Ka Nānā ʻana i ka Regression Linear Maʻalahi
ʻO ka regression linear maʻalahi kahi ʻenehana helu i hoʻohana ʻia e kālailai i ka pilina ma waena o ʻelua mau loli quantitative. ʻO ka loli a mākou e hoʻāʻo nei e wānana ua kapa ʻia ʻo ia ka loli dependent a i ʻole ka pane, ʻoiai ʻo ka loli i hoʻohana ʻia e hana i ka wānana ua kapa ʻia ʻo ia ka loli kūʻokoʻa a i ʻole ka wānana. Ma ka regression linear maʻalahi, hoʻāʻo mākou e ʻimi i ka laina pololei maikaʻi loa e wehewehe ana i ka pilina ma waena o kēia mau loli ʻelua.
Nā Manaʻo Kumu o ka Regression Linear Maʻalahi
Hoʻokumu ʻia ke ʻano regression linear maʻalahi ma ke kuhiakau ʻana aia kahi pilina linear ma waena o ke ʻano hilinaʻi \(Y\) a me ke ʻano kūʻokoʻa \(X\). ʻO ke ʻano maʻamau o kahi kumu hoʻohālike regression linear maʻalahi:
\[ Y = \beta_0 + \beta_1 X + \epsilon \]
Ma hea:
– ʻO \( Y \) ke loli hilinaʻi.
– ʻO \( X \) ke ʻano loli kūʻokoʻa.
– ʻO \( \beta_0 \) ka intercept, ʻo ia ka waiwai o \(Y\) i ka wā \(X = 0\).
– ʻO \( \beta_1 \) ka pali a i ʻole ka gradient, ʻo ia ka loli awelika ma \(Y\) no kēlā me kēia loli ʻāpana ma \(X\).
– ʻO \( \epsilon \) ka hewa a i ʻole ka huaʻōlelo koena e hōʻike ana i ka loli ma \(Y\) ʻaʻole hiki ke wehewehe ʻia e \(X\).
ʻO ka pahuhopu o ka regression linear maʻalahi, ʻo ia ke kuhi ʻana i nā palena \(\beta_0\) a me \(\beta_1\) i hiki ke hoʻohana ʻia ke kumu hoʻohālike e wānana i ka waiwai o \(Y\) e pili ana me ka waiwai o \(X\).
Ke ʻAno Hana Kuea Liʻiliʻi Loa
ʻO kekahi o nā ʻano hana i hoʻohana pinepine ʻia no ke kau ʻana i kahi kumu hoʻohālike regression linear maʻalahi ʻo ia ke ʻano Least Squares. ʻO ka pahuhopu o kēia ʻano hana e hōʻemi i ka huina o nā huinahā o nā ʻokoʻa kū pololei ma waena o nā nānā maoli a me nā waiwai i wānana ʻia e ke kumu hoʻohālike. Manaʻo mākou he n nā nānā i loaʻa i nā hui \((x_i, y_i)\) no \(i = 1, 2, …, n\). ʻO ka hana e hōʻemi ʻia:
S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – (\beta_0 + \beta_1 x_i))^2 \]
No ka loaʻa ʻana o \(\beta_0\) a me \(\beta_1\) e hoʻemi ana i kēia hana, lawe mākou i nā derivatives hapa o \(S(\beta_0, \beta_1)\) e pili ana i kēlā me kēia palena a hoʻonohonoho i kēia mau derivatives i ka ʻole. Hiki ke hoʻomaʻalahi ʻia ka helu makemakika penei:
\[ \beta_1 = \frac{\sum_{i=1}^{n} (x_i – \bar{x})(y_i – \bar{y})}{\sum_{i=1}^{n} (x_i – \bar{x})^2} \]
\[ \beta_0 = \bar{y} – \beta_1 \bar{x} \]
Ma hea:
– ʻO \(\bar{x}\) ka awelika o \(X\)
– ʻO \(\bar{y}\) ka awelika o \(Y\)
Ma hope o ka loaʻa ʻana o nā palena \(\beta_0\) a me \(\beta_1\), hiki ke hoʻohana ʻia kahi kumu hoʻohālike regression linear maʻalahi e wānana i ka waiwai o \(Y\) no kēlā me kēia waiwai o \(X\).
Nā Manaʻo ma ka Regression Linear Maʻalahi
No nā hopena kūpono a hilinaʻi hoʻi, manaʻo ka regression linear maʻalahi i kekahi mau mea:
1. Linearity: Pono ka pilina ma waena o ke loli hilinaʻi a me ke loli kūʻokoʻa e linear.
2. Kūʻokoʻa: Pono e kūʻokoʻa nā nānā ʻana kekahi i kekahi.
3. Homoscedasticity: Pono ke ʻano loli o ke koena e mau ma waena o ka laulā o nā waiwai o ke ʻano kūʻokoʻa.
4. Ke Koena Maʻamau: Pono nā koena (nā hewa) e hahai i kahi hoʻolaha maʻamau.
Inā ʻaʻole i hoʻokō ʻia kēia mau kuhiakau, ʻaʻole hiki ke hilinaʻi ʻia nā hopena o kahi kumu hoʻohālike regression linear maʻalahi a ʻaʻole paha e hiki ke hana i nā wānana pololei.
