Ke ʻAno Hana Kuea Liʻiliʻi Loa: He ʻAno Makemakika no ka Kuhi ʻana
Pendahuluan
ʻO ke ʻano o nā huinahā liʻiliʻi loa he ʻano hana helu i hoʻohana ʻia e kuhi i nā palena i loko o kahi kumu hoʻohālike regression ma ka hoʻēmi ʻana i ka huina o nā hewa huinahā ma waena o nā waiwai maoli a me nā waiwai i wānana ʻia e ke kumu hoʻohālike. He mea kaulana loa kēia ʻano hana a hoʻohana pinepine ʻia ma nā ʻano like ʻole e like me ka hoʻokele waiwai, ka ʻenekinia, ka biology, a me nā ʻepekema pili kanaka. Ua hāpai mua ʻia ke kumumanaʻo o nā huinahā liʻiliʻi loa e Adrien-Marie Legendre i ka hoʻomaka ʻana o ke kenekulia 19 a ua hoʻomohala hou ʻia e Carl Friedrich Gauss.
Ka Hoʻomaopopo Kumu
Ma keʻano laulā, ʻo ke ʻano o nā huinaha liʻiliʻi loa e manaʻo e loaʻa ka laina regression kūpono loa no kahi ʻikepili ma ka hoʻemi ʻana i ka huina o nā huinaha o nā koena, a i ʻole nā hewa wānana. ʻO ke koena ka ʻokoʻa ma waena o ka waiwai i ʻike ʻia a me ka waiwai i wānana ʻia.
Inā loaʻa iā mākou kahi ʻikepili i haku ʻia me nā hui o nā nānā ʻana \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)\), a laila ʻo kā mākou pahuhopu ka loaʻa ʻana o ka laina \(y = mx + b\) e hoʻēmi ana i ka huina o nā hewa kuea sum\( \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \).
Hiki ke hoʻopili ʻia kēia ʻano hana i ka regression linear maʻalahi a me ka regression linear maha. I ka regression linear maʻalahi, hoʻokahi wale nō loli kūʻokoʻa (x) kā mākou, ʻoiai ʻo ka regression linear maha e pili ana i nā loli kūʻokoʻa he nui.
Ka Hoʻihoʻi Laina Maʻalahi
E hoʻomaka kākou me ka regression linear maʻalahi. Manaʻo mākou he ʻikepili kā mākou \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)). ʻO ke kumu hoʻohālike regression linear maʻalahi a mākou e makemake ai e hoʻokomo:
\[ y = mx + b + \epsilon \]
kahi ʻo \( m \) ka pali, ʻo \( b \) ka intercept, a ʻo \( \epsilon \) ka hewa random.
Ma ka hoʻohana ʻana i ke ʻano hana liʻiliʻi loa, hiki iā mākou ke loaʻa nā kuhi o nā palena \( m \) a me \( b \) ma ka hoʻēmi ʻana i ka hana hewa huinahā:
S(m, b) = \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \]
No ka hoʻēmi ʻana iā \( S(m, b) \), loaʻa iā mākou nā derivatives hapa o \( S \) e pili ana iā \( m \) a me \( b \), a laila e hoʻoponopono i kēia kaulike no \( m \) a me \( b \):
\[ \begin{aligned}
\frac{\partial S}{\partial m} &= -2 \sum_{i=1}^{n} x_i (y_i – (mx_i + b)) = 0 \\
\frac{\partial S}{\partial b} &= -2 \sum_{i=1}^{n} (y_i – (mx_i + b)) = 0
\end{aligned} \]
Ma hope o ka hoʻomaʻalahi ʻana, loaʻa iā mākou nā ʻelua mau kaulike maʻamau:
\[ \begin{aligned}
n\bar{y} &= m \sum_{i=1}^{n} x_i + nb \\
\sum_{i=1}^{n}x_i y_i &= m \sum_{i=1}^{n}x_i^2 + b \sum_{i=1}^{n}x_i
\end{aligned} \]
Ma ka hoʻoponopono ʻana i ka ʻōnaehana o nā kaulike ma luna, hiki iā mākou ke loaʻa nā waiwai o \( m \) a me \( b \) e hōʻemi ana i ka hewa huinahā.
