Numerical Methods

Data analytics starts with numerical foundations. Before fitting models or interpreting results, you need fluency with the computational tools and mathematical structures that underlie every machine learning algorithm.

This module builds those foundations using Python. We begin with the core scientific Python stack — NumPy, SciPy, Matplotlib — and work up through linear algebra, least-squares regression, and numerical optimization. Chemical engineering examples (Arrhenius kinetics, spectroscopic data, reaction rate fitting) illustrate each concept throughout.

Topics

• Topic 1.1 — Python Basics: NumPy arrays, vectorized operations, and the scientific Python ecosystem. Covers array creation, slicing, broadcasting, and common pitfalls (copy vs. view semantics).

• Topic 1.2 — Linear Algebra: Matrix operations, solving linear systems, orthogonalization (Gram–Schmidt), rank, and condition numbers. Foundational for understanding regression and dimensionality reduction.

• Topic 1.3 — Linear Regression: Least-squares fitting as a linear algebra problem. Normal equations, polynomial features, residual analysis, and scikit-learn’s LinearRegression interface.

• Topic 1.4 — Numerical Optimization: Gradient descent, quasi-Newton methods (L-BFGS-B), and constrained optimization via scipy.optimize. Explores loss surfaces, convergence behavior, and regularization penalties.

• Topic 1.5 — Nonlinear Parameter Estimation: Least squares as maximum likelihood, Hessian-based confidence intervals via autograd, parameter correlations and confidence ellipses, and sloppy (unidentifiable) parameters when model complexity outruns the data.