Tikanga Jackknife i roto i ngā Tatauranga
He tikanga tauira anō nui te tikanga jackknife i roto i ngā tatauranga, inā koa mō te ine i te koretake o tētahi whakatau tata. He maha ngā wā ka whakamahia te jackknife hei whakatau tata i te pānga me te rerekētanga o tētahi tātai tata, tae atu ki te hanga i ngā inenga tika pērā i te hapa paerewa. He tino māmā tēnei tikanga, kāore e hiahiatia he whakapae tohatoha tino pakari, ā, ka taea te whakamahi ki te whānuitanga o ngā raruraru, mai i ngā tatauranga tawhito ki te tātari raraunga hou.
Papamuri me ngā ariā taketake
Nā Maurice Quenouille i whakauru te maripi tākihi, ā, nō muri mai ka rongonuihia e John Tukey. I ahu mai te ingoa "jackknife" i te maripi pōke maha-whakamahinga, nā te mea he ngāwari te tikanga, ā, ka taea te whakamahi i roto i ngā horopaki maha. Ko te whakaaro matua ko tēnei: mēnā he tauira kei a tātou te rahi n, ka hangaia e tātou ētahi "tauira rūpahu" mā te tango i tētahi kitenga i ia wā, kātahi ka tatau anō i te tātai i runga i ia tauira. Mā te tirotiro i te huringa o te tātai ina tangohia tētahi kitenga, ka whiwhi tātou i te māramatanga ki te pumau o te tātai i runga i te rerekētanga o ngā raraunga.
Hei tauira, me kī kei a tātou ngā raraunga \(x_1, x_2, \dots, x_n\) ā, e hiahia ana tātou ki te whakatau tata i tētahi tawhā \(\theta\) mā te whakamahi i te whakatau tata \( \hat{\theta}=t(x_1,\dots,x_n)\). I roto i te jackknife, ka hangaia e tātou he n tauira iti o te rahi \(n-1\), arā, ko te tauira iti \(i\)th e muku ana i te \(x_i\). Kātahi ka tatauhia e tātou:
\[
\hat{\theta}_{(i)} = t(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)
\]
Ko te uara \(\hat{\theta}_{(i)}\) ka kiia ko te whakatau tata waiho-kotahi-ki-waho.
Ngā taahiraa o te tikanga Jackknife
I roto i te tukanga, ka taea te whakamārama i te maripi mā roto i ngā mahi e whai ake nei:
1. Tātaihia te tātai i runga i ngā raraunga katoa
Tātaihia \(\hat{\theta}\) te tauira katoa.
2. Waihangahia he tauira iti kāore e hiahiatia he kotahi
Mō ia \(i = 1,2,\dots,n\), tangohia te kitenga \(x_i\) ka tatau i te tātai \(\hat{\theta}_{(i)}\).
3. Tātaihia te toharite o te tātaitanga maripi-kākā
Kotahi te wā whakatā toharite:
\[
\bar{\theta}_{(\cdot)} = \frac{1}{n}\sum_{i=1}^n \hat{\theta}_{(i)}
\]
4. Whakatau i te rerekētanga (te hapa paerewa rānei)
Ko te rerekētanga o te maripi kau ka tatauhia i te nuinga o te wā me:
\[
\whānuipōtae{\mathrm{Var}}_{J}(\hat{\theta}) = \frac{n-1}{n}\sum_{i=1}^n \left(\hat{\theta}_{(i)} – \bar{\theta}_{(\cdot)}\right)^2
\]
Ko te hapa paerewa ko te pūtake tapawhā o te rerekētanga.
5. Te aromatawai i te pānga me te whakatikatika i te pānga (whiringa)
Ka taea hoki e Jackknife te whakatau tata i te whakaaro pākiki mā te:
\[
\widehat{\mathrm{Bias}}_{J}(\hat{\theta}) = (n-1)\maui(\bar{\theta}_{(\cdot)} – \hat{\theta}\right)
\]
Ka taea te whakatikatika i te taha whakarara mā te:
\[
\hat{\theta}_{J} = \hat{\theta} – \widehat{\mathrm{Bias}}_{J}(\hat{\theta})
\]
Te Whakamārama: mēnā he rerekē te toharite waiho-kotahi-i-waho i te tātaitanga katoa, he tohu tēnei o te whakaaro hē ka taea te whakatika.
