xhat

xhat, written as x̂, is a notation used in several mathematical and scientific fields to indicate a value associated with estimation, projection, or direction. The symbol appears across disciplines and its precise meaning depends on context. In general, the hat over a variable signals that the quantity is derived, predicted, or normalized relative to a standard form.

In statistics and data analysis, x̂ commonly denotes an estimator of a parameter x or the estimated value of a variable based on observed data. It is the predicted or inferred quantity obtained from a model, data sample, or algorithm, as opposed to the true but unknown x.

In signal processing and state estimation, x̂_k often denotes the estimated state at time k, produced by filters such as the Kalman filter or Bayesian estimators. The notation emphasizes that the value is an estimate rather than a measured quantity.

In physics, x̂ is widely used to represent the position operator in quantum mechanics, acting on wavefunctions in the position basis. The same notation can appear in classical mechanics as a direction indicator or unit vector in the x direction, depending on the convention used.

In vector notation, x̂ is occasionally used to denote a unit vector in the x direction or a normalized version of a vector x. Other common conventions use bold letters or explicit unit vectors, so interpretation depends on the surrounding context.

In machine learning and regression, X̂ is used to denote predicted targets or outputs produced by a trained model, such as X̂ = f(data).