Tikanga Tapawhā Iti Rawa: He Tātaritanga Whānui
Ko te Tikanga Tapawhā Iti rawa tētahi o ngā tikanga tino taketake o te tātari tatauranga, he mahi nui tana i roto i te whakauru raraunga, te whakatauira, me te tātari matapae. He huarahi pāngarau e whakaiti ana i te tapeke o ngā tapawhā o ngā rerekētanga i waenga i ngā uara i kitea me ngā uara i matapaetia. Mā te mahi pera, ka whakaratohia te rārangi, te kōpiko rānei e tino pai ana ki tētahi huinga raraunga kua hoatu. I roto i tēnei tuhinga, ka tūhuratia e mātou ngā pūtake, ngā tono, ngā putanga, me ngā herenga o te Tikanga Tapawhā Iti rawa.
Papakupu Aamu
I whakaurua tuatahitia te Tikanga Tapawhā Iti e Carl Friedrich Gauss i te tau 1795 i te 18 o ōna tau, ahakoa kāore i whakaputaina tae noa ki te tau 1809. Nā Adrien-Marie Legendre, tētahi atu tohunga pāngarau rongonui, i whakawhanake me te whakaputa motuhake i tētahi tikanga ōrite i te tau 1805. Ahakoa tēnei, ko te tuhi a Gauss me te whakawhānui ake i te tikanga i whakatakoto i te turanga mō te whakamahinga o ēnei rā.
Tuāpapa Pāngarau
I tōna āhua māmā rawa, ka whakamahia te Tikanga Tapawhā Iti hei whakauru i tētahi tauira rārangi ki tētahi huinga pūwāhi raraunga. Mēnā he huinga pūwāhi raraunga `n` \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)\, ko te whāinga he kimi i tētahi rārangi \(y = mx + c\) e whakaiti ana i te tapeke o ngā tapawhā o ngā tawhiti poutū (ngā toenga) i waenganui i ngā uara i kitea \(y_i\) me ngā uara i matapaetia e te rārangi \(\hat{y}_i\).
I roto i te pāngarau, ka taea te whakaatu i tēnei whāinga penei:
\[ S = \sum_{i=1}^{n} (y_i – \hat{y}_i)^2 = \sum_{i=1}^{n} (y_i – (mx_i + c))^2 \]
Ko tēnei whārite te tapeke o ngā tapawhā o ngā toenga, \(S\), ā, ko tā mātou whāinga he kimi i ngā uara o \(m\) (pikinga) me \(c\) (haukotinga) e whakaiti ana i a \(S\).
Te Whakaputa
Hei kimi i ngā uara o \(m\) me \(c\) e whakaiti ana i te tapeke o ngā tapawhā \(S\), ka tangohia e mātou ngā taupatupatu ā-wāhanga o \(S\) e pā ana ki \(m\) me \(c\) ka whakatakotoria ki te kore.
1. Te Whakapūtanga Wāhanga e pā ana ki \(m\):
\[ \frac{\partial S}{\partial m} = \sum_{i=1}^{n} 2(y_i – (mx_i + c))(-x_i) = 0 \]
2. Te Whakapūtanga Wāhanga e pā ana ki \(c\):
\[ \frac{\partial S}{\partial c} = \sum_{i=1}^{n} 2(y_i – (mx_i + c))(-1) = 0 \]
Mā te whakaoti i ēnei whārite tukutahi ka puta ngā whārite noa:
\[ m = \frac{n(\tapeke x_i y_i) – (\tapeke x_i)(\tapeke y_i)}{n (\tapeke x_i^2) – (\tapeke x_i)^2} \]
\[ c = \frac{(\tapeke y_i)(\tapeke x_i^2) – (\tapeke x_i)(\tapeke x_i y_i)}{n(\tapeke x_i^2) – (\tapeke x_i)^2} \]
Mā ēnei whārite ka homai ngā uara o \(m\) me \(c\) e whakaiti ana i te tapeke o ngā toenga tapawhā.
tono
He whānui ngā whakamahinga o te Tikanga Tapawhā Iti rawa puta noa i ngā momo mara:
1. Ōhanga me te Pūtea: Hei whakatauira me te matapae i ngā tohu ōhanga pēnei i te GDP, ngā reiti pikinga utu, me ngā utu hea.
2. Hangarau: Whakamahia i roto i te tukatuka tohu, ngā pūnaha whakahaere, me te whakamātautau pono hei whakauru i ngā tauira ki ngā raraunga empirika.
3. Rongoā: Āwhina ki te whakauru i ngā pihi tipu, ngā pihi urupare-horopeta, me ētahi atu tukanga koiora ki ngā raraunga whakamātautau.
