Muenzaniso wemubvunzo wekukurukurirana pamusoro pe coefficient of determination
Chiyero chekusarudza (R²) chinhu chakakosha pakuongorora kwekudzokorora, zvichiratidza kuti modhi yekudzokorora inotsanangura sei kusiyana kwedata. Muchinyorwa chino, tichatsanangura pfungwa yechiyero chekusarudza kuburikidza nemuenzaniso wakadzama nekukurukurirana.
Pfungwa Yekutanga yeCoefficient yeKusarudza
Chiyero chekutsanangura kana \( R^2 \) chinoyerwa pachikero che0 kusvika 1, apo:
– \( R^2 = 0 \) zvinoratidza kuti modhi yekudzokorora haikwanise kutsanangura kusiyana kwedata zvachose.
– \( R^2 = 1 \) inoratidza kuti modhi yekudzokorora inokwanisa kutsanangura kusiyana kwese kwedata zvakakwana.
Nzira yekutanga yekuverenga coefficient of determination ndeiyi:
\[ R^2 = 1 – \frac{SSR}{SST} \]
Di mana:
– SSR (Square of Squared Residuals) ihuwandu hwemaseko emusiyano uripo pakati pemakoshero akafanotaurwa nemuenzaniso nemakoshero chaiwo.
– SST (Total Sum of Squares) ihuwandu hwese hwezvikwere zvemusiyano uripo pakati pehuwandu chaihwo neavhareji yehuwandu chaihwo.
Muenzaniso wezvinetso
Ngatikurukurei muenzaniso wedambudziko kuti tinzwisise kuverenga kwe coefficient of determination zvakadzama.
Muenzaniso wematambudziko:
Ngatitii tine ruzivo rwehuwandu hwemaawa ekudzidza (X) uye mamakisi ebvunzo (Y) evadzidzi gumi:
| Vadzidzi | Maawa Ekudzidza (X) | Chibodzwa Chebvunzo (Y) |
|——-|——————–|———————–|
| 1 | 2 | 58 |
| 2 | 3 | 64 |
| 3 | 4 | 70 |
| 4 | 5 | 85 |
| 5 | 2 | 57 |
| 6 | 3 | 68 |
| 7 | 4 | 72 |
| 8 | 5 | 90 |
| 9 | 3 | 62 |
| 10 | 4 | 78 |
Tichagadzira muenzaniso wekudzokorora mutsara uri nyore apo mamakisi ebvunzo (Y) anofanotaurwa zvichienderana nemaawa ekudzidza (X).
Kukurukurirana
1. Kuvaka Muenzaniso Wakareruka weKudzoreredza Mutsetse
Muenzaniso wekuregera mutsara uri nyore une chimiro:
\[ Y = a + bX \]
Di mana:
– \( Y \) ndiyo bvunzo yakafanotaurwa.
– \( X \) ndiyo nhamba yemaawa ekudzidza.
– \( a \) ndiyo intercept (nzvimbo yekusangana paY axis apo X = 0).
– \( b \) ndiyo nzira yekutsvedza (kutsvedza kwemutsetse wekudzokera shure).
Kuti tiverenge ma parameters \( a \) uye \( b \), tinoshandisa fomura inotevera:
\[ b = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{n(\sum{X^2}) – (\sum{X})^2} \]
\[ a = \frac{\sum{Y} – b(\sum{X})}{n} \]
Apo \( n \) iri nhamba yedata (munyaya iyi n = 10).
