Faʻataʻitaʻiga o se fesili e talanoaina i le fua faʻatatau o le faʻamautuina
O le fua fa'atatau o le fa'amautuina (R²) o se parakalafa taua i le au'ili'iliga o le regression, e fa'aalia ai le lelei o le fa'amatalaina e se fa'ata'ita'iga regression o le fesuia'iga moni o fa'amaumauga. I totonu o lenei tusiga, o le a matou fa'amatalaina le manatu o le fua fa'atatau o le fa'amautuina e ala i se fa'ata'ita'iga auiliili ma se talanoaga.
Manatu Fa'avae o le Coefficient of Determination
O le fua fa'atatau o le fa'amoemoega po'o le \( R^2 \) e fuaina i luga o se fua fa'atatau o le 0 i le 1, lea:
– \( R^2 = 0 \) o loʻo faʻaalia ai e le mafai e le faʻataʻitaʻiga regression ona faʻamatalaina le fesuiaʻiga o faʻamaumauga.
– \( R^2 = 1 \) e faʻaalia ai e mafai e le faʻataʻitaʻiga regression ona faʻamatalaina atoatoa uma fesuiaʻiga o faʻamaumauga.
O le fua fa'atatau mo le fuafuaina o le coefficient of determination o le:
\[ R^2 = 1 – \frac{SSR}{SST} \]
O fea:
– SSR (Aofa'iga o Totoega Fa'atafafa) o le aofa'i lea o sikuea o eseesega i le va o tau na valoia e le fa'ata'ita'iga ma tau moni.
– SST (Aofa'i Atoa o Sikuea) o le aofa'i atoa lea o le aofa'i o sikuea o eseesega i le va o le tau moni ma le averesi o tau moni.
Faʻataʻitaʻiga o faʻafitauli
Se'i o tatou talanoaina se fa'ata'ita'iga o se fa'afitauli ina ia malamalama atili ai i le fuafuaina o le coefficient of determination.
Faʻataʻitaʻiga o faʻafitauli:
Faapea la ua ia i tatou faʻamaumauga i le aofaʻi o itula suʻesuʻe (X) ma togi o suʻega (Y) a tamaiti aʻoga e toʻa 10:
| Tamaiti Aʻoga | Itula Suesue (X) | Sikoa o le Suʻega (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 |
O le a matou fatuina se faʻataʻitaʻiga faigofie o le linear regression lea e valoia ai togi o suʻega (Y) e faʻavae i luga o itula suʻesuʻe (X).
Talanoaga
1. Fausiaina o se Faʻataʻitaʻiga Faʻasolosolo Faʻasolosolo Faigofie
O se faʻataʻitaʻiga faigofie o le linear regression e iai le foliga:
\[ Y = a + bX \]
O fea:
– \( Y \) o le togi o le suega ua valoia.
– \( X \) o le aofaʻi o itula e suʻesuʻe ai.
– \( a \) o le intercept (le tulaga o le fetaulaiga i luga o le Y axis pe a X = 0).
– \( b \) o le fa’asolo (fa’asolo o le laina fa’asolo).
Mo le fuafuaina o parakalafa \( a \) ma le \( b \), matou te faʻaaogaina le fua faʻatatau lenei:
\[ b = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{n(\sum{X^2}) – (\sum{X})^2} \]
\[ a = \frac{\sum{Y} – b(\sum{X})}{n} \]
O fea o le \(n \) o le aofaʻi o faʻamatalaga (i lenei tulaga n = 10).
Mai le laulau e mafai ona tatou fuafuaina:
– \(\aofa'i{X} = 36\)
– \(\aofa'i{Y} = 704\)
– \(\aofa'i{X^2} = 140\)
– \(\aofa'i{Y^2} = 50428\)
– \(\aofa'i{XY} = 2576\)
Se'i o tatou fuafuaina muamua le b:
\[ b = \frac{10(2576) – (36)(704)}{10(140) – (36)^2} \]
\[ b = \frac{25760 – 25344}{1400 – 1296} \]
\[ b = \frac{416}{104} \]
\[ b = 4 \]
Ona tatou fuafuaina lea o se:
\[ a = \frac{704 – 4(36)}{10} \]
\[ a = \frac{704 – 144}{10} \]
\[ a = \frac{560}{10} \]
\[ a = 56 \]
O lea la, o le faʻataʻitaʻiga linear regression tatou te mauaina o le:
\[ Y = 56 + 4X \]
2. Fuafua le Tau Vavaloina (Y')
Sosoo ai, matou te fuafuaina le tau na valoiaina \( Y' \) mo \( X \) taʻitasi:
| Tamaiti Aʻoga | Itula Suesue (X) | Sikoa o le Suʻega (Y) | Sikoa Vavaloina (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. Fuafuaina o le SSR ma le SST
Sosoo ai, matou te fuafuaina le SSR ma le SST e maua ai le \( R^2 \).
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:
\[ SST = \sum{(Y – \bar{Y})^2} \]
O fea:
\[ \bar{Y} = \frac{\sum{Y}}{n} = \frac{704}{10} = 70.4 \]
\[ SST = (58 – 70.4)^2 + (64 – 70.4)^2 + (70 – 70.4)^2 + (85 – 70.4)^2 + (57 – 70.4)^2 + (68 – 70.4)^2 + (72 – 70.4)^2 + (90 – 70.4)^2 + (62 – 70.4)^2 + (78 – 70.4)^2 \]
\[ SST = 153.76 + 40.96 + 0.16 + 213.16 + 178.56 + 5.76 + 2.56 + 384.16 + 70.56 + 57.76 \]
\[ SST = 1107.44 \]
4. Fuafuaina o le Coefficient of Determination \( R^2 \):
\[ R^2 = 1 – \frac{SSR}{SST} \]
\[ R^2 = 1 – \frac{454}{1107.44} \]
\[ R^2 = 1 – 0.41 \]
\[ R^2 = 0.59 \]
I'uga
Mai le fuafuaga o loʻo i luga, na matou maua ai se fua faʻatatau o le faʻamoemoe (R2 = 0.59). O loʻo faʻaalia ai o le faʻataʻitaʻiga linear regression na matou fatuina e mafai ona faʻamatalaina pe tusa ma le 59% o le fesuiaʻiga i togi o suʻega e faʻavae i luga o itula suʻesuʻe. O le 41% o totoe o le fesuiaʻiga e ono mafua mai i isi mea e le o aofia i le faʻataʻitaʻiga.
I le malamalama i laasaga ma fuafuaga o loʻo i luga, e mafai ona tatou iloa ai le taua o le coefficient of determination i le iloiloina o le lelei o le faʻataʻitaʻiga regression tatou te fausia e faʻamatalaina ai le fesuiaʻiga moni o faʻamaumauga. O se meafaigaluega aoga tele i le auʻiliʻiliga faʻamaumauga ma le faʻataʻitaʻiga o faʻamaumauga.
