Homocedasticity of the error : Error has the same variance, is additive, and does not depend of any specific value

Independence of errors

Non-linearity of X and Y in the considered interval

Error proportional to the value of the dependent variable

Conditional dependence of errors

The values are random variables

The values are not random variables. They are constants

Linearity of X and Y in the considered interval

Is a process to be avoided because it can only generate errors and difficults the understanding of problems

May help to the fulfilment of some premisses that may not be fulfilled without the application of logarithms. Some examples are linearity between X and Y, and normality and/or homocedasticity of errorsors

We define the loss function (error between current predicted value and reality):

Replacing the predicted value with the formula of the estimation  we obtain

Now we define a Learning rate . It should be a small number, e.g. 0,01.

We start with two values for and . Suppose and .

At each step:

We calculate the partial derivative of the loss function with respect to , and replace the current values of to obtain the derivative value :

Similarly, we calculate the partial derivative with respect to , :

Now we update the current value of and using the equations

We repeat this process until the loss function is a very small value, or ideally 0. The final values of and are the optimum values.

Solving a minimization problem

from which we found the estimated values