Piv txwv ntawm Linear Regression Cov Lus Nug thiab Kev Sib Tham
Kev txheeb cais linear yog ib txoj kev suav lej siv los txiav txim siab txog kev sib raug zoo ntawm ob lossis ntau qhov hloov pauv. Txoj kev no siv dav hauv ntau qhov chaw, suav nrog kev lag luam, kev lag luam, kev sib raug zoo hauv zej zog, thiab kev tshawb fawb txog ntuj tsim. Hauv tsab xov xwm no, peb yuav tham txog kev txheeb cais linear, yuav ua li cas xam nws, thiab muab ntau qhov piv txwv teeb meem nrog cov lus piav qhia los pab cov nyeem ntawv nkag siab lub tswv yim no tob.
Nkag Siab Txog Linear Regression
Kev txheeb xyuas kab ncaj (linear regression) yog ib txoj kev siv los ua qauv qhia txog kev sib raug zoo ntawm ib lossis ntau tus hloov pauv ywj pheej (cov lus kwv yees) thiab ib tus hloov pauv ywj pheej (cov lus teb). Kev txheeb xyuas kab ncaj (linear regression) yooj yim suav nrog ib tus hloov pauv ywj pheej thiab ib tus hloov pauv ywj pheej, thaum kev txheeb xyuas kab ncaj ntau (multiple linear regression) suav nrog ntau dua ib tus hloov pauv ywj pheej.
Cov qauv ntawm kab linear regression yooj yim yog:
\[ Y = a + bX \]
Qhov twg:
-\(Y\) yog qhov hloov pauv nyob ntawm.
- \( X \) yog qhov hloov pauv ywj pheej.
- \( a \) yog qhov intercept, uas yog tus nqi ntawm Y thaum X = 0.
- \(b \) yog tus coefficient regression, uas yog, Y hloov pauv ntau npaum li cas yog tias X hloov pauv los ntawm ib chav tsev.
Cov Kauj Ruam Linear Regression
1. Sau Cov Ntaub Ntawv: Ua ntej, sau cov ntaub ntawv uas yuav tsum tau soj ntsuam.
2. Cov Ntaub Ntawv Plot: Tsim ib daim duab scatter kom pom seb puas muaj kev sib raug zoo ntawm cov variables.
3. Xam Tus lej Regression: Siv txoj kev least squares los txiav txim seb kab zoo tshaj plaws yog dab tsi.
4. Kev Ntsuas Tus Qauv: Sim qhov tseem ceeb ntawm cov coefficients regression nrog t-test thiab txiav txim siab tus nqi R-squared kom pom tias tus qauv haum rau cov ntaub ntawv zoo npaum li cas.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Piv txwv lus nug 1: Kev hloov pauv yooj yim
Lo lus nug:
Ib tug kws tshawb fawb xav paub txog kev sib raug zoo ntawm tus naj npawb ntawm cov teev kawm (X) thiab cov qhab nia xeem ntawm cov tub ntxhais kawm (Y). Cov ntaub ntawv tau txais yog raws li nram no:
| Cov Sijhawm Kawm (X) | Cov Qhab Nia Xeem (Y) |
|——————–|———————–|
| 2 | 70 |
| 3 | 75 |
| 5 | 80 |
| 7 | 85 |
| 8 | 90 |
Ua ib qho linear regression equation los ntawm cov ntaub ntawv no!
