In the previous chapter, we introduced correlation concepts, and it is now necessary to deepen these concepts. This will help us understand how to use this information in advance and find possible relationships between variables. Let's start with a real example: a company at the launch of a new printer model wants to analyze sales at a number of stores to determine the best price. The following table shows the sales of the product in the last month and the sale price for these stores:

In the previous section, we learned that the coefficient of correlation between two quantitative variables X and Y provides information on the existence of a linear relation between the two variables. This index, however, does not allow determining whether it is X that affects Y, if it is Y that affects X, or whether both X and Y are consequences of a phenomenon that affects both of them. Only more knowledge of the problem under study can allow some hypothesis of the dependence of one variable on another.

Calculating correlation and covariance is a useful way to investigate whether there exists a linear relationship between variables, without the need to assume or fit a specific model...

In the previous section, we saw an example of simple linear regression, built the model, and now have a brief description of it. Next, we will explain the results in detail. We will get started by introducing the key concepts, with another simple linear regression example; we will just use data in the form of a spreadsheet containing the number of vehicles registered in Italy and the population of the different regions. Using this data we will try to determine the line that best estimates the relationship between the population and number of registered vehicles. We can do this in various different ways; we will begin with the simplest. Previously, we said that a linear relationship is represented by the following formula:

If we have a set of observations in the form (x₁, y₁), (x₂, y₂), ... (x_n, y_n), for each of these pairs we can write an equation of...

In the previous section, we adopted an algebraic approach to calculating the regression line. More generally, to create a linear regression model, we use the lm() function. This function creates a LinearModel object. The object of class lm has a series of properties that can be immediately viewed by simply clicking on it. These types of objects can be used for residual analysis and regression diagnosis. 

Models for the lm() function are specified symbolically. In fact, the first argument of the function is an object of class formula. A typical formula object has the following form:

response ~ terms

response represents the (numeric) response vector and terms is a series of terms specifying a linear...

So far, we have explored several real cases for which we have searched linear associations, and therefore we have built models of simple linear regression. Next, we tried to analyze the results to confirm the goodness of fit in the simulation of the real system. At this point, it is reasonable to wonder what results of a model perfectly fit a linear system. In this way we will know how to distinguish between a model with a good approximation to what is wrong. In this last case, clearly indicating a nonlinear relationship remains the best solution.

Previously, we said that a simple linear relationship is represented by the following formula:

Here, α and β, represent, respectively, the slope and the intercept with the y axis of the regression line. That being said, we build a dummy system by deciding a priori...

In this chapter, we introduced regression with the simplest algorithm: simple linear regression. We first described a regression problem and where to fit a regressor, and then provided some intuitions underneath the math formulation. Then, we learned how to tune the model for higher performance, and came to deeply understand every parameter of it. In addition, some tricks were described to lower the complexity and scaling of the approach.