Nzira yeJackknife muStatistics
Nzira yejackknife inzira yakakosha yekuenzanisazve nhamba, kunyanya pakuyera kusava nechokwadi kwekufungidzira. Jackknife inowanzo shandiswa kufungidzira kusarurama uye kusiyana kwemuongorori, pamwe nekugadzira zviyero zvekururama zvakaita sekukanganisa kwakajairwa. Nzira iyi iri nyore, haidi fungidziro dzakaoma dzekugovera, uye inogona kushandiswa kumatambudziko akasiyana-siyana, kubva kuhuwandu hwekare kusvika kuongororo yedata yemazuva ano.
Maonero ekare uye pfungwa huru
Bhani rejack rakaunzwa naMaurice Quenouille uye rakazokurumbira naJohn Tukey. Zita rekuti "jackknife" rakafuridzirwa nebanga rehomwe rinoshandisika zvakasiyana-siyana, sezvo nzira yacho ichichinjika uye inogona kushandiswa mumamiriro akasiyana-siyana. Pfungwa huru ndeiyi: kana tine sampuro yehukuru n, tinogadzira "samples dzedummy" dzakawanda nekubvisa ongororo imwe chete nguva imwe neimwe, tobva taverengazve ongororo pamuenzaniso wega wega. Nekucherechedza kuti ongororo inochinja sei kana ongororo imwe chete yabviswa, tinowana ruzivo rwekugadzikana kwemuongorori pamusoro pekusiyana kwedata.
Semuenzaniso, ngatitii tine data \(x_1, x_2, \dots, x_n\) uye tinoda kufungidzira parameter \(\theta\) tichishandisa estimator \( \hat{\theta}=t(x_1,\dots,x_n)\). Mu jackknife, tinogadzira n subsamples dzehukuru \(n-1\), kureva \(i\)th subsample inobvisa \(x_i\). Zvadaro tinoverenga:
\[
\hat{\theta}_{(i)} = t(x_1,\madotsi,x_{i-1},x_{i+1},\madotsi,x_n)
\]
Kukosha \(\hat{\theta}_{(i)}\) kunonzi leave-one-out estimate .
Matanho ekushandisa Jackknife
Nenzira, jackknife inogona kutsanangurwa mumatanho anotevera:
1. Verenga muongorori padata rakazara
Verenga \(\hat{\theta}\) pamusoro pemuenzaniso wese.
2. Gadzira mienzaniso midiki ye n leave-one-out
Pa \(i = 1,2,\dots,n\), bvisa zvaonekwa \(x_i\) woverenga muongorori \(\hat{\theta}_{(i)}\).
3. Verenga avhareji yemuongorori webanga rejackknife
Avhareji yekurega kamwe chete:
\[
\bar{\theta}_{(\cdot)} = \frac{1}{n}\sum_{i=1}^n \hat{\theta}_{(i)}
\]
4. Fungidzira musiyano (kana chikanganiso chakajairwa)
Kusiyana kwejackknife kunowanzo verengerwa ne:
\[
\widehat{\mathrm{Var}}_{J}(\hat{\theta}) = \frac{n-1}{n}\sum_{i=1}^n \left(\hat{\theta}_{(i)} – \bar{\theta}_{(\cdot)}\right)^2
\]
Chikanganiso chakajairika ndicho chivharo chemusiyano.
5. Kufungidzira rusaruro uye kugadzirisa rusaruro (sarudzo)
Jackknife inogonawo kufungidzira rusaruro kuburikidza ne:
\[
\widehat{\mathrm{Bias}}_{J}(\hat{\theta}) = (n-1)\left(\bar{\theta}_{(\cdot)} - \heti{\theta}\kurudyi)
\]
Kugadzirisa rusaruro kunogona kuitwa ne:
\[
\heti{\theta}_{J} = \heti{\theta} – \widehat{\mathrm{Bias}}_{J}(\hat{\theta})
\]
Dudziro: kana avhareji yekusiya-one-out yakasiyana neyakazara inoongorora zvinhu nenzira yakarongwa, pane chiratidzo chekusarura kunogona kugadziriswa.
