Linear Regression: Modeling Relationships Between Variables
If you've ever looked at data and wondered if there's a connection between two things - like weather and ice cream sales or studying and grades - then you're on your way to understanding linear regression. Linear regression is a way to model the relationship between two variables, like temperature and ice cream sales or study time and grades. It helps you see what happens to one variable when the other changes.
Least Squares: Finding the Best Fit
One way to fit a line to the data is to use the method of least squares. In this method, the line is fitted to minimize the distance between the line's predicted values and the actual observed values. The values of the line, or coefficients, are adjusted until the error, or difference between the predicted and observed values, is minimized. By minimizing the differences between the predicted and observed values of the dependent variable, the line estimated by the method of least squares is said to "fit" the data better than any other straight line.
Here is an example of how the least squares method works. Say we measured a person's height and their weight. If we plot the data, we might see that taller people tend to weigh more. We can fit a line to the data that best represents this relationship. By doing so, we will be able to predict the weight of a tall person or the height of a heavy person.
The equation for the line would be:
Weight = Height x Slope + Intercept
We use least squares to find the line that minimizes the sum of the squared differences between each point and the line. This gives us the values for the slope and y-intercept that best describe the relationship between weight and height. With these values, we can now use the line to estimate the weight of an individual given their height or the height of an individual given their weight.
Generalized Linear Models: A Probabilistic Perspective
Another way to approach linear regression is through the use of generalized linear models. This method views the relationship between variables as a probabilistic one. The goal is to estimate the parameters of the probability distribution that best describes the data. In the case of linear regression, the probability distribution is a Gaussian (or normal) distribution.
The parameters of the Gaussian distribution include the mean and standard deviation. The mean represents the value that is most likely to occur, and the standard deviation represents how spread out the data is around the mean. The goal of linear regression is to estimate the mean value of the dependent variable given the independent variable(s).
The process of estimating the parameters of the probability distribution is called maximum likelihood estimation. The idea is to choose the values for the parameters that make the observed data most likely to occur. Once this is done, we can use the estimated parameters to make predictions about new data.
Linear Regression in Practice
Linear regression is used in many different fields. For example, in finance, the technique can be used to model relationships between stock prices and interest rates. In healthcare, it can be used to understand the relationship between diet and the development of certain diseases. In education, it can be used to understand the relationship between study habits and academic performance.
Regardless of the field of application, linear regression is a powerful tool that can provide insight into the relationships between variables. Whether you are trying to predict the price of a stock or estimate the height of a child based on their parents' height, linear regression provides a useful framework for analyzing data and making predictions.
