Discretize

Discretize is to convert a continuous quantity, problem, or model into a discrete form that can be represented or computed with a finite set of values or grid points. In mathematics and numerical analysis, discretization replaces continuous variables by a finite collection of points, enabling computer simulation. For differential equations, time discretization introduces a grid in time and approximates derivatives by finite differences, while spatial discretization replaces a continuous domain by a mesh and uses methods such as finite difference, finite element, or finite volume to obtain a system of algebraic equations. The quality of the discretization is evaluated in terms of consistency, stability, and convergence as the grid is refined.

In data science and statistics, discretization (or binning) converts continuous features or responses into discrete categories. Methods include equal-width binning, equal-frequency binning, k-means-based discretization, and entropy-based approaches. Discretization can simplify models, improve robustness to noise, or satisfy algorithmic requirements, but it also incurs information loss and potential bias.

In signal processing, sampling is a form of time discretization, governed by the Nyquist criterion to avoid aliasing. Discretization error is a key concern across applications and is analyzed to ensure that discrete solutions approximate the continuous ones as resolution increases. The term is often used interchangeably with quantization in informal settings, though discretization typically refers to domain breadth (time, space, or features) rather than amplitude quantization.