Laʻana o nā nīnau kūkākūkā no ke ʻano o ka Least Squares Method
ʻO ke ʻano hana Least Squares Method (LEM) kahi ʻano hana helu i hoʻohana ʻia e ʻimi i ka laina kūpono loa e wānana pono ana i ka ʻikepili. Hoʻohana pinepine ʻia kēia ʻano hana i ka loiloi regression linear e ʻike i ka pilina ma waena o nā loli kūʻokoʻa a me nā loli hilinaʻi. E uhi kēia ʻatikala i nā manaʻo kumu o ke ʻano hana least squares, me nā laʻana a me nā wehewehe ʻanuʻu no ka hoʻomaopopo hohonu ʻana i ke ʻano o ka hana ʻana o kēia ʻano hana.
Nā Manaʻo Kumu o ke ʻAno Hana Kūikawā Liʻiliʻi Loa
ʻO ka pahuhopu o ke ʻano hana liʻiliʻi loa, ʻo ia ka hoʻemi ʻana i ka huina o nā huina o nā ʻokoʻa ma waena o nā waiwai i ʻike ʻia a me nā waiwai i wānana ʻia e ke kumu hoʻohālike regression. Hiki ke kākau ʻia ka hoohalike o kahi laina regression linear maʻalahi penei:
\[ y = a + bx \]
Ma hea:
– ʻO \( y \) ke loli hilinaʻi,
– ʻO \( x \) ke loli kūʻokoʻa,
– ʻO \( a \) ka intercept (ka waiwai o \( y \) i ka wā \( x = 0 \)),
– ʻO \( b \) ka piʻo o ka laina (piʻo, a i ʻole ke koina regression).
Kuhi ke ʻano hana liʻiliʻi loa i nā palena \( a \) a me \( b \) e hōʻemi ana i ka hana aʻe:
\[ \text{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
ʻO SSE ka huina o nā hewa kuea, ʻo \( y_i \) ka waiwai maoli, a ʻo \( \hat{y_i} = a + bx_i \) ka waiwai i wānana ʻia.
Nā ʻanuʻu hana o nā huinahā liʻiliʻi loa
No ka hoʻākāka ʻana i ke kumumanaʻo, e hoʻoponopono mākou i kahi pilikia hoʻohālike e pili ana i ka hoʻohana ʻana i ke ʻano hana liʻiliʻi loa.
Laʻana pilikia
Hāʻawi ʻia i ka ʻikepili ma lalo nei:
| x (Nā hola aʻo) | y (Helu hoʻokolohua) |
|———————–|———————–|
2 | 81 |
4 | 93 |
6 | 91 |
8 | 97 |
10 | 103 |
E hoʻoholo i ka laina regression linear e kūpono loa i ka ʻikepili.
Pahana
1. Ke helu nei i ka awelika o \( \bar{x} \) a me \( \bar{y} \)
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Ke helu ʻana i ka Palena \( b \) (Pīʻina)
Ua helu ʻia ke ʻano \( b \) e:
\[
b = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sum (x_i – \bar{x})^2}
\]
Ke helu nei i kēlā me kēia ʻāpana:
\[
huina (x_i – x})(y_i – y}) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
huina (x_i – \bar{x})^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
No laila:
\[
b = \frac{96}{40} = 2.4
\]
3. Ke helu ʻana i ka Parameter \( a \) (Intercept)
Ke hoʻohana nei i ka awelika o \( \bar{x} \) a me \( \bar{y} \):
\[
a = \bar{y} – b\bar{x} = 93 – 2.4 \times 6 = 93 – 14.4 = 78.6
\]
4. Ke kākau ʻana i ka hoʻohālikelike laina regression
Me nā palena i loaʻa, hiki iā mākou ke kākau i ka hoohalike o ka laina regression:
\[
y = 78.6 + 2.4x
\]
Ka Wehewehe ʻana a me ka Hōʻoia
I mea e kūpono ai kēia laina regression, hiki iā mākou ke helu i ka waiwai-y i wānana ʻia (\(\hat{y}\)) no kēlā me kēia x i ka ʻikepili mua, a me ka helu ʻana i ka Sum of Squared Errors (SSE) e hōʻoia i ka pololei o ka wānana.
| x | y | \(\papale{y}\) | \((y – \papale{y})^2\) |
|—|—-|—————|——————–|
| 2 | 81 | 83.4 | (81-83.4)^2 = 5.76 |
| 4 | 93 | 88.2 | (93-88.2)^2 = 23.04|
| 6 | 91 | 93.0 | (91-93.0)^2 = 4.00 |
| 8 | 97 | 97.8 | (97-97.8)^2 = 0.64 |
|10 |103 |102.6 | (103-102.6)^2= 0.16|
SSS:
\[
SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Me kahi SSE liʻiliʻi, hiki iā mākou ke hoʻoholo he kūpono maikaʻi ka laina regression i hana ʻia e ke ʻano o nā squares liʻiliʻi loa no kēia ʻikepili.
Ka hopena
ʻO ke ʻano hana Least Squares kahi mea hana loiloi helu ikaika no ka hoʻoholo ʻana i ka laina kūpono loa no kahi ʻikepili, e hoʻemi ana i ka hewa wānana ma muli o ka huinahā o nā deviations. Ma ka hoʻohana ʻana i nā ʻanuʻu o ka helu ʻana i ka mean, slope a me intercept, a me ke kākau ʻana a me ka hōʻoia ʻana i ka hoʻohālikelike laina regression, hiki iā mākou ke wānana pololei i ka waiwai o ka loli hilinaʻi mai nā loli kūʻokoʻa.
He mea pono loa ka hoʻomaopopo maikaʻi ʻana i kēia ʻano hana ma nā kahua e like me ka hoʻokele waiwai, biostatistics, ʻenekinia, a me nā ʻepekema pili kanaka kahi e hoʻopili pinepine ʻia ai ka loiloi regression. Hōʻike kēia ʻatikala, me nā laʻana paʻa, i ke koʻikoʻi a me ka pono o kēia ʻano hana i ka loiloi ʻikepili.
