The next step of our analysis will be towards multiple regression. This basically involves using more than one explanatory variable at once to predict the output of our response variable. If you are wondering about making a distinct linear regression for each explanatory variable, I definitely have to discourage you. How would you compute the final expected value? Would you employ a mean? A simple mean or a weighted mean assigning to each variable a different weight? How would you define this weight?

Those kinds of questions directly lead us to the multivariate linear regression. Couldn't we choose just the most influential variable and fit a simple regression model with that variable? Of course, we could, but how would you define which is the most influential one? Nevertheless, we shouldn't throw away the idea of selecting the most influential variables, since we are going to employ it when talking about dimensional reduction for multivariate linear regression.

But let's place things in order:

• First of all, I will show you a bit of mathematical notation about how to pass from univariate to multivariate analysis
• We will then resonate about the assumption of this extended model
• The last formal step will be talking about dimensional reduction, that is, the family of techniques employed to select the most significant variables, or generally speaking, reducing the number of variables employed

Once all of this is done, we will fit the multiple models, validating assumptions and visualizing results, as you are getting used to doing. 

No more time wasting now, let's start with the mathematical notation, but I promise it will be short.