Muenzaniso wekunzwisisa: muenzaniso wepakati
Kuti unzwisise jackknife nenzira intuitively, funga nezvemuenzaniso wekufungidzira mean:
\[
\hat{\mu} = \frac{1}{n}\sum_{i=1}^n x_i
\]
Kana tikabvisa chimwe chekucherechedza \(x_i\), avhareji inova:
\[
\hat{\mu}_{(i)} = \frac{1}{n-1}\sum_{j\ne i} x_j
\]
Kana iri avhareji, jackknife haipe "kushamisika" kukuru nekuti avhareji yakagadzikana uye bias idiki (mumamiriro ezvinhu akawanda). Zvisinei, kune vanoongorora zvakanyanya kuomarara—senge median, particular regression coefficient, a correlation, kana nonlinear statistical—shanduko inobva mukubvisa imwe data point inogona kuratidza kunzwisisika kweanoongorora uye kugadzira fungidziro inobatsira ye standard error yayo.
Pseudovalue: pfungwa inokosha mujackknife
Mune dzimwe hurukuro, jackknife inounza pseudovalue yekutarisa kwega kwega:
\[
\theta_i^{ } = n\hat{\theta} – (n-1)\hat{\theta}_{(i)}
\]
Ipapo muyeredzi wejackknife anogona kunyorwa seavhareji ye pseudovalues:
\[
\hat{\theta}_{J} = \frac{1}{n}\sum_{i=1}^n \theta_i^{ }
\]
Nzira yekushandisa pseudovalue inobatsira kutsanangura kuti ongororo yega yega "inobatsira" sei pakufungidzira kwekupedzisira uye inobatsira pakuongorora rusaruro.
Hukama huripo pakati pejackknife nebootstrap
Jackknife inowanzo fananidzwa nebootstrap, sezvo ese ari maviri ari nzira dzekugadzirisa sampuli. Zvisinei, pane misiyano yakakosha:
– Jackknife inoshandisa subsampling nekubvisa data rimwe chete (siya rimwe chete). Huwandu hwema replications hwakasiyana: exactly n .
- Bootstrapping inogadzira muenzaniso mutsva une chinotsiva, kazhinji kakawanda (semuenzaniso ka1000 kana 10.000), nokudaro ichipa fungidziro yekugoverwa kwemuongorori.
Kazhinji, bootstrap inochinjika uye inowanzova yakarurama pamatambudziko akaomarara, asi jackknife iri nyore uye inodhura zvishoma pakuverenga. Pamadata makuru, jackknife inogona kuva imwe nzira inokurumidza yekuwana zvikanganiso zvakajairwa, kunyanya kana kuverenga estimator kuchidhura asi kuchigoneka nguva n.
Mabhenefiti enzira yejackknife
Zvimwe zvakanakira jackknife zvinosanganisira:
1. Zviri nyore uye zviri nyore kushandisa
Pfungwa yekuti munhu abve pane chimwe chinhu inonzwisisika, uye fomura yekuti munhu asiyane iri nyore.
2. Zvishoma zvekufungidzira nezvekugoverwa
Jackknife haidi nguva dzose kufungidzira kuti zvinhu zvakajairika kana kuti fomu rekugovera chokwadi.
3. Inoshanda zvakanaka pakuverenga mamwe
Nekuti zvinongoda nguva n dzekuverenga estimator, jackknife inowanzo kuve yakareruka kupfuura bootstrapping iyo inoda zviuru zvekudzokorora.
4. Inobatsira pakufungidzira rusaruro
Kunyanya muzviyereso zvisiri zvemutsara izvo zvinowanzova zvisiri nyore kuverenga nekuongorora.
Zvipingamupinyi nezvinhu zvekuchenjerera
Kunyange zvazvo banga racho rine simba, rine zvarinogumira:
1. Hazvina kunyatsojeka kune vanoongorora zvisina kutsetseka
Semuenzaniso, pakati kana kuti quantiles mune mamwe mamiriro ezvinhu, kana nhamba dzinoenderana nehukuru hwakanyanya, jackknife dzimwe nguva inopa fungidziro dzisina kunyatsojeka dzemusiyano.
