8 Fast Python Linear Regression Techniques Compared for Speed and Complexity

The author discusses eight algorithms for performing simple linear regression in Python, focusing on their relative computational complexity rather than accuracy.

GitHub repository: https://github.com/tirthajyoti/PythonMachineLearning/blob/master/Linear_Regression_Methods.ipynb

Linear regression is often the starting point for statistical modeling and predictive analysis; understanding various fitting methods is crucial for data scientists.

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

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

Method 2: stats.linregress()

A highly specialized linear regression function in SciPy’s stats module, limited to two‑variable least‑squares. It is one of the fastest options for simple regression and returns slope, intercept, R² and standard error.

Method 3: optimize.curve_fit()

This function from scipy.optimize performs general curve fitting via least‑squares minimization, allowing any user‑defined function (e.g., mx + c) to be fitted to data, returning fitted parameters and the covariance matrix.

Method 4: numpy.linalg.lstsq

Computes the least‑squares solution of a linear system using matrix factorization. Works for under‑, exactly‑, or over‑determined systems; add a column of ones to the design matrix to estimate the intercept.

Method 5: statsmodels.OLS()

Statsmodels provides a comprehensive OLS implementation with detailed statistical output. Users must manually add a constant term for the intercept. The result includes full regression diagnostics comparable to R or Julia.

Method 6 & 7: Analytic solution via matrix inverse and Moore‑Penrose pseudoinverse

For well‑conditioned problems a closed‑form solution exists: method 6 uses a simple matrix inverse, while method 7 computes the Moore‑Penrose pseudoinverse via SVD, offering robustness on ill‑conditioned data at the cost of speed.

Method 8: sklearn.linear_model.LinearRegression()

Widely used in scikit‑learn; can be extended with cross‑validation and regularization (Lasso, Ridge). The core algorithm is essentially OLS.

Speed and time‑complexity measurement: Experiments on synthetic datasets growing up to ten million samples show that stats.linregress and the simple matrix‑inverse analytic solution are the fastest, even outperforming scikit‑learn’s LinearRegression.

Conclusion: Data scientists should explore multiple linear regression implementations, understand their computational trade‑offs, and select the method that best fits the dataset size and required statistical information.