ʻO ke ʻano hana Jackknife ma ka helu helu
He ʻano hana resampling koʻikoʻi ke ʻano jackknife i nā helu helu, ʻoiai no ke ana ʻana i ka maopopo ʻole o kahi kuhi. Hoʻohana pinepine ʻia ka jackknife e kuhi i ka bias a me ka variance o kahi estimator, a me ke kūkulu ʻana i nā ana pololei e like me ka hewa maʻamau. He maʻalahi kēia ʻano hana, ʻaʻole koi i nā kuhi hoʻolaha koʻikoʻi loa, a hiki ke hoʻopili ʻia i nā pilikia like ʻole, mai nā helu helu kuʻuna a hiki i ka loiloi ʻikepili hou.
Ka Moolelo a me nā Manaʻo Kumu
Ua hoʻolauna ʻia ka jackknife e Maurice Quenouille a ma hope ua hoʻolaha ʻia e John Tukey. Ua hoʻoulu ʻia ka inoa "jackknife" e kahi pahi ʻeke hiki ke hoʻohana ʻia, no ka mea, he maʻalahi ke ʻano hana a hiki ke hoʻohana ʻia i nā ʻano like ʻole. ʻO ke kumu nui kēia: inā loaʻa iā mākou kahi hāpana o ka nui n, hana mākou i kekahi mau "dummy samples" ma ka wehe ʻana i hoʻokahi nānā ʻana i kēlā me kēia manawa, a laila e helu hou i ka mea kuhi ma kēlā me kēia hāpana. Ma ka nānā ʻana i ke ʻano o ka loli ʻana o ka mea kuhi ke wehe ʻia kekahi nānā ʻana, loaʻa iā mākou ka ʻike i ke kūpaʻa o ka mea kuhi ma luna o ka loli o ka ʻikepili.
Eia kekahi laʻana, ke manaʻo nei he ʻikepili kā mākou \(x_1, x_2, \dots, x_n\) a makemake mākou e kuhi i kahi palena \(\theta\) me ka hoʻohana ʻana i ka mea kuhi \( \hat{\theta}=t(x_1,\dots,x_n)\). Ma ka jackknife, hana mākou i n mau subsamples o ka nui \(n-1\), ʻo ia hoʻi ka \(i\)th subsample e holoi ana iā \(x_i\). A laila helu mākou:
\[
\hat{\theta}_{(i)} = t(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)
\]
Ua kapa ʻia ka waiwai \(\hat{\theta}_{(i)}\) ʻo ka kuhi haʻalele-hoʻokahi-waho.
Nā ʻanuʻu hana Jackknife
Ma ke kaʻina hana, hiki ke wehewehe ʻia ke kākā ʻokiʻoki ma nā ʻanuʻu aʻe:
1. E helu i ka mea kuhi ma ka ʻikepili piha
E helu i ka \(\hat{\theta}\) ma luna o ka hāpana holoʻokoʻa.
2. E hana i nā laʻana liʻiliʻi waiho-hoʻokahi-waho
No kēlā me kēia \(i = 1,2,\dots,n\), e wehe i ka nānā ʻana \(x_i\) a e helu i ka mea kuhi \(\hat{\theta}_{(i)}\).
3. E helu i ka awelika o ka mea kuhi jackknife
ʻAwelike haʻalele hoʻokahi:
\[
\bar{\theta}_{(\cdot)} = \frac{1}{n}\sum_{i=1}^n \hat{\theta}_{(i)}
\]
4. Kuhi ʻokoʻa (a i ʻole hewa maʻamau)
Hoʻomaulia pinepine ʻia ka ʻokoʻa jackknife e:
\[
\widehat{\mathrm{Var}}_{J}(\hat{\theta}) = \frac{n-1}{n}\sum_{i=1}^n \left(\hat{\theta}_{(i)} – \bar{\theta}_{(\cdot)}\right)^2
\]
ʻO ke kuhihewa maʻamau ke kumu huinaha o ka ʻokoʻa.
5. Ka loiloi ʻana i ka manaʻo pāʻokoʻa a me ka hoʻoponopono ʻana i ka manaʻo pāʻokoʻa (koho)
Hiki iā Jackknife ke kuhi i ka manaʻo pāʻokoʻa ma o:
\[
\widehat{\mathrm{Bias}}_{J}(\hat{\theta}) = (n-1)\left(\bar{\theta}_{(\cdot)} – \hat{\theta}\right)
\]
Hiki ke hana i ka hoʻoponopono bias ma o:
\[
\hat{\theta}_{J} = \hat{\theta} – \widehat{\mathrm{Bias}}_{J}(\hat{\theta})
\]
Wehewehena: inā ʻokoʻa ka awelika waiho-hoʻokahi-waho mai ka mea kuhi piha ma ke ʻano ʻōnaehana, aia kahi hōʻailona o ka manaʻo pāʻokoʻa hiki ke hoʻoponopono ʻia.
