1. What is the Least Squares Method?

The least squares method is a statistical procedure to find the best-fitting curve to a given set of points by minimizing the sum of the squares of the residuals (differences between observed and estimated values). In linear algebra, it's used to solve overdetermined systems of linear equations.

2. How Does the Calculator Work?

The calculator uses the matrix formula:

Where:

• \( \beta \) — Vector of coefficients
• \( X \) — Design matrix (independent variables)
• \( X^T \) — Transpose of matrix X
• \( y \) — Vector of observed values (dependent variable)

Explanation: The formula calculates the coefficients that minimize the sum of squared differences between the observed and predicted values.

3. Importance of Least Squares

Details: Least squares is fundamental in regression analysis, machine learning, and statistics. It provides the best linear unbiased estimator (BLUE) under the Gauss-Markov theorem assumptions.

4. Using the Calculator

Tips: Enter matrix X with rows separated by semicolons and columns separated by commas. Enter vector y as comma-separated values. The number of rows in X must match the length of y.

5. Frequently Asked Questions (FAQ)

Q1: When should I use least squares?
A: Use it when you have more equations than unknowns (overdetermined system) and want the best approximate solution.

Q2: What are the assumptions of least squares?
A: Key assumptions include linearity, independence, homoscedasticity (constant variance), and normality of residuals.

Q3: What if X^T X is not invertible?
A: This indicates multicollinearity. You might need regularization (ridge regression) or to remove redundant variables.

Q4: How accurate is this calculator?
A: It provides exact algebraic solution, but numerical precision depends on matrix conditioning and implementation.

Q5: Can I use this for nonlinear relationships?
A: Not directly, but you can sometimes linearize nonlinear relationships through transformations.