Isibonelo Semibuzo Yengxoxo Yendlela Yezikwele Ezincane
Indlela Yezikwele Ezincane (i-LEM) iyindlela yezibalo esetshenziselwa ukuthola umugqa wokulingana okungcono kakhulu obikezela idatha ngempumelelo kakhulu. Le ndlela ivame ukusetshenziswa ekuhlaziyweni kokuhlehla okuqondile ukuthola ubudlelwano phakathi kweziguquguquko ezizimele nezixhomeke kuzo. Lesi sihloko sizomboza imiqondo eyisisekelo yendlela yezikwele ezincane, kanye nezibonelo kanye nezincazelo zesinyathelo ngesinyathelo ukuze kuqondwe ngokujulile ukuthi le ndlela isebenza kanjani.
Imiqondo Eyisisekelo Yendlela Yezikwele Ezincane
Umgomo wendlela ye-least squares ukunciphisa isamba sezikwele somehluko phakathi kwamanani abonwe kanye namanani abikezelwe yimodeli yokubuyela emuva. Isibalo somugqa wokubuyela emuva olula singabhalwa kanje:
\[ y = a + bx \]
Di mana:
– \( y \) iyi-variable exhomeke kuyo,
– \( x \) yi-variable ezimele,
– \( a \) yi-intercept (inani lika \( y \) lapho \( x = 0 \)),
– \( b \) ukuthambeka komugqa (ukuthambeka, noma i-regression coefficient).
Indlela yesikwele esincane ilinganisela amapharamitha \( a \) kanye \( b \) anciphisa umsebenzi olandelayo:
\[ \umbhalo{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
Lapho i-SSE iyi-Sum of Squared Errors, i-\( y_i \) iyinani langempela, kanti i-\( \hat{y_i} = a + bx_i \) iyinani elibikezelwe.
Izinyathelo Zendlela Yezikwele Ezincane
Ukuze sicacise umqondo, sizoxazulula inkinga eyisibonelo ehilela ukusetshenziswa kwendlela ye-least squares.
Isibonelo sezinkinga
Njengoba kunikezwe idatha elandelayo:
| x (Amahora okufunda) | y (Amaphuzu okuhlolwa) |
|———————–|——————–|
| 2 | 81 |
| 4 | 93 |
| 6 | 91 |
| 8 | 97 |
| 10 | 103 |
Nquma umugqa wokuhlehla oqondile ofanelana kahle nedatha.
Ingxoxo
1. Ukubala Isilinganiso se-\( \bar{x} \) kanye ne-\( \bar{y} \)
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Ukubala ipharamitha \( b \) (Ukuthambeka)
Ipharamitha \( b \) ibalwa ngo:
\[
b = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sum (x_i – \bar{x})^2}
\]
Ukubala ingxenye ngayinye:
\[
\isamba (x_i – \ibha{x})(y_i – \ibha{y}) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
\sum (x_i – \bar{x})^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
Ukuze:
\[
b = \frac{96}{40} = 2.4
\]
3. Ukubala ipharamitha \( a \) (Intercept)
Kusetshenziswa isilinganiso se-\( \bar{x} \) kanye ne-\( \bar{y} \):
\[
a = \bar{y} – b\bar{x} = 93 – 2.4 \izikhathi 6 = 93 – 14.4 = 78.6
\]
4. Ukubhala Isibalo Somugqa Wokugoba
Ngamapharamitha atholakele, singabhala i-equation yomugqa wokubuyela emuva:
\[
y = 78.6 + 2.4x
\]
Ukuhunyushwa Nokuqinisekiswa
Ukuqinisekisa ukuthi lo mugqa wokubuyela emuva uyalingana, singabala inani lika-y elibikezelwe (\(\hat{y}\)) ku-x ngayinye kudatha yokuqala, kanye nokubala i-Sum of Squared Errors (SSE) ukuqinisekisa ukunemba kwesibikezelo.
| x | y | \(\igqoko{y}\) | \((y – \hat{y})^2\) |
|—|——|—————|———————–|
| 2 | 81 | 83.4 | (81-83.4)^2 = 5.76 |
| 4 | 93 | 88.2 | (93-88.2)^2 = 23.04|
| 6 | 91 | 93.0 | (91-93.0)^2 = 4.00 |
| 8 | 97 | 97.8 | (97-97.8)^2 = 0.64 |
|10 |103 |102.6 | (103-102.6)^2= 0.16|
I-SSS:
\[
I-SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Nge-SSE encane kakhulu, singaphetha ngokuthi umugqa wokubuyela emuva okhiqizwa yindlela ye-least squares ulungele kahle le datha.
Isiphetho
Indlela Yezikwele Ezincane iyithuluzi lokuhlaziya izibalo elinamandla lokunquma umugqa ofaneleka kakhulu wedathasethi, ukunciphisa iphutha lokubikezela ngokusekelwe esikweleni sokuphambuka. Ngokusebenzisa izinyathelo zokubala isilinganiso, ukulinganisa ukuthambeka kanye nokunqamula, kanye nokubhala nokuqinisekisa i-regression line equation, singabikezela ngokunembile inani le-dependent variable kusuka ku-independent variables.
Ukuqonda kahle le ndlela kuwusizo kakhulu emikhakheni efana nezomnotho, i-biostatistics, ubunjiniyela, kanye nesayensi yezenhlalo lapho kusetshenziswa khona ukuhlaziywa kokuhlehla. Lesi sihloko, esinezibonelo eziqondile, sibonisa ukubaluleka kanye nokusebenza kwale ndlela ekuhlaziyweni kwedatha.
