Check out this apartment price prediction script that utilizes simple linear regression and this database to predict the price of an apartment. For explanation of all the mathematics scroll down.

To predict the price of an apartment, linear regression has to be used. Linear regression is easy to employ when 2 variable functions are used; therefore, in this script the area of an apartment is presented as $X$ and price as $Y$.

To calculate the optimal line of regression, minimizing the total sum of squares of $Y$ is required. $SSy$ shows how big is the sum of all the differences of $Yi$ and $\overline{Y}$, how precise is the linear function. The parameters of the linear function are $\hat{\beta_0}$ and $\hat{\beta_1}$. The optimal parameters can be calculated in 3 steps.

1. Deriving the formula for partial derivatives of $\hat{\beta_0}$ and $\hat{\beta_1}$
   -> It is done to determine the slope of the function at a given moment
2. Calculating zeroes of the two derivatives
   -> $\frac{\partial f}{\partial x} = 0$ is a local minimum of a function
3. Solving the simultanious equations for $\hat{\beta_0}$ and $\hat{\beta_1}$
   -> To derive the complete forumla for the linear regression $Y = \hat{\beta_1}X + \hat{\beta_0}$

To simplify $\sum_{i = 0}^{n}$ => $\sum$

Formulas for calculating the mean:

$\overline{X} = \frac{\sum Xi}{n}$

$\overline{Y} = \frac{\sum Yi}{n}$

For $\hat{\beta_0}$

$\frac{\partial }{\partial \beta_0} \sum (Yi - \hat{Y})^2 = \sum \frac{\partial }{\partial \beta_0} (Yi - \hat{\beta_0} - \hat{\beta_1}Xi)^2 = -2\sum (Yi - \hat{\beta_0} - \hat{\beta_1}Xi)$

For $\hat{\beta_1}$

$\frac{\partial }{\partial \beta_1} \sum (Yi - \hat{Y})^2 = \sum \frac{\partial }{\partial \beta_1} (Yi - \hat{\beta_0} - \hat{\beta_1}Xi)^2 = -2\sum Xi(Yi - \hat{\beta_0} - \hat{\beta_1}Xi)$

$-2\sum (Yi - \hat{\beta_0} - \hat{\beta_1}Xi = 0$
$-2\sum Xi(Yi - \hat{\beta_0} - \hat{\beta_1}Xi) = 0$

$\sum(Yi - \hat{\beta_0} - \hat{\beta_1}Xi) = 0$
$\sum Xi(Yi - \hat{\beta_0} - \hat{\beta_1}Xi) = 0$

Now derive the formula for $\hat{\beta_0}$ from the first equation to then substitute in the second equation

$\sum Yi - \sum \hat{\beta_0} - \sum \hat{\beta_1}Xi = 0$

$\hat{\beta_0}$ and $\hat{\beta_1}$ are treated as constants

$\sum Yi - \hat{\beta_0}\sum 1 - \hat{\beta_1}\sum Xi = 0$

$\sum Yi - \hat{\beta_0}n - \hat{\beta_1}\sum Xi = 0$

$\sum Yi - \hat{\beta_1}\sum Xi = \hat{\beta_0}n$

$\hat{\beta_0} = \frac{\sum Yi - \hat{\beta_1}\sum Xi}{n}$

$\hat{\beta_0} = \hat{Y} - \hat{\beta_1}\overline{X}$

We take the second equation and substitue $\hat{\beta_0}$

$\sum Xi(Yi - \hat{Y} - \hat{\beta_1}\overline{X} - \hat{\beta_1}Xi) = 0$

$\sum Xi((Yi - \hat{Y}) - \hat{\beta_1}(\overline{X} - Xi)) = 0$

$\sum Xi(Yi - \hat{Y}) - \sum Xi \hat{\beta_1}(\overline{X} - Xi) = 0$

$\sum Xi(Yi - \hat{Y}) - \hat{\beta_1}\sum Xi (\overline{X} - Xi) = 0$

$\sum Xi(Yi - \hat{Y}) = \hat{\beta_1}\sum Xi (\overline{X} - Xi)$

$\hat{\beta_1} = \frac{\sum Xi(Yi - \hat{Y})}{\sum Xi (\overline{X} - Xi)} = \frac{\sum (Xi-\overline{X})(Yi - \overline{Y})}{\sum (\overline{X} - Xi)^2}$

$\hat{\beta_1} = \frac{\sum (Xi-\overline{X})(Yi - \overline{Y})}{\sum (\overline{X} - Xi)^2}$

$\hat{\beta_0} = \hat{Y} - \hat{\beta_1}\overline{X}$

$Y = X\hat{\beta_1} + \hat{\beta_0}$

This is the optimal linear regression formula.