A "hat" (circumflex (ˆ)), placed over a symbol is a mathematical notation with various uses.
In statistics, a circumflex (ˆ), called a "hat", is used to denote an estimator or an estimated value.
[1] For example, in the context of errors and residuals, the "hat" over the letter
indicates an observable estimate (the residuals) of an unobservable quantity called
Another example of the hat operator denoting an estimator occurs in simple linear regression.
, with observations of independent variable data
and dependent variable data
, the estimated model is of the form
is commonly minimized via least squares by finding optimal values of
In statistics, the hat matrix H projects the observed values y of response variable to the predicted values ŷ: In screw theory, 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.
A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in
[2][1] This is especially common in physics context.
The Fourier transform of a function
In quantum mechanics, operators are denoted with hat notation.
For instance, see the time-independent Schrödinger equation, where the Hamiltonian operator is denoted
This algebra-related article is a stub.