Loiloi Hoʻohālike Regression
ʻO kahi ala e loiloi ai i ka maikaʻi o ka wānana ʻana o kahi kumu hoʻohālike regression linear maʻalahi, ʻo ia ka hoʻohana ʻana i ka Coefficient of Determination (\(R^2\)). Hōʻike ka coefficient of determination i ka hapa o ka loli i loko o ka loli hilinaʻi i hiki ke wehewehe ʻia e ka loli i loko o nā loli kūʻokoʻa.
\[ R^2 = \frac{\sum_{i=1}^{n} (\hat{y}_i – \bar{y})^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2} \]
Ma hea:
– ʻO \(\hat{y}_i\) ka waiwai i wānana ʻia o \(Y\).
– ʻO \(y_i\) ka waiwai maoli o \(Y\).
– ʻO \(\bar{y}\) ka awelika o nā waiwai o \(Y\).
Aia ka waiwai \(R^2\) mai ka 0 a i ka 1. ʻO kahi waiwai \(R^2\) kokoke i ka 1 e hōʻike ana hiki i ke kumu hoʻohālike ke wehewehe i ka hapa nui o ka loli i loko o ka loli hilinaʻi.
Hoʻokō ʻana ma ka ʻŌlelo Polokalamu
No ka hoʻokō ʻana i ka regression linear maʻalahi, hiki iā mākou ke hoʻohana i nā polokalamu helu like ʻole a i ʻole nā ʻōlelo papahana. Aia ma lalo kahi hoʻokō hoʻohālike ma Python me ka hoʻohana ʻana i ka waihona puke `scikit-learn`:
“`python
lawe i numpy e like me np
lawe mai i ka matplotlib.pyplot as plt
mai sklearn.linear_model lawe mai i ka LinearRegression
mai sklearn.metrics lawe mai i ka mean_squared_error, r2_score
'Ikepili
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)
kükohu
kumu hoʻohālike = LinearRegression()
kumu hoʻohālike.kūpono(X, y)
wānana
y_pred = kumu hoʻohālike.predict(X)
Ka helu hana
beta_0 = kumu hoʻohālike.intercept_
beta_1 = kumu hoʻohālike.coef_[0]
paʻi(f'Intercept: {beta_0}')
paʻi (f'Slope: {beta_1}')
paʻi(f'Hewa huinahā awelika: {mean_squared_error(y, y_pred)}')
paʻi(f'Coefficient o ka hoʻoholo (R^2): {r2_score(y, y_pred)}')
Ka hoʻolālā ʻikepili a me ka laina regression
plt.hoʻopuehu(X, y, kala='uliuli')
plt.plot(X, y_pred, kala='ʻulaʻula')
plt.xlabel('X')
plt.ylabel('Y')
plt.hōʻike()
".
Ma ka laʻana ma luna, lawe mua mākou i nā hale waihona puke e pono ai, wehewehe i ka ʻikepili \(X\) a me \(Y\), a laila hoʻohana i ka mea `LinearRegression` mai `scikit-learn` e hoʻopili i kahi kumu hoʻohālike i ka ʻikepili. Ke hoʻopili ʻia ke kumu hoʻohālike, hana mākou i nā wānana a helu i nā coefficients, a me ka mean squared error a me ka coefficient of determination. ʻO ka hope loa, hoʻolālā mākou i ka ʻikepili a me ka laina regression.
Ka hopena
He mea hana loiloi helu ikaika ka regression linear maʻalahi i hoʻohana ʻia e wehewehe i ka pilina ma waena o ʻelua mau loli quantitative. Me kekahi mau kuhiakau kumu e pili ana i ka linearity, ke kūʻokoʻa, ka homoscedasticity, a me ka normality, hiki iā mākou ke wānana i ka waiwai o ka loli hilinaʻi e pili ana i nā waiwai o nā loli kūʻokoʻa. Hāʻawi ke ʻano Least Squares i kahi ala kūpono e hoʻokomo i kahi laina regression a hoʻoholo i nā palena kūpono. Hāʻawi ka loiloi kumu hoʻohālike ma o ka coefficient of determination (R2) i ka ʻike i ka maikaʻi o ka hana a kā mākou kumu hoʻohālike.
ʻOiai he mau palena ko ka regression linear maʻalahi, e like me ka hiki ke lawelawe i ʻelua mau loli a me nā kuhiakau e pono e hoʻokō ʻia, ke mau nei kēia ʻenehana he kahua koʻikoʻi i nā helu helu a me ka nānā ʻikepili, a hoʻohana pinepine ʻia ma ke ʻano he hana mua i ka hoʻomaopopo ʻana i ka pilina ma waena o nā loli ma mua o ka neʻe ʻana i nā ʻano hana paʻakikī.
