Linear Regression

statistics

Definition

Linear Regression

Linear regression is a statistical technique that models the relationship between a dependent variable and one or more independent variables using a linear equation. It is commonly used to predict the value of the dependent variable based on the values of the independent variables.

Plot

The following shows a scatterplot with a linear regression function:

Formula

$$\begin{array}{r}f(x)=\underset{=k}{\underbrace{\frac{{\sum }_{i=1}^n(x_i-\bar{x})\cdot (y_i-\bar{y})}{{\sum }_{i=1}^n(x_i-\bar{x}{)}^2}}}x+\underset{=d}{\underbrace{\left(\bar{y}-k\cdot \bar{x}\right)}}\end{array}$$

Slope

$$k=\frac{{\sum }_{i=1}^n(x_i-\bar{x})\cdot (y_i-\bar{y})}{{\sum }_{i=1}^n(x_i-\bar{x}{)}^2}$$

$y$-Intercept

$$d=\bar{y}-k\cdot \bar{x}$$