Zarrachalar filtri sahifasining manbasini koʻrish

'''Zarrachalar filtrlari''' yoki '''ketma-ket Monte-Karlo''' usullari - signalni qayta ishlash va Bayes statistik xulosasi kabi chiziqli boʻlmagan holat-kosmik tizimlar uchun filtrlash muammolari uchun taxminiy yechimlarni topish uchun ishlatiladigan [[Montekarlo usuli|Monte-Karlo]] algoritmlari toʻplami hisoblanadi.<ref name="Wills">{{jurnal manbasi|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-yil|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 boʻladigan 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 boʻlib,1960-yillarning boshidan suyuqlik mexanikasida qoʻllanadigan oʻrtacha maydon oʻzaro taʼsir qiluvchi zarrachalar usullari haqida kiritilgan.<ref name="dm962"/> Keyinchalik "Sequential Monte Carlo" atamasi 1998-yilda Jun S. Liu va Rong Chen tomonidan kiritilgan<ref>{{jurnal manbasi|last=Liu|first1=Jun S.|last2=Chen|first2=Rong|date=1-sentabr 1998-yil|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 toʻplamidan (namunalar deb ham ataladi) foydalanadi. Holat-kosmos modeli chiziqli boʻlmagan boʻlishi mumkin va boshlangʻich 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 oʻrnatilgan metodologiyani taʼminlaydi<ref name="dm962"/><ref name=":22">{{jurnal manbasi|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 oʻzlarining bashoratlarini taxminiy (statistik) tarzda yangilaydi. Tarqatishdan olingan namunalar zarrachalar toʻplami bilan ifodalanadi; har bir zarrachaga tayinlangan ehtimollik ogʻirligi mavjud boʻlib, u zarrachaning [[Ehtimolik zichligi funksiyasi|ehtimollik zichligi funksiyasidan]] namuna olish [[Ehtimollik|ehtimolini]] ifodalaydi. Ogʻirlikning pasayishiga olib keladigan vazn nomutanosibligi ushbu filtrlash algoritmlarida tez-tez uchraydigan muammodir. Biroq, ogʻirliklar notekis boʻlishidan oldin, qayta namuna olish bosqichini kiritish orqali uni yumshatish mumkin. Bir nechta moslashtirilgan qayta namuna olish mezonlaridan foydalanish mumkin, shu jumladan ogʻirliklarning oʻzgarishi va bir xil taqsimotga tegishli nisbiy [[entropiya]] .<ref name=":0">{{jurnal manbasi|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> 

Statistik va ehtimollik nuqtai nazaridan, zarracha filtrlari [[Feynman–Kac formulasi|Feynman-Kac]] ehtimollik oʻlchovlarining oʻrtacha maydon zarracha talqini sifatida talqin qilinishi ham mumkin.<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"/><ref name="dmm00m2">{{jurnal manbasi|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="dp13">{{Kitob manbasi|last=Del Moral|first=Pierre|title=Mean field simulation for Monte Carlo integration|year=2013|publisher=Chapman & Hall/CRC Press|quote=Monographs on Statistics & Applied Probability|url=http://www.crcpress.com/product/isbn/9781466504059|pages=626}}</ref><ref>{{jurnal manbasi|title=Particle methods: An introduction with applications|journal=ESAIM: Proc.|doi=10.1051/proc/201444001|volume=44|pages=1–46|year=2014|last=Moral|first1=Piere Del|last2=Doucet|first2=Arnaud}}</ref> Ushbu zarrachalarni integratsiyalash usullari [[Kimyo|molekulyar kimyo]] va elementar zarrachalar [[Hisoblash fizikasi|hisoblash fizikasida]] 1951-yilda ikki fizik Teodor E. Xarris va Herman Kan, 1955-yilda Marshall N. Rozenblyut va Arianna V. Rozenblyut,<ref name=":5">{{jurnal manbasi|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> va yaqinda 1984-yilda Jek X. Xeterington tomonidan ishlab chiqilgan<ref name="h84">{{jurnal manbasi|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> Hisoblash fizikasida ushbu Feynman-Kac tipidagi zarrachalarni birlashtirish usullari Kvant Monte-Karloda, aniqrogʻi Diffuziya Monte-Karlo usullarida ham qoʻllanadi.<ref name="dm-esaim032">{{jurnal manbasi|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="caffarel12">{{jurnal manbasi|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="caffarel22">{{jurnal manbasi|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> Feynman-Kac oʻzaro taʼsir qiluvchi zarrachalar usullari, shuningdek, murakkab optimallashtirish muammolarini hal qilish uchun evolyutsion hisoblashda hozirda qoʻllanadigan mutatsiya-tanlash genetik algoritmlari bilan kuchli bogʻliq boʻladi.

