Kuongororwa kweKuderedzwa kweLinear kuri Nyore
Kudzokorora mutsara kuri nyore inzira yekuverenga inoshandiswa kuongorora hukama huripo pakati pezvinhu zviviri zvinoumba huwandu hwehuwandu. Chinochinja chatiri kuedza kufanotaura chinonzi dependent kana response variable, nepo chinochinja chinoshandiswa kuita kufanotaura chichinzi independent kana predictor variable. Mukudzokorora mutsara kuri nyore, tinoedza kuwana mutsetse wakatwasuka wakanakisisa unotsanangura hukama huripo pakati pezvinhu zviviri izvi.
Pfungwa Dzekutanga dzeKudzoreredza Kurongeka Kwakareruka
Kudzoreredza mutsara kuri nyore kwakavakirwa pafungidziro yekuti pane hukama hwakatsetseka pakati pekuchinja kunoenderana \(Y\) uye kuchinja kwakazvimiririra \(X\). Chimiro chakajairika chemuenzaniso wekudzokorora mutsara uri nyore ndeichi:
\[ Y = \beta_0 + \beta_1 X + \epsilon \]
Di mana:
– \( Y \) ishanduro inoenderana.
– \( X \) ishanduro yakazvimiririra.
– \( \beta_0 \) ndiyo intercept, inova kukosha kwe \(Y\) kana \(X = 0\).
– \( \beta_1 \) ndiyo nzira yekutsvedza kana kuti gradient, inova ndiyo avhareji yekuchinja mu \(Y\) kwekuchinja kweyuniti yega yega mu \(X\).
– \( \epsilon \) ishoko rekuti kukanganisa kana kuti residual term rinomiririra shanduko iri mu \(Y\) isingagone kutsanangurwa ne \(X\).
Chinangwa chekudzoreredza mutsara kuri nyore ndechekufungidzira maparamita \(\beta_0\) uye \(\beta_1\) kuitira kuti modhi igone kushandiswa kufanotaura kukosha kwe \(Y\) kwakabatana nekukosha kwe \(X\).
Nzira yeZvikwere Zvidiki
Imwe yenzira dzinonyanya kushandiswa pakuisa muenzaniso wekuregera mutsara uri nyore ndiyo nzira yeLeast Squares. Nzira iyi ine chinangwa chekuderedza huwandu hwemasekonzi ekutsauka kwakamira pakati pezvakaonekwa chaizvo uye kukosha kwakafanotaurwa nemuenzaniso. Ngatitii tine n observations ine mapeya \((x_i, y_i)\) ye \(i = 1, 2, …, n\). Basa rinofanira kuderedzwa ndeiri:
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – (\beta_0 + \beta_1 x_i))^2 \]
Kuti tiwane \(\beta_0\) uye \(\beta_1\) zvinoderedza basa iri, tinotora zvikamu zve \(S(\beta_0, \beta_1)\) maererano neparameter yega yega toisa izvi zvinova zero. Kuverenga kwemasvomhu kunogona kurerutswa seizvi:
\[ \beta_1 = \frac{\sum_{i=1}^{n} (x_i – \bar{x})(y_i – \bar{y})}{\sum_{i=1}^{n} (x_i – \bar{x})^2} \]
\[ \beta_0 = \mba{y} – \beta_1 \mba{x} \]
Di mana:
– \(\bar{x}\) ndiyo avhareji ye \(X\)
– \(\bar{y}\) ndiyo avhareji ye \(Y\)
Mushure mekuwana ma parameters \(\beta_0\) uye \(\beta_1\), muenzaniso we regression wakapfava unogona kushandiswa kufanotaura kukosha kwe \(Y\) kwemutengo wega wega we \(X\).
Mafungiro muKudzokorora Kwakareruka Kwemutsetse
Kuti uwane mhedzisiro inoshanda uye yakavimbika, kudzokororwa kuri nyore kwemutsara kunotora zvinhu zvakati wandei:
1. Kurongeka: Hukama huripo pakati pechinhu chinotsamira pane chimwe chinhu nechinhu chinotsamira pachimwe chinhu chinofanira kunge chakatwasuka.