Hoʻololi Linear Nui
I loko o ka regression linear maha, ke kū nei mākou i kahi kūlana kahi i loaʻa ai iā mākou ma mua o hoʻokahi loli kūʻokoʻa. Manaʻo mākou he ʻikepili kā mākou ma ke ʻano o kahi tuple \((x_{i1}, x_{i2}, …, x_{ik}, y_i)\). ʻO ke kumu hoʻohālike regression a mākou e hoʻohana ai:
\[ y = b_0 + b_1 x_1 + b_2 x_2 + … + b_k x_k + \epsilon \]
Hiki ke kākau ʻia kēia kaulike ma ke ʻano matrix penei:
\[ \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{\epsilon} \]
Ma hea:
– ʻO \( \mathbf{y} \) he vector kolamu o nā waiwai y i nānā ʻia.
– ʻO \( \mathbf{X} \) he matrix o nā waiwai x i nānā ʻia (me ke kolamu 1 no ka intercept).
– ʻO \( \mathbf{b} \) he vector kolamu o nā palena (me \( b_0 \)).
ʻO ka pahuhopu o ke ʻano hana liʻiliʻi loa, ʻo ia ka hoʻēmi ʻana i ka hana hewa quadratic ma lalo nei:
\[ S(\mathbf{b}) = (\mathbf{y} – \mathbf{Xb})^T (\mathbf{y} – \mathbf{Xb}) \]
No ka hoʻēmi ʻana i kēia hana, lawe mākou i ka derivative hapa o S e pili ana iā \( \mathbf{b} \) a hoʻonohonoho iā ia i ka ʻole. Hāʻawi kēia i ka hoohalike maʻamau no ka regression linear maha:
\[ \mathbf{X}^T \mathbf{Xb} = \mathbf{X}^T \mathbf{y} \]
Ma ka hoʻoponopono ʻana i ka ʻōnaehana o nā kaulike ma luna, hiki iā mākou ke loaʻa kahi kuhi o ka palena \( \mathbf{b} \):
\[ \mathbf{b} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} \]
Nā Pōmaikaʻi a me nā Palena
He nui nā pono o ke ʻano hana liʻiliʻi loa. He ʻano hana kūpono a maʻalahi hoʻi e hoʻohana. Hāʻawi ia i kahi hopena kū hoʻokahi inā hiki ke hoʻohuli ʻia ʻo \( \mathbf{X}^T \mathbf{X} \), e hilinaʻi ai no nā noi hana he nui.
Eia nō naʻe, he mau palena ko ke ʻano hana liʻiliʻi loa. He mea koʻikoʻi loa ia i nā outliers no ka mea ʻoi aku ka nui o ka hewa squared ma mua o nā ʻokoʻa nui ma mua o nā mea liʻiliʻi. Eia kekahi, pono e hoʻokō ʻia ke kuhiakau kuʻuna he mahele maʻamau ko nā hewa me ka ʻai ʻole a me ka loli mau no nā hopena maikaʻi.
Nā Hoʻohana Kūpono
Hoʻohana pinepine ʻia ke ʻano hana liʻiliʻi loa i ka nānā ʻana i nā ʻano ʻikepili, ka wānana ʻana, a me ke aʻo ʻana i ka mīkini e kūkulu i nā hiʻohiʻona wānana. I ka ʻoihana kālā, hoʻohana ʻia ke ʻano hana liʻiliʻi loa e wānana i nā kumukūʻai kūʻai a i ʻole ka hana mākeke. I ka lāʻau lapaʻau, hoʻohana ʻia ia e hoʻohālike i ka pilina ma waena o ka nui o ka lāʻau a me ka pane a ka mea maʻi. I loko o nā ʻepekema pilikanaka, kōkua ia e hoʻomaopopo i ka pilina ma waena o nā loli e like me ka hoʻonaʻauao a me ka loaʻa kālā.
Ka hopena
ʻO ke ʻano hana liʻiliʻi loa kekahi o nā ʻano hana kumu i nā helu helu a me ka nānā ʻikepili. ʻOiai he maʻalahi ke kumumanaʻo, hāʻawi kēia ʻano hana i ka mana koʻikoʻi i ke kumu hoʻohālike a me ka hoʻomaopopo ʻana i nā pilina ma waena o nā loli. Me nā noi ākea ma nā ʻano kahua like ʻole, he mea waiwai nui ka ʻike paʻa o kēia ʻano hana no nā poʻe loea a me nā mea noiʻi. I ka wā e hiki mai ana, me ka nui o ka ʻikepili i loaʻa i ka wā ʻikepili nui, ʻo ka hoʻololi ʻana a me ka hoʻopili ʻana o nā ʻano hana kuʻuna e like me nā liʻiliʻi loa e lilo wale nō i mea pili.