Zarrachalarni filtrlash metodologiyasi Yashirin Markov Modeli (HMM) va chiziqli boʻlmagan filtrlash muammolarini hal qilish uchun ishlatiladi. Chiziqli-Gauss signal-kuzatish modellari (Kalman filtri) yoki modellarning kengroq sinflari (Benes filtri<ref>{{jurnal manbasi|title=Asymptotic stability of beneš filters|journal=Stochastic Analysis and Applications|date=1-yanvar 1999-yil|issn=0736-2994|pages=1053–1074|volume=17|issue=6|doi=10.1080/07362999908809648|first1=D. L.|last=Ocone}}</ref>)ni isbotlashdan  tashqari, Mirey Chaleyat-Maurel va Dominik Mishel 1984-yilda tasodifiy holatlarning posterior taqsimoti ketma-ketligini isbotladilar. Berilgan kuzatishlarda yaʼni optimal filtr berilgan signalda chekli rekursiya yoʻq.<ref>{{jurnal manbasi|title=Des resultats de non existence de filtre de dimension finie|journal=Stochastics|date=1-yanvar 1984-yil|issn=0090-9491|pages=83–102|volume=13|issue=1–2|doi=10.1080/17442508408833312|first1=Mireille Chaleyat|last=Maurel|first2=Dominique|last2=Michel}}</ref> Ruxsat etilgan tarmoqni taxmin qilish, Markov zanjiri Monte-Karlo texnikasi, anʼanaviy chiziqlilashtirish, kengaytirilgan Kalman filtrlari yoki eng yaxshi chiziqli tizimni aniqlash (kutilayotgan xarajat-xato maʼnosida) keng koʻlamli tizimlar, beqaror jarayonlar bilan kurashishga qodir emas., yoki etarli darajada silliq boʻlmagan chiziqli.

Zarrachalar filtrlari va Feynman-Kac zarrachalar metodologiyasi signal va tasvirni qayta ishlash, Bayes xulosasi, mashinani oʻrganish, xavf tahlili va noyob hodisalardan namuna olish, [[muhandislik]] [[Robototexnika|va robototexnika]], [[Sunʼiy intellekt|sunʼiy intellekt]], bioinformatika,<ref name=":PFOBC">{{jurnal manbasi|doi=10.1186/s12864-019-5720-3|pmid=31189480|pmc=6561847|arxiv=1902.03188|title=Scalable optimal Bayesian classification of single-cell trajectories under regulatory model uncertainty|journal=BMC Genomics|volume=20|issue=Suppl 6|pages=435|year=2019|last=Hajiramezanali|first1=Ehsan|last2=Imani|first2=Mahdi|last3=Braga-Neto|first3=Ulisses|last4=Qian|first4=Xiaoning|last5=Dougherty|first5=Edward R.|bibcode=2019arXiv190203188H}}</ref> filogenetika, hisoblash fanlari, [[Iqtisodiyot tushunchasi|iqtisod]] va matematika moliyasida ham keng qoʻllanadi., bundan tashqari [[Kimyo|molekulyar kimyo]], [[hisoblash fizikasi]], farmakokinetika va boshqa sohalarda ham.

== Tarixi ==

=== Evristik algoritmlar ===
Statistik va ehtimollik nuqtai nazaridan, zarracha filtrlari tarmoqlanuvchi / genetik turdagi algoritmlar va oʻrtacha maydon tipidagi oʻzaro taʼsir qiluvchi zarrachalar metodologiyalari sinfiga kiradi. Ushbu zarracha usullarini talqin qilish ilmiy intizomga bogʻliqdir. Evolyutsion hisoblashda oʻrtacha maydon genetik tipidagi zarrachalar metodologiyalari koʻpincha evristik va tabiiy qidiruv algoritmlari sifatida ishlatiladi Metaevristik). [[Hisoblash fizikasi]] va [[Kimyo|molekulyar kimyoda]] evristik algaritmlar Feynman-Kac yoʻlini integratsiyalash muammolarini hal qilish yoki Boltsmann-Gibbs oʻlchovlari, eng yuqori xos qiymatlari va [[Shredinger tenglamasi|Shredinger]] operatorlarining asosiy holatlarini hisoblash uchun ishlatiladi. [[Biologiya]] va [[Genetika|genetikada]] ular baʼzi bir muhitda individlar yoki genlar populyatsiyasining evolyutsiyasini ham ifodalashi mumkin.


