Tikanga Tapawhā Iti Rawa: He Huarahi Pāngarau ki te Whakatau Tatau
Pendahuluan
Ko te tikanga o ngā tapawhā iti rawa he tikanga tatauranga e whakamahia ana hei whakatau tata i ngā tawhā i roto i tētahi tauira whakatau tata mā te whakaiti i te tapeke o ngā hapa tapawhā i waenga i ngā uara tūturu me ngā uara i matapaetia e te tauira. He tino rongonui tēnei tikanga, ā, he maha ngā whakamahinga i roto i ngā momo mara pēnei i te ōhanga, te hangarau, te koiora, me ngā pūtaiao pāpori. I tuatahitia te ariā o ngā tapawhā iti rawa e Adrien-Marie Legendre i te tīmatanga o te rautau 19, ā, i muri mai ka whakawhanakehia anō e Carl Friedrich Gauss.
Te Māramatanga Taketake
I te nuinga o te wā, ko te whāinga o te tikanga tapawhā iti rawa ko te kimi i te rārangi whakatau tata pai rawa atu mō tētahi huinga raraunga mā te whakaiti i te tapeke o ngā tapawhā o ngā toenga, ngā hapa matapae rānei. Ko te toenga ko te rerekētanga i waenga i te uara i kitea me te uara i matapaetia.
Mena he huinga raraunga tā tātou kei roto ko ngā takirua o ngā kitenga \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)\), ko tā tātou whāinga he kimi i te rārangi \(y = mx + b\) e whakaiti ana i te tapeke o ngā hapa tapawhā tapeke\( \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \).
Ka taea te whakamahi i tēnei tikanga ki ngā taunga whakatautau rārangi māmā me ngā taunga whakatautau rārangi maha. I roto i te whakatautau rārangi māmā, kotahi anake te taurangi motuhake (x), ko te whakatautau rārangi maha ia he maha atu i te kotahi te taurangi motuhake.
Whakamuritanga Raina Māmā
Me tīmata tātou ki te whakatauira rārangi māmā. Me kī he huinga raraunga tā tātou \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)). Ko te tauira whakatauira rārangi māmā e hiahia ana tātou ki te whakauru ko:
\[ y = mx + b + \epsilon \]
ko \( m \) te pikinga, ko \( b \) te haukoti, ā, ko \( \epsilon \) te hapa matapōkere.
Mā te whakamahi i te tikanga tapawhā iti rawa, ka kitea ngā whakatau tata o ngā tawhā \( m \) me \( b \) mā te whakaiti i te mahi hapa tapawhā:
\[ S(m, b) = \tapeke_{i=1}^{n} (y_i – (mx_i + b))^2 \]
Hei whakaiti i te \( S(m, b) \), ka kitea e mātou ngā pānga ā-wāhanga o \( S \) e pā ana ki \( m \) me \( b \), kātahi ka whakaoti i tēnei whārite mō \( m \) me \( b \):
\[ \begin{aligned}
\frac{\partial S}{\partial m} &= -2 \sum_{i=1}^{n} x_i (y_i – (mx_i + b)) = 0 \\
\frac{\partial S}{\partial b} &= -2 \sum_{i=1}^{n} (y_i – (mx_i + b)) = 0
\end{arārangi} \]
I muri i te whakangawari, ka whiwhi tātou i ngā whārite noa e rua e whai ake nei:
\[ \begin{aligned}
n\bar{y} &= m \sum_{i=1}^{n} x_i + nb \\
\sum_{i=1}^{n}x_i y_i &= m \sum_{i=1}^{n}x_i^2 + b \sum_{i=1}^{n}x_i
\end{arārangi} \]
Mā te whakaoti i te pūnaha whārite i runga ake nei, ka kitea ngā uara o \( m \) me \( b \) e whakaiti ana i te hapa tapawhā.
