Njira ya Jackknife mu Ziwerengero
Njira ya jackknife ndi njira yofunika kwambiri yowerengera ziwerengero, makamaka poyesa kusatsimikizika kwa kuyerekezera. Jackknife nthawi zambiri imagwiritsidwa ntchito kuwerengera kusagwirizana ndi kusiyana kwa wowerengera, komanso kupanga miyeso yolondola monga cholakwika chokhazikika. Njirayi ndi yosavuta, sifunikira malingaliro okhwima kwambiri ogawa, ndipo ingagwiritsidwe ntchito pamavuto osiyanasiyana, kuyambira ziwerengero zakale mpaka kusanthula deta yamakono.
Chiyambi ndi malingaliro oyambira
Mpeni wa jack unayambitsidwa ndi Maurice Quenouille ndipo pambuyo pake unatchuka ndi John Tukey. Dzina lakuti "jackknife" linachokera ku mpeni wa m'thumba wosiyanasiyana, chifukwa njirayo ndi yosinthasintha ndipo ingagwiritsidwe ntchito m'njira zosiyanasiyana. Lingaliro lofunikira ndi ili: ngati tili ndi chitsanzo cha kukula n, timapanga "zitsanzo zopusa" zingapo pochotsa kuwona kamodzi nthawi iliyonse, kenako nkuwerengeranso woyerekeza pa chitsanzo chilichonse. Mwa kuwona momwe woyerekeza amasinthira pamene kuwona kamodzi kachotsedwa, timapeza chidziwitso cha kukhazikika kwa woyerekeza pakusintha kwa deta.
Mwachitsanzo, tiyerekeze kuti tili ndi deta \(x_1, x_2, \madontho, x_n\) ndipo tikufuna kuyerekeza parameter \(\theta\) pogwiritsa ntchito estimator \( \hat{\theta}=t(x_1,\madontho,x_n)\). Mu jackknife, timapanga zitsanzo zazing'ono za n za kukula \(n-1\), zomwe ndi chitsanzo chaching'ono cha \(i\)th chomwe chimachotsa \(x_i\). Kenako timawerengera:
\[
\hat{\theta}_{(i)} = t(x_1,\madontho,x_{i-1},x_{i+1},\madontho,x_n)
\]
Mtengo \(\hat{\theta}_{(i)}\) umatchedwa leave-one-out estimate .
Njira zogwiritsira ntchito Jackknife
Mwa njira, jackknife ikhoza kufotokozedwa m'njira zotsatirazi:
1. Werengani wowerengera deta yonse
Werengerani \(\hat{\theta}\) pa chitsanzo chonse.
2. Pangani zitsanzo zazing'ono za n leave-one-out
Pa \(i = 1,2,\madontho,n\), chotsani zomwe zawonedwa \(x_i\) ndikuwerengera wowerengera \(\hat{\theta}_{(i)}\).
3. Werengerani avareji ya chowerengera cha jackknife
Avereji ya kuchotsedwa kamodzi:
\[
\bar{\theta}_{(\cdot)} = \frac{1}{n}\sum_{i=1}^n \hat{\theta}_{(i)}
\]
4. Yerekezerani kusiyana (kapena cholakwika chokhazikika)
Kusiyana kwa jackknife nthawi zambiri kumawerengedwa ndi:
\[
\widehat{\mathrm{Var}}_{J}(\hat{\theta}) = \frac{n-1}{n}\sum_{i=1}^n \left(\hat{\theta}_{(i)} – \bar{\theta}_{(\cdot)}\right)^2
\]
Cholakwika chokhazikika ndi muzu wa kusiyana.
5. Kuyerekeza tsankho ndi kukonza tsankho (ngati mukufuna)
Jackknife imathanso kuwerengera tsankho kudzera mu:
\[
\widehat{\mathrm{Bias}}_{J}(\hat{\theta}) = (n-1)\left(\bar{\theta}_{(\cdot)} - \hat{\theta}\right)
\]
Kukonza tsankho kungachitike ndi:
\[
\chipewa{\theta}_{J} = \chipewa{\theta} – \widehat{\mathrm{Bias}}_{J}(\chipewa{\theta})
\]
Kutanthauzira: ngati avhareji ya leave-one-out ikusiyana ndi yowerengera yonse mwadongosolo, pali chizindikiro cha tsankho lomwe lingakonzedwe.
