Ukuhlaziywa Kokuhlehla Okulula Komugqa
Ukuhlehla okulula komugqa kuyindlela yezibalo esetshenziswa ukuhlaziya ubudlelwano phakathi kweziguquguquko ezimbili zobuningi. Iguquguquko esizama ukulibikezela libizwa ngokuthi i-dependent noma i-response variable, kanti i-variable esetshenziswa ukwenza isibikezelo ibizwa ngokuthi i-independent noma i-predictor variable. Ku-regression elula yomugqa, sizama ukuthola umugqa oqondile ongcono kakhulu ochaza ubudlelwano phakathi kwalezi ziguquguquko ezimbili.
Imiqondo Eyisisekelo Yokuhlubuka Okulula Komugqa
Ukuhlehla okulula okuqondile kusekelwe ekucabangeni ukuthi kukhona ubudlelwano obuqondile phakathi kwe-variable exhomeke ku-\(Y\) kanye ne-variable ezimele \(X\). Uhlobo olujwayelekile lwemodeli elula yokuhlehla okuqondile yilolu:
\[ Y = \beta_0 + \beta_1 X + \epsilon \]
Di mana:
– \( Y \) iyi-variable exhomeke kuyo.
– \( X \) yi-variable ezimele.
– \( \beta_0 \) yi-intercept, okuyinani lika-\(Y\) lapho \(X = 0\).
– \( \beta_1 \) ukuthambeka noma i-gradient, okuyishintsho elimaphakathi ku-\(Y\) loshintsho ngalunye lweyunithi ku-\(X\).
– \( \epsilon \) yiphutha noma igama elisele elimele ukuguquguquka ku-\(Y\) okungenakuchazeka yi-\(X\).
Umgomo wokuhlehla okulula okuqondile ukulinganisa amapharamitha \(\beta_0\) kanye \(\beta_1\) ukuze imodeli ingasetshenziswa ukubikezela inani \(Y\) elihlotshaniswa nenani \(X\).
Indlela Yezikwele Ezincane
Enye yezindlela ezisetshenziswa kakhulu zokufaka imodeli elula yokuqondisa eqondile yindlela ye-Least Squares. Le ndlela ihlose ukunciphisa isamba sezikwele zokuphambuka okuqondile phakathi kokubonwa kwangempela kanye namanani abikezelwe yimodeli. Ake sithi sinokubonwa okungu-n okuhlanganisa amabhangqa \((x_i, y_i)\) kwe-\(i = 1, 2, …, n\). Umsebenzi okufanele uncishiswe yilo:
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – (\beta_0 + \beta_1 x_i))^2 \]
Ukuze sithole i-\(\beta_0\) kanye ne-\(\beta_1\) ezinciphisa lo msebenzi, sithatha ama-derivative angaphelele e-\(S(\beta_0, \beta_1)\) maqondana nepharamitha ngayinye bese sibeka la ma-derivatives ku-zero. Ukubalwa kwezibalo kungenziwa lula kanje:
\[ \beta_1 = \frac{\sum_{i=1}^{n} (x_i – \bar{x})(y_i – \bar{y})}{\sum_{i=1}^{n} (x_i – \bar{x})^2} \]
\[ \beta_0 = \ibha{y} – \beta_1 \ibha{x} \]
Di mana:
– \(\bar{x}\) isilinganiso se-\(X\)
– \(\bar{y}\) isilinganiso se-\(Y\)
Ngemva kokuthola amapharamitha \(\beta_0\) kanye \(\beta_1\), imodeli elula yokuqondisa eqondile ingasetshenziswa ukubikezela inani \(Y\) lenani ngalinye \(X\).
Izinkolelo ku-Simple Linear Regression
Ukuze uthole imiphumela evumelekile nethembekile, ukuhlehla okulula okuqondile kucabanga izinto eziningana:
1. Ukulingana: Ubudlelwano phakathi kwe-variable exhomeke kuyo kanye ne-variable ezimele kumele bube yi-linear.
2. Ukuzimela: Izinto ezibonwayo kumele zingabi ngezabanye.
3. Ukuguquguquka: Ukuguquguquka okusele kumele kuhlale njalo kulo lonke ububanzi bamanani e-variable ezimele.
4. Ukulingana Okusele: Ama-Residuals (amaphutha) kumele alandele ukusatshalaliswa okuvamile.
Uma lokhu kuqagela kungahlangatshezwana nakho, imiphumela yemodeli elula yokubuyela emuva eqondile ngeke ithembeke futhi ingase ingakwazi ukwenza izibikezelo ezinembile.
