Weighted

Weighted describes a quantity to which weights are assigned to reflect relative importance, frequency, or likelihood. Weights are nonnegative numbers, often normalized so that their sum equals one, but normalization is not always required. In mathematics and data analysis, weights allow combining values, costs, or probabilities in a way that emphasizes certain components.

In statistics and data analysis, the weighted average of values x_i with weights w_i is the sum of w_i x_i divided by the sum of w_i. Weights can reflect measurement reliability, time duration, or frequency of occurrence. For example, combining grades with more credit hours yields a weighted mean.

In graph theory and networks, a weighted graph assigns a weight to each edge (or sometimes to vertices) to represent distance, capacity, or cost. The length of a path is the sum of its edge weights, and algorithms such as Dijkstra's use weights to find shortest paths.

In voting and decision making, weights can assign different influence to participants or options, as in weighted voting systems or juries. In statistics and machine learning, weights adjust for heterogeneity, measurement error, or class imbalance, and appear in weighted least squares and loss functions.

Weighting also appears in time series and signal processing as weighted averages or moving averages, where recent observations may be given more weight. Weighted sampling and importance weighting are used in estimation and simulation to emphasize more informative observations.

In practice, weights should be nonnegative and properly interpreted; improper weighting can bias results. Weights may be normalized, rescaled, or applied as a function of features to model varying influence.