Chitsanzo chodziwikiratu: chitsanzo chapakati
Kuti mumvetse bwino jackknife, ganizirani chitsanzo cha mean estimator:
\[
\hat{\mu} = \frac{1}{n}\sum_{i=1}^n x_i
\]
Ngati tichotsa chinthu chimodzi chomwe chawonedwa \(x_i\), avhareji imakhala:
\[
\hat{\mu}_{(i)} = \frac{1}{n-1}\sum_{j\ne i} x_j
\]
Pankhani ya avareji, jackknife sipereka "kudabwitsa" kwakukulu chifukwa avareji ndi yokhazikika ndipo bias ndi yaying'ono (m'malo ambiri). Komabe, kwa owerengera ovuta kwambiri—monga apakati, coefficient inayake ya regression, correlation, kapena nonlinear statistical—kusintha komwe kumachitika chifukwa chochotsa mfundo imodzi ya data kungawulule kukhudzika kwa owerengera ndikupanga kuyerekezera kothandiza kwa cholakwika chake chokhazikika.
Pseudovalue: lingaliro lofunika kwambiri mu jackknife
Mu zokambirana zina, jackknife imayambitsa chizindikiro cha chinthu china pa zomwe zawonedwa:
\[
\theta_i^{ } = n\hat{\theta} – (n-1)\hat{\theta}_{(i)}
\]
Kenako chowerengera cha jackknife chingalembedwe ngati avareji ya ma pseudovalues:
\[
\hat{\theta}_{J} = \frac{1}{n}\sum_{i=1}^n \theta_i^{ }
\]
Njira ya pseudovalue imathandiza kufotokoza momwe kuwona kulikonse "kumathandizira" ku kuyerekezera komaliza ndikuthandizira kusanthula kwa tsankho.
Ubale pakati pa jackknife ndi bootstrap
Jackknife nthawi zambiri imayerekezeredwa ndi bootstrap, chifukwa zonse ziwiri ndi njira zosinthira zitsanzo. Komabe, pali kusiyana kwakukulu:
– Jackknife amagwiritsa ntchito subsampling pochotsa deta imodzi (kusiya imodzi). Chiwerengero cha ma replications ndi chotsimikizika: ndendende n .
- Bootstrapping imapanga chitsanzo chosinthidwa ndi chosinthidwa, nthawi zambiri (monga nthawi 1000 kapena 10.000), motero imapereka kuyerekezera kwa kugawa kwa estimator.
Kawirikawiri, bootstrap imakhala yosinthasintha komanso nthawi zambiri imakhala yolondola kwambiri pamavuto ovuta, koma jackknife ndi yosavuta komanso yotsika mtengo kwambiri. Pa ma data akuluakulu, jackknife ikhoza kukhala njira ina yofulumira yopezera zolakwika zokhazikika, makamaka pamene kuwerengera koyerekeza kumakhala kokwera mtengo koma nthawi zina n ndizotheka.
Ubwino wa njira ya jackknife
Ubwino wina wa jackknife ndi monga:
1. Zosavuta komanso zosavuta kugwiritsa ntchito
Lingaliro la leave-one-out ndi losavuta kumva, ndipo njira yosinthira ndi yosavuta.
2. Malingaliro ochepa okhudza kugawa
Jackknife si nthawi zonse imafuna kuganiza kuti ndi yachibadwa kapena mawonekedwe enaake ogawa.
3. Yogwira ntchito bwino pa mawerengedwe ena
Chifukwa zimangofunika nthawi n zokha zowerengera zowerengera, jackknife nthawi zambiri imakhala yopepuka kuposa bootstrapping yomwe imafuna masauzande ambiri.
4. Zothandiza pakuyerekeza tsankho
Makamaka mu zowerengera zopanda mzere zomwe nthawi zambiri zimakhala zovuta kuziwerengera mozama.
Zofooka ndi zinthu zoti muzisamala nazo
Ngakhale kuti mpeni wa jackknife ndi wamphamvu, uli ndi zofooka zake:
1. Zosalondola kwenikweni kwa owerengera osakhwima kwambiri
Mwachitsanzo, apakatikati kapena ma quantiles m'mikhalidwe ina, kapena ziwerengero zomwe zimadalira kwambiri, jackknife nthawi zina imapereka ziwerengero zosalondola kwenikweni za kusiyana.