Laʻana maʻalahi: awelika laʻana
No ka hoʻomaopopo pono ʻana i ke ʻano o ka jackknife, e noʻonoʻo i ka mea hoʻohālikelike awelika laʻana:
\[
\hat{\mu} = \frac{1}{n}\sum_{i=1}^n x_i
\]
Inā mākou e wehe i hoʻokahi nānā ʻana \(x_i\), lilo ka awelika i:
\[
\hat{\mu}_{(i)} = \frac{1}{n-1}\sum_{j\ne i} x_j
\]
I ke ʻano o nā awelika, ʻaʻole hāʻawi ka jackknife i kahi "kahaha" nui no ka mea ua paʻa ka awelika a ua liʻiliʻi ka bias (ma nā ʻano he nui). Eia nō naʻe, no nā mea kuhi paʻakikī—e like me ka median, kahi coefficient regression kikoʻī, kahi pilina, a i ʻole kahi helu helu nonlinear—hiki i ka loli i loaʻa mai ka wehe ʻana i kahi kiko ʻikepili hoʻokahi ke hōʻike i ka ʻike o ka mea kuhi a hana i kahi kuhi pono o kāna hewa maʻamau.
Pseudovalue: he manaʻo koʻikoʻi ma ka jackknife
Ma kekahi mau kūkākūkā ʻana, hoʻolauna ʻo jackknife i kahi pseudovalue no kēlā me kēia nānā ʻana:
\[
\theta_i^{ } = n\hat{\theta} – (n-1)\hat{\theta}_{(i)}
\]
A laila hiki ke kākau ʻia ka mea kuhi jackknife ma ke ʻano he awelika o nā pseudovalues:
\[
\hat{\theta}_{J} = \frac{1}{n}\sum_{i=1}^n \theta_i^{ }
\]
Kōkua ke ʻano pseudovalue i ka wehewehe ʻana pehea e "hāʻawi" ai kēlā me kēia nānā ʻana i ka manaʻo hope loa a hoʻomaʻalahi i ka nānā ʻana i ka manaʻo pāʻonioni.
ʻO ka pilina ma waena o ka jackknife a me ka bootstrap
Hoʻohālikelike pinepine ʻia ʻo Jackknife me ka bootstrap, ʻoiai he mau ʻano hana resampling ʻelua. Eia nō naʻe, aia nā ʻokoʻa koʻikoʻi:
– Hoʻohana ʻo Jackknife i ka subsampling ma ka wehe ʻana i hoʻokahi ʻikepili (waiho-hoʻokahi-waho). He deterministic ka helu o nā replications: pololei ʻo n.
- Hoʻokumu ʻo Bootstrapping i kahi resample me ka hoʻololi ʻana, maʻamau i nā manawa he nui (e.g. 1000 a i ʻole 10.000 mau manawa), no laila e hāʻawi ana i kahi kuhi o ka hoʻolaha empirical o ka estimator.
Ma keʻano laulā, ʻoi aku ka maʻalahi o ka bootstrap a ʻoi aku ka pololei no nā pilikia paʻakikī, akā ʻoi aku ka maʻalahi o ka jackknife a emi ke kumukūʻai o ka helu ʻana. Ma nā ʻikepili nui, hiki i ka jackknife ke lilo i koho wikiwiki no ka loaʻa ʻana o nā hewa maʻamau, ʻoiai ke kumukūʻai o ka helu ʻana i ka mea kuhi akā hiki nō ke hana i nā manawa n.
Nā Pōmaikaʻi o ke ʻano hana jackknife
ʻO kekahi o nā pono o kahi jackknife:
1. Maʻalahi a maʻalahi hoʻi e hoʻokō
He mea maʻalahi ke ʻano o ka waiho-hoʻokahi-waho, a he maʻalahi ke ʻano hoʻohālikelike.
2. Nā kuhi hoʻolaha liʻiliʻi
ʻAʻole koi mau ʻo Jackknife i ka manaʻo o ke ʻano maʻamau a i ʻole ke ʻano hoʻolaha kikoʻī.
3. Kūpono no kekahi mau helu ʻana
Ma muli o ka mea e pono ai nā manawa n o nā helu estimator wale nō, ʻoi aku ka māmā o ka jackknife ma mua o ka bootstrapping e koi ai i nā tausani o nā replications.
4. Pono no ka loiloi pānaʻi
ʻOi loa aku hoʻi i nā mea kuhi nonlinear ʻaʻole maʻalahi ka helu ʻana ma ke ʻano analytical.
Nā palena a me nā mea e makaʻala ai
ʻOiai he ikaika, he mau palena ko ka jackknife:
1. ʻAʻole pololei loa no nā mea kuhi ʻaʻole maʻalahi loa
No ka laʻana, ka median a i ʻole nā quantiles ma kekahi mau kūlana, a i ʻole nā helu helu e hilinaʻi nei i nā waiwai koʻikoʻi, hāʻawi pinepine ka jackknife i nā kuhi kūpono ʻole o ka variance.
