Linear and Polynomial Regression Calculator

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This interactive web tool lets you perform linear and polynomial regression analysis quickly and visually. Simply enter your datasets (X, Y), and the application will automatically generate a scatter plot. You can instantly calculate the best-fit trend line (linear or polynomial of various degrees), visualize it on the graph, and obtain the precise equation and coefficient of determination (R²).

What is Regression?

Regression (or Regression Analysis) is a powerful statistical tool used to model and analyze the relationship between two or more variables. Its main objective is to find the function that best fits a set of observed data, thus allowing us to predict future values or understand how a change in one variable impacts the others.

In simple terms, if you have a collection of points on a graph (input data X and output Y), regression analysis plots the "best line" or "best curve" that describes that trend.

Common Types

Linear Regression: Looks for the simplest and most direct relationship between the variables (y = mx + c), represented by a straight line.
Polynomial Regression: Uses a curve (a polynomial of degree 2, 3, 4, etc.) to describe more complex and nonlinear relationships between the data.

Brief History
The concept was formalized by the British polymath Sir Francis Galton in the late 19th century. Galton studied the relationship between the heights of parents and children and noted that extreme heights tended to "regress" toward the population mean in the next generation, hence the term "regression." Today, the technique is applied to any type of predictive relationship, regardless of its biological origin.

Key Uses and Applications
Regression analysis is fundamental in almost any field that requires prediction and modeling:

Economics and Finance:
- Predicting stock or commodity prices.
- Estimating the impact of interest rates on consumer spending.
Science and Technology:
- Engineering: Modeling the performance of materials under different temperature or pressure conditions.
- Data Science: Training machine learning models to predict continuous variables (such as temperature or energy consumption).
Health and Biology:
- Determine the necessary medication dosage to achieve a specific effect.
- Predict the progression of a disease based on demographic factors.
Marketing and Business:
- Estimate future sales based on the advertising budget.
- Analyze which factors (price, location, reviews) most influence the demand for a product.

In short, if you need to answer the question: If I do X, what result will I get?, Regression Analysis is the ideal statistical tool.

The linear regression formula is:

Where:

y: is the predicted value (dependent variable).
x: is the input value (independent variable).
m: is the slope of the line.
b: is the y-intercept.

The formula for polynomial regression of degree k:

The optimal solution for the β coefficients is found using the Normal Equation, which minimizes the total squared error:

Where:

β: is the vector of coefficients (β0, β1,..., βk) that we are looking for.
X: is the Design Matrix, which includes the X data raised to the powers x^0, x^1, x^2,..., x^k.
y: is the vector of observed Y values.
X^T: is the transpose of the design matrix.
(X^T•X)^-1: is the inverse of the resulting matrix.