Hanyar Mafi Sauƙi: Hanyar Lissafi don Kimantawa
Pendahuluan
Hanyar mafi ƙarancin murabba'i wata dabara ce ta ƙididdiga da ake amfani da ita don kimanta sigogi a cikin tsarin koma-baya ta hanyar rage jimlar kurakuran murabba'i tsakanin ainihin ƙimomin da ƙimar da samfurin ya annabta. Wannan hanyar tana da matuƙar shahara kuma ana amfani da ita akai-akai a fannoni daban-daban kamar tattalin arziki, injiniyanci, ilmin halitta, da kimiyyar zamantakewa. Adrien-Marie Legendre ne ya fara gabatar da manufar mafi ƙarancin murabba'i a farkon ƙarni na 19 kuma daga baya Carl Friedrich Gauss ya ƙara haɓaka ta.
Fahimtar Asali
Gabaɗaya, hanyar mafi ƙarancin murabba'ai tana nufin nemo layin komawa baya mafi dacewa don saita bayanai ta hanyar rage jimlar murabba'ai na ragowar, ko kurakuran hasashen. Sauran shine bambanci tsakanin ƙimar da aka lura da ƙimar da aka annabta.
Idan muna da saitin bayanai wanda ya ƙunshi nau'i-nau'i na lura \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)\), to burinmu shine mu nemo layin \(y = mx + b\) wanda ke rage jimlar kurakuran murabba'i sum\( \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \).
Ana iya amfani da wannan hanyar ga duka juyi mai sauƙi na layi da juyi mai yawa. A cikin juyi mai sauƙi na layi, muna da juyi mai zaman kansa ɗaya kawai (x), yayin da juyi mai yawa na layi ya ƙunshi juyi mai zaman kansa fiye da ɗaya.
Sauƙin Layi Mai Sauƙi
Bari mu fara da sauƙin komawa ga layi. A ce muna da saitin bayanai \((x_1, y_1), (x_2, y_2), …, (x_n, y_n)). Tsarin komawa ga layi mai sauƙi da muke son dacewa da shi shine:
\[y = mx + b + \epsilon \]
inda \( m \) shine gangaren, \( b \) shine tsangwama, kuma \( \epsilon \) shine kuskuren bazata.
Ta amfani da hanyar mafi ƙarancin murabba'i, za mu iya samun kimantawa na sigogi \( m \) da \( b \) ta hanyar rage aikin kuskuren murabba'i:
\[ S(m, b) = \sum_{i=1}^{n} (y_i – (mx_i + b))^2 \]
Domin rage girman \( S(m, b) \), mun sami ɓangarorin da suka samo asali na \( S \) dangane da \( m \) da \( b \), sannan mu warware wannan daidaito don \( m \) da \( b \):
\[ \begin{aligned}
\frac{\partial S}{\partial m} &= -2 \sum_{i=1}^{n} x_i (y_i – (mx_i + b)) = 0 \\
\frac{\partial S}{\partial b} &= -2 \sum_{i=1}^{n} (y_i – (mx_i + b)) = 0
\end{aligned} \]
Bayan sauƙaƙewa, za mu sami waɗannan daidaitattun daidaito guda biyu na yau da kullun:
\[ \begin{aligned}
n\bar{y} &= m \sum_{i=1}^{n} x_i + nb \\
\sum_{i=1}^{n}x_i y_i &= m \sum_{i=1}^{n}x_i^2 + b \sum_{i=1}^{n}x_i
\end{aligned} \]
Ta hanyar warware tsarin daidaito da ke sama, za mu iya samun ƙimar \( m \) da \( b \) waɗanda ke rage kuskuren murabba'i.
