Tātari Whakamuri Raina Māmā
Ko te whakatauira rārangi māmā he tikanga tatauranga e whakamahia ana hei tātari i te whanaungatanga i waenga i ngā taurangi tauanga e rua. Ko te taurangi e ngana ana mātou ki te matapae ka kiia ko te taurangi whakawhirinaki, ko te taurangi urupare rānei, ko te taurangi e whakamahia ana hei matapae ka kiia ko te taurangi motuhake, ko te taurangi matapae rānei. I roto i te whakatauira rārangi māmā, ka ngana mātou ki te kimi i te rārangi tika pai rawa atu e whakaahua ana i te whanaungatanga i waenga i ēnei taurangi e rua.
Ngā Ariā Taketake o te Whakamuritanga Raina Māmā
Ko te whakatauira raina māmā e ahu mai ana i te whakaaro he whanaungatanga raina kei waenganui i te taurangi whakawhirinaki \(Y\) me te taurangi motuhake \(X\). Ko te āhua whānui o te tauira whakatauira raina māmā ko:
\[ Y = \beta_0 + \beta_1 X + \epsilon \]
Kei hea:
– Ko te taurangi whakawhirinaki ko \( Y \).
– Ko te taurangi motuhake ko \( X \).
– Ko te \( \beta_0 \) te haukoti, koinei te uara o \(Y\) ina \(X = 0\).
– Ko \( \beta_1 \) te pikinga, te pikinga rānei, koinei te huringa toharite i roto i te \(Y\) mō ia huringa waeine i roto i te \(X\).
– Ko \( \epsilon \) te hapa, te toenga rānei e tohu ana i te rerekētanga o \(Y\) kāore e taea te whakamārama e \(X\).
Ko te whāinga o te whakatauira rārangi māmā he whakatau tata i ngā tawhā \(\beta_0\) me \(\beta_1\) kia taea ai te whakamahi i te tauira hei matapae i te uara o \(Y\) e pā ana ki te uara o \(X\).
Tikanga Tapawhā Iti Rawa
Ko tētahi o ngā tikanga e whakamahia whānuitia ana mō te whakauru i tētahi tauira whakatauira rārangi māmā ko te tikanga Tapawhā Iti rawa. Ko te whāinga o tēnei tikanga he whakaiti i te tapeke o ngā tapawhā o ngā rerekētanga poutū i waenga i ngā kitenga tūturu me ngā uara i matapaetia e te tauira. Me kī he n ā tātou kitenga kei roto ko ngā takirua \((x_i, y_i)\) mō \(i = 1, 2, …, n\). Ko te mahi hei whakaiti ko:
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – (\beta_0 + \beta_1 x_i))^2 \]
Hei kimi i a \(\beta_0\) me \(\beta_1\) e whakaiti ana i tēnei mahi, ka tangohia e mātou ngā pānga taha o \(S(\beta_0, \beta_1)\) e pā ana ki ia tawhā, ā, ka whakatakotoria ēnei pānga ki te kore. Ka taea te whakangawari i te tatau pāngarau penei:
\[ \beta_1 = \frac{\sum_{i=1}^{n} (x_i – \bar{x})(y_i – \bar{y})}{\sum_{i=1}^{n} (x_i – \bar{x})^2} \]
\[ \beta_0 = \bar{y} – \beta_1 \bar{x} \]
Kei hea:
– Ko te toharite o \(\bar{x}\) te \(X\)
– Ko te toharite o \(\bar{y}\) te \(Y\)
I muri i te whiwhinga o ngā tawhā \(\beta_0\) me \(\beta_1\), ka taea te whakamahi i tētahi tauira whakatauira rārangi māmā hei matapae i te uara o \(Y\) mō ia uara o \(X\).
Ngā Whakaaro i roto i te Whakamuritanga Raina Māmā
Mō ngā hua tika me te pono, e whakaarohia ana e te whakatauira raina māmā ētahi mea:
1. Te Raina: Me raina te whanaungatanga i waenga i te taurangi whakawhirinaki me te taurangi motuhake.
2. Te Tū motuhake: Me tū motuhake ngā tirohanga tetahi i tetahi.
3. Te Homocedasticity: Me pumau tonu te rerekētanga toenga puta noa i te whānuitanga o ngā uara o te taurangi motuhake.
4. Te Noa o te Toenga: Me whai ngā toenga (ngā hapa) i tētahi tohatoha noa.
Ki te kore ēnei whakapae e tutuki, kāore e pono ngā hua o tētahi tauira whakatauira rārangi māmā, ā, kāore pea e taea te whakaputa matapae tika.
