Hat operator
Encyclopedia
The hat operator is a mathematical notation
with various uses in different branches of science and mathematics.
, one use of the hat
operator is to represent the cross product
operation. Since the cross product is a linear transformation
, it can be represented as a matrix
. The hat operator takes a vector
and transforms it into its equivalent matrix.
For example, in three dimensions,
, the hat is used to denote an estimator
or an estimated value, as opposed to its theoretical counterpart. For example, in the context of errors and residuals
, the "hat" over the letter ε indicates an observable estimate (the residuals) of an unobservable quantity called ε (the statistical errors).
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
with various uses in different branches of science and mathematics.
Hat matrix
In statistics, the hat matrix H projects the observed values y of response variable to the predicted values ŷ:Cross product
In screw theoryScrew theory
Screw theory refers to the algebra and calculus of pairs of vectors, such as forces and moments and angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies....
, one use of the hat
Circumflex
The circumflex is a diacritic used in the written forms of many languages, and is also commonly used in various romanization and transcription schemes. It received its English name from Latin circumflexus —a translation of the Greek περισπωμένη...
operator is to represent the cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
operation. Since the cross product is a linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
, it can be represented as a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. The hat operator takes a vector
Coordinate vector
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....
and transforms it into its equivalent matrix.
For example, in three dimensions,
Estimated Value
In statisticsStatistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the hat is used to denote an estimator
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
or an estimated value, as opposed to its theoretical counterpart. For example, in the context of errors and residuals
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
, the "hat" over the letter ε indicates an observable estimate (the residuals) of an unobservable quantity called ε (the statistical errors).