Piv txwv ntawm Cov Lus Nug Sib Tham Txog Txoj Kev Least Squares
Txoj Kev Ntsuas Tsawg Kawg (LEM) yog ib txoj kev suav lej siv los nrhiav txoj kab uas haum tshaj plaws uas kwv yees cov ntaub ntawv tau zoo tshaj plaws. Txoj kev no feem ntau siv rau hauv kev tshuaj xyuas linear regression los nrhiav kev sib raug zoo ntawm cov hloov pauv ywj pheej thiab cov hloov pauv nyob ntawm seb muaj dab tsi. Tsab xov xwm no yuav tham txog cov ntsiab lus yooj yim ntawm txoj kev ntsuas tsawg kawg, nrog rau cov piv txwv thiab cov lus piav qhia ib kauj ruam zuj zus kom nkag siab tob dua txog txoj kev no ua haujlwm li cas.
Cov Ntsiab Lus Tseem Ceeb ntawm Txoj Kev Least Squares
Lub hom phiaj ntawm txoj kev tsawg kawg nkaus yog kom txo qhov sib npaug ntawm cov plaub fab ntawm qhov sib txawv ntawm cov nqi pom thiab cov nqi kwv yees los ntawm tus qauv regression. Cov kab zauv ntawm kab linear regression yooj yim tuaj yeem sau ua:
\[ y = a + bx \]
Qhov twg:
- \( y \) yog qhov hloov pauv nyob ntawm,
- \( x \) yog tus hloov pauv ywj pheej,
- \( a \) yog qhov intercept (tus nqi ntawm \( y \) thaum \( x = 0 \)),
- \( b \) yog qhov nqes hav ntawm txoj kab (qaum hav, lossis coefficient regression).
Txoj kev ntsuas tsawg tshaj plaws kwv yees cov kev cai \(a\) thiab \(b\) uas txo cov haujlwm hauv qab no:
\[ \text{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
Qhov twg SSE yog Qhov Sib Npaug ntawm Squared Yuam Kev, \( y_i \) yog tus nqi tiag tiag, thiab \( \hat{y_i} = a + bx_i \) yog tus nqi kwv yees.
Cov Kauj Ruam ntawm Txoj Kev Least Squares
Yuav kom piav qhia lub tswv yim no kom meej, peb yuav daws ib qho piv txwv teeb meem uas cuam tshuam nrog kev siv txoj kev least squares method.
Piv txwv txog teeb meem
Muab cov ntaub ntawv hauv qab no:
| x (Cov sijhawm kawm) | y (Qhab nia xeem) |
|——————–|———————–|
| 2 | 81 |
| 4 | 93 |
| 6 | 91 |
| 8 | 97 |
| 10 | 103 |
Txheeb xyuas kab linear regression uas haum rau cov ntaub ntawv zoo tshaj plaws.
Kev Sib Tham
1. Xam Qhov Nruab Nrab ntawm \( \bar{x} \) thiab \( \bar{y} \)
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Xam Cov Txheej Txheem \( b \) (Qhov Nce)
Tus parameter \(b\) yog xam los ntawm:
\[
b = \frac{\sum (x_i - x)(y_i - y)}{\sum (x_i - x)^2}
\]
Xam txhua feem:
\[
\sum (x_i – x)(y_i – y) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
\sum (x_i – x)^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
Yog li ntawd:
\[
b = \frac{96}{40} = 2.4
\]
3. Xam Tus Qauv \( a \) (Kev Cuam Tshuam)
Siv qhov nruab nrab ntawm \( \bar{x} \) thiab \( \bar{y} \):
\[
a = y – b x = 93 – 2.4 x 6 = 93 – 14.4 = 78.6
\]
4. Sau Cov Kab Rov Tav Regression
Nrog cov kev ntsuas pom, peb tuaj yeem sau cov kab zauv ntawm kab regression:
\[
y = 78.6 + 2.4x
\]
Kev Txhais Lus thiab Kev Txheeb Xyuas
Yuav kom paub tseeb tias kab regression no haum, peb tuaj yeem xam tus nqi y uas kwv yees tau (\(\hat{y}\)) rau txhua x hauv cov ntaub ntawv pib, thiab xam tus lej Sum of Squared Errors (SSE) los xyuas qhov tseeb ntawm qhov kev kwv yees.
| 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:
\[
SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Nrog rau SSE me me, peb tuaj yeem xaus lus tias kab regression tsim los ntawm txoj kev least squares yog qhov haum rau cov ntaub ntawv no.
Xaus
Txoj Kev Ntsuas Tsawg Tshaj Plaws yog ib qho cuab yeej tshuaj xyuas kev suav lej muaj zog rau kev txiav txim siab qhov kab zoo tshaj plaws rau cov ntaub ntawv teeb tsa, txo qhov yuam kev kwv yees raws li qhov square ntawm qhov sib txawv. Los ntawm kev siv cov kauj ruam ntawm kev xam qhov nruab nrab, kwv yees qhov nqes hav thiab qhov intercept, thiab sau thiab txheeb xyuas qhov kab sib npaug regression, peb tuaj yeem kwv yees qhov tseeb ntawm tus nqi ntawm cov hloov pauv nyob ntawm tus kheej los ntawm cov hloov pauv ywj pheej.
Kev nkag siab zoo txog txoj kev no muaj txiaj ntsig zoo heev rau hauv cov teb xws li kev lag luam, biostatistics, engineering, thiab kev sib raug zoo hauv zej zog uas kev tshuaj xyuas regression feem ntau siv. Tsab xov xwm no, nrog rau cov piv txwv tseeb, qhia txog qhov tseem ceeb thiab kev siv tau ntawm txoj kev no hauv kev tshuaj xyuas cov ntaub ntawv.