Aromatawai Tauira Whakamuri
Ko tētahi huarahi hei aromatawai i te pai o te matapae a tētahi tauira whakatautau rārangi māmā ko te whakamahi i te Tauwehenga Whakatau (\(R^2\)). E whakaatu ana te tauwehenga whakatau i te ōwehenga o te rerekētanga i roto i te taurangi whakawhirinaki ka taea te whakamārama mā te rerekētanga o ngā taurangi motuhake.
\[ R^2 = \frac{\sum_{i=1}^{n} (\hat{y}_i – \bar{y})^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2} \]
Kei hea:
– Ko te uara matapae o \(\hat{y}_i\) ko \(Y\).
– Ko te uara tuturu o te \(y_i\) ko \(Y\).
– Ko te \(\bar{y}\) te toharite o ngā uara o \(Y\).
Kei waenganui i te 0 ki te 1 te uara \(R^2\). Ko te uara \(R^2\) e tata ana ki te 1 e tohu ana ka taea e te tauira te whakamārama i te nuinga o ngā rerekētanga o te taurangi whakawhirinaki.
Te Whakatinanatanga i roto i te Reo Hōtaka
Hei whakatinana i te whakatauira rārangi māmā, ka taea e tātou te whakamahi i ngā pūmanawa tatauranga, i ngā reo hōtaka rānei. Kei raro nei tētahi tauira whakatinanatanga i roto i te Python mā te whakamahi i te whare pukapuka `scikit-learn`:
"`Pitoni
kawemai numpy hei np
kawemai matplotlib.pyplot hei plt
mai i te sklearn.linear_model kawemai i te LinearRegression
mai i te sklearn.metrics kawemai hapa_tapawhā_mean, r2_score
Raraunga
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)
tauira
tauira = WhakahokingaRaina()
tauira.taurite(X, y)
Matapae
y_pred = tauira.matapae(X)
Tauwehenga
beta_0 = tauira.tautoko_
beta_1 = tauira.tauira_[0]
tā(f'Whakataunga: {beta_0}')
tā(f'Pīnakitanga: {beta_1}')
tā(f'Hapa tapawhā toharite: {hara_tapawhā_mean(y, y_pred)}')
tāia(f'Tauwehenga whakatau (R^2): {r2_score(y, y_pred)}')
Kauwhata raraunga me te rārangi whakatauira
plt.marara(X, y, tae='kikorangi')
plt.plot(X, y_pred, tae='whero')
plt.xlabel('X')
plt.ylabel('Y')
plt.whakaatu()
""
I te tauira i runga ake nei, ka kawemai tuatahi mātou i ngā whare pukapuka e tika ana, ka tautuhi i ngā raraunga \(X\) me \(Y\), kātahi ka whakamahi i te mea `LinearRegression` mai i `scikit-learn` hei whakauru i tētahi tauira ki ngā raraunga. Kia whakaurua te tauira, ka matapae mātou, ka tatau i ngā taunga, tae atu ki te hapa tapawhā toharite me te taunga whakatau. Hei whakamutunga, ka tuhia e mātou ngā raraunga me te rārangi whakatau.
Whakamutunga
He taputapu tātari tatauranga kaha te whakatauira rārangi māmā e whakamahia ana hei whakamārama i te whanaungatanga i waenga i ngā taurangi tauanga e rua. Mā te whakamahi i ētahi whakaaro taketake mō te rārangi, te tū motuhake, te ōritetanga, me te āhua noa, ka taea e tātou te matapae i te uara o te taurangi whakawhirinaki i runga i ngā uara o ngā taurangi motuhake. Mā te tikanga Tapawhā Iti rawa e whakarato he huarahi whai hua hei whakauru i tētahi rārangi whakatau me te whakatau i ngā tawhā tino pai. Mā te aromatawai tauira mā te tauwehenga whakatau (R2) ka kitea te pai o te mahi a tā tātou tauira.
Ahakoa he iti noa ngā herenga o te whakatauira rārangi māmā, pērā i te kaha ki te whakahaere i ngā taurangi e rua me ngā whakapae me tutuki, he tūāpapa nui tonu tēnei tikanga i roto i ngā tatauranga me te tātari raraunga, ā, he maha ngā wā ka whakamahia hei taahiraa tuatahi ki te mārama ki te whanaungatanga i waenga i ngā taurangi i mua i te neke atu ki ngā tikanga uaua ake.