Biologiya va [[Genetika|genetikada]] avstraliyalik genetik Aleks Freyzer 1957-yilda organizmlarning [[Chatishtirish|sunʼiy tanlanishining]] genetik turini simulyatsiya qilish boʻyicha bir qator maqolalarni nashr etdi.<ref>{{jurnal manbasi|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|url=https://archive.org/details/computermodelsin00unse|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|url=https://archive.org/details/computersimulati0000cros|publisher=John Wiley & Sons|location=London|isbn=978-0-471-18880-3}}</ref> Freyzerning simulyatsiyalari zamonaviy mutatsiya-tanlash genetik zarrachalar algoritmlarining barcha muhim elementlarini oʻz ichiga olgan.Matematik nuqtai nazardan, baʼzi bir qisman va shovqinli kuzatuvlar berilgan signalning tasodifiy holatlarining shartli taqsimlanishi ehtimollik potentsial funksiyalari ketma-ketligi bilan ogʻirlikdagi signalning tasodifiy traektoriyalari boʻyicha Feynman-Kac ehtimolligi bilan tavsiflanadi.<ref name="dp042"/><ref name="dmm002"/> Kvant Monte-Karlo va aniqrogʻi, Diffuziya Monte-Karlo usullari, shuningdek, Feynman-Kac yoʻl integrallarining oʻrtacha maydon genetik turi zarrachalar yaqinlashishi sifatida talqin qilinishi mumkin.<ref name="dp042" /><ref name="dmm002" /><ref name="dmm00m2"/><ref name="h84"/><ref name="dm-esaim032"/><ref name="caffarel1">{{jurnal manbasi|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">{{jurnal manbasi|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 koʻpincha Enriko Fermi va Robert Richtmyer bilan bogʻliq boʻlib, ular 1948-yilda neytron zanjiri reaksiyalarining oʻrtacha maydon zarrachalari talqinini<ref>{{jurnal manbasi|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 boʻyicha 1951-yilda nashr etilgan, zarracha uzatish energiyasini baholash uchun oʻrtacha maydon, lekin evristik genetik usullardan foydalangan holda oldingi muhim ishlaridan iqtibos keltirish mumkin.<ref>{{jurnal manbasi|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 oʻxshash 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"/> Ilgʻor 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">{{jurnal manbasi|last=Kitagawa|first1=G.|date=1993-yil yanvar|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 oʻzgartirilgan versiyasi 1996-yilda paydo boʻldi<ref>{{jurnal manbasi|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 oʻzlarining asosiy ishlarida<ref name="Gordon1993">{{jurnal manbasi|title=Novel approach to nonlinear/non-Gaussian Bayesian state estimation|journal=IEE Proceedings F - Radar and Signal Processing|date=1993-yil aprel|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 qoʻllashni nashr etdilar. Mualliflar oʻzlarining 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 koʻrsatdi. Mustaqil ravishda, Per Del Moral<ref name="dm962">{{jurnal manbasi|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>{{jurnal manbasi|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=1997-yil iyul|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|qaralgan sana=2023-05-30|arxiv-sana=2022-11-10|arxiv-url=https://web.archive.org/web/20221110053359/https://homepages.laas.fr/monin/Version_anglaise/Publications_files/GPS.pdf|url-status=dead}}</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. 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 />
Convention DRET no. 89.34.553.00.470.75.01, Research report no.2 (54p.), January (1992).</ref><ref>P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. Theoretical results <br />
Convention DRET no. 89.34.553.00.470.75.01, Research report no.3 (123p.), October (1992).</ref><ref>P. Del Moral, J.-Ch. Noyer, G. Rigal, and G. Salut. Particle filters in radar signal processing : detection, estimation and air targets recognition.<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>

=== 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 qoʻllanadigan tabiiy va evristik algoritmlarni taqdim etadi., shuningdek, hisob-kitoblarning notoʻgʻriligi va genealogik va ajdodlar daraxtiga asoslangan algoritmlar haqida munozara ham yoʻq. Ushbu zarracha algoritmlarining matematik asoslari va birinchi qatʼiy tahlili 1996-yilda Per Del Moral<ref name="dm962"/><ref name=":22"/> tomonidan amalga oshirilgan. Maqolada<ref name="dm962" />, shuningdek, zarrachaning ehtimollik funksiyalarining yaqinlashuvi va normallashtirilmagan shartli ehtimollik oʻlchovlarining xolis xususiyatlarining isboti mavjud. Ushbu maqolada keltirilgan ehtimollik funksiyalarining xolis zarracha baholovchisi bugungi kunda Bayes statistik xulosasida qoʻllanadi.


Den Crisan, Jessica Gaines va Terri Lyons<ref name=":42">{{jurnal manbasi|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>{{jurnal manbasi|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>{{jurnal manbasi|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">{{jurnal manbasi|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 birgalikda 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}}</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">{{jurnal manbasi|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 boʻyicha birinchi yagona yaqinlashuv natijalari 1990-yillarning oxirida Per Del Moral va Elis Guionnet tomonidan bir qancha kuzatuvlar natijasida ishlab chiqilgan.<ref name="dg99">{{jurnal manbasi|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">{{jurnal manbasi|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 qatʼiy tahlili 2001-yilda P. Del Moral va L. Miklo tomonidan amalga oshirilgan. 

