Misali na Tambayoyin Tattaunawa na Hanyar Mafi Sauƙi
Hanyar Mafi Sauƙi (LEM) hanya ce ta ƙididdiga da ake amfani da ita don nemo layin mafi dacewa wanda ya fi hasashen bayanai yadda ya kamata. Sau da yawa ana amfani da wannan hanyar a cikin nazarin komawar layi don gano alaƙar da ke tsakanin masu canji masu zaman kansu da waɗanda suka dogara da su. Wannan labarin zai rufe mahimman ra'ayoyi na hanyar mafi ƙarancin murabba'i, tare da misalai da bayani mataki-mataki don fahimtar yadda wannan hanyar ke aiki.
Ka'idoji na Asali na Hanyar Mafi Karancin Murabba'i
Manufar hanyar mafi ƙarancin murabba'i ita ce rage jimlar murabba'ai na bambance-bambancen da ke tsakanin ƙimar da aka lura da ƙimar da samfurin koma-baya ya annabta. Ana iya rubuta daidaiton layin komawa-baya mai sauƙi kamar haka:
\[y = a + bx \]
Ina:
– \(y \) shine ma'aunin da ya dogara,
– \( x \) shine mai canzawa mai zaman kansa,
– \( a \) shine tsangwama (ƙimar \( y \) lokacin da \( x = 0 \)),
– \( b \) shine gangaren layin (ganga, ko ma'aunin komawa baya).
Hanyar mafi ƙarancin murabba'ai tana ƙiyasta sigogi \( a \) da \( b \) waɗanda ke rage aikin da ke tafe:
\[ \text{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
Inda SSE shine Jimlar Kurakurai Masu Zurfi, \( y_i \) shine ainihin ƙimar, kuma \( \hat{y_i} = a + bx_i \) shine ƙimar da aka annabta.
Matakai na Hanyar Mafi Karancin Murabba'i
Don fayyace manufar, za mu magance matsalar misali da ta shafi amfani da hanyar ƙananan murabba'ai.
Misalin matsalolin
Dangane da waɗannan bayanai:
| x (Lokutan karatu) | y (Maki na jarrabawa) |
|——————–|———————–|
| 2 | 81 |
| 4 | 93 |
| 6 | 91 |
| 8 | 97 |
| 10 | 103 |
Kayyade layin komawar layi wanda ya fi dacewa da bayanai.
Tattaunawa
1. Lissafin Matsakaicin \( \bar{x} \) da \( \bar{y} \)
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Lissafin Siga \( b \) (Girman)
Ana ƙididdige siga \( b \) ta hanyar:
\[
b = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sum (x_i – \bar{x})^2}
\]
Lissafin kowane ɓangare:
\[
\sum (x_i – \bar{x})(y_i – \bar{y}) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
\sum (x_i – \bar{x})^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
Don haka:
\[
b = \frac{96}{40} = 2.4
\]
3. Lissafin Siga \( a \) (Canzawa)
Yin amfani da matsakaicin \( \bar{x} \) da \( \bar{y} \):
\[
a = \bar{y} – b\bar{x} = 93 – 2.4 \sau 6 = 93 – 14.4 = 78.6
\]
4. Rubuta Lissafin Layin Komawa Baya
Da sigogin da aka samo, za mu iya rubuta lissafin layin komawa baya:
\[
y = 78.6 + 2.4x
\]
Fassara da Tabbatarwa
Domin tabbatar da cewa wannan layin komawa baya ya dace, za mu iya ƙididdige ƙimar y da aka annabta (\(\hat{y}\)) ga kowane x a cikin bayanan farko, da kuma ƙididdige Jimlar Kurakurai Masu Zurfi (SSE) don tabbatar da daidaiton hasashen.
| x | y | \(\hat{y}\) | \((y - \hat{y})^2\) |
|—|—-|—————|——————–|
| 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:
\[
SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Tare da ƙaramin SSE, za mu iya yanke hukuncin cewa layin juyawa da aka samar ta hanyar mafi ƙarancin murabba'i ya dace da wannan bayanan.
Kammalawa
Hanyar Mafi Sauƙi Kayan aiki ne mai ƙarfi na nazarin ƙididdiga don tantance layin da ya fi dacewa da bayanai, rage kuskuren hasashen bisa ga murabba'in karkacewar. Ta hanyar amfani da matakan ƙididdige matsakaicin, kimanta gangara da tsangwama, da kuma rubutawa da tabbatar da daidaiton layin komawa baya, za mu iya yin hasashen ƙimar canjin da ya dogara da shi daidai daga masu canji masu zaman kansu.
Fahimtar wannan hanyar tana da matuƙar amfani a fannoni kamar tattalin arziki, kididdigar halittu, injiniyanci, da kuma kimiyyar zamantakewa inda ake yawan amfani da nazarin koma-baya. Wannan labarin, tare da misalai na gaske, ya nuna mahimmanci da amfanin wannan hanyar a cikin nazarin bayanai.