Tauira māmā: toharite tauira
Hei mārama ake ki te jackknife, whakaarohia te tauira toharite whakatau tata:
\[
\hat{\mu} = \frac{1}{n}\sum_{i=1}^n x_i
\]
Ki te tangohia tētahi kitenga \(x_i\), ka noho te toharite hei:
\[
\hat{\mu}_{(i)} = \frac{1}{n-1}\sum_{j\ne i} x_j
\]
Mō ngā toharite, kāore te jackknife e whakarato i tētahi "ohorere" nui nā te mea he pumau te toharite, ā, he iti te pānga (i roto i ngā horopaki maha). Heoi, mō ngā tātaitanga uaua ake—pērā i te tau waenga, i tētahi taunga whakatautau motuhake, i tētahi hononga, i tētahi tatauranga kore-raina rānei—ka taea e te huringa e puta mai ana i te tango i tētahi pūwāhi raraunga kotahi te whakaatu i te aro o te tātaitanga me te whakaputa i tētahi tātaitanga whai hua mō tōna hapa paerewa.
Pseudovalue: he ariā nui i roto i te jackknife
I ētahi kōrero, ka whakaurua e jackknife he uara rūpahu mō ia kitenga:
\[
\theta_i^{ } = n\hat{\theta} – (n-1)\hat{\theta}_{(i)}
\]
Kātahi ka taea te tuhi i te tātaitanga jackknife hei toharite o ngā uara rūpahu:
\[
\hat{\theta}_{J} = \frac{1}{n}\sum_{i=1}^n \theta_i^{ }
\]
Mā te huarahi uara rūpahu ka whakamārama i te "whai wāhi" o ia kitenga ki te whakatau tata whakamutunga, ā, ka āwhina hoki i te tātaritanga pīnakitanga.
Te whanaungatanga i waenga i te jackknife me te bootstrap
He maha ngā wā ka whakaritea a Jackknife ki te bootstrap, nā te mea he tikanga tauira anō ēnei e rua. Heoi anō, he rerekētanga nui ēnei:
– Ka whakamahia e Jackknife te tauira iti mā te tango i tētahi raraunga (waiho-kotahi-i waho). He mea whakatau te maha o ngā tāruarua: ko te n tonu.
– Mā te Bootstrapping ka hangaia he tauira anō me te whakakapinga, he maha ngā wā (hei tauira, 1000, 10.000 rānei), ā, ka puta he whakatau tata mō te tohatoha ā-wheako o te kaiwhakatau tata.
I te nuinga o te wā, he ngāwari ake te whakamahi i te bootstrap, ā, he maha ngā wā he tika ake mō ngā raruraru uaua, engari he māmā ake te whakamahi i te jackknife, ā, he iti ake te utu tātai. I runga i ngā huinga raraunga nui, he huarahi tere te whakamahi i te jackknife hei tiki hapa paerewa, inā koa he utu nui te tātai i te tātai engari he taea tonu te whakamahi i ngā wā n.
Ngā painga o te tikanga jackknife
Ko ētahi o ngā painga o te maripi koi ko:
1. Māmā, ngāwari hoki te whakatinana
He māmā noa te ariā waiho-kotahi-i waho, ā, he māmā noa iho te tātai rerekētanga.
2. Ngā whakapae tohatoha iti
Kāore a Jackknife e hiahia ana i ngā wā katoa ki te whakaaro he mea noa, he mea rānei puka tohatoha tino
3. Whai hua mō ētahi tātaitanga
Nā te mea e hiahiatia ana he n ngā wā o ngā tātaitanga whakatau tata, he māmā ake te jackknife i te bootstrapping e hiahia ana kia mano ngā tāruaruatanga.
4. He whai hua mō te aromatawai i te pānga
Tērā pea mō ngā tātaitanga kore-raina, he uaua hoki te tatau mā te tātari.
Ngā herenga me ngā mea hei tūpato
Ahakoa te kaha, he iti noa ngā ngoikoretanga o te maripi koi:
1. Kāore i te tino tika mō ngā tātaitanga tino kore-maeneene
Hei tauira, ko te tau waenga, ko ngā inenga rānei i ētahi āhuatanga, ko ngā tatauranga rānei e whakawhirinaki ana ki ngā uara tino nui, i ētahi wā ka whakaratohia e te jackknife ngā whakatau tata iti ake o te rerekētanga.
