Hanyar Jackknife a Kididdiga
Hanyar wukar jackknife wata muhimmiyar dabara ce ta sake yin samfuri a kididdiga, musamman don auna rashin tabbas na kimantawa. Sau da yawa ana amfani da wukar jackknife don kimanta son zuciya da bambancin mai kimantawa, da kuma gina ma'aunin daidaito kamar kuskuren da aka saba. Wannan dabarar tana da sauƙi, ba ta buƙatar tsauraran zato na rarrabawa, kuma ana iya amfani da ita ga matsaloli iri-iri, tun daga kididdigar gargajiya zuwa nazarin bayanai na zamani.
Bayani da ra'ayoyi na asali
Maurice Quenouille ne ya gabatar da wukar jackknife, daga baya John Tukey ya shahara da ita. Sunan "jackknife" ya samo asali ne daga wukar aljihu mai amfani, domin hanyar tana da sassauƙa kuma ana iya amfani da ita a yanayi daban-daban. Babban ra'ayin shine wannan: idan muna da samfurin girman n, muna ƙirƙirar "samfuran banza" da yawa ta hanyar cire lura ɗaya a kowane lokaci, sannan mu sake lissafin mai kimantawa akan kowane samfurin. Ta hanyar lura da yadda mai kimantawa ke canzawa lokacin da aka cire lura ɗaya, muna samun fahimta game da daidaiton mai kimantawa akan bambancin bayanai.
Misali, a ce muna da bayanai \(x_1, x_2, \dots, x_n\) kuma muna son kimanta siga \(\theta\) ta amfani da mai kimantawa \( \hat{\theta}=t(x_1,\dots,x_n)\). A cikin jackknife, muna samar da ƙananan samfuran n na girman \(n-1\), wato ƙaramin samfurin \(i\) wanda ke goge \(x_i\). Sannan muna ƙididdigewa:
\[
\hat{\theta}_{(i)} = t(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)
\]
Ana kiran ƙimar \(\hat{\theta}_{(i)}\) ƙimar fita-ɗaya.
Matakan Hanyar Jackknife
A tsari, ana iya bayanin jackknife a cikin matakai masu zuwa:
1. Lissafa mai kimantawa akan cikakken bayanai
Lissafi \(\hat{\theta}\) akan dukkan samfurin.
2. Ƙirƙiri samfuran da aka bari-ɗaya-ɗaya
Ga kowanne \(i = 1,2,\dits,n\), cire abin lura \(x_i\) sannan ka lissafa mai kimantawa \(\hat{\theta}_{(i)}\).
3. Lissafa matsakaicin ma'aunin wukake
Matsakaicin hutu ɗaya:
\[
\bar{\theta}_{(\cdot)} = \frac{1}{n}\sum_{i=1}^n \hat{\theta}_{(i)}
\]
4. Bambancin kimantawa (ko kuskuren yau da kullun)
Yawancin lokaci ana ƙididdige bambancin jackknife tare da:
\[
\widehat{\mathrm{Var}}_{J}(\hat{\theta}) = \frac{n-1}{n}\sum_{i=1}^n \left(\hat{\theta}_{(i)} – \bar{\theta}_{(\cdot)}\dama)^2
\]
Kuskuren da aka saba yi shine tushen murabba'in bambancin.
5. Kimanta son zuciya da gyaran son zuciya (zaɓi ne)
Jackknife kuma zai iya kimanta nuna bambanci ta hanyar:
\[
\widehat{\mathrm{Bias}}_{J}(\hat{\theta}) = (n-1) \hagu(\bar{\theta}_{(\cdot)} - \hat{\theta} \ dama)
\]
Ana iya yin gyaran fuska ta hanyar:
\[
\hat{\theta}_{J} = \hat{\theta} – \widehat{\mathrm{Bias}}_{J}(\hat{\theta})
\]
Fassara: idan matsakaicin barin-ɗaya-ɗaya ya bambanta da cikakken kimantawa ta hanyar tsari, akwai alamar son zuciya da za a iya gyarawa.
