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Estimation method that fits a model by minimizing the sum of squared residuals between predictions and observed values.
DESCRIPTION
Least Squares underpins linear regression and many fitting problems by optimizing squared error. It has closed-form solutions for OLS and extends to weighted and generalized forms. Squared loss emphasizes large errors, making the method sensitive to outliers. Regularized variants like ridge and lasso address overfitting and multicollinearity. Proper diagnostics include residual plots, assumptions checks, and goodness-of-fit metrics.
ALTERNATIVE DEFINITIONS
Wikipedia
Least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals.
Defines least squares broadly for overdetermined systems.
Encyclopaedia Britannica
Least squares method, in statistics, a technique for estimating unknown quantities by minimizing the sum of the squares of the deviations between observed and estimated values.
Emphasizes estimation of unknown quantities via squared deviations.
Wolfram MathWorld
The method of least squares is a procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets.
Highlights curve fitting perspective and offset minimization.
NIST
Least squares fitting is a method of determining the best fit to a set of data points by minimizing the sum of the squares of the residuals.
Government handbook emphasizing practical fitting approach.
RELATED TERMS
Linear Model Linear Regression Residuals Maximum Likelihood L1 Loss Ridge Regression Variance Systematic Error RMSE
COMPETING TERMS
Least Absolute Deviations Robust Regression
KNOWLEDGE AREAS
Mathematics Data Science Data Modeling
TARGET AUDIENCE
Data Scientist Data Analyst Data Modeler AI Engineer
EXTERNAL SOURCES
- Wikipedia - Least squares
- NIST - Least squares in NIST handbook
- StatsModels - OLS implementation and examples
- scikit-learn - OLS in scikit-learn
TOP QUESTIONS
Why is Least Squares widely used?
It’s computationally efficient, has closed-form solutions in OLS, and aligns with Gaussian noise assumptions for maximum likelihood estimation.
Category: Relevancy
What are assumptions behind Least Squares?
Linearity, independence, homoscedasticity, and normality of errors underpin classical inference in OLS.
Category: Definition
How do outliers affect Least Squares?
Squared errors overweight large deviations, making Least Squares sensitive to outliers; robust methods mitigate this.
Category: Risks
What is the difference between OLS and WLS?
OLS treats all residuals equally; WLS assigns weights to account for heteroscedasticity or varying reliability.
Category: Definition
When should you use Ridge instead of Least Squares?
In multicollinearity or overfitting scenarios, Ridge regularization stabilizes estimates at the cost of introducing bias.
Category: Alternatives
How to evaluate a Least Squares model?
Use residual diagnostics, R²/adjusted R², RMSE, and cross-validation for predictive performance.
Category: HowTo
What are common mistakes with Least Squares?
Ignoring assumptions, not scaling features, and using inappropriate model forms can mislead inference and predictions.
Category: Mistakes
How does Least Squares relate to maximum likelihood?
Under Gaussian noise, minimizing squared error is equivalent to maximizing the likelihood of the data.
Category: Definition
Is Least Squares suitable for large-scale problems?
Yes, with iterative solvers and distributed linear algebra, but careful numerical stability and conditioning are important.
Category: Opinion