Whakamuritanga Raina Maha
I roto i te whakatauira rārangi maha, ka tūtaki tātou ki tētahi āhuatanga kei reira he maha atu i te kotahi te taurangi motuhake. Me kī he raraunga kei a tātou i te āhua o tētahi taupū \((x_{i1}, x_{i2}, …, x_{ik}, y_i)\). Ko te tauira whakatauira e whakamahia ana e tātou ko:
\[ y = b_0 + b_1 x_1 + b_2 x_2 + … + b_k x_k + \epsilon \]
Ka taea te tuhi i tēnei whārite ki te āhua matihiko penei:
\[ \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{\epsilon} \]
kāore i te mana:
– Ko \( \mathbf{y} \) he whārite pou o ngā uara y kua kitea.
– Ko \( \mathbf{X} \) he matihiko o ngā uara x kua kitea (tae atu ki te pou 1 mō te haukoti).
– Ko \( \mathbf{b} \) he whārite pou o ngā tawhā (tae atu ki a \( b_0 \)).
Ko te whāinga o te tikanga tapawhā iti rawa ko te whakaiti i te mahi hapa tapawhā e whai ake nei:
\[ S(\mathbf{b}) = (\mathbf{y} – \mathbf{Xb})^T (\mathbf{y} – \mathbf{Xb}) \]
Hei whakaiti i tēnei mahi, ka tangohia e mātou te pānga ā-wāhanga o S e pā ana ki \( \mathbf{b} \) ka whakatakotoria ki te kore. Mā tēnei ka puta te whārite noa mō te whakatautautanga rārangi maha:
\[ \mathbf{X}^T \mathbf{Xb} = \mathbf{X}^T \mathbf{y} \]
Mā te whakaoti i te pūnaha whārite i runga ake nei, ka taea e tātou te whiwhi i tētahi whakatau tata mō te tawhā \( \mathbf{b} \):
\[ \mathbf{b} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} \]
Ngā Painga me ngā Here
He maha ngā painga o te tikanga tapawhā iti rawa. He tikanga tino whai hua, he ngāwari hoki te whakamahi. Ka tukuna he otinga ahurei mēnā he mea taea te huri i te \( \mathbf{X}^T \mathbf{X} \), ā, he pono mō ngā tono mahi maha.
Heoi anō, he iti noa iho ngā herenga o te tikanga tapawhā iti rawa. He tino aro ki ngā mea o waho nā te mea ka nui ake te whakanui a te hapa tapawhā i ngā rerekētanga nui i ngā mea iti. Hei tāpiri, me tutuki te whakaaro matarohia he tohatoha noa ngā hapa me te kore toharite me te rerekētanga pumau kia pai ai ngā hua.
Ngā Whakamahinga Whai Hua
He maha ngā wā ka whakamahia te tikanga tapawhā iti rawa i roto i te tātaritanga ia o ngā raraunga, te matapae, me te ako mīhini hei hanga tauira matapae. I roto i te umanga pūtea, ka whakamahia te tikanga tapawhā iti rawa hei matapae i ngā utu hea, i te mahi mākete rānei. I roto i te rongoā, ka whakamahia hei whakatauira i te whanaungatanga i waenga i te horopeta rongoā me te urupare a te tūroro. I roto i ngā pūtaiao pāpori, ka āwhina i te mārama ki te whanaungatanga i waenga i ngā taurangi pēnei i te mātauranga me te moni whiwhi.
Whakamutunga
Ko te tikanga tapawhā iti rawa tētahi o ngā tikanga matua o te tatauranga me te tātari raraunga. Ahakoa he māmā noa te ariā, he nui te mana o tēnei tikanga ki te whakatauira me te mārama ki ngā whanaungatanga i waenga i ngā taurangi. Nā te whānui o ngā tono puta noa i te whānuitanga o ngā mara, he mea tino nui te māramatanga pakari ki tēnei tikanga mō ngā tohunga me ngā kairangahau. I te heke mai, me te piki haere o te nui o ngā raraunga e tūtakihia ana i te wā raraunga nui, ka nui haere te whai tikanga o te urutau me te tono o ngā tikanga tawhito pēnei i te tapawhā iti rawa.
