Regressione Lineare © 2023 erreeffe

Formule dei coefficienti

Regressione lineare semplice di y rispetto a x \[y = mx + q\]

• Coefficiente angolare (pendenza) \(m\): \[ m = \frac{n \sum_j x_j y_j - \sum_j x_j \sum_j y_j}{n \sum_j x_j^2 - (\sum_j x_j)^2} \]
• Intercetta \(q\): \[ q = \frac{\sum_j y_j}{n} - m \frac{\sum_j x_j}{n} \]

Regressione lineare semplice di x rispetto a y \[x = my + q\]

• Coefficiente angolare (pendenza) \(m\): \[ m = \frac{n \sum_j x_j y_j - \sum_j x_j \sum_j y_j}{n \sum_j y_j^2 - (\sum_j y_j)^2} \]
• Intercetta \(q\): \[ q = \frac{\sum_j x_j}{n} - m \frac{\sum_j y_j}{n} \]

Regressione lineare polinomiale di y rispetto ad x \[y = a_2 x^2 + a_1 x + a_0\]

\[ \pmatrix{\sum_j 1 & \sum_j x_j & \sum_j x_j^2 \\ \sum_j x_j & \sum_j x_j^2 & \sum_j x_j^3 \\ \sum_j x_j^2 & \sum_j x_j^3 & \sum_j x_j^4 } \pmatrix{ a_0 \\ a_1 \\ a_2 } = \pmatrix{ \sum_j y_j \\ \sum_j y_j x_j \\ \sum_j y_j x_j^2 } \]