Layin Layi Mai Yawa
A cikin koma-baya mai layi da yawa, muna fuskantar yanayi inda muke da fiye da ma'auni mai zaman kansa ɗaya. A ce muna da bayanai a cikin nau'in tuple \((x_{i1}, x_{i2}, …, x_{ik}, y_i)\). Tsarin koma-baya da muke amfani da shi shine:
\[ y = b_0 + b_1 x_1 + b_2 x_2 + … + b_k x_k + \epsilon \]
Ana iya rubuta wannan lissafin a cikin nau'in matrix kamar haka:
\[ \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{\epsilon} \]
Ina:
– \( \mathbf{y} \) ginshiƙi ne mai nuna ƙimar y da aka lura.
– \( \mathbf{X} \) matrix ne na ƙimar x da aka lura (gami da ginshiƙi na 1 don tsangwama).
– \( \mathbf{b} \) ginshiƙi ne na sigogi (gami da \( b_0 \)).
Manufar hanyar mafi ƙarancin murabba'i ita ce rage aikin kuskuren kwata-kwata mai zuwa:
\[ S(\mathbf{b}) = (\mathbf{y} - \mathbf{Xb})^T (\mathbf{y} - \mathbf{Xb}) \]
Domin rage wannan aikin, mun ɗauki ɓangaren da aka samo daga S dangane da \( \mathbf{b} \) kuma muka saita shi zuwa sifili. Wannan yana samar da daidaito na yau da kullun don komawar layi mai yawa:
\[ \mathbf{X}^T \mathbf{Xb} = \mathbf{X}^T \mathbf{y} \]
Ta hanyar warware tsarin daidaito da ke sama, za mu iya samun kimantawa na siga \( \mathbf{b} \):
\[ \mathbf{b} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} \]
Fa'idodi da Iyakoki
Hanyar mafi ƙarancin murabba'i tana da fa'idodi da yawa. Hanya ce mai inganci kuma mai sauƙi don amfani. Tana ba da mafita ta musamman idan \( \mathbf{X}^T \mathbf{X} \) ba ta juyawa, wanda hakan ke sa ta zama abin dogaro ga aikace-aikace da yawa masu amfani.
Duk da haka, hanyar mafi ƙarancin murabba'ai ita ma tana da iyakoki. Yana da matuƙar sauƙi ga waɗanda ba su da bambanci saboda kuskuren murabba'i yana jaddada manyan bambance-bambance fiye da ƙananan. Bugu da ƙari, dole ne a cika zato na gargajiya cewa kurakuran suna da rarraba ta al'ada tare da sifili matsakaici da bambancin da ke ci gaba don samun sakamako mai kyau.
Aikace-aikace Masu Amfani
Ana amfani da hanyar mafi ƙarancin murabba'i akai-akai a cikin nazarin yanayin bayanai, hasashen yanayi, da koyon injin don gina samfuran hasashen yanayi. A cikin masana'antar kuɗi, ana amfani da hanyar mafi ƙarancin murabba'i don hasashen farashin hannun jari ko aikin kasuwa. A cikin magani, ana amfani da ita don yin kwaikwayon alaƙar da ke tsakanin yawan shan magani da amsawar marasa lafiya. A cikin kimiyyar zamantakewa, yana taimakawa wajen fahimtar alaƙar da ke tsakanin masu canji kamar ilimi da samun kuɗi.
Kammalawa
Hanyar mafi ƙarancin murabba'i tana ɗaya daga cikin manyan dabarun kididdiga da nazarin bayanai. Duk da cewa tana da sauƙi a cikin ra'ayi, wannan hanyar tana ba da ƙarfi mai yawa wajen yin ƙira da fahimtar alaƙa tsakanin masu canji. Tare da aikace-aikacen da aka yi amfani da su a fannoni daban-daban, fahimtar wannan hanyar tana da matuƙar amfani ga ƙwararru da masu bincike. A nan gaba, tare da ƙaruwar yawan bayanai da aka ci karo da su a zamanin manyan bayanai, daidaitawa da amfani da hanyoyin gargajiya kamar ƙananan murabba'i za su ƙara zama masu mahimmanci.