Misalin fahimta: matsakaicin samfurin
Don fahimtar jackknife cikin fahimta, yi la'akari da ƙimar samfurin:
\[
\hat{\mu} = \frac{1}{n}\sum_{i=1}^n x_i
\]
Idan muka cire wani abu da aka lura da shi \(x_i\), matsakaicin zai zama:
\[
\hat{\mu}_{(i)} = \frac{1}{n-1}\sum_{j\ne i} x_j
\]
A yanayin matsakaici, wukar jackknife ba ta bayar da babban "abin mamaki" ba saboda matsakaicin yana da karko kuma son zuciya ƙarami ne (a cikin mahallin da yawa). Duk da haka, ga masu ƙididdigewa masu rikitarwa - kamar matsakaici, takamaiman ma'aunin komawa baya, haɗin kai, ko ƙididdigar da ba ta layi ba - canjin da ya samo asali daga cire wurin bayanai guda ɗaya na iya bayyana yanayin mai kimantawa kuma ya samar da ƙiyasin da ya dace na kuskurensa na yau da kullun.
Pseudovalue: muhimmin ra'ayi a cikin jackknife
A wasu tattaunawa, jackknife ya gabatar da wani abu mai ban mamaki ga kowane lura:
\[
\theta_i^{ } = n\hat{\theta} – (n-1)\hat{\theta}_{(i)}
\]
Sannan ana iya rubuta ma'aunin jackknife a matsayin matsakaicin ƙimar ƙarya:
\[
\hat{\theta}_{J} = \frac{1}{n}\sum_{i=1}^n \theta_i^{ }
\]
Tsarin ƙima na ƙarya yana taimakawa wajen bayyana yadda kowace lura ke "ba da gudummawa" ga kimantawa ta ƙarshe kuma yana sauƙaƙa nazarin son zuciya.
Alaƙa tsakanin jackknife da bootstrap
Sau da yawa ana kwatanta wukar Jack da bootstrap, domin duka hanyoyin sake yin samfuri ne. Duk da haka, akwai muhimman bambance-bambance:
– Jackknife yana amfani da ƙarin samfuri ta hanyar cire bayanai ɗaya (bar ɗaya). Adadin kwafi yana da ƙayyadaddun bayanai: daidai n.
– Bootstrapping yana ƙirƙirar sake samfuri tare da maye gurbinsa, yawanci sau da yawa (misali sau 1000 ko 10.000), don haka yana ba da kimantawar rarrabawar mai kimantawa.
Gabaɗaya, takalmin ya fi sassauƙa kuma sau da yawa ya fi daidai ga matsaloli masu rikitarwa, amma wukar jack ta fi sauƙi kuma ba ta da tsada a lissafi. A manyan bayanai, wukar jack na iya zama madadin sauri don samun kurakurai masu tsauri, musamman lokacin da lissafin mai ƙididdigewa yana da tsada amma har yanzu yana yiwuwa a wasu lokutan.
Amfanin hanyar jackknife
Wasu daga cikin fa'idodin jackknife sun haɗa da:
1. Mai sauƙi kuma mai sauƙin aiwatarwa
Manufar barin-ɗaya-ɗaya tana da sauƙin fahimta, kuma dabarar bambancin abu ce mai sauƙi.
2. Kaɗan zato game da rarrabawa
Jackknife ba koyaushe yana buƙatar ɗaukar al'ada ko fom ɗin rarrabawa tabbata.
3. Inganci ga wasu ƙididdiga
Domin yana buƙatar lokutan n kawai na lissafin kimantawa, jackknife sau da yawa ya fi sauƙi fiye da bootstrapping wanda ke buƙatar dubban kwafi.
4. Yana da amfani wajen kimanta son zuciya
Musamman a cikin masu kimantawa marasa layi waɗanda yawanci ba su da sauƙin ƙididdigewa ta hanyar nazari.
