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 $\hat{\epsilon }$ indicates an observable estimate (the residuals) of an unobservable quantity called $\epsilon$ (the statistical errors).

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

$\widehat{\mathbf{𝐲}}=H\mathbf{𝐲}.$

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.

$\mathbf{𝐚}\times \mathbf{𝐛}=\hat{\mathbf{a}}\mathbf{𝐛}$

For example, in three dimensions,

$\mathbf{𝐚}\times \mathbf{𝐛}=\left[\begin{array}{c}{a}_x\\ a_y\\ a_z\end{array}\right]\times \left[\begin{array}{c}{b}_x\\ b_y\\ b_z\end{array}\right]=\left[\begin{array}{ccc}0& -a_z& a_y\\ a_z& 0& -a_x\\ -a_y& a_x& 0\end{array}\right]\left[\begin{array}{c}{b}_x\\ b_y\\ b_z\end{array}\right]=\hat{\mathbf{a}}\mathbf{𝐛}$

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 $\widehat{\mathbf{𝐯}}$ (pronounced "v-hat").

Fourier transform

The Fourier transform of a function $f$ is traditionally denoted by $\hat{f}$.

See also

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