Hat notation

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

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

Another example of the hat operator denoting an estimator occurs in simple linear regression. Assuming a model of $$y_i = \beta_0+\beta_1 x_i+\varepsilon_i$$, 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:
 * $$\hat{\mathbf{y}} = H \mathbf{y}.$$

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.


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

For example, in three dimensions,


 * $$\mathbf{a} \times \mathbf{b} = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} \times \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} = \begin{bmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{bmatrix} \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} = \mathbf{\hat{a}} \mathbf{b} $$

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

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