Iyakoki da abubuwan da za a kula da su
Ko da yake yana da ƙarfi, jackknife yana da ƙuntatawa:
1. Ba daidai ba ne ga masu kimantawa marasa santsi sosai
Misali, matsakaicin ko quantules a wasu yanayi, ko kididdigar da ta dogara da matsanancin ƙima, jackknife wani lokacin yana ba da kimantawar bambance-bambance marasa daidaito.
2. Ba koyaushe ya dace da bayanai masu dogaro ba
A cikin jerin lokaci ko bayanai na sarari, lura ba ta da wani tasiri. Cire maki ɗaya zai iya karya tsarin dogaro. Ga irin waɗannan lamuran, ana amfani da bambance-bambance kamar toshewar jackknife (cire toshe ɗaya na bayanai a lokaci guda).
3. Mai saurin kamuwa da abubuwan da suka shafi tasirin abubuwa masu yawa
Idan akwai bayanai marasa amfani ko kuma bayanai masu "amfani", ƙiyasin barin-ɗaya-ɗaya na iya canzawa sosai. Wannan ba koyaushe rauni bane - a zahiri, yana iya zama muhimmiyar alama - amma bambancin da ke haifar da hakan na iya zama babba kuma yana buƙatar fassarar da aka yi da kyau.
4. Ƙarfin daidaitawa a babban n
Ko da yake ya fi rahusa fiye da yin amfani da takalmin bootstrapping, har yanzu yana buƙatar kimantawa na n estimator. Idan n yana cikin miliyoyi kuma masu kimantawa suna da tsada, wannan na iya zama matsala.
Bambancin: goge-d jackknife da toshe jackknife
Baya ga barin-ɗaya, akwai bambance-bambance:
– Delete-d jackknife: yana goge abubuwan lura na d a kowane kwafi (maimakon 1 kawai). Wannan na iya inganta daidaito a wasu yanayi, musamman ga masu kimantawa marasa santsi.
– Toshe jackknife: yana cire wani toshe wanda ke ɗauke da wasu abubuwan lura da ke kusa, wanda ya dace da bayanan da ke da alaƙa ta atomatik (misali bayanai na yau da kullun, na mako-mako, ko na sarari).
Zaɓin d ko girman tubalan ya dogara ne akan tsarin bayanai da kuma burin da aka nufa.
Amfani da wukake a aikace
Ana amfani da wukake na Jack a fannoni daban-daban:
– Nazarin Halittu da Cututtukan da ke Yaɗuwa: ƙiyasta kurakuran da aka saba yi don ma'aunin haɗari ko sigogin samfuri lokacin da dabarun nazari suka yi wahala.
– Tattalin Arziki: kimanta daidaiton sigogi, musamman a cikin samfuran da aka iyakance.
– Kimiyyar kwamfuta da koyon injina: ra'ayin barin-daya-daya yana da alaƙa da haɗin gwiwa, kodayake manufofin sun bambanta (tabbatar da hasashen da kimanta daidaiton sigogi).
– Ilimin Halittu da bincike: kimanta bambancin ra'ayi ko wasu fihirisa da rashin tabbas na ƙididdiga masu rikitarwa.
Penutup
Hanyar wukar jackknife wata dabara ce ta sake yin samfuri ta gargajiya wadda ta ci gaba da dacewa a yau. Ta hanyar amfani da wata dabara mai sauƙi - barin wani abu da aka lura da shi da kuma sake lissafin mai kimantawa - wukar jackknife na iya samar da kimantawa na bambance-bambance, kuskuren yau da kullun, da son zuciya ba tare da lissafin lissafi mai rikitarwa ba. Duk da haka, amfani da shi yana buƙatar la'akari da yanayin mai kimantawa, girman samfurin, da tsarin dogaro da bayanai. A aikace, wukar jackknife sau da yawa zaɓi ne mai sauri da bayyananne, ko kuma a matsayin ƙari ga amfani da hanyoyin sake yin samfuri masu ƙarfi kamar bootstrapping.
Idan kuna so, zan iya ƙara ƙaramin misalin lissafin lambobi (misali don haɗin kai ko komawa baya) ko kuma haɗa da aiwatar da jackknife a cikin R/Python don fayyace aikace-aikacen.
