8 Python Linear Regression Techniques Compared for Speed and Complexity

Linear regression is often the starting point for statistical modeling and predictive analysis in data science, yet its implementation details can greatly affect performance on large datasets.

Python has become the language of choice for many data scientists, making it important to understand the variety of linear‑model fitting methods available and how to evaluate their computational complexity.

Method 1: Scipy.polyfit() or numpy.polyfit()

This general least‑squares polynomial fitting function works for any degree; for simple linear regression set the degree to 1. It returns an array of regression coefficients that minimize variance.

Method 2: stats.linregress()

A highly specialized function from SciPy’s statistics module that fits a simple two‑variable least‑squares regression. It returns the slope, intercept, R², and standard error, making it one of the fastest options for basic linear regression.

Method 3: optimize.curve_fit()

From SciPy’s optimize module, this function performs least‑squares minimization to fit any user‑defined function to data, making it suitable for both simple linear and multivariate regression.

Method 4: numpy.linalg.lstsq

This matrix‑factorization approach solves the least‑squares problem by minimizing the Euclidean 2‑norm of the residuals, handling under‑, exactly‑, or over‑determined systems.

Method 5: Statsmodels.OLS()

Statsmodels provides a comprehensive suite for statistical modeling; its OLS function yields full regression statistics and requires manually adding a constant column for the intercept.

Method 6 & 7: Analytic solutions using matrix inversion

For well‑conditioned problems, the closed‑form solution (XᵀX)⁻¹Xᵀy can be computed directly (Method 6) or via the Moore‑Penrose pseudoinverse with SVD (Method 7), the latter being slower but more robust on ill‑conditioned data.

Method 8: sklearn.linear_model.LinearRegression()

Widely used by machine‑learning engineers, this scikit‑learn estimator can be extended with cross‑validation and regularization techniques such as Lasso or Ridge, though the core algorithm remains the same.

Speed and time‑complexity measurement

Experiments on synthetic datasets growing up to ten million samples show that the simple matrix‑inversion analytic solution and stats.linregress are the fastest, outperforming the commonly used scikit‑learn LinearRegression.

Conclusion

Data scientists should be aware of multiple linear‑regression implementations, as different methods offer trade‑offs in speed, flexibility, and statistical output. Most of the discussed techniques can also be extended to multivariate or polynomial regression, and their performance differences become significant on large‑scale data.