Why linear regression choose to minimize the residual on y-axis but not the absolute error between prediction and measurement?

Question

Does anyone know why linear regression choose to minimize only $(y_{predict} - y_{measured})^2$, but not $(y_{predict} - y_{measured})^2 + (x_{predict} - x_{measured})^2 $?

Answer 1

The linear regression belongs to the supervised learning methods: the methods that try to generate predictions of a variable (Y) from values of one or more variables(the Xs). X is consequently not predicted, but only observed, as a regressor. Y is the only variable that will be predicted, so you have to minimize the RSS that you wrote.