2. Hazviwanzo kukodzera data rine zvinhu zvinoenderana naro
Munhevedzano yenguva kana data renzvimbo, kucherechedza hakuna kuzvimiririra. Kubvisa poindi imwe chete kunogona kuputsa chimiro chekuvimba. Pazviitiko zvakaita seizvi, misiyano yakaita se block jackknife (kubvisa block imwe yedata panguva imwe chete) inoshandiswa.
3. Inonzwa zvakanyanya kana paine zvinhu zvinokanganisa zvakanyanya
Kana paine data risingashande kana kuti "rinoshandiswa", fungidziro yekuti munhu haashande inogona kuchinja zvakanyanya. Izvi hazvisi nguva dzose kushaya simba—kutaura chokwadi, zvinogona kuva chiratidzo chakakosha—asi musiyano unobuda unogona kuva mukuru uye unoda kududzirwa kwakanyatsonaka.
4. Kukwanisa kukura pahukuru hwakakura kwazvo
Kunyange zvazvo yakachipa pane bootstrapping, jackknife ichiri kuda kuongororwa kwen estimator. Kana n iri mumamiriyoni uye estimators ichidhura, izvi zvinogona kuva dambudziko.
Kusiyana: delete-d jackknife uye block jackknife
Kunze kwekusiya-kubuda kamwe chete, kune misiyano:
– Delete-d jackknife: inobvisa kucherechedza kwe d pakukopa (panzvimbo pe1 chete). Izvi zvinogona kuvandudza kururama mune mamwe mamiriro ezvinhu, kunyanya kune vanoongorora vasina kutsetseka.
– Block jackknife: inobvisa block ine zvakati wandei zviri pedyo, zvakakodzera data rine autocorrelation (semuenzaniso data rezuva nezuva, revhiki nevhiki, kana renzvimbo).
Sarudzo yehukuru hwe d kana block inoenderana nechimiro chedata uye chinangwa chekufungidzira.
Kushandisa jackknife mukuita
Jackknife inoshandiswa mumhando dzakasiyana-siyana:
– Biostatistics ne epidemiology: kufungidzira zvikanganiso zvakajairwa zvezviyero zvengozi kana ma parameter emuenzaniso kana mafomura ekuongorora akaoma.
– Econometrics: kuongorora kugadzikana kwema parameter, kunyanya mumienzaniso mishoma.
– Sainzi yekombuta nekudzidza kwemuchina: pfungwa yekuti munhu arege kuita chimwe chinhu ine chekuita nekuongorora zvinhu zvakasiyana, kunyangwe zvinangwa zvacho zvakasiyana (kufungidzira kuti munhu afanotaura chokwadi chechinhu ichi kana kuti kuenzanisa kuti zvinhu zvinofamba sei).
– Ecology neongororo: kufungidzira kusiyana-siyana kana zvimwe zviratidzo uye kusava nechokwadi kwenhamba dzakaoma.
Penutup
Nzira yejackknife inzira yekare yekuenzanisa mhedzisiro ichiri kushanda nanhasi. Nekushandisa pfungwa iri nyore—kusiya ongororo imwe chete uye kuverengazve estimator—jackknife inogona kupa fungidziro yekusiyana, kukanganisa kwakajairwa, uye kusarura pasina kuverenga kwakaoma kwemasvomhu. Zvisinei, kushandiswa kwayo kunoda kutariswa kwehunhu hweestimator, saizi yemuenzaniso, uye chimiro chedata. Mukuita, jackknife inowanzo kuve sarudzo inokurumidza uye yakajeka, kana sekuwedzera pakushandisa nzira dzakasimba dzekuenzanisa mhedzisiro dzakadai sebootstrapping.
Kana muchida, ndinogona zvakare kuwedzera muenzaniso mudiki wekuverenga nhamba (semuenzaniso wekubatanidza kana kudzokorora) kana kusanganisira jackknife implementation muR/Python kuti ndijekese application.
