Ke ʻAno Hana Kuea Liʻiliʻi Loa: He Loiloi Holoʻokoʻa
ʻO ke ʻano hana Least Squares kekahi o nā ʻano hana koʻikoʻi loa i ka loiloi regression helu, e pāʻani ana i kahi kuleana koʻikoʻi i ka hoʻopili ʻikepili, ke kumu hoʻohālike, a me ka loiloi wānana. He ʻano makemakika ia e hoʻemi ana i ka huina o nā huina o nā ʻokoʻa ma waena o nā waiwai i nānā ʻia a me nā waiwai i wānana ʻia. Ma ka hana ʻana pēlā, hāʻawi ia i ka laina kūpono loa a i ʻole ke piʻo i kahi hoʻonohonoho ʻikepili i hāʻawi ʻia. Ma kēia ʻatikala, e ʻimi mākou i nā kumu, nā noi, nā derivations, a me nā palena o ke ʻano hana Least Squares.
Ka Hanana Hōʻike
Ua hoʻolauna mua ʻia ke ʻano hana Least Squares e Carl Friedrich Gauss i ka makahiki 1795 i kona makahiki 18, ʻoiai ʻaʻole i paʻi ʻia a hiki i ka makahiki 1809. ʻO Adrien-Marie Legendre, kekahi mea makemakika kaulana ʻē aʻe, ua hoʻomohala kūʻokoʻa a paʻi i kahi ʻano hana like i ka makahiki 1805. ʻOiai naʻe kēia, ʻo ka hoʻopaʻa ʻana a Gauss a me ka hoʻonui ʻana i ke ʻano hana i hoʻokumu i ke kumu no ka hoʻohana hou ʻana.
Kumu Pākē
Ma kona ʻano maʻalahi loa, hoʻohana ʻia ke ʻano hana Least Squares e hoʻopili i kahi kumu hoʻohālike linear i kahi hoʻonohonoho o nā kiko ʻikepili. Hāʻawi ʻia i kahi hoʻonohonoho o nā kiko ʻikepili `n` \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)\), ʻo ka pahuhopu ka loaʻa ʻana o kahi laina \(y = mx + c\) e hoʻemi ana i ka huina o nā huina o nā mamao kū pololei (nā koena) ma waena o nā waiwai i ʻike ʻia \(y_i\) a me nā waiwai i wānana ʻia e ka laina \(\hat{y}_i\).
Ma ke ʻano makemakika, hiki ke hōʻike ʻia kēia pahuhopu penei:
S = \sum_{i=1}^{n} (y_i – \hat{y}_i)^2 = \sum_{i=1}^{n} (y_i – (mx_i + c))^2 \]
Hōʻike kēia kaulike i ka huina o nā huinahā o nā koena, \(S\), a ʻo kā mākou pahuhopu ka loaʻa ʻana o nā waiwai o \(m\) (slope) a me \(c\) (intercept) e hoʻemi ana iā \(S\).
Hoʻopuka
No ka loaʻa ʻana o nā waiwai o \(m\) a me \(c\) e hoʻemi ana i ka huina o nā kuea \(S\), lawe mākou i nā derivatives hapa o \(S\) e pili ana iā \(m\) a me \(c\) a hoʻonohonoho iā lākou i ka ʻole.
1. ʻĀpana Hoʻoili e pili ana iā \(m\):
\[ \frac{\partial S}{\partial m} = \sum_{i=1}^{n} 2(y_i – (mx_i + c))(-x_i) = 0 \]
2. ʻĀpana Hoʻoili e pili ana iā \(c\):
\[ \frac{\partial S}{\partial c} = \sum_{i=1}^{n} 2(y_i – (mx_i + c))(-1) = 0 \]
ʻO ka hoʻoponopono ʻana i kēia mau kaulike like e hopena i nā kaulike maʻamau:
\[ m = \frac{n(\sum x_i y_i) – (\sum x_i)(\sum y_i)}{n (\sum x_i^2) – (\sum x_i)^2} \]
\[ c = \frac{(\sum y_i)(\sum x_i^2) – (\sum x_i)(\sum x_i y_i)}{n(\sum x_i^2) – (\sum x_i)^2} \]
Hāʻawi kēia mau kaulike iā mākou i nā waiwai o \(m\) a me \(c\) e hōʻemi ana i ka huina o nā koena huinahā.
noi
He nui nā noi o ke ʻano hana Least Squares ma nā ʻano like ʻole:
1. Hoʻokele waiwai a me ke kālā: Hoʻohana ʻia e hoʻohālike a wānana i nā ʻōkuhi hoʻokele waiwai e like me ka GDP, nā helu hoʻonui, a me nā kumukūʻai kūʻai.
2. ʻEnekinia: Hoʻohana ʻia i ka hana hōʻailona, nā ʻōnaehana kaohi, a me ka hoʻāʻo hilinaʻi e hoʻopili i nā kumu hoʻohālike i ka ʻikepili empirical.
