Muenzaniso weMibvunzo yeKukurukurirana yeMaitiro eZvikwere Zvishoma
Nzira yeLeast Squares (LEM) inzira yekuverenga nhamba inoshandiswa kuwana mutsara wezvakanakira zvinofanotaura data zvinobudirira. Nzira iyi inowanzo shandiswa mukuongorora kwemutsara wekuregera kuti ione hukama huripo pakati pezvinhu zvakazvimiririra nezvinoenderana. Chinyorwa chino chichafukidza pfungwa huru dzenzira yeleast squares, pamwe chete nemienzaniso netsananguro dzenhanho nenhanho kuti tinzwisise zvakadzama mashandiro anoita nzira iyi.
Pfungwa Dzekutanga dzeNzira Yezvikwere Zvidiki
Chinangwa chenzira ye least squares ndechekuderedza huwandu hwesquares hwemusiyano uripo pakati pezvakawanikwa uye zvafanotaurwa ne regression model. Equation yemutsetse we regression wakapfava inogona kunyorwa seizvi:
\[ y = a + bx \]
Di mana:
– \( y \) ndiyo shanduko inoenderana,
– \( x \) ndiyo shanduko yakazvimiririra,
– \( a \) ndiyo intercept (kukosha kwe \( y \) apo \( x = 0 \)),
– \( b \) ndiyo nzira yekutsvedza kwemutsetse (slope, kana regression coefficient).
Nzira ye "least squares" inofungidzira ma "parameters" \( a \) uye \( b \) ayo anoderedza basa rinotevera:
\[ \text{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
Apo SSE iri Sum of Squared Errors, \( y_i \) ndiyo kukosha chaiko, uye \( \hat{y_i} = a + bx_i \) ndiyo kukosha kwakafanotaurwa.
Matanho Ekuita Masekonzi Madiki
Kuti tijekese pfungwa iyi, tichagadzirisa dambudziko remuenzaniso rinosanganisira kushandiswa kwenzira ye "least squares".
Muenzaniso wezvinetso
Zvichienderana nedata rinotevera:
| x (Maawa ekudzidza) | y (Mapoinzi ebvunzo) |
|———————–|——————–|
| 2 | 81 |
| 4 | 93 |
| 6 | 91 |
| 8 | 97 |
| 10 | 103 |
Sarudza mutsetse wekudzokera shure unonyatsoenderana nedata.
Kukurukurirana
1. Kuverenga Avhareji ye \( \bar{x} \) uye \( \bar{y} \)
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Kuverenga Paramita \( b \) (Kutsveyama)
Paramita \( b \) inoverengwa ne:
\[
b = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sum (x_i – \bar{x})^2}
\]
Kuverenga chikamu chimwe nechimwe:
\[
\sum (x_i – \bar{x})(y_i – \bar{y}) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
\sum (x_i – \bar{x})^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
Kuti:
\[
b = \frac{96}{40} = 2.4
\]
3. Kuverenga Parameter \( a \) (Intercept)
Uchishandisa avhareji ye \( \bar{x} \) uye \( \bar{y} \):
\[
a = \bar{y} – b\bar{x} = 93 – 2.4 \kawanza 6 = 93 – 14.4 = 78.6
\]
4. Kunyora Regression Line Equation
Nema parameter akawanikwa, tinogona kunyora equation yemutsara we regression:
\[
y = 78.6 + 2.4x
\]
Dudziro uye Kusimbiswa
Kuti tive nechokwadi chekuti mutsetse uyu wekudzoka unoenderana, tinogona kuverenga y-value yakafanotaurwa (\(\hat{y}\)) ye x yega yega mudata rekutanga, pamwe nekuverenga Sum of Squared Errors (SSE) kuti tisimbise kururama kwekufanotaura.
| 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:
\[
SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Nekunge paine SSE diki, tinogona kugumisa kuti mutsetse we regression unogadzirwa nenzira ye least squares wakakodzera data iri.
Mhedziso
Nzira yeLeast Squares chishandiso chine simba chekuongorora nhamba chekuona mutsetse wakakodzera dataset, kuderedza kukanganisa kwekufanotaura zvichibva pasquare yezvakatsauka. Nekushandisa matanho ekuverenga avhareji, kufungidzira slope ne intercept, uye kunyora nekusimbisa regression line equation, tinogona kufanotaura nemazvo kukosha kwe dependent variable kubva kune independent variables.
Kunzwisisa nzira iyi zvakanaka kunobatsira zvikuru muzvikamu zvakaita sezvehupfumi, biostatistics, engineering, uye social sciences uko kuongorora kwekudzoka kwedata kunowanzo shandiswa. Chinyorwa chino, chine mienzaniso chaiyo, chinoratidza kukosha uye kushanda kwenzira iyi mukuongorora data.
