Fa'ata'ita'iga o Fesili ma Talanoaga e uiga i le Linear Regression
O le linear regression o se metotia fa'amaufa'ailoga e fa'aaogaina e fuafua ai le sootaga i le va o ni fesuia'iga se lua pe sili atu. O lenei metotia e fa'aaogaina lautele i matā'upu eseese, e aofia ai le tamaoaiga, pisinisi, saienisi lautele, ma saienisi fa'alenatura. I totonu o lenei tusiga, o le a matou talanoaina ai le linear regression, pe fa'apefea ona fuafua, ma tu'uina atu ni fa'ata'ita'iga o fa'afitauli fa'atasi ai ma fa'amatalaga e fesoasoani ai i le au faitau ia malamalama loloto i lenei manatu.
Malamalama i le Linear Regression
O le linear regression o se metotia fa'ata'ita'i e fa'aaogaina e fa'ata'ita'i ai le sootaga i le va o le tasi pe sili atu fesuia'iga tuto'atasi (predictors) ma se fesuia'iga fa'alagolago (response). O le simple linear regression e aofia ai le tasi fesuia'iga tuto'atasi ma le tasi fesuia'iga fa'alagolago, ae o le multiple linear regression e aofia ai le sili atu ma le tasi fesuia'iga tuto'atasi.
O le fua fa'atatau o se laina fa'asolosolo faigofie e fa'apea:
\[ Y = a + bX \]
O fea:
– \( Y \) o le fesuia'iga fa'alagolago.
– \( X \) o le fesuia'iga tuto'atasi.
– \( a \) o le intercept, o le tau lea o le Y pe a X = 0.
– \( b \) o le fua fa'atatau o le regression, o lona uiga, o le tele o suiga o le Y pe afai e suia le X i le tasi le iunite.
Laasaga o le Fa'asolosolo Fa'asolosolo
1. Aoina Fa'amaumauga: Muamua, ao mai fa'amaumauga e su'esu'eina.
2. Fa'asologa o Fa'amaumauga: Fausia se fa'asologa fa'asalalau e va'ai pe i ai se sootaga laina i le va o fesuia'iga.
3. Fuafua le Regression Coefficient: Faaaoga le metotia o le least squares e fuafua ai le laina sili ona lelei.
4. Su'eina o le Fa'ata'ita'iga: Su'e le taua o le regression coefficients i se t-test ma fuafua le tau R-squared e va'ai pe fa'apefea ona fetaui lelei le fa'ata'ita'iga ma fa'amaumauga.
Fesili Fa'ata'ita'i ma Talanoaga
Fa'ata'ita'iga Fesili 1: Fa'asologa Fa'asolosolo Fa'alava Faigofie
Fesili:
E fia iloa e se tagata suʻesuʻe le sootaga i le va o le aofaʻi o itula e suʻesuʻe ai (X) ma togi o suʻega a tamaiti aʻoga (Y). O faʻamatalaga na maua e faʻapea:
| Itula Suesue (X) | Sikoa o le Suega (Y) |
|———————–|——————–|
| 2 | 70 |
| 3 | 75 |
| 5 | 80 |
| 7 | 85 |
| 8 | 90 |
Fai se fua fa'atatau o le regression linear mai nei fa'amatalaga!
Talanoaga:
1. Fuafuaina o le Averesi:
\[
\bar{X} = \frac{2 + 3 + 5 + 7 + 8}{5} = 5
\]
\[
\pa{Y} = \frac{70 + 75 + 80 + 85 + 90}{5} = 80
\]
2. Fuafuaina o le Regression Coefficient \( b \):
\[
b = \frac{\sum (X_i – \bar{X})(Y_i – \bar{Y})}{\sum (X_i – \bar{X})^2}
\]
\[
\aofa'i (X_i – \bar{X})(Y_i – \bar{Y}) = (2 – 5)(70 – 80) + (3 – 5)(75 – 80) + (5 – 5)(80 – 80) + (7 – 5)(85 – 80) + (8 – 5)(90 – 80)
\]
\[
= (-3)(-10) + (-2)(-5) + (0)(0) + (2)(5) + (3)(10) = 30 + 10 + 0 + 10 + 30 = 80
\]
\[
\aofa'i (X_i – \bar{X})^2 = (2 – 5)^2 + (3 – 5)^2 + (5 – 5)^2 + (7 – 5)^2 + (8 – 5)^2
\]
\[
= 9 + 4 + 0 + 4 + 9 = 26
\]
\[
b = \frac{80}{26} \approx 3.08
\]
3. Fuafuaina o le Intercept \( a \):
\[
a = \bar{Y} – b\bar{X}
\]
\[
a = 80 – 3.08 \fa'ateleina 5 = 80 – 15.4 = 64.6
\]
4. Fa'atusatusaga o le Regression:
\[
Y = 64.6 + 3.08X
\]
O lea la, o le fua fa'atatau o le linear regression mo fa'amaumauga o le \( Y = 64.6 + 3.08X \). O lona uiga o itula fa'aopoopo ta'itasi o le su'esu'ega e fa'amoemoeina e fa'ateleina ai le togi o le suega i le 3.08 points.
