• What is Regression Anaylsis?

Regression analysis is a way to understand the relationship between two or more things. It helps us determine how one data point can predict or explain another data point.

As an example, if you have a bunch of data showing the number of people who text while driving and the number of car accidents they’ve gotten in, regression analysis can help find a line that best fits those points. The line might predict that the more you text while driving, the more likely you are to get into an accident in your car.

• How is regression analysis useful in computer science?

Computer scientists use regression analysis for all sorts of things.

One such example is performance optimization. By analyzing how different variables affect the speed or efficiency of an algorithm or software, a computer scientist can make improvements.

Regression analysis can also be used in pattern recognition, by finding relationships between features in data and patterns we want to recognize.

It can also be used in data analysis to understand relationships between different data variables.

Regression models involve the following components:

• the unknown parameters \(\beta\)
• the independent variables \(X_{i}\)
• the dependent variable \(Y_{i}\)
• the error terms \(e_{i}\)

Most models propose that \(Y_{i}\) is a function of \(X_{i}\) and \(\beta\) with \(e_{i}\) representing an additive error term. That would then look like:

\(Y_{i} = f( X_{i}, \beta ) + e_{i}\)

A very simple form of regression is linear regression, which can be expressed as:

\(Y = a X + b\)

Where \(Y\) is the dependent variable we want to predict, \(X\) is the independent variable we use to make predictions, \(a\) is the slope of the line, representing how much \(Y\) changes for a one unit change in \(X\), and \(b\) is the intercept, representing the value of \(Y\) when \(X\) is 0.

\(a\) and \(b\) are the unknown parameters in this example, and the analysis is about finding values for these parameters that best describe the relationship between X and Y in the given data.

## `geom_smooth()` using formula = 'y ~ x'

The regression line can tell you various things about the relationship of the independent variable X and the dependent variable Y.

• The slope of the line indicates the direction of the relationship. a positive slope suggests a positive relationship, meaning as X increases, Y tends to increase. A negative slope suggests a negative relationship.

• The steepness of the slope indicates the strength of the relationship. The steeper the slope, the stronger the relationship, as a steep slope indicates a small change in X corresponds to a large change in Y.

You can use the interactive plot to look more closely at the values for each of the data points and for the regression line. See if you can answer these questions:

• Is there a relationship between car speed and stopping distance?

• What is the direction of the relationship?

• How strong is the relationship?

• What is the intercept of the regression line?

• How well does the regression line fit the data?

ggplot(swiss, aes(x=Infant.Mortality, y = Fertility))+
  geom_point()+
  geom_smooth(method = "lm") +
  theme(aspect.ratio = .9) +
  labs(x = "Infant Mortality", y = "Fertility") +
  ggtitle("Linear Regression: Infant Mortality vs Fertility")

## `geom_smooth()` using formula = 'y ~ x'

Regression analysis is a powerful tool in a broader field of statistical analysis, used to analyze the relationships between variables and make predictions.

It can be used to determine the direction and strength of a relationship between variables. It can be used to make predictions about the dependent variable based on the values of the independent variables.

Overall, while this is a powerful tool, it’s important to also remember to keep context in mind when analyzing data and also to be aware of bias. Interpret regression results carefully and remember that its use depends on the questions and data at hand.