Njira Yowerengera Magawo Ocheperako: Njira Yowerengera Masamu
Pendauluan
Njira ya ma squares ang'onoang'ono ndi njira yowerengera yomwe imagwiritsidwa ntchito poyesa magawo mu regression model pochepetsa kuchuluka kwa zolakwika za squared pakati pa zenizeni ndi zomwe zanenedweratu ndi chitsanzocho. Njirayi ndi yotchuka kwambiri ndipo imagwiritsidwa ntchito nthawi zambiri m'magawo osiyanasiyana monga zachuma, uinjiniya, sayansi ya zamoyo, ndi sayansi ya chikhalidwe cha anthu. Lingaliro la ma squares ang'onoang'ono linaperekedwa koyamba ndi Adrien-Marie Legendre kumayambiriro kwa zaka za m'ma 19 ndipo pambuyo pake linapitilizidwa ndi Carl Friedrich Gauss.
Kumvetsetsa Koyambira
Kawirikawiri, njira ya least squares ikufuna kupeza mzere woyenera bwino wa regression wa deta pochepetsa kuchuluka kwa ma squares a zotsalira, kapena zolakwika zolosera. Chotsalira ndi kusiyana pakati pa mtengo wowonedwa ndi mtengo wolosera.
Ngati tili ndi deta yokhala ndi ma peya a zomwe zawonedwa \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)\), ndiye kuti cholinga chathu ndikupeza mzere \(y = mx + b\) womwe umachepetsa kuchuluka kwa zolakwika zokwana masikweya sikweya sum\( \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \).
Njira iyi ingagwiritsidwe ntchito pa njira yosavuta yosinthira mzere ndi njira zambiri zosinthira mzere. Mu njira yosavuta yosinthira mzere, tili ndi njira imodzi yokha yosinthira yokha (x), pomwe njira zambiri zosinthira mzere zimaphatikizapo njira zingapo zosinthira zokha.
Kuchepetsa Kosavuta kwa Linear
Tiyeni tiyambe ndi mzere wolunjika wosavuta. Tiyerekeze kuti tili ndi deta yosungidwa \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)). Chitsanzo chosavuta cholunjika chomwe tikufuna kuti chigwirizane ndi ichi:
\[ y = mx + b + \epsilon \]
kumene \( m \) ndi malo otsetsereka, \( b \) ndi njira yolumikizira, ndipo \( \epsilon \) ndi cholakwika chosasinthika.
Pogwiritsa ntchito njira ya least squares, titha kupeza ziwerengero za magawo \( m \) ndi \( b \) mwa kuchepetsa ntchito ya squared error:
\[ S(m, b) = \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \]
Kuti tichepetse \( S(m, b) \), timapeza zotumphukira pang'ono za \( S \) poyerekeza ndi \( m \) ndi \( b \), kenako timathetsa equation iyi ya \( m \) ndi \( b \):
\[ \begin{aligned}
\frac{\partial S}{\partial m} &= -2 \sum_{i=1}^{n} x_i (y_i – (mx_i + b)) = 0 \\
\frac{\partial S}{\partial b} &= -2 \sum_{i=1}^{n} (y_i – (mx_i + b)) = 0
\end{aligned} \]
Pambuyo posavuta, timapeza ma equation awiri otsatirawa:
\[ \begin{aligned}
n\bar{y} &= m \sum_{i=1}^{n} x_i + nb \\
\sum_{i=1}^{n}x_i y_i &= m \sum_{i=1}^{n}x_i^2 + b \sum_{i=1}^{n}x_i
\end{aligned} \]
Mwa kuthetsa dongosolo la ma equation pamwambapa, titha kupeza ma values a \( m \) ndi \( b \) omwe amachepetsa cholakwika cha sikweya.
