Nazarin Sauƙin Layi na Layi
Sauƙin komawa kan layi mai sauƙi wata dabara ce ta ƙididdiga da ake amfani da ita don nazarin dangantakar da ke tsakanin masu canji guda biyu masu yawa. Ana kiran mai canjin da muke ƙoƙarin yin hasashen mai dogaro ko mai amsawa, yayin da mai canjin da ake amfani da shi don yin hasashen ana kiransa mai zaman kansa ko mai hasashen yanayi. A cikin mai sauƙin komawa kan layi, muna ƙoƙarin nemo mafi kyawun layi madaidaiciya wanda ke bayyana alaƙar da ke tsakanin waɗannan masu canji guda biyu.
Ka'idoji na Asali na Sauƙin Layi na Komawa Baya
Sauƙin komawar layi mai sauƙi ya dogara ne akan zato cewa akwai dangantaka ta layi tsakanin mai dogara \(Y\) da mai zaman kansa mai canji \(X\). Tsarin gabaɗaya na samfurin komawar layi mai sauƙi shine:
\[ Y = \beta_0 + \beta_1 X + \epsilon \]
Ina:
– \( Y \) shine ma'aunin da ya dogara.
– \( X \) shine mai canzawa mai zaman kansa.
– \( \beta_0 \) shine katsewar, wanda shine ƙimar \(Y\) lokacin da \(X = 0\).
– \( \beta_1 \) shine gangara ko juzu'i, wanda shine matsakaicin canji a cikin \(Y\) ga kowane canjin raka'a a cikin \(X\).
– \( \epsilon \) shine kuskuren ko kalmar da ta rage wadda ke wakiltar bambancin da ke cikin \(Y\) wanda ba za a iya bayyana shi ta hanyar \(X\) ba.
Manufar sauƙin komawa ga layi mai sauƙi shine a kimanta sigogi \(\beta_0\) da \(\beta_1\) ta yadda za a iya amfani da samfurin don annabta ƙimar \(Y\) da ke da alaƙa da ƙimar \(X\).
Hanyar Mafi Karancin Murabba'i
Ɗaya daga cikin hanyoyin da aka fi amfani da su don daidaita samfurin juyi mai sauƙi shine hanyar Least Squares. Wannan hanyar tana da nufin rage jimlar murabba'ai na karkacewar tsaye tsakanin abubuwan da aka lura da kuma ƙimar da samfurin ya annabta. A ce muna da abubuwan da aka lura da n waɗanda suka ƙunshi nau'i-nau'i \((x_i, y_i)\) don \(i = 1, 2, …, n\). Aikin da za a rage shine:
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – (\beta_0 + \beta_1 x_i))^2 \]
Domin nemo \(\beta_0\) da \(\beta_1\) waɗanda suka rage wannan aikin, mun ɗauki ɓangarorin da suka samo asali na \(S(\beta_0, \beta_1)\) dangane da kowane siga kuma muka saita waɗannan abubuwan da suka samo asali zuwa sifili. Ana iya sauƙaƙe lissafin lissafi kamar haka:
\[ \beta_1 = \frac{\sum_{i=1}^{n} (x_i – \bar{x})(y_i – \bar{y})}{\sum_{i=1}^{n} (x_i – \bar{x})^2} \]
\[ \beta_0 = \bar{y} - \beta_1 \bar{x} \]
Ina:
– \(\bar{x}\) shine matsakaicin \(X\)
– \(\bar{y}\) shine matsakaicin \(Y\)
Bayan samun sigogin \(\beta_0\) da \(\beta_1\), ana iya amfani da samfurin juyi mai sauƙi don yin hasashen ƙimar \(Y\) ga kowane ƙimar \(X\).
Zato a cikin Sauƙin Layi Mai Sauƙi
Don samun sakamako masu inganci da inganci, sauƙin komawa kan layi yana ɗaukar abubuwa da yawa:
1. Layi: Alaƙar da ke tsakanin ma'aunin da ya dogara da ma'aunin da ba shi da 'yanci dole ne ta kasance layi.
2. 'Yancin Kai: Dole ne lura ta kasance ba ta da alaƙa da juna.
3. Hulɗa da Mutum: Dole ne bambancin da ya rage ya kasance mai daidaito a cikin kewayon ƙimar canjin mai zaman kansa.
4. Daidaiton Ragowa: Dole ne ragowar (kurakurai) su bi rarrabawar da aka saba.
Idan ba a cika waɗannan zato ba, sakamakon samfurin juyi mai sauƙi na layi ba zai zama abin dogaro ba kuma ƙila ba zai iya yin hasashen da ya dace ba.
