September (1991).
  33. ↑ P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. Experimental results.
    Convention DRET no. 89.34.553.00.470.75.01, Research report no.2 (54p.), January (1992).
  34. ↑ P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. Theoretical results
    Convention DRET no. 89.34.553.00.470.75.01, Research report no.3 (123p.), October (1992).
  35. ↑ P. Del Moral, J.-Ch. Noyer, G. Rigal, and G. Salut. Particle filters in radar signal processing : detection, estimation and air targets recognition.
    LAAS-CNRS, Toulouse, Research report no. 92495, December (1992).
  36. ↑ P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation.
    Studies on: Filtering, optimal control, and maximum likelihood estimation. Convention DRET no. 89.34.553.00.470.75.01. Research report no.4 (210p.), January (1993).
  37. ↑ Crisan, Dan; Gaines, Jessica; Lyons, Terry (1998). "Convergence of a branching particle method to the solution of the Zakai". SIAM Journal on Applied Mathematics 58 (5): 1568–1590. doi:10.1137/s0036139996307371. https://semanticscholar.org/paper/99e8759a243cd0568b0f32cbace2ad0525b16bb6. 
  38. ↑ Crisan, Dan; Lyons, Terry (1997). "Nonlinear filtering and measure-valued processes". Probability Theory and Related Fields 109 (2): 217–244. doi:10.1007/s004400050131. 
  39. ↑ Crisan, Dan; Lyons, Terry (1999). "A particle approximation of the solution of the Kushner–Stratonovitch equation". Probability Theory and Related Fields 115 (4): 549–578. doi:10.1007/s004400050249. 
  40. ↑ Crisan, Dan; Del Moral, Pierre; Lyons, Terry (1999). "Discrete filtering using branching and interacting particle systems". Markov Processes and Related Fields 5 (3): 293–318. http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf. 
  41. ↑ 41,0 41,1 Del Moral, Pierre; Guionnet, Alice (1999). "On the stability of Measure Valued Processes with Applications to filtering". C. R. Acad. Sci. Paris 39 (1): 429–434. 
  42. ↑ 42,0 42,1 Del Moral, Pierre; Guionnet, Alice (2001). "On the stability of interacting processes with applications to filtering and genetic algorithms". Annales de l'Institut Henri Poincaré 37 (2): 155–194. doi:10.1016/s0246-0203(00)01064-5. Archived from the original on 2014-11-07. https://web.archive.org/web/20141107004539/https://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps. 
  43. ↑ Del Moral, P.; Guionnet, A. (1999). "Central limit theorem for nonlinear filtering and interacting particle systems". The Annals of Applied Probability 9 (2): 275–297. doi:10.1214/aoap/1029962742. ISSN 1050-5164.  
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