Hat notation

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 "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 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:

$\widehat{𝐲}=H𝐲.$

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.

$𝐚\times 𝐛=\widehat{𝐚}𝐛$

For example, in three dimensions,

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

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 $\widehat{𝐯}$ (pronounced "v-hat"). This is especially common in physics context.

Fourier transform

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

Operator

In quantum mechanics, operators are denoted with hat notation. For instance, see the time-independent Schrödinger equation, where the Hamiltonian operator is denoted $\hat{H}$.

$\hat{H}\psi =E\psi$

See also

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References

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