Laʻana o kahi nīnau kūkākūkā ma ke koina hoʻoholo
ʻO ke koina hoʻoholo (R²) kahi palena koʻikoʻi i ka loiloi regression, e hōʻike ana i ka maikaʻi o ka wehewehe ʻana o kahi kumu hoʻohālike regression i ka loli maoli o ka ʻikepili. Ma kēia ʻatikala, e wehewehe mākou i ke kumumanaʻo o ke koina hoʻoholo ma o kahi laʻana kikoʻī a me ke kūkākūkā ʻana.
Manaʻo Kumu o ke Koepili o ka Hoʻoholo
Ua ana ʻia ke koina hoʻoholo a i ʻole \( R^2 \) ma kahi pālākiō o 0 a 1, kahi:
– Hōʻike ʻo \( R^2 = 0 \) ʻaʻole hiki i ke kumu hoʻohālike regression ke wehewehe i ka loli o ka ʻikepili.
– Hōʻike ʻo \( R^2 = 1 \) hiki i ke kumu hoʻohālike regression ke wehewehe pono i nā ʻano loli āpau o ka ʻikepili.
ʻO ke ʻano kumu no ka helu ʻana i ke koina hoʻoholo:
\[ R^2 = 1 – \frac{SSR}{SST} \]
Ma hea:
- ʻO SSR (Hum of Squared Residuals) ka huina o nā huinahā o nā ʻokoʻa ma waena o nā waiwai i wānana ʻia e ke kumu hoʻohālike a me nā waiwai maoli.
– ʻO ka SST (Huina Huina o nā Kuea) ka huina o ka huina o nā kuea o nā ʻokoʻa ma waena o ka waiwai maoli a me ka awelika o nā waiwai maoli.
Laʻana pilikia
E kūkākūkā kākou i kahi pilikia hoʻohālike e hoʻomaopopo hohonu ai i ka helu ʻana o ke koina hoʻoholo.
Laʻana pilikia:
Manaʻo mākou he ʻikepili kā mākou e pili ana i ka helu o nā hola aʻo (X) a me nā helu hoʻokolohua (Y) o 10 mau haumāna:
| Nā Haumāna | Nā Hola Haʻawina (X) | Ka helu hoʻokolohua (Y) |
|——-|———————–|———————–|
1 | 2 | 58 |
2 | 3 | 64 |
3 | 4 | 70 |
4 | 5 | 85 |
5 | 2 | 57 |
6 | 3 | 68 |
7 | 4 | 72 |
8 | 5 | 90 |
9 | 3 | 62 |
| 10 | 4 | 78 |
E hana mākou i kahi kumu hoʻohālike regression linear maʻalahi kahi i wānana ʻia ai nā helu hoʻokolohua (Y) ma muli o nā hola aʻo (X).
Pahana
1. Ke kūkulu ʻana i kahi kumu hoʻohālike regression linear maʻalahi
ʻO ke ʻano o ke kumu hoʻohālike regression linear maʻalahi:
\[ Y = a + bX \]
Ma hea:
– ʻO \( Y \) ka helu hoʻāʻo i wānana ʻia.
– ʻO \( X \) ka helu o nā hola aʻo.
– ʻO \( a \) ke kiko intercept (ke kiko o ka hui ʻana ma ke axis Y ke X = 0).
– ʻO \( b \) ka pali (ʻano o ka laina regression).
No ka helu ʻana i nā palena \( a \) a me \( b \), hoʻohana mākou i kēia haʻilula:
\[ b = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{n(\sum{X^2}) – (\sum{X})^2} \]
\[ a = \frac{\sum{Y} – b(\sum{X})}{n} \]
Ma kahi o \(n \) ka helu o ka ʻikepili (i kēia hihia n = 10).
