Indlela Yezikwele Ezincane: Indlela Yezibalo Yokulinganisa
I-Pendahuluan
Indlela yezikwele ezincane iyindlela yezibalo esetshenziswa ukulinganisa amapharamitha kumodeli yokubuyela emuva ngokunciphisa inani lamaphutha aphindwe kabili phakathi kwamanani angempela kanye namanani abikezelwe yimodeli. Le ndlela ithandwa kakhulu futhi ivame ukusetshenziswa emikhakheni ehlukahlukene njengezomnotho, ubunjiniyela, i-biology, kanye nesayensi yezenhlalo. Umqondo wezikwele ezincane wahlongozwa okokuqala ngu-Adrien-Marie Legendre ekuqaleni kwekhulu le-19 futhi kamuva wathuthukiswa nguCarl Friedrich Gauss.
Ukuqonda Okuyisisekelo
Ngokuvamile, indlela ye-least squares ihlose ukuthola umugqa wokuhlehlisa ofaneleka kakhulu wesethi yedatha ngokunciphisa isamba sezikwele zezinsalela, noma amaphutha okubikezela. I-residual umehluko phakathi kwenani elibonwe kanye nenani elibikezelwe.
Uma sinesethi yedatha equkethe amabhangqa okubonwa \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)\), khona-ke umgomo wethu ukuthola umugqa \(y = mx + b\) onciphisa isamba samaphutha ayisikwele sum\( \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \).
Le ndlela ingasetshenziswa kokubili ekubuyiseleni okuqondile okulula kanye nokubuyela emuva okuqondile okuningi. Kukubuyela emuva okuqondile okulula, sine-variable eyodwa ezimele (x), kuyilapho ukubuyela emuva okuqondile okuningi kuhilela okungaphezu kwe-variable eyodwa ezimele.
Ukuhlehliswa Okulula Komugqa
Ake siqale ngokuhlehla okulula komugqa. Ake sithi sinesethi yedatha \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)). Imodeli elula yokuhlehla komugqa esifuna ukuyifanisa yile:
\[ y = mx + b + \epsilon \]
lapho \( m \) kuyi-slope, \( b \) kuyi-intercept, kanye \( \epsilon \) kuyiphutha elingahleliwe.
Sisebenzisa indlela ye-least squares, singathola izilinganiso zamapharamitha \( m \) kanye \( b \) ngokunciphisa umsebenzi wephutha eliyisikwele:
\[ S(m, b) = \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \]
Ukuze sinciphise i-\( S(m, b) \), sithola ama-derivative angaphelele e-\( S \) maqondana ne-\( m \) kanye ne-\( b \), bese sixazulula lesi sibalo se-\( m \) kanye ne-\( b \):
\[ \begin{aligned}
\frac{\partial S}{\partial m} &= -2 \sum_{i=1}^{n} x_i (y_i – (mx_i + b)) = 0 \\
\frac{\partial S}{\partial b} &= -2 \sum_{i=1}^{n} (y_i – (mx_i + b)) = 0
\end{aligned} \]
Ngemva kokwenza lula, sithola izilinganiso ezimbili ezijwayelekile ezilandelayo:
\[ \begin{aligned}
n\bar{y} &= m \sum_{i=1}^{n} x_i + nb \\
\sum_{i=1}^{n}x_i y_i &= m \sum_{i=1}^{n}x_i^2 + b \sum_{i=1}^{n}x_i
\end{aligned} \]
Ngokuxazulula uhlelo lwezibalo ezingenhla, singathola amanani ka-\( m \) kanye no-\( b \) anciphisa iphutha eliyisikwele.