== Filtrlash muammosi ==

=== Maqsad ===
Zarrachalar filtrining maqsadi kuzatuv oʻzgaruvchilari berilgan holat oʻzgaruvchilarining orqa zichligini baholashdir. Zarrachalar filtri yashirin Markov modeli bilan foydalanish uchun moʻljallangan boʻlib, unda tizim yashirin va kuzatilishi mumkin boʻlgan oʻzgaruvchilarni oʻz ichiga oladi. Kuzatish mumkin boʻlgan oʻzgaruvchilar (kuzatish jarayoni) maʼlum funktsional shakl orqali yashirin oʻzgaruvchilar (holat-jarayon) bilan bogʻlanadi. Xuddi shunday, holat oʻzgaruvchilari evolyutsiyasini aniqlaydigan dinamik tizimning ehtimollik tavsifi maʼlum. Umumiy zarrachalar filtri kuzatuvni oʻlchash jarayonidan foydalanib, yashirin holatlarning keyingi taqsimotini baholaydi. Quyidagi kabi davlat makoniga nisbatan:

: <math>\begin{array}{cccccccccc}
X_0&\to &X_1&\to &X_2&\to&X_3&\to &\cdots&\text{signal}\\
\downarrow&&\downarrow&&\downarrow&&\downarrow&&\cdots&\\
Y_0&&Y_1&&Y_2&&Y_3&&\cdots&\text{observation}
\end{array}</math>

filtrlash muammosi yashirin holatlarning qiymatlarini '''ketma-ket''' baholash <math>X_k</math>, kuzatish jarayonining qiymatlarini hisobga olgan holdagi qiymati <math>Y_0,\cdots,Y_k,</math> istalgan vaqtda ''k'' qadam.

Barcha Bayes hisob-kitoblari <math>X_k</math> orqa zichlikdan kelib chiqadi <math>p(x_k|y_0,y_1,...,y_k)</math> . Zarrachalarni filtrlash metodologiyasi genetik turdagi zarrachalar algoritmi bilan bogʻliq boʻlgan empirik oʻlchov yordamida ushbu shartli ehtimolliklarning taxminiyligini taʼminlaydi. Bundan farqli oʻlaroq, Markov zanjiri Monte-Karlo yoki ahamiyatli namuna olish yondashuvi toʻliq posteriorni modellashtiradi va u quyidagicha korinishga egadir <math>p(x_0,x_1,...,x_k|y_0,y_1,...,y_k)</math> .

=== Signal-kuzatish modeli ===
Zarrachalar usullari koʻpincha <math>X_k</math> qiymatni taxmin qiladiva kuzatishlar <math>Y_k</math> Ushbu shaklda modellashtirish mumkin:

* <math>X_0, X_1, \cdots</math> Markov jarayonidir <math>\mathbb R^{d_x}</math> (baʼzilar uchun <math>d_x\geqslant 1</math>) oʻtish ehtimoli zichligiga koʻra rivojlanadi <math>p(x_k|x_{k-1})</math> . Ushbu model ham koʻpincha sintetik tarzda yoziladi
*: <math>X_k|X_{k-1}=x_k \sim p(x_k|x_{k-1})</math>

: dastlabki ehtimollik zichligi bilan <math>p(x_0)</math> .

* Kuzatishlar <math>Y_0, Y_1, \cdots</math> baʼzi shtat maydonidagi qiymatlarni qabul qilish <math>\mathbb{R}^{d_y}</math> (baʼzilar uchun <math>d_y\geqslant 1</math>) va shartli ravishda mustaqildirlar <math>X_0, X_1, \cdots</math> maʼlum. Boshqacha aytganda, har biri <math>Y_k</math> faqat bogʻliq <math>X_k</math> . Bundan tashqari, biz shartli taqsimlashni nazarda tutamiz <math>Y_k</math> berilgan <math>X_k=x_k</math> mutlaq uzluksiz va sintetik tarzda bizda mavjud
*: <math>Y_k|X_k=y_k \sim p(y_k|x_k)</math>

Ushbu xususiyatlarga ega tizimga misol:

: <math>X_k = g(X_{k-1}) + W_{k-1}</math>
: <math>Y_k = h(X_k) + V_k</math>

qayerda ikkalasi <math>W_k</math> va <math>V_k</math> [[Ehtimolik zichligi funksiyasi|ehtimollik zichlik funksiyalari]] maʼlum boʻlgan oʻzaro mustaqil ketma-ketliklar va ''g'' va ''h'' maʼlum funksiyalardir. Ushbu ikkita tenglamani holat fazosi tenglamalari sifatida koʻrish mumkin va Kalman filtri uchun holat fazosi tenglamalariga oʻxshaydi. Yuqoridagi misoldagi ''g'' va ''h'' funksiyalar chiziqli boʻlsa va ikkalasi ham boʻlsa <math>W_k</math> va <math>V_k</math> Gauss boʻlsa, Kalman filtri aniq Bayes filtrlash taqsimotini topadi. Agar yoʻq boʻlsa, Kalman filtriga asoslangan usullar birinchi darajali yaqinlashish (EKF) yoki ikkinchi darajali yaqinlashish boʻladi.

