# Regression

Regression analysis is a form of predictive modeling technique which investigates the relationship between a dependent (target) and independent variable(s) (predictor). This technique is used for forecasting, time series modeling, and finding the causal effect relationship between the variables.

The simplest form of regression is linear regression, where we predict the dependent variable Y based on a linear combination of one or more independent variables X.

The general equation of a simple (one-variable) linear regression model is:

Y = β0 + β1 X + ε

Here:

Y is the dependent variable that we want to predict or explain.
X is the independent variable that we are using to predict Y.
β0 is the y-intercept of the regression line. It's the predicted value of Y when X=0.
β1 is the slope of the regression line. It represents the predicted change in Y for each one-unit change in X.
ε is the residual error term which accounts for the variation in Y that cannot be explained by the X variables in our model.
For multiple regression, where we have more than one independent variable, the equation extends to:

Y = β0 + β1X1 + β2X2 + ... + βnXn + ε

Here, β1 through βn are the coefficients of the independent variables X1 through Xn in the regression equation.

These coefficients β0, β1, ..., βn are determined using a method called "least squares", which minimizes the sum of the squared residuals ε. This method calculates the best-fitting line through the data points in your dataset.

In practical application, regression analysis provides a way to make quantitative predictions and to understand how a set of predictor variables impact the outcome variable. It's a common and valuable tool in statistics and machine learning.

Example

Let's say we are interested in how the number of study hours affects the final exam score of students. We collect the following data from five students:

StudentStudy Hours (X)Exam Score (Y)
1250
2360
3470
4580
5690

We would like to model the relationship between Study Hours (X) and Exam Score (Y) using a simple linear regression.

Assume from our calculations (which would normally be done using a software package or statistical formulas), we have estimated the following regression equation:

Y = 20 + 10X

Here, β0 (the y-intercept) is 20 and β1 (the slope) is 10. This means that for each additional hour spent studying, we expect a student's exam score to increase by 10 points.

With this model, we can predict the exam score for a student who studies for a certain number of hours. For instance, if a student studies for 7 hours, we predict their score as:

Y = 20 + 10.7 = 90

This basic example illustrates how a simple linear regression model can be applied. In more complex scenarios with multiple predictor variables, the interpretation of the coefficients and prediction becomes more intricate. In these cases, you also have to account for potential interactions and correlation between the predictor variables.