Fesili Faataitaiga 2: Suega Faataitaiga ma le Faauigaga
Fesili:
Fa'aauau pea i fa'amaumauga lava e tasi, fuafua le tau o le R-squared (R²) e fua ai pe o le a le fetaui lelei o le fa'ata'ita'iga ma fa'amaumauga. Fa'ata'ita'i fo'i le taua o le regression coefficient \( b \).
Talanoaga:
1. Fuafua le Aofa'i Atoa o Sikuea (SST), le Aofa'i Fa'asolosolo o Sikuea (SSR), ma le Aofa'i Fa'aletonu o Sikuea (SSE):
\[
SST = \aofa'i (Y_i – \bar{Y})^2
\]
\[
SST = (70 – 80)^2 + (75 – 80)^2 + (80 – 80)^2 + (85 – 80)^2 + (90 – 80)^2 = 100 + 25 + 0 + 25 + 100 = 250
\]
\[
SSR = \aofa'i (\hat{Y}_i – \bar{Y})^2
\]
O fea o le \( \hat{Y}_i \) o le tau valoia o le fua fa'atatau o le regression:
\[
\hat{Y}_i = 64.6 + 3.08X_i
\]
\[
\hat{Y} = [67.76, 70.84, 76.0, 82.16, 85.24]
\]
\[
\bar{Y} = 80
\]
\[
SSR = (67.76 – 80)^2 + (70.84 – 80)^2 + (76.0 – 80)^2 + (82.16 – 80)^2 + (85.24 – 80)^2
\]
\[
SSR = (-12.24)^2 + (-9.16)^2 + (-4.0)^2 + 2.16^2 + 5.24^2 = 149.8
\]
2. Fuafuaina o le SSE:
\[
SSE = SST – SSR = 250 – 149.8 = 100.2
\]
3. Fuafuaina o le R-squared:
\[
R^2 = \frac{SSR}{SST} = \frac{149.8}{250} \approx 0.6
\]
O le tau R-squared e 0.6 e faʻaalia ai o lenei faʻataʻitaʻiga e faʻamatalaina pe tusa ma le 60% o le eseesega i faʻamaumauga. O loʻo faʻaalia ai o le laina regression e fetaui lelei ma faʻamaumauga.
4. Su'ega-t mo le Taua o le Coefficient \( b \):
\[
t = \frac{b}{SE(b)}
\]
\[
SE(b) = \sqrt{\frac{SSE}{n-2}} / \sqrt{\sum (X_i – \bar{X})^2}
\]
\[
SE(b) = \sqrt{\frac{100.2}{5-2}} / \sqrt{26}
\]
\[
SE(b) = \sqrt{33.4} / \sqrt{26} \approx 1.13
\]
\[
t = \frac{3.08}{1.13} \approx 2.73
\]
Faatasi ai ma le \( t-statistic \approx 2.73 \), afai tatou te faaaogaina se tapulaa masani mo le taua (α = 0.05), tatou te faatusatusa i le t-table. Mo se faataitaiga, mo le \( df = 3 \), o le taua \( t \) e tusa ma le 2.353. Ona \( t-observed > t-critical \), o lona uiga o le coefficient e taua.
I'uga
I totonu o lenei tusiga, ua matou talanoaina ai mea faavae o le linear regression, le auala e fuafua ai le regression coefficient ma le intercept, ma le auala e faʻamatalaina ai taunuuga e faʻaaoga ai faʻataʻitaʻiga faʻafitauli. O le faʻataʻitaʻi soo i seti faʻamaumauga eseese e taua tele ina ia tomai i le faʻaaogaina o lenei metotia. O le linear regression o se meafaigaluega taua i le auʻiliʻiliga o faʻamaumauga ma e mafai ona maua ai ni malamalamaaga loloto i sootaga i le va o fesuiaʻiga.
