Mohlala oa Mokhoa oa Lisekwere tse Nyenyane Lipotso tsa Puisano
Mokhoa oa Least Squares (LEM) ke mokhoa oa lipalo-palo o sebelisoang ho fumana mola oa ho lekana hantle o bolelang esale pele data ka katleho. Mokhoa ona o atisa ho sebelisoa tlhahlobong ea linear regression ho lemoha kamano pakeng tsa mefuta-futa e ikemetseng le e itšetlehileng ka eona. Sehlooho sena se tla akaretsa likhopolo tsa motheo tsa mokhoa oa least squares, hammoho le mehlala le litlhaloso tsa mohato ka mohato bakeng sa kutloisiso e tebileng ea hore na mokhoa ona o sebetsa joang.
Mehopolo ea Motheo ea Mokhoa oa Likwere tse Nyenyane
Sepheo sa mokhoa oa lisekoere tse nyane ke ho fokotsa kakaretso ea lisekoere tsa liphapang lipakeng tsa boleng bo bonoang le boleng bo boletsoeng esale pele ke mohlala oa regression. Tekanyo ea mola o bonolo oa regression o ka ngoloa tjena:
\[ y = a + bx \]
Di mana:
– \( y \) ke phetoho e itshetlehileng,
– \( x \) ke phetoho e ikemetseng,
– \( a \) ke khaotso (boleng ba \( y \) ha \( x = 0 \)),
– \( b \) ke leralla la mola (leralla, kapa coefficient ea regression).
Mokhoa oa lisekoere tse nyane o hakanya liparamente \( a \) le \( b \) tse fokotsang ts'ebetso e latelang:
\[ \text{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
Moo SSE e leng Kakaretso ea Liphoso tse Sekwere, \( y_i \) ke boleng ba 'nete, 'me \( \hat{y_i} = a + bx_i \) ke boleng bo boletsoeng esale pele.
Mehato ea Mokhoa oa Likwere tse Nyenyane
Ho hlakisa mohopolo ona, re tla rarolla bothata ba mohlala bo kenyeletsang tshebediso ya mokgwa wa dikwere tse nyane.
Mohlala oa mathata
Ka lebaka la lintlha tse latelang:
| x (Lihora tsa ho ithuta) | y (Lintlha tsa tlhahlobo) |
|———————–|——————–|
| 2 | 81 |
| 4 | 93 |
| 6 | 91 |
| 8 | 97 |
| 10 | 103 |
Fumana mola o otlolohileng oa ho khutlela morao o lumellanang hantle le data.
Puisano
1. Ho Bala Karolelano ea \( \bar{x} \) le \( \bar{y} \)
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Ho Bala Paramethara \( b \) (Moedi)
Paramethara \( b \) e balwa ka:
\[
b = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sum (x_i – \bar{x})^2}
\]
Ho bala karolo ka 'ngoe:
\[
\sum (x_i – \bar{x})(y_i – \bar{y}) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
\sum (x_i – \bar{x})^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
E le hore:
\[
b = \frac{96}{40} = 2.4
\]
3. Ho bala Paramethara \( a \) (Khetholla)
Ho sebelisoa karolelano ea \( \bar{x} \) le \( \bar{y} \):
\[
a = \bar{y} – b\bar{x} = 93 – 2.4 \makgetlo a 6 = 93 – 14.4 = 78.6
\]
4. Ho Ngola Tekanyo ea Mola oa Regression
Ka diparamitha tse fumanweng, re ka ngola equation ya mola wa regression:
\[
y = 78.6 + 2.4x
\]
Tlhaloso le Netefatso
Ho netefatsa hore mola ona wa ho kgutlisa o a lekana, re ka bala boleng ba y bo boletsweng esale pele (\(\hat{y}\)) bakeng sa x ka nngwe ho data ya pele, hammoho le ho bala Kakaretso ya Diphoso tse Sekwere (SSE) ho netefatsa ho nepahala ha ponelopele.
| x | y | \(\hat{y}\) | \((y – \hat{y})^2\) |
|—|——|—————|———————–|
| 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|
SSS:
\[
SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Ka SSE e nyane haholo, re ka etsa qeto ea hore mola oa regression o hlahisoang ke mokhoa oa lisekoere tse nyane o loketse data ena hantle.
Qetello
Mokhoa oa Least Squares ke sesebelisoa se matla sa tlhahlobo ea lipalo-palo bakeng sa ho fumana mola o loketseng hantle bakeng sa sete ea data, ho fokotsa phoso ea ho bolela esale pele ho latela sekwere sa liphapang. Ka ho sebelisa mehato ea ho bala karolelano, ho hakanya leralla le ho thibela, le ho ngola le ho netefatsa equation ea mola oa regression, re ka bolela esale pele ka nepo boleng ba phetoho e itšetlehileng ka eona ho tsoa ho li-variable tse ikemetseng.
Kutloisiso e ntle ea mokhoa ona e molemo haholo masimong a kang moruo, lipalo-palo tsa baeloji, boenjiniere le mahlale a kahisano moo tlhahlobo ea regression e sebelisoang khafetsa. Sengoloa sena, se nang le mehlala e tiileng, se bontša bohlokoa le molemo oa mokhoa ona tlhahlobong ea data.