Ukuhlehliswa Okunemigqa Okuningi
Ku-regression eqondile eminingi, sibhekene nesimo lapho sineziguquguquko ezizimele ezingaphezu kweyodwa. Ake sithi sinedatha ngesimo se-tuple \((x_{i1}, x_{i2}, …, x_{ik}, y_i)\). Imodeli yokubuyisela esiyisebenzisayo yile:
\[ y = b_0 + b_1 x_1 + b_2 x_2 + … + b_k x_k + \epsilon \]
Lesi sibalo singabhalwa ngesimo se-matrix kanje:
\[ \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{\epsilon} \]
Kuphi:
– \( \mathbf{y} \) iyivektha yekholomu yamanani ka-y aqashelwe.
– \( \mathbf{X} \) iyi-matrix yamanani ka-x aqashelwe (kufaka phakathi ikholomu 1 ye-intercept).
– \( \mathbf{b} \) iyivektha yekholomu yamapharamitha (kufaka phakathi \( b_0 \)).
Umgomo wendlela ye-least squares ukunciphisa umsebenzi olandelayo we-quadratic error:
\[ S(\mathbf{b}) = (\mathbf{y} – \mathbf{Xb})^T (\mathbf{y} – \mathbf{Xb}) \]
Ukuze sinciphise lo msebenzi, sithatha i-partial derivative ka-S maqondana ne-\( \mathbf{b} \) bese siyibeka ku-zero. Lokhu kuveza i-equation evamile yokubuyela emuva okuqondile okuningi:
\[ \mathbf{X}^T \mathbf{Xb} = \mathbf{X}^T \mathbf{y} \]
Ngokuxazulula uhlelo lwezibalo olungenhla, singathola isilinganiso sepharamitha \( \mathbf{b} \):
\[ \mathbf{b} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} \]
Izinzuzo kanye nokulinganiselwa
Indlela ye-least squares inezinzuzo eziningi. Iyindlela ephumelela kakhulu futhi elula ukuyisebenzisa. Inikeza ikhambi eliyingqayizivele uma \( \mathbf{X}^T \mathbf{X} \) ingaguquki, okwenza ithembeke ezinhlelweni eziningi ezisebenzayo.
Kodwa-ke, indlela ye-least squares nayo inemikhawulo. Izwela kakhulu kuma-outliers ngoba iphutha le-squared ligcizelela umehluko omkhulu kakhulu kunemincane. Ngaphezu kwalokho, ukucabanga kwakudala kokuthi amaphutha anokusatshalaliswa okuvamile okune-zero mean kanye nokwehluka okuqhubekayo kumele kuhlangatshezwane nakho ukuze kube nemiphumela emihle.
Izicelo Eziwusizo
Indlela ye-least squares ivame ukusetshenziswa ekuhlaziyweni kwedatha, ekubikezeleni, nasekufundeni komshini ukwakha amamodeli okubikezela. Embonini yezezimali, indlela ye-least squares isetshenziselwa ukubikezela amanani esitoko noma ukusebenza kwemakethe. Kwezokwelapha, isetshenziselwa ukulingisa ubudlelwano phakathi komthamo wemithi kanye nempendulo yesiguli. Kwezesayensi yezenhlalo, kusiza ukuqonda ubudlelwano phakathi kwezinto eziguquguqukayo ezifana nemfundo kanye nemali engenayo.
Isiphetho
Indlela ye-least squares ingenye yezindlela eziyisisekelo ekuhlaziyeni izibalo kanye nedatha. Nakuba ilula ngokomqondo, le ndlela inikeza amandla amakhulu ekubumbeni nasekuqondeni ubudlelwano phakathi kwezinto eziguquguqukayo. Njengoba kunezinhlelo zokusebenza eziningi emikhakheni eminingi, ukuqonda okuqinile kwale ndlela kubaluleke kakhulu kochwepheshe kanye nabacwaningi ngokufanayo. Kusukela manje, njengoba inani ledatha elikhulayo lihlangana nenkathi yedatha enkulu, ukuzivumelanisa nokusetshenziswa kwezindlela zakudala ezifana ne-least squares kuzobe sekufanelekela kakhulu.