=== Taxminiy Bayes hisoblash modellari ===
Muayyan baʼzi masalalarda kuzatuvlarni shartli taqsimlash, signalning tasodifiy holatlarini hisobga olgan holda, zichlikka ega boʻlmasligi mumkin; ikkinchisini hisoblash imkonsiz yoki juda murakkab boʻlishi mumkin.<ref name=":PFOBC"/> Bunday holda, qoʻshimcha darajaga yaqinlashtirish kerak. Bitta strategiya signalni almashtirishdir yaʼni <math>X_k</math> , Markov zanjiri tomonidan <math>\mathcal X_k=\left(X_k,Y_k\right)</math> va shaklning virtual kuzatuvini joriy etish quyidagicha

: <math>\mathcal Y_k=Y_k+\epsilon \mathcal V_k\quad\mbox{for some parameter}\quad\epsilon\in [0,1]</math>

mustaqil tasodifiy oʻzgaruvchilarning ayrim ketma-ketligi uchun <math>\mathcal V_k</math> maʼlum [[Ehtimolik zichligi funksiyasi|ehtimollik zichligi funksiyalari]] bilan aniqlanadi. Asosiy gʻoya buni kuzatishdir yaʼni

: <math>\text{Law}\left(X_k|\mathcal Y_0=y_0,\cdots, \mathcal Y_k=y_k\right)\approx_{\epsilon\downarrow 0} \text{Law}\left(X_k|Y_0=y_0,\cdots, Y_k=y_k\right)</math>

=== Nochiziqli filtrlash tenglamasi ===
Shartli ehtimollik uchun Bayes qoidasi :

: <math>p(x_0, \cdots, x_k|y_0,\cdots,y_k) =\frac{p(y_0,\cdots,y_k|x_0, \cdots, x_k) p(x_0,\cdots,x_k)}{p(y_0,\cdots,y_k)}</math>

qayerda

: <math>\begin{align}
p(y_0,\cdots,y_k) &=\int p(y_0,\cdots,y_k|x_0,\cdots, x_k) p(x_0,\cdots,x_k) dx_0\cdots dx_k \\
p(y_0,\cdots, y_k|x_0,\cdots ,x_k) &=\prod_{l=0}^{k} p(y_l|x_l) \\
p(x_0,\cdots, x_k) &=p_0(x_0)\prod_{l=1}^{k} p(x_l|x_{l-1})
\end{align}</math>

{{NumBlk|:|<math>\begin{align} 
p(x_k|y_0,\cdots,y_{k-1}) &\stackrel{\text{updating}}{\longrightarrow} p (x_k|y_0,\cdots, y_k)=\frac{p (y_k|x_k)p (x_k|y_0,\cdots, y_{k-1})}{\int p (y_k|x'_k)p (x'_k|y_0,\cdots, y_{k-1})dx'_k}  \\
&\stackrel{\text{prediction}}{\longrightarrow}p (x_{k+1}|y_0,\cdots, y_k)=\int p (x_{k+1}|x_k) p (x_k|y_0,\cdots, y_k) dx_k
\end{align}</math>|Eq. 1}}

=== Feynman-Kac formulasi ===
Biz vaqt gorizonti n va kuzatishlar ketma-ketligini tuzamiz <math>Y_0=y_0,\cdots,Y_n=y_n</math> va har bir ''k'' = 0, ..., ''n'' uchun biz quyidagilarni oʻrnatamiz:

: <math>G_k(x_k)=p(y_k|x_k).</math>

Bu belgida, traektoriyalar toʻplami boʻyicha har qanday cheklangan ''F'' funksiyasi uchun <math>X_k</math> kelib chiqishi ''k'' = 0 dan ''k'' = ''n'' vaqtgacha, bizda Feynman-Kac formulasi mavjud

: <math>\begin{align}
\int F(x_0,\cdots,x_n) p(x_0,\cdots,x_n|y_0,\cdots,y_n) dx_0\cdots dx_n &= \frac{\int F(x_0,\cdots,x_n) \left\{\prod\limits_{k=0}^{n} p(y_k|x_k)\right\}p(x_0,\cdots,x_n) dx_0\cdots dx_n}{\int \left\{\prod\limits_{k=0}^{n} p(y_k|x_k)\right\}p(x_0,\cdots,x_n) dx_0\cdots dx_n}\\
&=\frac{E\left(F(X_0,\cdots,X_n)\prod\limits_{k=0}^{n} G_k(X_k)\right)}{E\left(\prod\limits_{k=0}^{n} G_k(X_k)\right)}
\end{align}</math>

=== Genealogik daraxtga asoslangan zarrachalarni tekislash ===
Ajdodlar yoʻlini oʻtmishda izlash

: <math>\left(\widehat{\xi}^i_{0,k},\widehat{\xi}^i_{1,k},\cdots,\widehat{\xi}^i_{k-1,k},\widehat{\xi}^i_{k,k}\right), \quad \left(\xi^i_{0,k},\xi^i_{1,k},\cdots,\xi^i_{k-1,k},\xi^i_{k,k}\right)</math>

shaxslardan <math>\widehat{\xi}^i_{k}\left(=\widehat{\xi}^i_{k,k}\right)</math> va <math>\xi^i_{k}\left(={\xi}^i_{k,k}\right)</math> har bir vaqt ''k'' qadamida bizda zarrachalarning yaqinlashuvlari ham mavjud