Kubva patafura tinogona kuverenga:
– \(\sum{X} = 36\)
– \(\sum{Y} = 704\)
– \(\sum{X^2} = 140\)
– \(\sum{Y^2} = 50428\)
– \(\sum{XY} = 2576\)
Ngatitangei kuverenga b:
\[ b = \frac{10(2576) – (36)(704)}{10(140) – (36)^2} \]
\[ b = \frac{25760 – 25344}{1400 – 1296} \]
\[ b = \frac{416}{104} \]
\[b = 4 \]
Zvadaro, tinoverenga a:
\[ a = \frac{704 – 4(36)}{10} \]
\[ a = \frac{704 – 144}{10} \]
\[ a = \frac{560}{10} \]
\[ a = 56 \]
Saka, muenzaniso wekudzokorora mutsara watinowana ndewekuti:
\[ Y = 56 + 4X \]
2. Verenga Kukosha Kwakafanotaurwa (Y')
Tevere, tinoverenga kukosha kwakafanotaurwa \( Y' \) kwega yega \( X \):
| Vadzidzi | Maawa Ekudzidza (X) | Chibodzwa Chebvunzo (Y) | Chibodzwa Chakafanotaurwa (Y') |
|——-|——————–|———————–|———————-|
| 1 | 2 | 58 | \( 56 + 4(2) = 64 \) |
| 2 | 3 | 64 | \( 56 + 4(3) = 68 \) |
| 3 | 4 | 70 | \( 56 + 4(4) = 72 \) |
| 4 | 5 | 85 | \( 56 + 4(5) = 76 \) |
| 5 | 2 | 57 | \( 56 + 4(2) = 64 \) |
| 6 | 3 | 68 | \( 56 + 4(3) = 68 \) |
| 7 | 4 | 72 | \( 56 + 4(4) = 72 \) |
| 8 | 5 | 90 | \( 56 + 4(5) = 76 \) |
| 9 | 3 | 62 | \( 56 + 4(3) = 68 \) |
| 10 | 4 | 78 | \( 56 + 4(4) = 72 \) |
3. Kuverenga SSR neSST
Tevere, tinoverenga SSR neSST kuti tiwane \( R^2 \).
SSR:
\[ SSR = \sum{(Y – Y')^2} \]
\[ SSR = (58 – 64)^2 + (64 – 68)^2 + (70 – 72)^2 + (85 – 76)^2 + (57 – 64)^2 + (68 – 68)^2 + (72 – 72)^2 + (90 – 76)^2 + (62 – 68)^2 + (78 – 72)^2 \]
\[ SSR = 36 + 16 + 4 + 81 + 49 + 0 + 0 + 196 + 36 + 36 \]
\[SSR = 454 \]
SST:
\[ SST = \sum{(Y – \ bha{Y})^2} \]
Di mana:
\[ \bar{Y} = \frac{\sum{Y}}{n} = \frac{704}{10} = 70.4 \]
\[ SST = (58 – 70.4)^2 + (64 – 70.4)^2 + (70 – 70.4)^2 + (85 – 70.4)^2 + (57 – 70.4)^2 + (68 – 70.4)^2 + (72 – 70.4)^2 + (90 – 70.4)^2 + (62 – 70.4)^2 + (78 – 70.4)^2 \]
\[ SST = 153.76 + 40.96 + 0.16 + 213.16 + 178.56 + 5.76 + 2.56 + 384.16 + 70.56 + 57.76 \]
\[ SST = 1107.44 \]
4. Kuverenga Coefficient yeKusarudza \( R^2 \):
\[ R^2 = 1 – \frac{SSR}{SST} \]
\[ R^2 = 1 – \frac{454}{1107.44} \]
\[ R^2 = 1 – 0.41 \]
\[ R^2 = 0.59 \]
Mhedziso
Kubva pakuverenga kuri pamusoro apa, takawana coefficient of determination (R2 = 0.59). Izvi zvinoratidza kuti linear regression model yatakagadzira inogona kutsanangura inenge 59% yekuchinja kwemapoinzi ebvunzo zvichibva pamaawa ekudzidza. 41% yasara yekuchinja inogona kunge yakakonzerwa nezvimwe zvinhu zvisina kuiswa mumuenzaniso.
Nekunzwisisa matanho nekuverenga zviri pamusoro apa, tinogona kuona kukosha kwe coefficient of determination mukuongorora kuti regression model yatinovaka inotsanangura sei kusiyana kwedata. Ichi chishandiso chinobatsira zvikuru mukuongorora nhamba uye kuenzanisa data.