2. ʻAʻole kūpono mau no ka ʻikepili me nā mea hilinaʻi
I loko o ka ʻikepili manawa a i ʻole ka ʻikepili spatial, ʻaʻole kūʻokoʻa nā nānā ʻana. ʻO ka wehe ʻana i hoʻokahi kiko hiki ke uhaki i ka ʻōnaehana hilinaʻi. No nā hihia e like me kēia, hoʻohana ʻia nā ʻano like ʻole e like me ka block jackknife (wehe i hoʻokahi poloka ʻikepili i ka manawa).
3. ʻIke i nā nānā ʻana i nā hopena koʻikoʻi
Inā he mau outliers a i ʻole ka ʻikepili "leveraged", hiki ke loli nui ka manaʻo haʻalele-hoʻokahi-out. ʻAʻole kēia he nāwaliwali mau—ʻoiai, hiki ke lilo i hōʻailona koʻikoʻi—akā hiki ke nui ka ʻokoʻa hopena a pono e wehewehe pono ʻia.
4. Ka hiki ke hoʻonui ʻia ma ka nui loa n
ʻOiai ʻoi aku ka liʻiliʻi o ke kumukūʻai ma mua o ka bootstrapping, koi mau ʻo jackknife i nā loiloi estimator n. Inā aia ʻo n i loko o nā miliona a he pipiʻi nā estimators, hiki i kēia ke lilo i mea pilikia.
Nā ʻano like ʻole: holoi-d jackknife a me ka poloka jackknife
Ma waho aʻe o ka haʻalele-hoʻokahi-waho, aia nā ʻano like ʻole:
– Delete-d jackknife: holoi i nā nānā ʻana d no kēlā me kēia hana hou ʻana (ma kahi o 1 wale nō). Hiki i kēia ke hoʻomaikaʻi i ka pololei i kekahi mau kūlana, ʻoi aku hoʻi no nā mea kuhi ʻaʻole laumania.
– ʻO ka ʻoki palaka poloka: wehe i kahi poloka e loaʻa ana kekahi mau nānā ʻana e pili kokoke ana, kūpono no ka ʻikepili nona ka autocorrelation (e.g. ʻikepili i kēlā me kēia lā, hebedoma, a i ʻole ka ʻikepili spatial).
ʻO ke koho ʻana o ka d a i ʻole ka nui o ka poloka e pili ana i ka hoʻonohonoho ʻikepili a me ka pahuhopu inference.
Ka hoʻohana ʻana o ka jackknife ma ka hana
Hoʻohana ʻia ka Jackknife ma nā ʻano like ʻole:
– Biostatistics a me epidemiology: ke kuhi ʻana i nā hewa maʻamau no nā ana pilikia a i ʻole nā palapala hoʻohālike ke paʻakikī nā ʻano loiloi.
– Econometrics: loiloi o ke kūpaʻa o nā palena, ʻoi aku hoʻi i nā laʻana i kaupalena ʻia.
– ʻEpekema kamepiula a me ke aʻo ʻana i ka mīkini: pili loa ka manaʻo haʻalele-hoʻokahi-waho i ka hōʻoia kea, ʻoiai ʻokoʻa nā pahuhopu (hōʻoia wānana vs kuhi pololei o ka palena).
– ʻEkolojia a me nā anamanaʻo: ka loiloi ʻana o ka ʻokoʻa a i ʻole kekahi mau ʻōkuhi a me ka maopopo ʻole o nā helu helu paʻakikī.
Pani
ʻO ke ʻano hana jackknife kahi ʻano hana resampling maʻamau e pili mau nei i kēia lā. Ma ka hoʻohana ʻana i kahi manaʻo maʻalahi—me ka haʻalele ʻana i hoʻokahi nānā ʻana a me ka helu hou ʻana i ka mea kuhi—hiki i ka jackknife ke hāʻawi i nā kuhi o ka variance, ka hewa maʻamau, a me ka bias me ka ʻole o nā helu makemakika paʻakikī. Eia nō naʻe, pono ka noʻonoʻo ʻana i ke ʻano o ka mea kuhi, ka nui o ka laʻana, a me ke ʻano hilinaʻi o ka ʻikepili i kona hoʻohana ʻana. I ka hana maoli, ʻo ka jackknife pinepine kahi koho wikiwiki a moakāka, a i ʻole he mea hoʻohui i ka hoʻohana ʻana i nā ʻano hana resampling ikaika e like me ka bootstrapping.
Inā makemake ʻoe, hiki iaʻu ke hoʻohui i kahi laʻana helu helu liʻiliʻi (e.g. no ka pilina a i ʻole ka regression) a i ʻole e hoʻokomo i kahi hoʻokō jackknife ma R/Python e wehewehe i ka noi.
