A "hat" (circumflex (ˆ)), placed over a symbol is a mathematical notation with various uses.

Estimated value

In statistics, a circumflex (ˆ), nicknamed a "hat", is used to denote an estimator or an estimated value. For example, in the context of errors and residuals (the statistical errors).

Another example of the hat denoting an estimator occurs in simple linear regression is commonly minimized via least squares for the observed data.

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 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.

For example, in three dimensions,

Unit vector

In mathematics, a unit vector in a normed vector space (pronounced "v-hat"). This is especially common in physics context.

Fourier transform

The Fourier transform.

Operator

In quantum mechanics, operators are denoted with hat notation. For instance, see the time-independent Schrödinger equation, where the Hamiltonian operator.

See also

References