Nzira yeMakwere Akaderera: Nzira yeMasvomhu yeKufungidzira
Pendauluan
Nzira ye least squares inzira inoshandiswa pakuongorora ma parameters mu regression model nekuderedza huwandu hwe squared errors pakati pema qualitudes chaiwo nema values akafanotaurwa nemodhi. Iyi nzira yakakurumbira uye inowanzoshandiswa munzvimbo dzakasiyana siyana dzakadai se economics, engineering, biology, uye social sciences. Pfungwa ye least squares yakatanga kutaurwa naAdrien-Marie Legendre pakutanga kwezana remakore rechi19 uye yakazosimudzirwazve naCarl Friedrich Gauss.
Kunzwisisa Kwekutanga
Kazhinji, nzira ye "least squares" inovavarira kuwana mutsetse we "regression" wakakodzera data reset nekuderedza huwandu hwe "squares" dze "residues", kana zvikanganiso zve "prediction errors". "Residual" ndiyo musiyano uripo pakati pe "value" yakaonekwa ne "value" yakagara yaonekwa.
Kana tine seti yedata ine mapeya ekucherechedza \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)\), saka chinangwa chedu ndechekuwana mutsetse \(y = mx + b\) unoderedza huwandu hwezvikanganiso zvesquared sum\( \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \).
Nzira iyi inogona kushandiswa pakudzokorora kusingachinji uye pakudzokorora kusingachinji. Mukudzokorora kusingachinji, tine shanduko imwe chete yakazvimirira (x), nepo kudzokorora kusingachinji kunosanganisira zvinopfuura imwe chete yakazvimirira.
Kudzoreredza Kwakareruka Kwemutsetse
Ngatitangei nekudzokorora mutsara kuri nyore. Ngatitii tine seti yedata \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)). Muenzaniso wekudzokorora mutsara uri nyore watinoda kuti ukwane ndewekuti:
\[ y = mx + b + \epsilon \]
apo \( m \) iri mukwidza, \( b \) iri intercept, uye \( \epsilon \) iri random error.
Tichishandisa nzira ye "least squares", tinogona kuwana fungidziro dzema parameters \( m \) uye \( b \) nekuderedza basa re "squared error":
\[ S(m, b) = \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \]
Kuti tideredze \( S(m, b) \), tinowana zvikamu zve \( S \) maererano ne \( m \) uye \( b \), tobva tagadzirisa equation iyi ye \( m \) uye \( b \):
\[ \begin{aligned}
\frac{\partial S}{\partial m} &= -2 \sum_{i=1}^{n} x_i (y_i – (mx_i + b)) = 0 \\
\frac{\partial S}{\partial b} &= -2 \sum_{i=1}^{n} (y_i – (mx_i + b)) = 0
\end{aligned} \]
Mushure mekurerutsa, tinowana maequations maviri akajairika anotevera:
\[ \begin{aligned}
n\bar{y} &= m \sum_{i=1}^{n} x_i + nb \\
\sum_{i=1}^{n}x_i y_i &= m \sum_{i=1}^{n}x_i^2 + b \sum_{i=1}^{n}x_i
\end{aligned} \]
Nekugadzirisa hurongwa hwema equation ari pamusoro apa, tinogona kuwana kukosha kwe \( m \) uye \( b \) kunoderedza kukanganisa kwakapetwa kaviri.
Kudzoreredzwa Kwemitsara Yakawanda
Mukudzokorora kwemutsara kwakawanda, tinosangana nemamiriro ezvinhu apo tine zvinopfuura chimwe chete zvakasiyana-siyana. Ngatitii tine data muchimiro che tuple \((x_{i1}, x_{i2}, …, x_{ik}, y_i)\). Muenzaniso wekudzokorora watinoshandisa ndewekuti:
\[ y = b_0 + b_1 x_1 + b_2 x_2 + … + b_k x_k + \epsilon \]
Equation iyi inogona kunyorwa muchimiro chematrix seizvi:
\[ \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{\epsilon} \]
di mana:
– \( \mathbf{y} \) ivhekitari yekoramu ine kukosha kwe y kwakacherechedzwa.
– \( \mathbf{X} \) inhamba ye x values dzakaonekwa (kusanganisira koramu 1 ye intercept).
– \( \mathbf{b} \) ikoramu vhekitari yemaparamita (kusanganisira \( b_0 \)).
Chinangwa chenzira ye least squares ndechekuderedza basa rinotevera re quadratic error:
\[ S(\mathbf{b}) = (\mathbf{y} – \mathbf{Xb})^T (\mathbf{y} – \mathbf{Xb}) \]
Kuti tideredze basa iri, tinotora chikamu cheS chinobva pachikamu tichienzanisa ne \( \mathbf{b} \) tochiisa pazero. Izvi zvinopa equation yakajairika ye multiple linear regression:
\[ \mathbf{X}^T \mathbf{Xb} = \mathbf{X}^T \mathbf{y} \]
Nekugadzirisa hurongwa hwema equation ari pamusoro apa, tinogona kuwana fungidziro yeparameter \( \mathbf{b} \):
\[ \mathbf{b} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} \]
Zvakanakira uye Zvisina Kukwana
Nzira ye "least squares" ine mabhenefiti akawanda. Inzira inoshanda uye iri nyore kushandisa. Inopa mhinduro yakasiyana kana \( \mathbf{X}^T \mathbf{X} \) isingachinjiki, zvichiita kuti ive yakavimbika kune akawanda mashandisirwo anoshanda.
Zvisinei, nzira ye "least squares" inewo miganhu. Inonyanya kunzwisiswa ne "outliers" nekuti "squared error" inosimbisa misiyano mikuru kupfuura midiki. Uyezve, fungidziro yekare yekuti zvikanganiso zvine "normal distribution" isina "zero mean" uye "constant variance" inofanira kuzadzikiswa kuti pave nemhedzisiro yakanaka.
Maitiro Anoshanda
Nzira ye "least squares" inowanzo shandiswa mukuongorora data trend, kufungidzira, uye kudzidza kwemuchina kuvaka mamodheru ekufanotaura. Muindasitiri yemari, nzira ye "least squares" inoshandiswa kufanotaura mitengo yemasheya kana mashandiro emusika. Mukurapa, inoshandiswa kutevedzera hukama huripo pakati pemuyero wemishonga nemhinduro yemurwere. Musainzi yemagariro evanhu, inobatsira kunzwisisa hukama huripo pakati pezvinhu zvakaita sedzidzo nemari inowanikwa.
Mhedziso
Nzira ye "least squares" ndeimwe yenzira dzakakosha mukuongorora manhamba nedata. Kunyange zvazvo iri nyore, nzira iyi inopa simba guru mukuita mamodheru nekunzwisisa hukama huripo pakati pezvinhu zvakasiyana-siyana. Nekushandiswa kwakapararira munzvimbo dzakasiyana-siyana, kunzwisisa kwakasimba kwenzira iyi kunokosha kune nyanzvi nevaongorori zvakafanana. Kuenderera mberi, nehuwandu hwedata huri kuwedzera munguva yedata guru, kugadziriswa nekushandiswa kwenzira dzekare dzakadai seleast squares zvichava zvakakosha chete.