4. Te Ako Mīhini: Ko ngā tauira whakatauira rārangi, e whakamahi pinepine ana i te Tikanga Tapawhā Iti mō te whakatau tatai tawhā, he mea tūāpapa i roto i ngā tikanga ako e tirotirohia ana.
5. Tātai whetū: I whakamahia e Gauss i ngā wā o mua hei tatau i ngā porowhita o ngā tinana o te rangi.
Ngā momo rerekētanga me ngā toronga
1. Ngā Tapawhā Iti Rawa Kua Taumaha: Mēnā he rerekē ngā rerekētanga o ngā kitenga, ka tohaina e ngā Tapawhā Iti Rawa Kua Taumaha ngā taumaha ki ia pūwāhi raraunga, ka whakaitihia te tapeke taumaha o ngā tapawhā.
2. Ngā Tapawhā Iti Kore-Rārangi: Ka whakawhānui i te tikanga kia uru ki ngā tauira kore-rārangi, mā roto i ngā tikanga tāruarua pēnei i ngā rauropi Gauss-Newton, Levenberg-Marquardt rānei.
3. Ngā Tapawhā Iti Kua Whakaritea (Ridge Regression): Ka tāpirihia he whiu ki te tapeke o ngā tapawhā hei ārai i te taunga nui, he mea tino whai hua ina pā ana ki te maha-whakapiripiri.
4. Ngā Tapawhā Iti Whānui: E whai whakaaro ana ki ngā kitenga e pā ana ki ngā āhuatanga e pā ana, e whakawhānui ana i te tikanga tapawhā iti hei whakahaere i ngā hapa e pā ana ki ngā āhuatanga e pā ana ki te hononga-ā-roto me ngā hapa heteroscedastic.
Ngā Utauta Tātari
He maha ngā taputapu pūmanawa me ngā taiao hōtaka e tuku ana i ngā mahi kua hangaia mō te mahi i te whakatauira tapawhā iti rawa:
1. Python: Ka whakaratohia e ngā whare pukapuka pēnei i a NumPy me SciPy ngā mahi `numpy.linalg.lstsq` me `scipy.optimize.curve_fit` mō te taunga tapawhā iti rawa raina me te kore-raina.
2. R: Ngā mahi pēnei i te `lm()` mō ngā tauira rārangi me te `nls()` mō ngā tauira kore-rārangi i roto i te taiao tatauranga whānui a R.
3. MATLAB: Ko te mahi `lsqcurvefit` i roto i te MATLAB e whakamahia ana mō te taunga kore-raina, me te tautoko ā-roto mō ngā mahi taurangi raina.
Nga whakataunga
Ahakoa tōna whānui o te whakamahinga, he iti noa ngā herenga o te Tikanga Tapawhā Iti:
1. Te aro ki ngā mea o waho: He tino aro te Least Squares ki ngā mea o waho, nā te mea ka whakanuia e te kupu tapawhā te pānga o ngā toenga nui.
2. Te Whakaaro o te Raina: E whakaaro ana he whanaungatanga raina i waenga i ngā taurangi, kāore pea e pono mō ngā huinga raraunga uaua.
3. Ngā Take o te Maha-whakapiripiri: I roto i te whakatauira rārangi maha, ka taea e te maha-whakapiripiri i waenga i ngā taurangi matapae te whakapohehe i te whakatau tata o ngā tauwehenga.
4. Te ōritetanga o te rerekētanga (Te ōritetanga): E whakaaro ana he pumau te rerekētanga o ngā kupu hapa puta noa i ngā kitenga, kāore pea e pērā i ngā wā katoa.
Opaniraa
Ahakoa tōna pakeketanga, ko te Tikanga Tapawhā Iti rawa te tūāpapa mō te tātari raraunga tatauranga me te whakatauira. Nā tōna māmā, tōna whai huatanga, me tōna ngāwari ki te urutau ki ngā momo āhua, mai i te whakatauira raina taketake ki ngā tauira uaua kore-raina me ngā tauira whānui, he mea tino nui. He mea nui te mārama ki ōna turanga pāngarau, ki ōna tono, me ōna herenga mō te hunga katoa e uru ana ki te tātari raraunga, kia tika ai te whakamahi i te tikanga hei whiwhi māramatanga whai tikanga. Ahakoa i roto i te ao mātauranga, i te ahumahi rānei, ko te Tikanga Tapawhā Iti rawa te taputapu pakari tonu mō te mārama ki ngā whanaungatanga i roto i ngā raraunga me te hanga matapae whai whakaaro.