Ukuhlolwa Kwemodeli Yokuhlehla
Enye indlela yokuhlola ukuthi imodeli elula yokuqondisa eqondile ibikezele kahle kangakanani ukusebenzisa i-Coefficient of Determination (\(R^2\)). I-coefficient of determination ikhombisa isilinganiso sokuguquguquka ku-variable exhomeke ku-dependent engachazwa ngokuguquguquka ku-variables ezimele.
\[ R^2 = \frac{\sum_{i=1}^{n} (\hat{y}_i – \bar{y})^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2} \]
Di mana:
– \(\hat{y}_i\) inani elibikezelwe lika-\(Y\).
– \(y_i\) inani langempela lika-\(Y\).
– \(\bar{y}\) isilinganiso samanani ka-\(Y\).
Inani le-\(R^2\) lisukela ku-0 kuya ku-1. Inani le-\(R^2\) eliseduze no-1 libonisa ukuthi imodeli ingachaza iningi lokuguquguquka ku-variable exhomeke kuyo.
Ukusetshenziswa kolimi lokuhlela
Ukuze sisebenzise ukuhlehla okulula okuqondile, singasebenzisa isofthiwe yezibalo noma izilimi zokuhlela ezahlukahlukene. Ngezansi isibonelo sokusetshenziswa ku-Python sisebenzisa umtapo wolwazi we-`scikit-learn`:
"`python
ngenisa i-numpy njenge-np
ngenisa i-matplotlib.pyplot njenge-plt
kusuka ku-sklearn.linear_model ngenisa i-LinearRegression
kusuka ku-sklearn.metrics ngenisa iphutha_eliyisikwele_eliphakathi, i-r2_score
Idatha
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)
imodeli
imodeli = LinearRegression ()
imodeli.fit(X, y)
Ukubikezela
y_pred = imodeli.ukubikezela(X)
I-Coefficient
i-beta_0 = imodeli.ukuvimbela_
i-beta_1 = imodeli.coef_[0]
phrinta(f'Intercept: {beta_0}')
phrinta(f'Slope: {beta_1}')
phrinta(f'Iphutha eliyisikwele eliphakathi: {mean_squared_error(y, y_pred)}')
phrinta(f'Isilinganiso sokunquma (R^2): {r2_score(y, y_pred)}')
Isakhiwo sedatha kanye nomugqa wokubuyela emuva
plt.scatter(X, y, umbala='oluhlaza okwesibhakabhaka')
plt.plot(X, y_pred, umbala='obomvu')
i-plt.xlabel('X')
i-plt.ylabel('Y')
i-plt.show()
``
Esibonelweni esingenhla, siqala ngokungenisa imitapo yolwazi edingekayo, sichaze idatha \(X\) kanye \(Y\), bese sisebenzisa into ethi `LinearRegression` evela ku-`scikit-learn` ukuze sifake imodeli kudatha. Uma imodeli isifakiwe, senza izibikezelo futhi sibale ama-coefficient, kanye nephutha eliyisikwele eliphakathi kanye ne-coefficient yokunquma. Okokugcina, sihlela idatha kanye nomugqa wokubuyela emuva.
Isiphetho
Ukuhlehla okulula komugqa kuyithuluzi lokuhlaziya izibalo elinamandla elisetshenziselwa ukuchaza ubudlelwano phakathi kweziguquguquko ezimbili zobuningi. Ngokucabanga okuyisisekelo mayelana nokulingana, ukuzimela, i-homoscedasticity, kanye nokujwayelekile, singabikezela inani leguquguquko elixhomeke kumanani eziguquguquko ezizimele. Indlela ye-Least Squares inikeza indlela ephumelelayo yokulingana nomugqa wokuhlehla nokunquma amapharamitha afanele. Ukuhlolwa kwemodeli nge-coefficient of determination (R2) kunikeza ukuqonda kokuthi imodeli yethu isebenza kahle kangakanani.
Nakuba ukuhlehla okulula okuqondile kunemikhawulo, njengokukwazi ukusingatha iziguquguquko ezimbili kanye nokucabanga okumele kuhlangatshezwane nakho, le ndlela ihlala iyisisekelo esibalulekile ekuhlaziyweni kwezibalo kanye nedatha, futhi ivame ukusetshenziswa njengesinyathelo sokuqala ekuqondeni ubudlelwano phakathi kweziguquguquko ngaphambi kokuqhubekela ezindleleni eziyinkimbinkimbi kakhulu.
