Hat operator

Short description: Mathematical notation

The hat operator is a mathematical notation with various uses in different branches of science and mathematics.

Estimated value

In statistics, a circumflex (ˆ), called a "hat", is used to denote an estimator or an estimated value. For example, in the context of errors and residuals, the "hat" over the letter [math]\displaystyle{ \hat{\varepsilon} }[/math] indicates an observable estimate (the residuals) of an unobservable quantity called [math]\displaystyle{ \varepsilon }[/math] (the statistical errors).

Another example of the hat operator denoting an estimator occurs in simple linear regression. Assuming a model of [math]\displaystyle{ y_i = \beta_0+\beta_1 x_i+\varepsilon_i }[/math], with observations of independent variable data [math]\displaystyle{ x_i }[/math] and dependent variable data [math]\displaystyle{ y_i }[/math], the estimated model is of the form [math]\displaystyle{ \hat{y}_i = \hat{\beta}_0+\hat{\beta}_1 x_i }[/math] where [math]\displaystyle{ \sum_i (y_i-\hat{y}_i)^2 }[/math] is commonly minimized via least squares by finding optimal values of [math]\displaystyle{ \hat{\beta}_0 }[/math] and [math]\displaystyle{ \hat{\beta}_1 }[/math] 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:

[math]\displaystyle{ \hat{\mathbf{y}} = H \mathbf{y}. }[/math]

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.

[math]\displaystyle{ \mathbf{a} \times \mathbf{b} = \mathbf{\hat{a}} \mathbf{b} }[/math]

For example, in three dimensions,

[math]\displaystyle{ \mathbf{a} \times \mathbf{b} = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} \times \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} = \begin{bmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{bmatrix} \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} = \mathbf{\hat{a}} \mathbf{b} }[/math]

Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in [math]\displaystyle{ \hat {\mathbf {v} } }[/math] (pronounced "v-hat").

Fourier transform

The Fourier transform of a function [math]\displaystyle{ f }[/math] is traditionally denoted by [math]\displaystyle{ \hat{f} }[/math].

See also

• Exterior algebra
• Top-hat filter
• Circumflex, noting that precomposed glyphs [letter-with-circumflex] do not exist for all letters.