Kimanta Tsarin Komawa Baya
Hanya ɗaya ta tantance yadda samfurin juyi mai sauƙi na layi ya annabta shi ne amfani da Coefficient of Determination (\(R^2\)). Coefficient of determination yana nuna rabon bambancin da ke cikin ma'aunin dogara wanda za a iya bayyana shi ta hanyar bambancin da ke cikin ma'aunin masu zaman kansu.
\[ R^2 = \frac{\sum_{i=1}^{n} (\hat{y}_i – \bar{y})^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2} \]
Ina:
– \(\hat{y}_i\) shine ƙimar da aka annabta ta \(Y\).
– \(y_i\) shine ainihin ƙimar \(Y\).
– \(\bar{y}\) shine matsakaicin ƙimar \(Y\).
Ƙimar \(R^2\) tana tsakanin 0 zuwa 1. Ƙimar \(R^2\) kusa da 1 tana nuna cewa samfurin zai iya bayyana yawancin bambancin da ke cikin canjin da ya dogara da shi.
Aiwatarwa a Harshen Shirye-shirye
Don aiwatar da sauƙin komawa ga layi mai sauƙi, za mu iya amfani da software na ƙididdiga ko harsunan shirye-shirye daban-daban. A ƙasa akwai misali aiwatarwa a cikin Python ta amfani da ɗakin karatu na 'scikit-learn':
"' Python
shigo da numpy as np
shigo da matplotlib.pyplot azaman plt
daga sklearn.linear_model shigo da LinearRegression
daga sklearn.metrics shigo da mean_squared_error, r2_score
data
X = np.array([[1], [2], [3], [4], [5]]).astype(np.float64)
y = np.array([1.5, 3.6, 3.5, 2.9, 5.5]).astype(np.float64)
model
model = LinearRegression ()
samfurin.daidaitacce(X, y)
Hasashen
y_pred = model.predict (X)
Mai daidaita
beta_0 = samfurin.tsangwama_
beta_1 = samfuri.coef_[0]
buga(f'Intercept: {beta_0}')
buga(f'Slope: {beta_1}')
buga(f'Matsakaicin kuskuren murabba'i: {mean_squared_error(y, y_pred)}')
bugawa(f'Ma'aunin tantancewa (R^2): {r2_score(y, y_pred)}')
Tsarin bayanai da layin komawa baya
plt.scatter(X, y, launi='shuɗi')
plt.plot(X, y_pred, launi='ja')
plt.xlabel('X')
alamar plt.ylabel('Y')
plt.ya nuna ()
““
A cikin misalin da ke sama, da farko za mu shigo da ɗakunan karatu da ake buƙata, mu ayyana bayanan \(X\) da \(Y\), sannan mu yi amfani da abin `LinearRegression` daga `scikit-learn` don dacewa da samfuri zuwa ga bayanai. Da zarar an haɗa samfurin, za mu yi hasashen kuma mu ƙididdige ma'aunin, da kuma matsakaicin kuskuren murabba'i da ma'aunin ƙuduri. A ƙarshe, za mu tsara bayanai da layin komawa baya.
Kammalawa
Sauƙin komawa kan layi kayan aiki ne mai ƙarfi na nazarin ƙididdiga wanda ake amfani da shi don bayyana alaƙar da ke tsakanin masu canji biyu na adadi. Tare da wasu zato na asali game da layi, 'yancin kai, homoscedasticity, da kuma daidaito, za mu iya annabta ƙimar mai canzawa mai dogaro bisa ga ƙimar masu canji masu zaman kansu. Hanyar Least Squares tana ba da hanya mai inganci don dacewa da layin komawa da ƙayyade sigogi mafi kyau. Kimanta samfura ta hanyar ma'aunin ƙuduri (R2) yana ba da haske game da yadda samfurinmu yake aiki.
Duk da cewa sauƙin komawa baya na layi yana da iyakoki, kamar iya sarrafa masu canji guda biyu kawai da kuma zato da dole ne a cika, wannan dabarar ta kasance muhimmin tushe a cikin kididdiga da nazarin bayanai, kuma galibi ana amfani da ita azaman matakin farko na fahimtar alaƙar da ke tsakanin masu canji kafin a ci gaba zuwa hanyoyin da suka fi rikitarwa.
