Regression analysis

We can look for patterns in things.

We see how one thing changes with another. This helps us guess what might happen next. It is like finding a path through dots.

People use this to make guesses. They can guess about the future. This is called prediction. It helps us learn about our world.

Long ago, people used tools to find these paths. Some tools took a whole day to work! Now, we use computers to do it fast.

We use math to see how things connect.

Imagine you have a bunch of dots on a page. These dots show how two things connect. Maybe they show how much a person weighs and how tall they are.

Regression analysis is a way to find the best path through those dots. We call this path a regression line. This line helps us see a pattern. It shows how one thing changes when another thing changes.

People use these lines for two main jobs. First, they use them for prediction. This means they guess what might happen next. Second, they use them to see if one thing causes another.

This math has a long history. Isaac Newton used early versions of it in the year 1700. Later, math experts named Legendre and Gauss used it to study stars. They wanted to find the paths of comets in space.

In the past, this work was very slow. In the 1950s, people used big desk calculators. One single math job could take 24 hours!

Today, we use computers and small calculators to do it fast.

Imagine you have many dots on a graph. Each dot represents a piece of information, like a person's height and weight.

These dots might look messy at first. Regression analysis is a way to find a clear pattern among them. It helps us see how one thing changes when another thing changes. We call the thing we want to understand the dependent variable. The thing we use to explain it is the independent variable.

By finding a mathematical path through the dots, we can see the relationship between them.

One common way to do this is called linear regression. This method tries to find a straight line that fits the data best. A popular way to do this is called ordinary least squares. This method looks at the distance between each dot and the line. It tries to make the total sum of these squared distances as small as possible. This creates a unique line that represents the average path.

This line helps us make smart guesses about new data. We can use it to predict what might happen in the future. It can also help us see if one thing might cause another.

This math has a long and interesting history. Isaac Newton used an early version of this in 1700. He used it while studying the equinoxes. Later, math experts used these ideas to look at the stars. Adrien-Marie Legendre published a method for it in 1805. Around the same time, Carl Friedrich Gauss used it in 1809. They both wanted to find the orbits of comets and planets.

Gauss even shared more developments of this theory in 1821.

Did you know the word "regression" comes from biology? In the 19th century, Francis Galton studied the heights of families. He noticed that children of very tall parents tended to be closer to the average height. This movement back toward the middle is called regression toward the mean. Later, other scientists like Udny Yule and Karl Pearson applied this to math. They assumed the data followed a shape called a Gaussian distribution.

This helped make the math work for many different types of problems.

In the past, doing this math was a very slow job. During the 1950s and 1960s, economists used large desk calculators. Before 1970, one single calculation could take 24 hours to finish!

Today, things are much faster and easier. We use computers and small handheld calculators to find these patterns instantly. We can even use regression to study complex things like images or graphs. It remains a very important tool for scientists and researchers everywhere.

Regression analysis is a powerful statistical method used to estimate relationships between variables. It helps researchers understand how a dependent variable changes when one or more independent variables change. The dependent variable is often called the outcome, response variable, or label. The independent variables are known as regressors, predictors, covariates, or features. By finding a mathematical function that best fits the observed data, scientists can model complex real-world processes.

To perform a regression, researchers must first choose a specific mathematical model. This model usually describes the dependent variable as a function of the independent variables plus an error term. The error term, or residual, represents random noise or other factors not included in the model. One of the most common methods is ordinary least squares. This technique calculates a unique line or hyperplane that minimizes the sum of the squared differences between the actual data points and the fitted line.

This process allows researchers to estimate the conditional expectation, which is the population average value for a given set of inputs.

There are several different types of regression depending on the goal of the study. Linear regression is the most frequent form, where the relationship is modeled as a straight line or a linear combination of variables. Other methods include quantile regression, which estimates different location parameters. Nonparametric regression is used when the relationship is non-linear and does not follow a specific shape. Researchers might also use robust regression to handle outliers or errors-in-variables models when predictors contain measurement errors.

The history of regression is filled with famous mathematicians and astronomers. Isaac Newton used an embryonic form of linear regression in 1700 while studying equinoxes. He performed data averaging and forced his regression line to pass through the average point by summing residuals to zero. Later, Adrien-Marie Legendre published the method of least squares in 1805. Carl Friedrich Gauss also published the method in 1809 to determine the orbits of comets and planets.

Gauss later expanded this theory in 1821 with the Gauss-Markov theorem.

The term "regression" actually comes from the field of biology. In the 19th century, Francis Galton studied the heights of human descendants. He observed that the children of very tall ancestors tended to move back toward the normal average. This biological phenomenon is known as regression toward the mean. Later, mathematicians Udny Yule and Karl Pearson extended this concept into a general statistical context. They assumed the variables followed a Gaussian distribution, a concept later refined by R.A. Fisher in the 1920s.

Regression analysis serves two primary purposes: prediction and causal inference. In prediction and forecasting, the method is used to guess future outcomes, which overlaps heavily with machine learning. In causal inference, researchers use it to try to understand if one variable actually causes a change in another. However, regression alone only shows relationships within a fixed dataset. To claim causality, a researcher must carefully justify why the relationship would hold in a new context.

In the mid-20th century, performing these calculations was a massive undertaking. During the 1950s and 1960s, economists used electromechanical desk calculators for their work. Before 1970, it could take up to 24 hours to receive the result from just one regression. Today, the process is nearly instant thanks to modern computers and handheld scientific calculators. Modern research now applies regression to complex data objects like curves, images, and graphs.