1. What is Least Squares Error?

The least squares error measures the discrepancy between observed values (y) and values predicted by a linear model (Xβ). It's the sum of squared differences between actual and predicted values, used in linear regression to find the best-fitting line.

2. How Does the Calculator Work?

The calculator uses the least squares error formula:

Where:

• \( y \) — Vector of observed values
• \( X \) — Design matrix of predictor variables
• \( \beta \) — Vector of coefficients
• The operation \( X \beta \) represents matrix multiplication

Explanation: The error is calculated by first computing the predicted values (Xβ), then summing the squared differences between these predictions and the actual values (y).

3. Importance of Least Squares Error

Details: Least squares error is fundamental in regression analysis, providing a measure of model fit. Minimizing this error yields the optimal coefficients for linear models.

4. Using the Calculator

Tips: Enter y as comma-separated values, X as semicolon-separated rows with comma-separated values, and β as comma-separated values. Dimensions must match (X rows = y length, X columns = β length).

5. Frequently Asked Questions (FAQ)

Q1: What units does the error have?
A: The error is in squared units of the response variable (y).

Q2: How is this related to R-squared?
A: R-squared is 1 minus (error/total variation), representing proportion of variance explained.

Q3: What if my dimensions don't match?
A: The calculator will show an error. Ensure X has same rows as y length and same columns as β length.

Q4: Can I use this for multiple regression?
A: Yes, the calculator handles multiple predictors through the X matrix.

Q5: What's the difference between error and residual?
A: Residuals are individual differences (y - Xβ), while error is the sum of squared residuals.