ʻO ke kumu o ka ʻatikala e hāʻawi i ke kākoʻo i nā ʻepekema data hoʻomaka. IN Ua wehewehe mākou i ʻekolu mau ala e hoʻoponopono ai i ka hoʻohālikelike hoʻihoʻi laina: hoʻonā analytical, gradient descent, stochastic gradient descent. A laila no ka hoʻonā analytical ua hoʻohana mākou i ke ʻano . Ma kēia ʻatikala, e like me ka manaʻo o ke poʻo, e hōʻoia mākou i ka hoʻohana ʻana i kēia ʻano hana a i ʻole, ʻo ia hoʻi, e loaʻa iā mākou iho.
No ke aha he mea maʻalahi ka nānā ʻana i ke ʻano ?
ʻO ia me ka matrix equation i ka nui o nā hihia e hoʻomaka ai ke kamaʻāina me ka regression linear. I ka manawa like, he kakaikahi nā helu kikoʻī o ka loaʻa ʻana o ka formula.
No ka laʻana, i nā papa aʻo mīkini mai Yandex, ke hoʻolauna ʻia nā haumāna i ka regularization, hāʻawi ʻia lākou e hoʻohana i nā hana mai ka waihona. sklearn, ʻoiai ʻaʻole i haʻi ʻia kahi huaʻōlelo e pili ana i ka hōʻike matrix o ka algorithm. I kēia manawa makemake paha kekahi poʻe hoʻolohe e hoʻomaopopo i kēia pilikia me ka kikoʻī - e kākau i nā code me ka ʻole o ka hoʻohana ʻana i nā hana i mākaukau. A no ka hana ʻana i kēia, pono ʻoe e hōʻike mua i ka hoʻohālikelike me kahi regularizer ma ke ʻano matrix. E ʻae kēia ʻatikala i ka poʻe e makemake ana e aʻo i kēlā mau mākau. E hoʻomaka kākou.
Nā kūlana mua
Nā hōʻailona pahuhopu
Loaʻa iā mākou ka laulā o nā kumu waiwai. No ka laʻana, hiki i ka mea hōʻailona ke kumukūʻai o kekahi waiwai: ʻaila, gula, palaoa, kālā, etc. I ka manawa like, ma ka helu o nā helu hōʻailona hōʻailona ke manaʻo nei mākou i ka helu o nā nānā. ʻO ia mau ʻike, ʻo ia hoʻi, nā kumukūʻai aila o kēlā me kēia mahina no ka makahiki, ʻo ia hoʻi, e loaʻa iā mākou he 12 mau kumu waiwai. E hoʻomaka kākou e hoʻokomo i ka notation. E hōʻike mākou i kēlā me kēia waiwai o ka hōʻailona pahuhopu e like me . Loaʻa iā mākou nā nānā ʻana, ʻo ia hoʻi, hiki iā mākou ke hōʻike i kā mākou nānā ʻana .
Hoʻihoʻi hou
E noʻonoʻo mākou aia kekahi mau kumu e wehewehe i nā waiwai o ka hōʻailona pahuhopu. Eia kekahi laʻana, ua hoʻoikaika nuiʻia ke kumukūʻai kālā / ruble e ke kumukūʻai o kaʻaila, ka Federal Reserve rate, a me nā mea'ē aʻe. I ka manawa like, pono e pili kēlā me kēia helu kuhikuhi i kahi waiwai regressor, ʻo ia hoʻi, inā loaʻa iā mākou 12 mau hōʻailona pahuhopu no kēlā me kēia mahina ma 2018, a laila pono mākou e loaʻa i nā waiwai regressor 12 no ka manawa like. E hōʻike mākou i nā waiwai o kēlā me kēia regressor e . E waiho i ko kakou hihia regressors (i.e. nā mea e hoʻopili ai i nā waiwai hōʻailona pahuhopu). ʻO ia ke ʻano o kā mākou regressors hiki ke hōʻike ʻia e like me kēia: no ka regressor 1st (no ka laʻana, ke kumu kūʻai o ka aila): , no ka 2nd regressor (no ka laʻana, ka Fed rate): , No "-th" regressor:
ʻO ka hilinaʻi ʻana o nā hōʻailona pahuhopu i nā regressors
E noʻonoʻo kākou i ka hilinaʻi o ka hōʻailona pahuhopu mai regressors "Hiki ke hōʻike ʻia ka nānā ʻana ma o ka hoʻohālikelike hoʻihoʻi laina o ke ʻano:
kahi - "-th" waiwai regressor mai 1 a ,
- ka helu o nā regressors mai ka 1 a hiki i
— nā ʻāpana angular, e hōʻike ana i ka nui e hoʻololi ai ka hōʻailona pahuhopu i helu ʻia ma ka awelika ke hoʻololi ka regressor.