: <math>\begin{align}
\widehat{p}(d(x_0,\cdots,x_k)|y_0,\cdots,y_k) &:=\frac{1}{N}\sum_{i=1}^N \delta_{\left(\widehat{\xi}^i_{0,k},\cdots,\widehat{\xi}^i_{0,k}\right)}(d(x_0,\cdots,x_k)) \\
&\approx_{N\uparrow\infty} p(d(x_0,\cdots,x_k)|y_0,\cdots,y_k) \\
&\approx_{N\uparrow\infty} \sum_{i=1}^N \frac{p(y_k|\xi^i_{k,k})}{\sum_{j=1}^Np(y_k|\xi^j_{k,k})} \delta_{\left(\xi^i_{0,k},\cdots,\xi^i_{0,k}\right)}(d(x_0,\cdots,x_k)) \\
& \ \\
\widehat{p}(d(x_0,\cdots,x_k)|y_0,\cdots,y_{k-1}) &:=\frac{1}{N}\sum_{i=1}^N \delta_{\left(\xi^i_{0,k},\cdots,\xi^i_{k,k}\right)}(d(x_0,\cdots,x_k)) \\
&\approx_{N\uparrow\infty} p(d(x_0,\cdots,x_k)|y_0,\cdots,y_{k-1}) \\
&:=p(x_0,\cdots,x_k|y_0,\cdots,y_{k-1}) dx_0,\cdots,dx_k
\end{align}</math>

Bu empirik yaqinlashuvlar zarracha integral yaqinliklariga ekvivalentdir

: <math>\begin{align}
\int F(x_0,\cdots,x_n) \widehat{p}(d(x_0,\cdots,x_k)|y_0,\cdots,y_k) &:=\frac{1}{N}\sum_{i=1}^N F\left(\widehat{\xi}^i_{0,k},\cdots,\widehat{\xi}^i_{0,k}\right) \\
&\approx_{N\uparrow\infty} \int F(x_0,\cdots,x_n) p(d(x_0,\cdots,x_k)|y_0,\cdots,y_k) \\
&\approx_{N\uparrow\infty} \sum_{i=1}^N \frac{p(y_k|\xi^i_{k,k})}{\sum_{j=1}^N p(y_k|\xi^j_{k,k})} F\left(\xi^i_{0,k}, \cdots,\xi^i_{k,k} \right) \\
& \ \\
\int F(x_0,\cdots,x_n) \widehat{p}(d(x_0,\cdots,x_k)|y_0,\cdots,y_{k-1}) &:=\frac{1}{N} \sum_{i=1}^N F\left(\xi^i_{0,k},\cdots,\xi^i_{k,k}\right) \\
&\approx_{N\uparrow\infty} \int F(x_0,\cdots,x_n) p(d(x_0,\cdots,x_k)|y_0,\cdots,y_{k-1})
\end{align}</math>

=== Ehtimollik funksiyalarining xolis zarracha baholari ===
Biz ushbu formuladan foydalanamiz

: <math>p(y_0,\cdots,y_n)=\prod_{k=0}^n p(y_k|y_0,\cdots,y_{k-1})</math>

bilan

: <math>p(y_k|y_0,\cdots,y_{k-1})=\int p(y_k|x_k) p(dx_k|y_0,\cdots,y_{k-1})</math>

=== Orqaga zarrachalar silliqlash ===
Bayes qoidasidan foydalanib, biz formulaga egamiz

: <math>p(x_0,\cdots,x_n|y_0,\cdots,y_{n-1}) = p(x_n | y_0,\cdots,y_{n-1}) p(x_{n-1}|x_n, y_0,\cdots,y_{n-1} ) \cdots p(x_1|x_2,y_0,y_1) p(x_0|x_1,y_0)</math>

Eʼtibor bering

: <math> \begin{align} 
p(x_{k-1}|x_{k},(y_0,\cdots,y_{k-1})) &\propto p(x_{k}|x_{k-1})p(x_{k-1}|(y_0,\cdots,y_{k-1})) \\
p(x_{k-1}|(y_0,\cdots,y_{k-1}) &\propto p(y_{k-1}|x_{k-1})p(x_{k-1}|(y_0,\cdots,y_{k-2})
\end{align}</math>

== Ilovalar ==
Zarrachalar filtrlari va Feynman-Kac zarrachalar metodologiyasi shovqinli kuzatuvlar yoki kuchli chiziqli boʻlmagan holatlarga qarshi kurashish uchun samarali vosita sifatida bir nechta kontekstlarda qoʻllanishi mumkin, masalan:

* Bayes xulosasi, mashinani oʻrganish, xavf tahlili va kamdan-kam uchraydigan hodisalar namunasi
* [[Bioinformatika|Bioinformatika]]<ref name=":PFOBC"/>
* Hisoblash fani
* [[Iqtisodiyot tushunchasi|Iqtisodiyot]], moliyaviy matematika va matematik moliya : zarracha filtrlari makroiqtisodiyotdagi dinamik stokastik umumiy muvozanat modellari va optsion bahosi kabi muammolar bilan bogʻliq yuqori oʻlchamli va/yoki murakkab integrallarni hisoblash uchun zarur boʻlgan simulyatsiyalarni amalga oshirishi mumkin<ref>{{jurnal manbasi|doi=10.1080/07474938.2011.607333|title=A Survey of Sequential Monte Carlo Methods for Economics and Finance|journal=Econometric Reviews|last=Creal|first1=Drew|volume=31|issue=2|year=2012|pages=245–296|url=https://research.vu.nl/en/publications/991e471a-a074-42a1-8206-0fbef56a3d93}}</ref>
* [[Muhandislik]]
* Xatolarni aniqlash va izolyatsiya qilish : kuzatuvchiga asoslangan sxemalarda zarracha filtri nosozliklarni izolyatsiya qilishga imkon beruvchi kutilgan sensorlar chiqishini bashorat qilishi mumkin<ref>{{jurnal manbasi|doi=10.1109/TIE.2015.2399396|title=Intelligent Particle Filter and Its Application to Fault Detection of Nonlinear System|journal=IEEE Transactions on Industrial Electronics|volume=62|issue=6|year=2015|last=Shen|first1=Yin|last2=Xiangping|first2=Zhu|page=1}}</ref><ref>{{jurnal manbasi|doi=10.3390/s21093066|title=A Particle Filtering Approach for Fault Detection and Isolation of UAV IMU Sensors: Design, Implementation and Sensitivity Analysis|journal=Sensors|volume=21|issue=9|year=2021|last=D'Amato|first1=Edigio|last2=Notaro|first2=Immacolata|last3=Nardi|first3=Vito Antonio|last4=Scordamaglia|first4=Valerio|page=3066|pmid=33924891|pmc=8124649|bibcode=2021Senso..21.3066D}}</ref><ref>{{jurnal manbasi|doi=10.1080/00207720110102566|title=Particle filtering-based fault detection in non-linear stochastic systems|journal=International Journal of Systems Science|volume=33|issue=4|year=2002|first1=V.|last=Kadirkamanathan|first2=P.|last2=Li|first3=M. H.|last3=Jaward|first4=S. G.|last4=Fabri|pages=259–265}}</ref>
* [[Kimyo|Molekulyar kimyo]] va [[hisoblash fizikasi]]
* Farmakokinetika<ref>Bonate P: Pharmacokinetic-Pharmacodynamic Modeling and Simulation. Berlin: Springer; 2011.</ref>
* Filogenetika
* [[Robototexnika]], [[Sunʼiy intellekt|sunʼiy intellekt]] : Monte-Karlo mahalliylashtirish mobil robotlarni lokalizatsiya qilishda de-fakto standartdir<ref name="aaai1999">
Dieter Fox, Wolfram Burgard, Frank Dellaert, and Sebastian Thrun, "[http://www.cs.washington.edu/ai/Mobile_Robotics/abstracts/sampling-aaai-99.abstract.html Monte Carlo Localization: Efficient Position Estimation for Mobile Robots]." ''Proc. of the Sixteenth National Conference on Artificial Intelligence'' John Wiley & Sons Ltd, 1999.</ref><ref name="pr">
Sebastian Thrun, Wolfram Burgard, Dieter Fox. [http://www.probabilistic-robotics.org/ ''Probabilistic Robotics''] MIT Press, 2005. Ch. 8.3 {{ISBN|9780262201629}}.</ref><ref name="robust">Sebastian Thrun, Dieter Fox, Wolfram Burgard, Frank Dellaert. "[http://robots.stanford.edu/papers/thrun.robust-mcl.html Robust monte carlo localization for mobile robots]." ''Artificial Intelligence'' 128.1 (2001): 99–141.
</ref>
* Signal va tasvirni qayta ishlash : vizual lokalizatsiya, kuzatish, xususiyatlarni aniqlash<ref>{{jurnal manbasi|url=https://link.springer.com/article/10.1007/s10723-019-09502-1|doi=10.1007/s10723-019-09502-1|title=A Robust and Accurate Particle Filter-Based Pupil Detection Method for Big Datasets of Eye Video|year=2020|last=Abbasi|first1=Mahdi|last2=Khosravi|first2=Mohammad R.|journal=Journal of Grid Computing|volume=18|issue=2|pages=305–325}}</ref>