3. Lāʻau lapaʻau: Kōkua i ke kūpono ʻana i nā piʻo ulu, nā piʻo pane-lāʻau, a me nā kaʻina hana olaola ʻē aʻe i ka ʻikepili hoʻokolohua.
4. Ke Aʻo ʻana i ka Mīkini: ʻO nā kumu hoʻohālike regression linear, kahi e hoʻohana pinepine ai i ke ʻano hana Least Squares no ka hoʻohālikelike ʻana i nā palena, he mea kumu ia i nā ʻano hana aʻo i kiaʻi ʻia.
5. Astronomy: Ua hoʻohana ʻia e Gauss e helu i nā kaapuni o nā kino lani.
Nā ʻano like ʻole a me nā hoʻonui
1. Nā Kuea Kaumaha Liʻiliʻi Loa: I nā hihia kahi i loaʻa ai nā ʻokoʻa like ʻole o nā nānā ʻana, hāʻawi nā Kuea Kaumaha Liʻiliʻi Loa i nā kaumaha i kēlā me kēia kiko ʻikepili, e hoʻemi ana i ka huina kaumaha o nā kuea.
2. Nā Kiʻikuhi Loa ʻAʻole Linear: Hoʻonui i ke ʻano hana e kūpono i nā kumu hoʻohālike ʻaʻole linear, pinepine ma o nā ʻano hana iterative e like me nā algorithm Gauss-Newton a i ʻole Levenberg-Marquardt.
3. Nā Kuea Liʻiliʻi i Hoʻonohonoho ʻia (Ridge Regression): Hoʻohui i kahi huaʻōlelo hoʻopaʻi i ka huina o nā kuea e pale aku i ka overfitting, he mea pono loa ia i ka wā e hana ai me ka multicollinearity.
4. Nā Kuea Liʻiliʻi Loa i Hoʻolaha ʻia: Hōʻike no nā nānā ʻana i pili, e hoʻonui ana i ke ʻano hana kuea liʻiliʻi loa e lawelawe i nā hewa autocorrelated a me heteroscedastic.
Nā Mea Hana Heluhelu
Hāʻawi nā ʻano polokalamu like ʻole a me nā wahi papahana i nā hana i kūkulu ʻia no ka hana ʻana i ka regression liʻiliʻi loa:
1. Python: Hāʻawi nā hale waihona puke e like me NumPy lāua ʻo SciPy i nā hana `numpy.linalg.lstsq` a me `scipy.optimize.curve_fit` no ka hoʻopili ʻana i nā huinahā liʻiliʻi loa linear a me non-linear.
2. R: Nā hana e like me `lm()` no nā kumu hoʻohālike linear a me `nls()` no nā kumu hoʻohālike non-linear i loko o ke kaiapuni helu piha o R.
3. MATLAB: Hoʻohana ʻia ka hana `lsqcurvefit` ma MATLAB no ke kūpono ʻole linear, me ke kākoʻo i kūkulu ʻia no nā hana algebra linear.
hoʻokau
ʻOiai kona hoʻohana ākea ʻia, aia nā palena o ke ʻano hana Least Squares:
1. Ka ʻike i nā Outliers: He ʻike nui ʻo Least Squares i nā outliers, ʻoiai ʻo ka huaʻōlelo squared e hoʻonui i ka hopena o nā koena nui.
2. Manaʻo o ka Linearity: Manaʻo i kahi pilina linear ma waena o nā loli, kahi mea paha ʻaʻole paʻa no nā ʻikepili paʻakikī.
3. Nā pilikia Multicollinearity: I ka regression linear maha, hiki i ka multicollinearity ma waena o nā loli wānana ke hoʻololi i ka manaʻo o nā coefficients.
4. Homogeneity of Variance (Homoscedasticity): Manaʻo ʻia he mau ka ʻokoʻa o nā huaʻōlelo hewa ma waena o nā nānā ʻana, ʻaʻole paha ia he hihia mau.
Panina
ʻOiai kona mau makahiki, ʻo ke ʻano hana Least Squares ke kumu nui o ka loiloi ʻikepili helu a me ke kumu hoʻohālike. ʻO kona maʻalahi, ka pono, a me ka hiki ke hoʻololi i nā ʻano like ʻole, mai ka regression linear kumu a hiki i nā kumu hoʻohālike paʻakikī non-linear a me nā kumu hoʻohālike laulā, e lilo ia i mea waiwai nui. ʻO ka hoʻomaopopo ʻana i kona mau kumu makemakika, nā noi, a me nā palena he mea nui ia no kekahi mea e komo ana i ka loiloi ʻikepili, e hōʻoia ana ua hoʻohana pono ʻia ke ʻano hana e loaʻa ai nā ʻike koʻikoʻi. Inā ma ke kula a i ʻole ka ʻoihana, ke hoʻomau nei ke ʻano hana Least Squares i mea hana ikaika no ka hoʻomaopopo ʻana i nā pilina i loko o ka ʻikepili a me ka hana ʻana i nā wānana naʻauao.
