Kev Tshawb Fawb Txog Kev Hloov Pauv Linear Yooj Yim
Kev txheeb xyuas kab ncaj nraim yooj yim yog ib txoj kev suav lej siv los tshuaj xyuas kev sib raug zoo ntawm ob qho kev hloov pauv ntawm cov lej. Cov hloov pauv uas peb tab tom sim kwv yees hu ua cov hloov pauv nyob ntawm seb muaj dab tsi lossis cov lus teb, thaum cov hloov pauv siv los ua qhov kev kwv yees hu ua cov hloov pauv ywj pheej lossis cov hloov pauv kwv yees. Hauv kev txheeb xyuas kab ncaj nraim yooj yim, peb sim nrhiav txoj kab ncaj nraim zoo tshaj plaws uas piav qhia txog kev sib raug zoo ntawm ob qho kev hloov pauv no.
Cov Ntsiab Lus Tseem Ceeb ntawm Kev Hloov Pauv Linear Yooj Yim
Kev txheeb xyuas kab ncaj nraim yooj yim yog raws li qhov kev xav tias muaj kev sib raug zoo ntawm cov hloov pauv nyob ntawm tus kheej \(Y\) thiab cov hloov pauv ywj pheej \(X\). Hom dav dav ntawm tus qauv txheeb xyuas kab ncaj nraim yooj yim yog:
\[ Y = \beta_0 + \beta_1 X + \epsilon \]
Qhov twg:
-\(Y\) yog qhov hloov pauv nyob ntawm.
- \( X \) yog qhov hloov pauv ywj pheej.
- \( \beta_0 \) yog qhov intercept, uas yog tus nqi ntawm \(Y\) thaum \(X = 0\).
- \( \beta_1 \) yog qhov nqes hav lossis qhov gradient, uas yog qhov hloov pauv nruab nrab hauv \(Y\) rau txhua qhov kev hloov pauv hauv \(X\).
- \( \epsilon \) yog qhov yuam kev lossis cov lus seem uas sawv cev rau qhov sib txawv hauv \(Y\) uas tsis tuaj yeem piav qhia los ntawm \(X\).
Lub hom phiaj ntawm kev txheeb xyuas kab ncaj ncaj yooj yim yog kwv yees cov kev cai \(\beta_0\) thiab \(\beta_1\) kom tus qauv siv tau los kwv yees tus nqi ntawm \(Y\) cuam tshuam nrog tus nqi ntawm \(X\).
Txoj Kev Ntsuas Tsawg Tshaj Plaws
Ib qho ntawm cov txheej txheem siv ntau tshaj plaws rau kev haum rau tus qauv linear regression yooj yim yog txoj kev Least Squares. Txoj kev no lub hom phiaj yog kom txo qhov sib npaug ntawm cov squares ntawm qhov sib txawv ntsug ntawm qhov kev soj ntsuam tiag tiag thiab cov nqi kwv yees los ntawm tus qauv. Xav tias peb muaj n kev soj ntsuam uas muaj cov khub \((x_i, y_i)\) rau \(i = 1, 2, …, n\). Lub luag haujlwm yuav tsum tau txo qis yog:
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – (\beta_0 + \beta_1 x_i))^2 \]
Yuav nrhiav tau \(\beta_0\) thiab \(\beta_1\) uas txo qhov kev ua haujlwm no, peb siv cov derivatives ib nrab ntawm \(S(\beta_0, \beta_1)\) nrog rau txhua qhov parameter thiab teeb tsa cov derivatives no rau xoom. Kev suav lej tuaj yeem ua kom yooj yim raws li hauv qab no:
\[ \beta_1 = \frac{\sum_{i=1}^{n} (x_i – x)(y_i – y)}{\sum_{i=1}^{n} (x_i – x)^2} \]
\[ \beta_0 = \bar{y} – \beta_1 \bar{x} \]
Qhov twg:
- \(\bar{x}\) yog qhov nruab nrab ntawm \(X\)
- \(\bar{y}\) yog qhov nruab nrab ntawm \(Y\)
Tom qab tau txais cov kev cai \(\beta_0\) thiab \(\beta_1\), ib qho qauv linear regression yooj yim siv tau los kwv yees tus nqi ntawm \(Y\) rau txhua tus nqi ntawm \(X\).
Cov Kev Xav Hauv Kev Hloov Pauv Linear Yooj Yim
Rau cov txiaj ntsig zoo thiab txhim khu kev qha, kev hloov pauv linear yooj yim xav tias muaj ntau yam:
1. Linearity: Kev sib raug zoo ntawm qhov hloov pauv uas vam khom thiab qhov hloov pauv ywj pheej yuav tsum yog linear.
2. Kev Ywj Pheej: Cov kev soj ntsuam yuav tsum ywj pheej ntawm ib leeg.
3. Homoscedasticity: Qhov hloov pauv seem yuav tsum tsis tu ncua thoob plaws hauv qhov ntau ntawm cov nqi ntawm cov hloov pauv ywj pheej.