2. Kuzvimiririra: Zvinoonekwa zvinofanira kunge zvakazvimiririra.
3. Homoscedasticity: Kusiyana kwasara kunofanirwa kugara kuripo muhuwandu hwezvinhu zvinoshanduka zvakazvimiririra.
4. Kugara Kwakajairwa: Zvikanganiso zvinosara zvinofanira kutevedzera kugoverwa kwakajairika.
Kana fungidziro idzi dzikasazadzikiswa, mhedzisiro yemuenzaniso wekuregera wakapfava hauzovimbike uye ungasakwanisa kufanotaura zvakarurama.
Kuongorora kweModhi yeKudzokera shure
Imwe nzira yekuongorora kuti muenzaniso wekuregera mutsara wakafanotaura zvakanaka sei ndeyekushandisa Coefficient of Determination (\(R^2\)). Coefficient of determination inoratidza huwandu hwekusiyana mu dependent variable inogona kutsanangurwa nekusiyana mu independent variables.
\[ R^2 = \frac{\sum_{i=1}^{n} (\hat{y}_i – \bar{y})^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2} \]
Di mana:
– \(\hat{y}_i\) ndiyo kukosha kwakafanotaurwa kwe \(Y\).
– \(y_i\) ndiyo kukosha kwayo chaiko kwe \(Y\).
– \(\bar{y}\) ndiyo avhareji yehuwandu hwe \(Y\).
Kukosha kwe \(R^2\) kunotangira pa0 kusvika pa1. Kukosha kwe \(R^2\) kuri pedyo ne1 kunoratidza kuti modhi inogona kutsanangura zvakawanda zvekuchinja kuri mu dependent variable.
Kuitwa kweMutauro weKuronga
Kuti tishandise nzira iri nyore yekudzokorora mutsara, tinogona kushandisa mapurogiramu akasiyana-siyana ekuverenga nhamba kana mitauro yepurogiramu. Pazasi pane muenzaniso wekushandisa muPython uchishandisa raibhurari ye`scikit-learn`:
"'python
import numpy se np
import matplotlib.pyplot as plt
kubva sklearn.linear_model pinza LinearRegression
kubva sklearn.metrics import mean_squared_error, r2_score
Data
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)
mhando
modhi = LinearRegression()
model.fit(X, y)
Prediction
y_pred = model.predict(X)
Koefficient
beta_0 = model.intercept_
beta_1 = model.coef_[0]
print(f'Intercept: {beta_0}')
print(f'Slope: {beta_1}')
print(f'Chikanganiso chepakati pemativi maviri: {mean_squared_error(y, y_pred)}')
print(f'Coefficient of determination (R^2): {r2_score(y, y_pred)}')
Kuronga kwedata uye mutsetse wekudzoka
plt.scatter(X, y, ruvara='bhuruu')
plt.plot(X, y_pred, ruvara='tsvuku')
plt.xlabel('X')
plt.ylabel('Y')
plt.show ()
``
Mumuenzaniso uri pamusoro apa, tinotanga tapinza maraibhurari anodiwa, totsanangura data \(X\) uye \(Y\), tobva tashandisa chinhu che`LinearRegression` kubva ku`scikit-learn` kuti tikwanise modhi nedata. Kana modhi yaiswa, tinofanotaura uye tinoverenga ma coefficients, pamwe ne mean squared error uye coefficient of determination. Chekupedzisira, tinoronga data ne regression line.
Mhedziso
Kudzoreredza mutsara kuri nyore chishandiso chine simba chekuongorora nhamba chinoshandiswa kutsanangura hukama huripo pakati pezvinhu zviviri zvinochinja huwandu. Nekufunga kwakakosha nezve kurongeka, kuzvimirira, homoscedasticity, uye kurongeka, tinogona kufanotaura kukosha kwechinhu chinochinja zvichienderana nehukuru hwezvinhu zvinochinja zvakazvimiririra. Nzira yeLeast Squares inopa nzira inoshanda yekukodzera mutsetse wekudzoka uye kuona maparameter akakodzera. Kuongororwa kwemuenzaniso kuburikidza ne coefficient of determination (R2) kunopa nzwisiso yekuti modhi yedu inoshanda sei.
Kunyangwe kudzokororwa kwemutsara kuri nyore kune zvipingamupinyi, zvakaita sekukwanisa kubata mavariable maviri chete nefungidziro dzinofanira kuzadzikiswa, nzira iyi inoramba iri hwaro hwakakosha muhuwandu nekuongorora data, uye inowanzoshandiswa sedanho rekutanga mukunzwisisa hukama huripo pakati pemavariable usati waenderera mberi kune nzira dzakaoma.