I nā huaʻōlelo ʻē aʻe, no kēlā me kēia mea mākou (koe ) o ka regressor mākou e hoʻoholo "ko mākou" coefficient , a laila e hoʻonui i nā coefficients me nā waiwai o nā regressors ""ka nānā ʻana, ma muli o ka loaʻa ʻana o kahi hoʻohālikelike"-th" hōʻailona pahuhopu.
No laila, pono mākou e koho i ia mau coefficients , kahi o nā waiwai o kā mākou hana kokoke e loaʻa i kahi kokoke loa i nā waiwai hōʻailona.
ʻO ka loiloi i ka maikaʻi o ka hana kokoke
E hoʻoholo mākou i ka loiloi maikaʻi o ka hana hoʻohālikelike me ka hoʻohana ʻana i ke ala liʻiliʻi loa. ʻO ka hana loiloi maikaʻi ma kēia hihia e lawe i kēia ʻano:
Pono mākou e koho i kēlā mau waiwai o nā coefficients $w$ nona ka waiwai ʻo ia ka mea liʻiliʻi loa.
Ka hoohuli ana i ka hoohalike i ke ano matrix
Hōʻike vector
I ka hoʻomaka ʻana, i mea e maʻalahi ai kou ola, pono ʻoe e hoʻolohe i ka hoʻohālikelike regression linear a ʻike i ka coefficient mua. ʻaʻole i hoʻonui ʻia e kekahi regressor. I ka manawa like, ke hoʻololi mākou i ka ʻikepili i loko o ke ʻano matrix, ʻo ke kūlana i ʻōlelo ʻia ma luna nei e hoʻopiʻi nui i nā helu. Ma keia mea, ua manaoia e hookomo i kekahi regressor no ka mua coefficient a hoohalike me ka hookahi. A i ʻole, kēlā me kēia "e hoʻohālikelike i ka waiwai o kēia regressor i hoʻokahi - ma hope o nā mea a pau, ke hoʻonui ʻia e ka mea hoʻokahi, ʻaʻohe mea e loli mai ka manaʻo o ka hopena o ka helu ʻana, akā mai ka manaʻo o nā lula no ka huahana o nā matrices, ko mākou ʻeha. e hoemi loa ia.
I kēia manawa, i mea e maʻalahi ai ka mea, e manaʻo mākou he hoʻokahi wale nō "-th" nānā. A laila, e noʻonoʻo i nā waiwai o nā regressors "-th" nā ʻike ma ke ʻano he vector . Vector he ana ,ʻo ia hoʻi nā lālani a me 1 kolamu:
E hōʻike i nā coefficient i makemake ʻia ma ke ʻano he vector , he ana :
Hoʻohālikelike hoʻihoʻi laina no "-th" e lawe i ke ʻano:
ʻO ka hana no ka loiloi ʻana i ka maikaʻi o kahi hoʻohālike laina e lawe i ke ʻano:
E ʻoluʻolu, e like me nā lula o ka hoʻonui matrix, pono mākou e hoʻololi i ka vector .
