Daga Dangantaka zuwa Hasashe
A darasin da ya gabata mun koyi dangantaka - cewa masu canjin yanayi biyu na iya tafiya tare. Regression yana daukan wannan mataki gaba: yana ba ka damar yin hasashe. Idan ka san dangantakar tsakanin sa'o'in karatu da makin JAMB, za ka iya hasashen makin dalibi bisa sa'o'in karatu da ya yi.
Layyin da Ya Fi Dacewa (Line of Best Fit)
Regression na layi yana neman layin da ya fi dacewa da bayanan - layin da ya fi kusa da dukan digogi a zanen tarwatsewar. Wannan layin yana da sifa:
Y = a + bX
- Y: Abin da kake son hasashewa (mai dogaro)
- X: Abin da kake amfani da shi don hasashe (mai zaman kansa)
- a: Intercept - darajar Y sa'ad da X = 0
- b: Slope - yawan canjin Y ga kowane karin X guda daya
Mai bincike ya nazari dangantakar sa'o'in karatu a mako (X) da makin JAMB (Y) na dalibai 50 ya sami:
Makin JAMB = 120 + 8 × Sa'o'in karatu
- Intercept (120): Dalibin da ba ya karatu kwata-kwata (sa'o'i 0) za a yi hasashen samun maki 120
- Slope (8): Kowace sa'a guda daya da aka kara karatu, ana tsammanin maki 8 karin a JAMB
Don haka, dalibin da ke karatu sa'o'i 15 a mako: 120 + (8 × 15) = 120 + 120 = maki 240
Dalibin da ke karatu sa'o'i 25: 120 + (8 × 25) = 120 + 200 = maki 320
R-Squared (R²): Ingancin Model
R² yana gaya maka yawan bambancin da model ɗinka ke bayyanawa. Yana tsakanin 0 da 1:
- R² = 0: Model ba ya bayyana komai - layin ba shi da amfani
- R² = 0.5: Model yana bayyana kashi 50% na bambancin
- R² = 1: Model yana bayyana duk bambancin - cikakkiyar hasashe
A misalinmu na sa'o'in karatu da makin JAMB, R² = 0.45. Wannan yana nufin sa'o'in karatu suna bayyana kashi 45% na bambancin makin JAMB. Sauran kashi 55% na bambancin yana zuwa ne daga wasu abubuwa - irin su baiwar dalibi, ingancin malami, yanayin gida, da sauransu.
R² na 0.45 yana da kyau don binciken ilimi. A kimiyyar jiki, ana sa ran R² kusa da 1. A kimiyyar zamantakewa da ilimi, R² na 0.3 zuwa 0.5 yana da kyau.
Fassara Slope da Hankali
Slope yana gaya maka dangantaka, ba dalili ba. Sa'ad da muka ce "kowace sa'a guda daya ta karin karatu tana nufin maki 8 karin a JAMB," ba muna cewa sa'ar karatu ce ta haifar da karuwar ba. Watakila daliban da ke da himma suna karatu da yawa kuma suna yin kyau a jaraba saboda dalilai daban-daban.
Residuals: Kuskuren Hasashe
Babu model da ya yi cikakkiyar hasashe. Bambancin tsakanin ainihin darajar da hasashen model ana kiransa residual.
Residual = Ainihin daraja - Hasashen daraja
Amina tana karatu sa'o'i 20 a mako. Model ya hasashe za ta sami maki: 120 + (8 × 20) = 280. Amma ainihi ta sami maki 310.
Residual = 310 - 280 = +30
Amina ta yi fiye da abin da model ya hasashe da maki 30. Watakila tana da malamin koyarwa na musamman, ko kuma tana da baiwa ta halitta a jarabar. Residual mai girma na iya nuna cewa akwai abubuwa da model bai kama ba.
Iyakokin Regression na Layi
- Yana neman dangantaka ta layi ne kawai: Idan dangantakar tana da sifa kamar U, regression na layi ba zai kama ta ba.
- Hasashe bayan kewayon bayanai (extrapolation) ba ya da aminci: Idan bayananku sun rufe sa'o'i 5 zuwa 30 na karatu, hasashen makin dalibi mai karatu sa'o'i 50 ba zai yi aiki ba.
- Darajoji matsananciya suna shafe shi: Dalibi daya mai maki matsananciya zai iya canza layin gaba daya.
- Ba ya nuna dalilin: Kamar dangantaka, regression ba ya tabbatar da cewa X yana haifar da Y.
Dan kasuwa a Lagos yana amfani da regression don hasashen tallace-tallace (Y) bisa kudin talla (X). Model ɗinsa:
Tallace-tallace = ₦200,000 + 3 × Kudin talla
Wannan yana nufin: ba tare da talla ba, ana tsammanin tallace-tallace na ₦200,000 (daga abokan ciniki na yau da kullum). Kowace Naira 1 da aka kashe a talla, ana tsammanin ₦3 na karin tallace-tallace.
Amma idan ya kashe ₦10,000,000 a talla, model ya ce zai sami ₦30,200,000. A zahiri, akwai iyaka - ba za ka iya ci gaba da kara tallace-tallace ba ba tare da iyaka ba ta karin talla. Wannan shine haɗarin extrapolation.
Regression Mai Masu Canjin Yanayi Da Yawa (Multiple Regression)
A aikace, yawancin abubuwa suna da masu canjin yanayi da yawa da ke shafe su. Multiple regression yana ba ka damar hada masu canjin yanayi da yawa:
Y = a + b₁X₁ + b₂X₂ + b₃X₃ + ...
Hasashen makin JAMB bisa abubuwa da yawa:
Maki = 80 + 6(sa'o'in karatu) + 15(malamin koyarwa: eh=1, a'a=0) + 0.5(makin WAEC)
Wannan model ya fi daidai saboda yana la'akari da abubuwa da yawa maimakon daya kawai.
Regression yana ba ka damar hasashen daraja daya (Y) bisa daraja daya ko fiye (X) ta amfani da layyin da ya fi dacewa. Slope yana gaya maka yawan canjin Y ga kowane karin X guda daya. R² yana gaya maka ingancin model - yawan bambancin da yake bayyanawa. Ka guji hasashe bayan kewayon bayanan (extrapolation), ka tuna cewa regression ba ya tabbatar da dalilin, kuma koyaushe ka duba residuals don ganin inda model bai yi aiki ba.