Chitsanzo cha Mafunso Okambirana pa Metropolis Stage
Ponena za kuyerekezera kwa Monte Carlo, gawo la Metropolis ndi njira yofunika kwambiri mu makina owerengera ndi magawo ena. Mu gawo lino, tikambirana makamaka njira ya Metropolis-Hastings, njira yomwe imagwiritsidwa ntchito poyesa kuchokera ku kugawa kwa kuthekera kovuta. Mwa kumvetsetsa masitepe omwe ali mu njira iyi, titha kuchita kuyerekezera kolondola komanso kogwira mtima.
Chiyambi cha Metropolis Algorithm
Njira ya Metropolis idayambitsidwa ndi Nicholas Metropolis ndi anzake mu 1953. Njirayi imagwiritsidwa ntchito poyesa ndikutsanzira momwe zinthu zilili, makamaka zomwe zimakhudza tinthu tambiri monga mpweya kapena zakumwa. Mtundu wamakono wa njira iyi, Metropolis-Hastings, ndi njira yodziwira zomwe zimalola zitsanzo kuti zitengedwe kuchokera ku kugawa kwa chandamale komwe sikuli koyenera.
Masitepe mu Metropolis Algorithm
Kuti mumvetse momwe njira ya Metropolis imagwirira ntchito, ndikofunikira kudziwa bwino njira izi:
1. Kuyambitsa: Yambani mwa kusankha mwachisawawa yankho loyambirira kuchokera pamalo a yankho kapena kugawa koyambirira. Mwachitsanzo, timayamba ndi kutentha kapena malo a tinthu.
2. Kupereka Gawo Latsopano: Perekani lingaliro la mkhalidwe watsopano (yankho latsopano) mwa kusintha pang'ono ku mkhalidwe womwe ulipo. Izi nthawi zambiri zimatchedwa gawo la "lingaliro". Kusintha kumeneku nthawi zambiri kumachokera ku kugawa kofanana, monga kugawa kwa Gaussian.
3. Kuwerengera Chiŵerengero Chovomerezeka: Werengerani chiŵerengero chovomerezeka, chomwe chimatsimikiza ngati tikuvomereza kapena kukana kusuntha komwe kwaperekedwa. Chiŵerengerochi ndi chiŵerengero cha kuthekera kwa mkhalidwe watsopano ku mkhalidwe womwe ulipo. Mu zolemba zamasamu, chiŵerengerochi chimaperekedwa ndi:
\[
A = \min\left(1, \frac{P(\text{new})}{P(\text{current})}\right)
\]
kumene \( P \) ndi mwayi wa mkhalidwe winawake.
4. Chisankho Pogwiritsa Ntchito Chiŵerengero Chovomerezeka: Yerekezerani chiŵerengero chovomerezeka ndi mtengo wosasinthika wochokera ku kugawa kofanana pakati pa 0 ndi 1. Ngati chiŵerengero chovomerezeka chili chachikulu kuposa mtengo wosasinthika, landirani kusuntha kwatsopano; apo ayi, kukani ndipo pitirizani kukhala momwe mulili pano.
5. Kubwerezabwereza: Bwerezani masitepe 2 mpaka 4 kuti mupeze chiwerengero chomwe mukufuna cha kubwerezabwereza kapena mpaka dongosolo litafika pamlingo woyenera.
Mafunso ndi Kukambirana Zitsanzo
Tiyeni tikambirane mafunso ena achitsanzo kuti timvetse bwino siteji ya Metropolis.
Chitsanzo cha Funso 1
Funso: Muli ndi tinthu tating'onoting'ono tomwe tili mu gawo limodzi la malo \( x \) lomwe limakhudzidwa ndi mphamvu yomwe ingagwiritsidwe ntchito \( U(x) = x^2 \). Gwiritsani ntchito njira ya Metropolis kuti muyerekezere kugawa kwa tinthu tating'onoting'ono.
Kukambirana:
1. Kuyambitsa: Yambani kuchokera pamalo \( x = 0 \).
2. Perekani Chisankho Chatsopano: Perekani malo atsopano \( x' = x + \Delta x \), ndi \( \Delta x \) yotengedwa kuchokera ku kugawa kwa Gaussian ndi zero yapakati.
3. Kuwerengera Chiŵerengero cha Mphamvu: Werengani chiŵerengero cha mphamvu:
\[
\Delta U = U(x') – U(x) = x'^2 – x^2
\]
Chifukwa chake, chiŵerengero chovomerezeka ndi:
\[
A = \min\left(1, e^{-\Delta U}\right)
\]
4. Chisankho: Ngati \( A \) ndi nambala yoposa yachisawawa pakati pa 0 ndi 1, landirani \( x' \); apo ayi, khalani pamalo \( x \).
5. Kubwerezabwereza: Bwerezani izi m'masitepe 10,000.
Kugawa malo komwe kudzachitika kudzatsatira kugawa kwa Gaussian komwe kuli zero wapakati ndi kusiyana kosiyana ndi kuthekera, komwe pankhaniyi, kumabweretsa kugawa komwe kumapangidwa ndi ntchito yamphamvu yomwe ingathe kuchitika.
Chitsanzo cha Funso 2
Funso: Gwiritsani ntchito njira ya Metropolis kuti mugwirizane ndi lingaliro la ntchito ya Bayesian. Tiyerekeze kuti tikufuna kuyika malo otsetsereka osavuta mu dataset pogwiritsa ntchito linear regression ndi MCMC.
Kukambirana:
1. Kuyambitsa: Khazikitsani magawo oyamba a chitsanzo \( \beta = (m, c) \).
2. Kupereka Gawo Latsopano: Perekani ma parameter atsopano a multivariate normal proposal distribution. Mwachitsanzo, gwiritsani ntchito Gaussian distribution ya ma variable \( m \) ndi \( c \).
3. Chiŵerengero Chovomerezeka: Werengerani chiŵerengero chovomerezeka ndi:
\[
A = \min\left(1, \frac{L(m', c'| \text{data})P(m', c')}{L(m, c| \text{data})P(m, c)}\right)
\]
Kumene \( L \) ndiye kuthekera, ndipo \( P \) ndiye choyambirira cha parameter.
4. Chisankho: Yerekezerani chiŵerengero ndi mtengo wosasinthika wa 0 mpaka 1 kuti muvomereze kapena kukana pempholo.
5. Kubwerezabwereza: Yendetsani kuyeserera ndi kubwerezabwereza kokwanira mpaka kuyanjana kukwaniritsidwe.
Ndi njira iyi, titha kupeza ma posterior distributions a regression parameters, zomwe zimatipatsa njira yodziwira ndikutanthauzira ubale mu data.
Mapeto
Gawo la Metropolis mu Monte Carlo simulations limatithandiza kuyesa kuchokera ku magawidwe ovuta ndipo limakhala ngati maziko a njira ya Metropolis-Hastings. Pogwiritsa ntchito njira imeneyi m'magawo osiyanasiyana, titha kupeza njira yolondola kwambiri yopangira zitsanzo komanso kumvetsetsa bwino dongosololi. Mu ntchito kuyambira pa fizikisi ndi biology mpaka sayansi ya makompyuta ndi ziwerengero, njira iyi imapereka mayankho abwino komanso ogwira mtima pamavuto ovuta.