2. Sizoyenera nthawi zonse pa deta yokhala ndi zodalira
Mu mndandanda wa nthawi kapena deta ya malo, zowonera sizidziyimira pawokha. Kuchotsa mfundo imodzi kungaswe kapangidwe ka kudalira. Pazochitika ngati izi, mitundu yosiyanasiyana monga block jackknife (kuchotsa block imodzi ya data nthawi imodzi) imagwiritsidwa ntchito.
3. Kuzindikira zinthu zomwe zingakhudze kwambiri
Ngati pali deta yosiyana kapena "yogwiritsidwa ntchito", kuyerekezera kwa leave-one-out kungasinthe kwambiri. Izi sizili nthawi zonse zofooka—kwenikweni, zitha kukhala chizindikiro chofunikira—koma kusiyana komwe kumachitika kungakhale kwakukulu ndipo kumafuna kutanthauzira mosamala.
4. Kukula kwakukulu kwambiri
Ngakhale kuti mtengo wake ndi wotsika kuposa bootstrapping, jackknife imafunabe kuwunika kwa n estimator. Ngati n ili mu mamiliyoni ndipo owerengera ndi okwera mtengo, izi zitha kukhala zovuta.
Kusiyana: delete-d jackknife ndi block jackknife
Kupatulapo kusiya chimodzi, palinso mitundu yosiyanasiyana:
– Delete-d jackknife: imachotsa mawonedwe a d pa replication iliyonse (m'malo mwa 1 yokha). Izi zitha kupititsa patsogolo kulondola pazochitika zina, makamaka kwa owerengera osasalala.
– Block jackknife: imachotsa block yomwe ili ndi mawonedwe angapo oyandikana nawo, oyenera deta yomwe ili ndi kulumikizana kwa auto (monga deta ya tsiku ndi tsiku, sabata iliyonse, kapena malo).
Kusankha d kapena kukula kwa block kumadalira kapangidwe ka deta ndi cholinga cha inference.
Kugwiritsa ntchito jackknife pochita
Jackknife imagwiritsidwa ntchito m'magawo osiyanasiyana:
- Biostatistics ndi epidemiology: kuyerekezera zolakwika zokhazikika pa miyeso ya zoopsa kapena magawo a chitsanzo pamene njira zowunikira zili zovuta.
– Econometrics: kuwunika kukhazikika kwa magawo, makamaka m'zitsanzo zochepa.
– Sayansi ya makompyuta ndi kuphunzira kwa makina: lingaliro loti munthu asagwiritse ntchito njira imodzi limagwirizana kwambiri ndi kutsimikizira zinthu, ngakhale kuti zolinga zake ndi zosiyana (kutsimikizira kuneneratu poyerekeza ndi kuyerekezera kulondola kwa magawo).
– Zachilengedwe ndi kafukufuku: kuyerekezera kusiyanasiyana kapena zizindikiro zina ndi kusatsimikizika kwa ziwerengero zovuta.
Kutseka
Njira ya jackknife ndi njira yakale yosinthira masampulo yomwe ikugwirabe ntchito masiku ano. Pogwiritsa ntchito lingaliro losavuta—kusiya kuwona kamodzi ndikuwerengeranso choyerekeza—jackknife imatha kupereka ziyerekezo za kusiyana, zolakwika wamba, ndi tsankho popanda kuwerengera kovuta kwa masamu. Komabe, kugwiritsidwa ntchito kwake kumafuna kuganizira za mtundu wa choyerekeza, kukula kwa chitsanzo, ndi kapangidwe ka deta. Mwachizolowezi, jackknife nthawi zambiri imakhala njira yachangu komanso yowonekera, kapena yowonjezera kugwiritsa ntchito njira zolimba zosinthira masampulo monga bootstrapping.
Ngati mukufuna, nditha kuwonjezera chitsanzo chaching'ono chowerengera manambala (monga kulumikizana kapena kubwereza) kapena kuphatikiza kukhazikitsa kwa jackknife mu R/Python kuti ndifotokoze bwino momwe pulogalamuyi ikuyendera.
