List of Mathematical Formulas: “Least Squares Method

List of basic formulas related to the least squares method:

1. The general form of linear regression is:
   [
   y = \beta_0 + \beta_1 x + \epsilon
   ]
   where (y) is the dependent variable, (x) is the independent variable, (\beta_0) is the intercept, (\beta_1) is the slope, and (\epsilon) is the error term.
2. Regression coefficients:

• Free term ((\beta_0)):
  [
  \beta_0 = \bar{y} – \beta_1 \bar{x}
  ]
• Slope coefficient ((\beta_1)):
  [
  \beta_1 = \frac{n(\sum xy) – (\sum x)(\sum y)}{n(\sum x^2) – (\sum x)^2}
  ]
  where (n) is the number of observations, (\sum xy) is the sum of the products of (x) and (y), (\sum x) and (\sum y) are the sums of the values ​​of (x) and (y), (\sum x^2) is the sum of the squares of the values ​​of (x).

1. Sum of squared errors (SSE):
   [
   SSE = \sum (y_i – \hat{y}_i)^2
   ]
   where (y_i) are the observed values, (\hat{y}_i) are the predicted values.
2. Total Sum of Squares (SST):
   [
   SST = \sum (y_i – \bar{y})^2
   ]
3. Sum of squares regression (SSR):
   [
   SSR = SST – SSE
   ]
4. Coefficient of determination (R²):
   [
   R^2 = \frac{SSR}{SST}
   ]
   This coefficient shows what proportion of the variation in the dependent variable is explained by the independent variable.