Kuchepetsa Mizere Yambiri
Mu regression yambiri yolunjika, timakumana ndi vuto lomwe tili ndi zosinthika zodziyimira pawokha zoposa chimodzi. Tiyerekeze kuti tili ndi deta mu mawonekedwe a tuple \((x_{i1}, x_{i2}, …, x_{ik}, y_i)\). Chitsanzo cha regression chomwe timagwiritsa ntchito ndi:
\[ y = b_0 + b_1 x_1 + b_2 x_2 + … + b_k x_k + \epsilon \]
Equation iyi ikhoza kulembedwa mu mawonekedwe a matrix motere:
\[ \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{\epsilon} \]
Kumene:
– \( \mathbf{y} \) ndi vekitala ya miyeso ya y yomwe yawonedwa.
– \( \mathbf{X} \) ndi matrix ya ma x values omwe awonedwa (kuphatikiza ndime 1 ya intercept).
– \( \mathbf{b} \) ndi vekitala ya magawo (kuphatikiza \( b_0 \)).
Cholinga cha njira ya least squares ndikuchepetsa ntchito yotsatirayi ya quadratic error:
\[ S(\mathbf{b}) = (\mathbf{y} – \mathbf{Xb})^T (\mathbf{y} – \mathbf{Xb}) \]
Kuti tichepetse ntchito iyi, timatenga gawo lochokera ku S poyerekeza ndi \( \mathbf{b} \) ndikuliyika pa zero. Izi zimapereka equation yachibadwa ya multiple linear regression:
\[ \mathbf{X}^T \mathbf{Xb} = \mathbf{X}^T \mathbf{y} \]
Mwa kuthetsa dongosolo la ma equation pamwambapa, titha kupeza kuyerekezera kwa parameter \( \mathbf{b} \):
\[ \mathbf{b} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} \]
Ubwino ndi Zofooka
Njira ya least squares ili ndi zabwino zambiri. Ndi njira yothandiza kwambiri komanso yosavuta kugwiritsa ntchito. Imapereka yankho lapadera ngati \( \mathbf{X}^T \mathbf{X} \) ndi yosasinthika, zomwe zimapangitsa kuti ikhale yodalirika pa ntchito zambiri zothandiza.
Komabe, njira ya least squares ilinso ndi zoletsa. Ndi yodziwika kwambiri ndi zinthu zopanda pake chifukwa cholakwika cha squared chimagogomezera kusiyana kwakukulu kuposa zazing'ono. Kuphatikiza apo, lingaliro lakale lakuti zolakwikazo zili ndi kugawa kwabwinobwino popanda kusiyana kwapakati komanso kosasintha kuyenera kukwaniritsidwa kuti pakhale zotsatira zabwino.
Kugwiritsa Ntchito Mwanzeru
Njira ya least squares imagwiritsidwa ntchito nthawi zambiri pofufuza zomwe zikuchitika pa data, kulosera, ndi kuphunzira makina kuti apange zitsanzo zolosera. Mu makampani azachuma, njira ya least squares imagwiritsidwa ntchito kulosera mitengo yamasheya kapena momwe msika umagwirira ntchito. Mu zamankhwala, imagwiritsidwa ntchito kuyerekezera ubale pakati pa mlingo wa mankhwala ndi momwe odwala amayankhira. Mu sayansi ya chikhalidwe cha anthu, zimathandiza kumvetsetsa ubale pakati pa zinthu monga maphunziro ndi ndalama.
Mapeto
Njira ya least squares ndi imodzi mwa njira zofunika kwambiri pa ziwerengero ndi kusanthula deta. Ngakhale kuti ndi yosavuta, njira iyi imapereka mphamvu yayikulu pakukonza ndi kumvetsetsa ubale pakati pa zinthu zosiyanasiyana. Ndi ntchito zambiri m'magawo osiyanasiyana, kumvetsetsa bwino njira iyi ndikofunikira kwambiri kwa akatswiri ndi ofufuza. Kupita patsogolo, ndi kuchuluka kwa deta komwe kukuchitika mu nthawi ya big data, kusintha ndi kugwiritsa ntchito njira zakale monga least squares kudzakhala kofunikira kwambiri.