Hōʻike matrix
Ma muli o ka hoʻonui ʻana i nā vectors, loaʻa iā mākou ka helu: , ka mea e manaoia. ʻO kēia helu ka hoʻohālikelike "-th" hōʻailona pahuhopu. Akā, pono mākou i kahi hoʻohālikelike ʻaʻole hoʻokahi wale nō kumu waiwai, akā ʻo lākou āpau. No ka hana ʻana i kēia, e kākau mākou i nā mea āpau ""th" regressors ma ke ano matrix . Loaʻa ka nui o ka matrix hopena :
I kēia manawa e lawe ʻia ka hoʻohālikelike linear regression i ke ʻano:
E hōʻike mākou i nā waiwai o nā kuhikuhi kuhikuhi (all ) no kēlā me kēia vector anana :
I kēia manawa hiki iā mākou ke kākau i ka hoʻohālikelike no ka loiloi ʻana i ka maikaʻi o kahi kŘkohu laina ma ke ʻano matrix:
ʻOiaʻiʻo, mai kēia kumulāʻau e loaʻa hou iā mākou ke ʻano i ʻike ʻia e mākou
Pehea ka hana ana? Wehe ʻia nā brackets, hoʻokō ʻia ka ʻokoʻa, hoʻololi ʻia nā ʻōlelo hopena, a pēlā aku, a ʻo ia ka mea a mākou e hana ai i kēia manawa.
Hoʻololi matrix
E wehe kākou i nā pale
E hoʻomākaukau kākou i hoʻohālikelike no ka hoʻokaʻawale ʻana
No ka hana ʻana i kēia, e hana mākou i kekahi mau hoʻololi. Ma ka helu ʻana aʻe e ʻoi aku ka maʻalahi iā mākou inā ʻo ka vector e hōʻike ʻia ma ka hoʻomaka o kēlā me kēia huahana i ka hoohalike.
Hoʻololi 1
Pehea i hiki mai ai? No ka pane ʻana i kēia nīnau, e nānā wale i ka nui o nā matrices e hoʻonui ʻia a ʻike i ka puka ʻana e loaʻa iā mākou kahi helu a i ʻole. .
E kākau kākou i ka nui o nā hōʻike matrix.
Hoʻololi 2
E kākau kākou ma ke ʻano like me ka hoʻololi 1
Ma ka hopena e loaʻa iā mākou kahi hoohalike e pono ai mākou e hoʻokaʻawale:
Hoʻokaʻawale mākou i ka hana loiloi maikaʻi o ke kumu hoʻohālike
E hoʻokaʻawale e pili ana i ka vector :
Nā nīnau no ke aha ʻaʻole pono, akā, e nānā mākou i nā hana no ka hoʻoholo ʻana i nā derivatives ma nā ʻōlelo ʻelua ʻelua i nā kikoʻī hou aku.
ʻokoʻa 1
E hoʻonui kākou i ka ʻokoʻa:
I mea e hoʻoholo ai i ka derivative o kahi matrix a i ʻole vector, pono ʻoe e nānā i ka mea i loko o lākou. E nānā kākou:
E hōʻike mākou i ka huahana o nā matrices ma o ka matrix . Matrix huinaha a oi aku, ua like ia. E lilo kēia mau waiwai iā mākou ma hope aku, e hoʻomanaʻo kākou iā lākou. Matrix he ana :
I kēia manawa, ʻo kā mākou hana e hoʻonui pono i nā vectors ma ka matrix a ʻaʻole e loaʻa "ʻelua ʻelua ʻelima," no laila e noʻonoʻo a makaʻala loa.
