Tusaalaha Habka Ugu Yar ee Saddexjibbaaran Su'aalaha Doodda
Habka Ugu Yar ee Labajibbaaran (LEM) waa hab tirakoob oo loo isticmaalo in lagu helo xariiqda ugu habboon ee ugu waxtarka badan ee saadaalinta xogta. Habkan waxaa badanaa loo isticmaalaa falanqaynta dib-u-celinta toosan si loo ogaado xiriirka ka dhexeeya doorsoomayaasha madaxbannaan iyo kuwa ku tiirsan. Maqaalkani wuxuu dabooli doonaa fikradaha aasaasiga ah ee habka ugu yar ee labajibbaaran, oo ay weheliso tusaalooyin iyo sharraxaad tallaabo-tallaabo ah si loo fahmo qoto dheer oo ku saabsan sida habkani u shaqeeyo.
Fikradaha Aasaasiga ah ee Habka Ugu Yar ee Labajibbaaran
Hadafka habka labajibbaaran ee ugu yar waa in la yareeyo wadarta labajibbaaran ee kala duwanaanshaha u dhexeeya qiimayaasha la arkay iyo qiimayaasha uu saadaaliyay qaabka dib-u-celinta. Isla'egta xariiqda dib-u-celinta toosan ee fudud waxaa loo qori karaa sidan:
\[ y = a + bx \]
Halkee:
– \( y \) waa doorsoomaha ku tiirsan,
– \( x \) waa doorsoomaha madaxbannaan,
– \( a \) waa dhexda (qiimaha \( y \) marka \( x = 0 \)),
– \( b \) waa jiirada xariiqda (jiirada, ama isku-xidhka dib-u-noqoshada).
Habka ugu yar ee labajibbaaran wuxuu qiyaasaa xuduudaha \( a \) iyo \( b \) ee yareeya shaqada soo socota:
\[ \text{SSE} = \sum_{i=1}^{n} (y_i – \hat{y_i})^2 \]
Halka SSE ay tahay Wadarta Khaladaadka Labajibbaaran, \( y_i \) waa qiimaha dhabta ah, iyo \( \hat{y_i} = a + bx_i \) waa qiimaha la saadaaliyay.
Tallaabooyinka Habka Ugu Yar ee Labajibbaaran
Si aan u caddayno fikradda, waxaan xallin doonnaa dhibaato tusaale ah oo ku lug leh isticmaalka habka ugu yar ee labajibbaaran.
Tusaalaha dhibaatooyinka
Marka la eego xogta soo socota:
| x (Saacadaha waxbarashada) | y (Dhibcaha imtixaanka) |
|——————–|———————–|
| 2 | 81 |
| 4 | 93 |
| 6 | 91 |
| 8 | 97 |
| 10 | 103 |
Go'aami xariiqda dib-u-celinta toosan ee ugu habboon xogta.
Dood
1. Xisaabinta Celceliska \( \bar{x} \) iyo \( \bar{y} \)
\[
\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6
\]
\[
\bar{y} = \frac{\sum y_i}{n} = \frac{81 + 93 + 91 + 97 + 103}{5} = 93
\]
2. Xisaabinta Halbeegga \( b \) (Jiilashada)
Halbeegga \( b \) waxaa lagu xisaabiyaa:
\[
b = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sum (x_i – \bar{x})^2}
\]
Xisaabinta qayb kasta:
\[
\sum (x_i – \bar{x})(y_i – \bar{y}) = (2-6)(81-93) + (4-6)(93-93) + (6-6)(91-93) + (8-6)(97-93) + (10-6)(103-93)
\]
\[
= (-4)(-12) + (-2)(0) + (0)(-2) + (2)(4) + (4)(10)
\]
\[
= 48 + 0 + 0 + 8 + 40 = 96
\]
\[
\sum (x_i – \bar{x})^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2
\]
\[
= (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2
\]
\[
= 16 + 4 + 0 + 4 + 16 = 40
\]
Sidaas darteed:
\[
b = \frac{96}{40} = 2.4
\]
3. Xisaabinta Halbeegga \( a \) (Ka-goynta)
Iyadoo la adeegsanayo celceliska \( \bar{x} \) iyo \( \bar{y} \):
\[
a = \bar{y} – b\bar{x} = 93 – 2.4 \jeer 6 = 93 – 14.4 = 78.6
\]
4. Qorista Isle'egta Khadka Dib-u-Celinta
Marka la helo xuduudaha la helay, waxaan qori karnaa isla'egta xariiqda dib-u-noqoshada:
\[
y = 78.6 + 2.4x
\]
Fasiraadda iyo Xaqiijinta
Si loo hubiyo in xariiqdan dib-u-celinta ay ku habboon tahay, waxaan xisaabin karnaa qiimaha y ee la saadaaliyay (\(\hat{y}\)) x kasta oo ku jira xogta bilowga ah, iyo sidoo kale waxaan xisaabin karnaa Wadarta Khaladaadka Labajibbaaran (SSE) si loo xaqiijiyo saxnaanta saadaasha.
| x | y | \(\koofi{y}\) | \((y - \ koofiyadda{y})^2\) |
|—|—-|—————|———————–|
| 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:
\[
SSE = 5.76 + 23.04 + 4.00 + 0.64 + 0.16 = 33.6
\]
Iyadoo SSE yar tahay, waxaan ku soo gabagabeyn karnaa in xariiqda dib-u-celinta ee lagu soo saaray habka ugu yar ee labajibbaaran ay ku habboon tahay xogtan.
Gabagabo
Habka Ugu Yar ee Labajibbaaran waa qalab falanqayn tirakoob oo awood badan si loo go'aamiyo xariiqda ugu habboon ee xog ururin, iyadoo la yareynayo khaladaadka saadaasha iyadoo lagu saleynayo labajibbaaran ee leexashada. Adigoo adeegsanaya tallaabooyinka xisaabinta celceliska, qiyaasidda jiirada iyo dhexda, iyo qorista iyo xaqiijinta isle'egta xariiqda dib u celinta, waxaan si sax ah u saadaalin karnaa qiimaha doorsoomaha ku tiirsan doorsoomayaasha madaxbannaan.
Faham wanaagsan oo ku saabsan habkan ayaa aad waxtar ugu leh dhinacyada sida dhaqaalaha, tirakoobka bayoolajiga, injineernimada, iyo cilmiga bulshada halkaas oo si joogto ah loo adeegsado falanqaynta dib-u-noqoshada. Maqaalkani, oo leh tusaalooyin la taaban karo, wuxuu muujinayaa muhiimadda iyo faa'iidada habkan ee falanqaynta xogta.