2. Kāore i te pai mō ngā raraunga me ngā whakawhirinakitanga
I roto i ngā raraunga raupapa wā, i ngā raraunga ā-wāhi rānei, kāore ngā tirohanga e tū motuhake. Mā te tango i tētahi pūwāhi kotahi ka pakaru te hanganga whakawhirinakitanga. Mō ngā take pēnei, ka whakamahia ngā momo rerekētanga pēnei i te block jackknife (te tango i tētahi poraka raraunga i te wā kotahi).
3. Tairongo ki ngā tirohanga pānga nui
Mena he raraunga kei waho, he raraunga "leveraged" rānei, ka tino rerekē te whakatau tata o te "let-one-out". Ehara tēnei i te ngoikore i ngā wā katoa—inaa, he tohu nui pea—engari ka nui te rerekētanga ka puta, ā, me āta whakamārama.
4. Te āheinga tauine i te nui rawa o te n
Ahakoa he iti ake te utu i te whakamahi i te bootstrapping, me whai tonu ngā aromatawai a te kaiwhakatau i te Jackknife i te n. Mena kei roto i ngā miriona te n, ā, he nui te utu o ngā kaiwhakatau, ka raru pea tēnei.
Ngā momo rerekē: delete-d jackknife me block jackknife
Haunga te waiho-kotahi-ki-waho, tera ano etahi momo rereke:
– Delete-d jackknife: ka muku i ngā kitenga d mō ia tukuruatanga (kaua ko te 1 anake). Ka taea e tēnei te whakapai ake i te tika i roto i ētahi āhuatanga, inā koa mō ngā tātaitanga kāore i te maeneene.
– Mā te maripi poraka: ka tangohia he poraka kei roto ētahi tirohanga tata, e tika ana mō ngā raraunga he hononga-ā-roto (hei tauira, raraunga ia rā, ia wiki, ia wāhi rānei).
Ko te whiriwhiri i te rahi o te "d", te rahi rānei o te poraka, ka whakawhirinaki ki te hanganga raraunga me te whāinga whakatau.
Te whakamahinga o te maripi kau i roto i te mahi
E whakamahia ana te maripi koi i roto i ngā momo mara:
– Tatauranga koiora me te mate urutā: te whakatau tata i ngā hapa paerewa mō ngā inenga mōrearea, ngā tawhā tauira rānei ina he uaua ngā tātai tātari.
– Ōhangarau: aromatawai i te pumau o ngā tawhā, inā koa i roto i ngā tauira iti.
– Pūtaiao rorohiko me te ako mīhini: he hononga tata te ariā waiho-kotahi-ki-te-kore ki te whakamana-whakawhiti, ahakoa he rerekē ngā whāinga (manatoko matapae vs aromatawai tika tawhā).
– Te rauropi me ngā rangahau: te aromatawai i te kanorau, i ētahi taupū rānei me te koretake o ngā tatauranga uaua.
Te Katinga
He tikanga tauira anō tawhito te tikanga jackknife e mau tonu ana te hiranga i ēnei rā. Mā te whakamahi i tētahi whakaaro māmā—me te kore e whakamahi i tētahi tirohanga, me te tatau anō i te tātaitanga—ka taea e te jackknife te whakarato i ngā tātaitanga mō te rerekētanga, te hapa paerewa, me te whakaaro pōturi, me te kore he tātaitanga pāngarau uaua. Heoi, me whai whakaaro ki te āhua o te tātaitanga, te rahi o te tauira, me te hanganga whakawhirinaki o ngā raraunga mō tōna whakamahinga. I roto i te mahi, he maha ngā wā he kōwhiringa tere, he mārama hoki te jackknife, he tāpiri rānei ki te whakamahi i ngā tikanga tauira anō pakari ake pērā i te bootstrapping.
Ki te hiahia koe, ka taea hoki e au te tāpiri i tētahi tauira tatau tau iti (hei tauira, mō te hononga, te whakatauira rānei) me te tāpiri rānei i tētahi whakatinanatanga jackknife ki roto i te R/Python hei whakamārama i te tono.