Mai ka papa hiki iā mākou ke helu:
– \(\sum{X} = 36\)
– \(\sum{Y} = 704\)
– \(\sum{X^2} = 140\)
– \(\sum{Y^2} = 50428\)
– \(\sum{XY} = 2576\)
E helu mua kākou iā b:
\[ b = \frac{10(2576) – (36)(704)}{10(140) – (36)^2} \]
\[ b = \frac{25760 – 25344}{1400 – 1296} \]
\[ b = \frac{416}{104} \]
\[ b = 4 \]
A laila, helu mākou i kahi:
\[ a = \frac{704 – 4(36)}{10} \]
\[ a = \frac{704 – 144}{10} \]
\[ a = \frac{560}{10} \]
\[ a = 56 \]
No laila, ʻo ke kumu hoʻohālike regression linear a mākou e loaʻa ai:
\[ Y = 56 + 4X \]
2. E helu i ka Waiwai i Wānana ʻia (Y')
A laila, helu mākou i ka waiwai i wānana ʻia \( Y' \) no kēlā me kēia \( X \):
| Nā Haumāna | Nā Hola Haʻawina (X) | Ka Helu Hoʻokolohua (Y) | Ka Helu i Wānana ʻia (Y') |
|——-|———————–|———————–|————————-|
| 1 | 2 | 58 | \( 56 + 4(2) = 64 \) |
| 2 | 3 | 64 | \( 56 + 4(3) = 68 \) |
| 3 | 4 | 70 | \( 56 + 4(4) = 72 \) |
| 4 | 5 | 85 | \( 56 + 4(5) = 76 \) |
| 5 | 2 | 57 | \( 56 + 4(2) = 64 \) |
| 6 | 3 | 68 | \( 56 + 4(3) = 68 \) |
| 7 | 4 | 72 | \( 56 + 4(4) = 72 \) |
| 8 | 5 | 90 | \( 56 + 4(5) = 76 \) |
| 9 | 3 | 62 | \( 56 + 4(3) = 68 \) |
| 10 | 4 | 78 | \( 56 + 4(4) = 72 \) |
3. Ke helu ʻana iā SSR a me SST
A laila, helu mākou iā SSR a me SST e loaʻa ai iā \( R^2 \).
SSR:
\[ SSR = \sum{(Y – Y')^2} \]
SSR = (58 – 64)^2 + (64 – 68)^2 + (70 – 72)^2 + (85 – 76)^2 + (57 – 64)^2 + (68 – 68)^2 + (72 – 72)^2 + (90 – 76)^2 + (62 – 68)^2 + (78 – 72)^2
\[ SSR = 36 + 16 + 4 + 81 + 49 + 0 + 0 + 196 + 36 + 36 \]
\[ SSR = 454 \]
SST:
\[ SST = \sum{(Y – \bar{Y})^2} \]
Ma hea:
\[ \bar{Y} = \frac{\sum{Y}}{n} = \frac{704}{10} = 70.4 \]
\[ SST = (58 – 70.4)^2 + (64 – 70.4)^2 + (70 – 70.4)^2 + (85 – 70.4)^2 + (57 – 70.4)^2 + (68 – 70.4)^2 + (72 – 70.4)^2 + (90 – 70.4)^2 + (62 – 70.4)^2 + (78 – 70.4)^2 \]
\[ SST = 153.76 + 40.96 + 0.16 + 213.16 + 178.56 + 5.76 + 2.56 + 384.16 + 70.56 + 57.76 \]
\[ SST = 1107.44 \]
4. Ke helu ʻana i ke Koehana Hoʻoholo \( R^2 \):
\[ R^2 = 1 – \frac{SSR}{SST} \]
\[ R^2 = 1 – \frac{454}{1107.44} \]
\[ R^2 = 1 – 0.41 \]
\[ R^2 = 0.59 \]
Ka hopena
Mai ka helu ʻana ma luna, ua loaʻa iā mākou kahi coefficient o ka hoʻoholo (R2 = 0.59). Hōʻike kēia e hiki i ke kumu hoʻohālike regression linear a mākou i hana ai ke wehewehe ma kahi o 59% o ka loli o nā helu hoʻokolohua e pili ana i nā hola aʻo. ʻO ke koena 41% o ka loli hiki ke ma muli o nā kumu ʻē aʻe i hoʻokomo ʻole ʻia i loko o ke kumu hoʻohālike.
Ma ka hoʻomaopopo ʻana i nā ʻanuʻu a me nā helu ma luna, hiki iā mākou ke ʻike i ke koʻikoʻi o ke koina o ka hoʻoholo ʻana i ka loiloi ʻana i ka maikaʻi o ke kumu hoʻohālike regression a mākou e kūkulu ai e wehewehe i ka loli maoli o ka ʻikepili. He mea hana pono loa ia i ka loiloi helu a me ke kumu hoʻohālike ʻikepili.