== Boshqa zarracha filtrlari ==

* Yordamchi zarrachalar filtri<ref name="apf1999">{{jurnal manbasi|last=Pitt, M.K.|last2=Shephard, N.|year=1999|title=Filtering Via Simulation: Auxiliary Particle Filters|journal=Journal of the American Statistical Association|volume=94|issue=446|pages=590–591|url=https://www.questia.com/PM.qst?a=o&se=gglsc&d=5002321997|access-date=2008-05-06|doi=10.2307/2670179|jstor=2670179|arxiv-sana=2007-10-16|arxiv-url=https://web.archive.org/web/20071016200646/http://www.questia.com/PM.qst?a=o|url-status=dead}}</ref>
* Xarajatlar boʻyicha zarracha filtri
* [[Zarrachalar filtri|Eksponensial tabiiy zarracha filtri]]<ref name="xnpf2015">{{ArXiv manbasi|author1=Zand, G.|author2=Taherkhani, M.|author3=Safabakhsh, R.|year=2015|title=Exponential Natural Particle Filter|eprint=1511.06603|class=cs.LG}}</ref>
* Feynman-Kac va oʻrtacha maydon zarrachalari metodologiyalari<ref name="dm962"/><ref name="dp13"/><ref name=":1"/>
* Gauss zarracha filtri
* Gauss-Germit zarrachalar filtri
* Ierarxik/Malajlanadigan zarrachalar filtri<ref name="Canton2011">{{jurnal manbasi|last=Canton-Ferrer, C.|last2=Casas, J.R.|last3=Pardàs, M.|year=2011|title=Human Motion Capture Using Scalable Body Models|journal=Computer Vision and Image Understanding|volume=115|issue=10|pages=1363–1374|doi=10.1016/j.cviu.2011.06.001}}</ref>
* Qoʻzgʻaluvchan zarrachalar filtri<ref>{{jurnal manbasi|last=Akyildiz|first1=Ömer Deniz|last2=Míguez|first2=Joaquín|date=1-mart 2020-yil|title=Nudging the particle filter|journal=Statistics and Computing|language=inglizcha|volume=30|issue=2|pages=305–330|doi=10.1007/s11222-019-09884-y|issn=1573-1375}}</ref>
* Zarracha Markov-Chain Monte-Karlo, qarang, masalan, pseudo-marginal Metropolis-Xastings algoritmi.
* Rao – qoralangan zarracha filtri<ref name="rbpf1999">{{Konferensiya manbasi|title=Rao–Blackwellised particle filtering for dynamic Bayesian networks|last=Doucet, A.|last2=De Freitas, N.|last3=Murphy, K.|last4=Russell, S.|year=2000|conference=Proceedings of the Sixteenth conference on Uncertainty in artificial intelligence|pages=176–183}}
</ref>
* [[Regularized auxiliary particle filter|Oddiy yordamchi zarrachalar filtri]]<ref name="jliu2011">{{jurnal manbasi|last=Liu, J.|last2=Wang, W.|last3=Ma, F.|year=2011|title=A Regularized Auxiliary Particle Filtering Approach for System State Estimation and Battery Life Prediction|journal=Smart Materials and Structures|volume=20|issue=7|pages=1–9|doi=10.1088/0964-1726/20/7/075021|bibcode=2011SMaS...20g5021L|url=https://escholarship.org/uc/item/0131z9gj}}</ref>
* Rad etish-namuna olishga asoslangan optimal zarracha filtri<ref name="optrj2008">{{Konferensiya manbasi|title=An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization|last=Blanco, J.L.|last2=Gonzalez, J.|last3=Fernandez-Madrigal, J.A.|year=2008|conference=IEEE International Conference on Robotics and Automation (ICRA'08)|pages=461–466}}
</ref><ref name="optrj2010">{{jurnal manbasi|title=Optimal Filtering for Non-Parametric Observation Models: Applications to Localization and SLAM|last=Blanco, J.L.|last2=Gonzalez, J.|last3=Fernandez-Madrigal, J.A.|year=2010|journal=The International Journal of Robotics Research|volume=29|number=14|pages=1726–1742|doi=10.1177/0278364910364165}}
</ref>
* Hidsiz zarrachalar filtri

== Yana qarang ==

* Kalman ansambli filtri
* Umumiy filtrlash
* Genetik algoritm
* Oʻrtacha maydonli zarrachalar usullari
* Monte-Karlo mahalliylashtirish
* Harakatlanuvchi ufqni baholash
* Rekursiv Bayes bahosi



== Manbalar ==
{{Manbalar}}

== Havolalar ==
 {{refbegin}}
* [http://web.maths.unsw.edu.au/~peterdel-moral/simulinks.html Feynman–Kac models and interacting particle algorithms (a.k.a. Particle Filtering)] Theoretical aspects and a list of application domains of particle filters
* [http://www-sigproc.eng.cam.ac.uk/smc/ Sequential Monte Carlo Methods (Particle Filtering)] homepage on University of Cambridge
* [https://web.archive.org/web/20060612210237/http://www.cs.washington.edu/ai/Mobile_Robotics/mcl/ Dieter Fox's MCL Animations]
* [https://web.archive.org/web/20140330053937/http://blogs.oregonstate.edu/hess/code/particles/ Rob Hess' free software]
* [http://www.jstatsoft.org/v30/i06/ SMCTC: A Template Class for Implementing SMC algorithms in C++]
* [http://www.oursland.net/projects/particlefilter/ Java applet on particle filtering]
* [https://zhouyan.github.io/vSMC/ vSMC : Vectorized Sequential Monte Carlo]
* [https://www.youtube.com/watch?v=bO_GajDgGJ4/ Particle filter explained in the context of self driving car]
{{refend}}{{Stochastic processes}}
[[Turkum:Pages with unreviewed translations]]