4. Qhov seem seem (normality): Cov seem seem (yuam kev) yuav tsum ua raws li kev faib tawm ib txwm muaj.
Yog tias cov kev xav no tsis raug ua tiav, cov txiaj ntsig ntawm tus qauv linear regression yooj yim yuav tsis txhim khu kev qha thiab tej zaum yuav tsis tuaj yeem ua qhov kev kwv yees raug.
Kev Ntsuam Xyuas Qauv Kev Hloov Pauv
Ib txoj kev los ntsuam xyuas seb tus qauv linear regression yooj yim tau kwv yees zoo npaum li cas yog siv Coefficient of Determination (\(R^2\)). Tus coefficient of determination qhia txog feem pua ntawm kev hloov pauv hauv cov variable uas nyob ntawm seb puas muaj peev xwm piav qhia tau los ntawm kev hloov pauv hauv cov variable ywj pheej.
\[ R^2 = \frac{\sum_{i=1}^{n} (y_i – \bar{y})^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2} \]
Qhov twg:
- \(\hat{y}_i\) yog tus nqi kwv yees ntawm \(Y\).
- \(y_i\) yog tus nqi tiag tiag ntawm \(Y\).
- \(\bar{y}\) yog qhov nruab nrab ntawm cov nqi ntawm \(Y\).
Tus nqi \(R^2\) yog txij li 0 txog 1. Tus nqi \(R^2\) uas nyob ze rau 1 qhia tau tias tus qauv no piav qhia tau feem ntau ntawm qhov sib txawv ntawm qhov sib txawv uas nyob ntawm tus neeg.
Kev siv hauv kev sau programming
Yuav kom siv tau yooj yim linear regression, peb siv tau ntau yam software lossis cov lus programming. Hauv qab no yog ib qho piv txwv ntawm kev siv Python siv lub tsev qiv ntawv 'scikit-learn':
"" python
import numpy as np
import matplotlib.pyplot as plt
los ntawm sklearn.linear_model import LinearRegression
los ntawm sklearn.metrics import qhov mean_squared_error, r2_score
Cov ntaub ntawv
X = np.array([[1], [2], [3], [4], [5]]).astype(np.float64)
y = np.array([1.5, 3.6, 3.5, 2.9, 5.5]).astype(np.float64)
qauv
qauv = LinearRegression()
qauv.haum(X, y)
Kev kwv yees
y_pred = qauv.predict(X)
Tus lej sib piv
beta_0 = model.intercept_
beta_1 = model.coef_[0]
luam tawm (f'Intercept: {beta_0}')
luam tawm (f'Qhov Siab: {beta_1}')
luam tawm (f'Qhov yuam kev squared nruab nrab: {mean_squared_error(y, y_pred)}')
luam tawm (f'Tus lej ntawm kev txiav txim siab (R^2): {r2_score(y, y_pred)}')
Cov duab kos thiab kab rov qab
plt.scatter(X, y, color='blue')
plt.plot(X, y_pred, color='liab')
plt.xlabel('X')
plt.ylabel('Y')
plt.qhia()
“
Hauv qhov piv txwv saum toj no, peb xub xa cov tsev qiv ntawv tsim nyog, txhais cov ntaub ntawv \(X\) thiab \(Y\), thiab tom qab ntawd siv cov khoom `LinearRegression` los ntawm `scikit-learn` los haum tus qauv rau cov ntaub ntawv. Thaum tus qauv tau haum, peb ua cov lus kwv yees thiab xam cov coefficients, nrog rau qhov yuam kev squared nruab nrab thiab coefficient ntawm kev txiav txim siab. Thaum kawg, peb plot cov ntaub ntawv thiab kab regression.
Xaus
Kev txheeb xyuas kab ncaj ncaj yooj yim yog ib qho cuab yeej tshuaj xyuas kev suav lej muaj zog uas siv los piav qhia txog kev sib raug zoo ntawm ob qho kev hloov pauv ntau. Nrog qee qhov kev xav yooj yim txog linearity, kev ywj pheej, homoscedasticity, thiab normality, peb tuaj yeem kwv yees tus nqi ntawm cov hloov pauv raws li cov nqi ntawm cov hloov pauv ywj pheej. Txoj kev Least Squares muab txoj hauv kev zoo los haum rau kab rov qab thiab txiav txim siab qhov zoo tshaj plaws. Kev ntsuam xyuas qauv los ntawm tus lej ntawm kev txiav txim siab (R2) muab kev nkag siab txog seb peb tus qauv ua tau zoo li cas.
Txawm hais tias kev hloov pauv linear yooj yim muaj cov kev txwv, xws li tsuas yog muaj peev xwm tswj hwm ob qho hloov pauv thiab cov kev xav uas yuav tsum tau ua tiav, cov txheej txheem no tseem yog lub hauv paus tseem ceeb hauv kev suav lej thiab kev tshuaj xyuas cov ntaub ntawv, thiab feem ntau yog siv ua thawj kauj ruam hauv kev nkag siab txog kev sib raug zoo ntawm cov hloov pauv ua ntej txav mus rau cov txheej txheem nyuaj dua.