Kev Sib Tham:
1. Xam Qhov Nruab Nrab:
\[
\bar{X} = \frac{2 + 3 + 5 + 7 + 8}{5} = 5
\]
\[
\bar{Y} = \frac{70+75+80+85+90}{5} = 80
\]
2. Xam Tus Coefficient Regression \( b \):
\[
b = \frac{\sum (X_i - X)(Y_i - Y)}{\sum (X_i - X)^2}
\]
\[
\sum (X_i – X)(Y_i – Y) = (2 – 5)(70 – 80) + (3 – 5)(75 – 80) + (5 – 5)(80 – 80) + (7 – 5)(85 – 80) + (8 – 5)(90 – 80)
\]
\[
= (-3)(-10) + (-2)(-5) + (0)(0) + (2)(5) + (3)(10) = 30 + 10 + 0 + 10 + 30 = 80
\]
\[
\sum (X_i – \bar{X})^2 = (2 – 5)^2 + (3 – 5)^2 + (5 – 5)^2 + (7 – 5)^2 + (8 – 5)^2
\]
\[
= 9 + 4 + 0 + 4 + 9 = 26
\]
\[
b = \frac{80}{26} \approx 3.08
\]
3. Xam qhov kev cuam tshuam \( a \):
\[
a = \bar{Y} – b \bar{X}
\]
\[
a = 80 – 3.08 \times 5 = 80 – 15.4 = 64.6
\]
4. Kab zauv rov qab los:
\[
Y = 64.6 + 3.08X
\]
Yog li, qhov sib npaug ntawm kev rov qab los ntawm kab rov tav rau cov ntaub ntawv yog \(Y = 64.6 + 3.08X \). Qhov no txhais tau tias txhua teev ntxiv ntawm kev kawm yuav tsum nce qhov qhab nia xeem los ntawm 3.08 cov ntsiab lus.
Piv txwv lus nug 2: Kev xeem qauv thiab kev txhais lus
Lo lus nug:
Txuas ntxiv nrog cov ntaub ntawv qub, xam tus nqi R-squared (R²) los ntsuas seb tus qauv haum rau cov ntaub ntawv zoo li cas. Tsis tas li ntawd, sim qhov tseem ceeb ntawm tus coefficient regression \(b\).
Kev Sib Tham:
1. Xam Tag Nrho Cov Naj Npawb ntawm Cov Plaub fab (SST), Cov Naj Npawb ntawm Cov Plaub fab rov qab (SSR), thiab Cov Naj Npawb ntawm Cov Plaub fab yuam kev (SSE):
\[
SST = \sum (Y_i – \bar{Y})^2
\]
\[
SST = (70 – 80)^2 + (75 – 80)^2 + (80 – 80)^2 + (85 – 80)^2 + (90 – 80)^2 = 100 + 25 + 0 + 25 + 100 = 250
\]
\[
SSR = \sum (\hat{Y}_i – \bar{Y})^2
\]
Qhov twg \( \hat{Y}_i \) yog tus nqi kwv yees ntawm qhov kev sib npaug 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. Xam SSE:
\[
SSE = SST – SSR = 250 – 149.8 = 100.2
\]
3. Xam R-squared:
\[
R^2 = \frac{SSR}{SST} = \frac{149.8}{250} \approx 0.6
\]
Tus nqi R-squared ntawm 0.6 qhia tau tias tus qauv no piav qhia txog kwv yees li 60% ntawm qhov sib txawv ntawm cov ntaub ntawv. Qhov no qhia tau tias kab regression haum rau cov ntaub ntawv zoo heev.
4. t-Kev Ntsuas Qhov Tseem Ceeb ntawm Coefficient \( b \):
\[
t = \frac{b}{SE(b)}
\]
\[
SE(b) = \sqrt{\frac{SSE}{n-2}} / \sqrt{\sum (X_i – X)^2}
\]
\[
SE(b) = \sqrt{\frac{100.2}{5-2}} / \sqrt{26}
\]
\[
SE(b) = \sqrt{33.4} / \sqrt{26} \approx 1.13
\]
\[
t = \frac{3.08}{1.13} \kwv yees li 2.73
\]
Nrog rau \(t-statistic \approx 2.73 \), yog tias peb siv qhov threshold rau qhov tseem ceeb (α = 0.05), peb piv rau t-rooj. Piv txwv li, rau \(df = 3 \), qhov tseem ceeb \(t \) yog kwv yees li 2.353. Tom qab ntawd \(t-observed > t-critical \), qhia tias tus coefficient yog qhov tseem ceeb.
Xaus
Hauv tsab xov xwm no, peb tau tham txog cov hauv paus ntawm linear regression, yuav ua li cas xam cov coefficient regression thiab intercept, thiab yuav ua li cas txhais cov txiaj ntsig siv cov teeb meem piv txwv. Kev xyaum ntau zaus nrog ntau cov ntaub ntawv yog qhov tseem ceeb kom paub siv txoj kev no. Linear regression yog ib qho cuab yeej muaj txiaj ntsig zoo hauv kev tshuaj xyuas cov ntaub ntawv thiab tuaj yeem muab kev nkag siab tob txog kev sib raug zoo ntawm cov hloov pauv.