Eia naʻe, ua hoʻokō mākou i kahi ʻōlelo paʻakikī! ʻOiaʻiʻo, loaʻa iā mākou kahi helu - he scalar. A i kēia manawa, no ka mea maoli, neʻe mākou i ka ʻokoʻa. Pono e huli i ka derivative o ka hua'ōlelo i loaʻa no kēlā me kēia coefficient a e kiʻi i ka vector dimension ma ke ʻano he puka . I ka hihia wale nō, e kākau wau i nā kaʻina hana ma ka hana:
1) hoʻokaʻawale e , loaʻa iā mākou:
2) hoʻokaʻawale e , loaʻa iā mākou:
3) hoʻokaʻawale e , loaʻa iā mākou:
ʻO ka mea hoʻopuka ka vector i hoʻohiki ʻia o ka nui :
Inā ʻoe e nānā pono i ka vector, e ʻike ʻoe e hiki ke hui pū ʻia nā ʻāpana hema a me nā ʻāpana ʻākau o ka vector i ke ʻano e hiki ai ke hoʻokaʻawale ʻia kahi vector mai ka vector i hōʻike ʻia. nui . Eia kekahi laʻana (mea hema o ka laina luna o ka vector) (ʻo ka mea ʻākau o ka laina kiʻekiʻe o ka vector) hiki ke hōʻike ʻia e like me a me ka - e like me etc. ma kēlā me kēia laina. E hui kākou:
E wehe kākou i ka vector a ma ka hopena e loaʻa iā mākou:
I kēia manawa, e nānā pono kākou i ka matrix hopena. ʻO ka matrix ka huina o ʻelua matrices :
E hoʻomanaʻo mākou ma mua iki ua ʻike mākou i kahi waiwai nui o ka matrix - ua like ia. Ma muli o kēia waiwai, hiki iā mākou ke ʻōlelo wiwo ʻole i ka ʻōlelo like . Hiki ke hōʻoia maʻalahi kēia ma o ka hoʻonui ʻana i ka huahana o ka matrices element ma ka element . ʻAʻole mākou e hana i kēia ma ʻaneʻi; hiki i ka poʻe hoihoi ke nānā iā lākou iho.
E hoʻi kāua i kā mākou ʻōlelo. Ma hope o kā mākou hoʻololi ʻana, ua ʻike ʻia ke ala a mākou i makemake ai e ʻike:
No laila, ua hoʻopau mākou i ka ʻokoʻa mua. E neʻe kākou i ka ʻōlelo ʻelua.
ʻokoʻa 2
E hahai kākou i ke ala i hahau ʻia. E ʻoi aku ka pōkole ma mua o ka mea ma mua, no laila mai hele mamao loa mai ka pale.
E hoʻonui i nā vectors a me ka matrix element ma ka element:
E wehe mākou i nā helu ʻelua no kekahi manawa - ʻaʻole ia he kuleana nui, a laila e hoʻihoʻi mākou i kona wahi. E hoonui kakou i na vectors me ka matrix. ʻO ka mea mua, e hoʻonui kākou i ka matrix i vector , ʻaʻohe o mākou kapu ma ʻaneʻi. Loaʻa iā mākou ka nui vector :
E hana kāua i kēia hana - hoʻonui i ka vector i ka vector hopena. Ma ka puka e kali ana ka helu iā mākou:
A laila e hoʻokaʻawale mākou. Ma ka hopena e loaʻa iā mākou kahi vector o ka nui :
Hoʻomanaʻo mai iaʻu i kekahi mea? Pololei kēnā! ʻO kēia ka huahana o ka matrix i vector .
No laila, ua hoʻokō pono ʻia ka ʻokoʻa ʻelua.
Ma kahi o ka hopena
I kēia manawa ua ʻike mākou i ka hiki ʻana mai o ke kaulike .
ʻO ka hope, e wehewehe mākou i kahi ala wikiwiki e hoʻololi i nā ʻano kumu.
E noʻonoʻo kākou i ka maikaʻi o ke kumu hoʻohālike e like me ke ʻano o nā ʻāpana liʻiliʻi loa:
E hoʻokaʻawale kākou i ka huaʻōlelo hopena:
Paipalapala
Nā kumu punaewele:
1)
2)
3)
4)
Nā puke haʻawina, nā hōʻiliʻili o nā pilikia:
1) Nā memo haʻi no ka makemakika kiʻekiʻe: papa piha / D.T. Kākau ʻia – 4th ed. – M.: Iris-press, 2006
2) Hoʻopili i ka loiloi regression / N. Draper, G. Smith - 2nd ed. – M.: Waiwai a me Heluhelu, 1986 (unuhi mai ka olelo Pelekania)
3) Nā pilikia no ka hoʻoholo ʻana i nā hoohalike matrix:
Source: www.habr.com