Chitsanzo cha Mafunso Okambirana Njira Yocheperako
Njira ya Least Squares (LEM) ndi njira yowerengera yomwe imagwiritsidwa ntchito kupeza mzere woyenera kwambiri womwe umaneneratu bwino deta. Njirayi nthawi zambiri imagwiritsidwa ntchito mu kusanthula kwa mzere kuti ipeze ubale pakati pa zosintha zodziyimira pawokha ndi zodalira. Nkhaniyi ifotokoza mfundo zoyambira za njira ya least squares, pamodzi ndi zitsanzo zamavuto ndi mafotokozedwe a sitepe ndi sitepe kuti mumvetsetse bwino momwe njira iyi imagwirira ntchito.
Malingaliro Oyambira a Njira Yocheperako
Cholinga cha njira ya ma squares ochepa ndikuchepetsa kuchuluka kwa ma squares a kusiyana pakati pa mitengo yomwe yawonedwa ndi mitengo yomwe yanenedweratu ndi chitsanzo cha regression. Equation ya mzere wosavuta wa regression ikhoza kulembedwa motere:
\[ y = a + bx \]
Kumene:
– \( y \) ndi chosinthika chodalira,
– \( x \) ndi chosinthika chodziyimira pawokha,
– \( a \) ndi intercept (mtengo wa \( y \) pamene \( x = 0 \)),
– \( b \) ndi kutsetsereka kwa mzere (kutsetsereka, kapena coefficient ya regression).
Njira ya least squares imawerengera magawo \( a \) ndi \( b \) omwe amachepetsa ntchito yotsatirayi:
\[ \text{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
Pamene SSE ndi Sum of Squared Errors, \( y_i \) ndi mtengo weniweni, ndipo \( \hat{y_i} = a + bx_i \) ndi mtengo wonenedweratu.
Masitepe a Njira Zocheperako
Kuti timvetse bwino lingaliroli, tithetsa vuto la chitsanzo chokhudza kugwiritsa ntchito njira ya ma squares ang'onoang'ono.
Chitsanzo cha mavuto
Popeza deta iyi ndi iyi:
| x (Maola ophunzirira) | y (Ziwerengero za mayeso) |
|———————–|——————–|
| 2 | 81 |
| 4 | 93 |
| 6 | 91 |
| 8 | 97 |
| 10 | 103 |
Dziwani mzere wobwerera m'mbuyo womwe ukugwirizana bwino ndi deta.
Zokambirana
1. Kuwerengera Avereji ya \( \bar{x} \) ndi \( \bar{y} \)
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Kuwerengera Parameter \( b \) (Slope)
Gawo \( b \) limawerengedwa ndi:
\[
b = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sum (x_i – \bar{x})^2}
\]
Kuwerengera chigawo chilichonse:
\[
\sum (x_i – \bar{x})(y_i – \bar{y}) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
\sum (x_i – \bar{x})^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
Ndicholinga choti:
\[
b = \frac{96}{40} = 2.4
\]
3. Kuwerengera Parameter \( a \) (Intercept)
Pogwiritsa ntchito avareji ya \( \bar{x} \) ndi \( \bar{y} \):
\[
a = \bar{y} – b\bar{x} = 93 – 2.4 \nthawi 6 = 93 – 14.4 = 78.6
\]
4. Kulemba Equation ya Mzere Wobwerera
Ndi magawo omwe apezeka, titha kulemba equation ya mzere wobwerera:
\[
y = 78.6 + 2.4x
\]
Kutanthauzira ndi Kutsimikizira
Kuti tiwonetsetse kuti mzere wobwerera uwu ukugwirizana, titha kuwerengera mtengo wa y woloseredwa (\(\hat{y}\)) wa x iliyonse mu deta yoyambirira, komanso kuwerengera Sum of Squared Errors (SSE) kuti titsimikizire kulondola kwa kulosera.
| | x | y | \(\chipewa{y}\) | \((y – \hat{y}))^2\) |
|—|—-|—————|———————–|
| 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:
\[
SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Ndi SSE yaying'ono, titha kunena kuti mzere wobwerera womwe umapangidwa ndi njira ya least squares ndi woyenera bwino pa data iyi.
Mapeto
Njira ya Least Squares ndi chida champhamvu chowunikira ziwerengero chodziwira mzere woyenera bwino wa deta, kuchepetsa cholakwika cholosera kutengera sikweya ya zolakwika. Pogwiritsa ntchito njira zowerengera mean, slope ndi intercept, ndikulemba ndikutsimikizira regression line equation, titha kulosera molondola mtengo wa dependent variable kuchokera ku zosinthika zodziyimira pawokha.
Kumvetsetsa bwino njira imeneyi n'kothandiza kwambiri m'magawo monga zachuma, ziwerengero za zinthu, uinjiniya, ndi sayansi ya chikhalidwe cha anthu komwe kusanthula kwa regression kumagwiritsidwa ntchito nthawi zambiri. Nkhaniyi, yokhala ndi zitsanzo zenizeni, ikuwonetsa kufunika ndi phindu la njira iyi pakusanthula deta.
