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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.^[1] For example, in the context of errors and residuals, the "hat" over the letter ${\displaystyle {\hat {\varepsilon }}}$ indicates an observable estimate (the residuals) of an unobservable quantity called ${\displaystyle \varepsilon }$ (the statistical errors).

Another example of the hat denoting an estimator occurs in simple linear regression. Assuming a model of ${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i}}$, with observations of independent variable data ${\displaystyle x_{i}}$ and dependent variable data ${\displaystyle y_{i}}$, the estimated model is of the form ${\displaystyle {\hat {y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}x_{i}}$ where ${\displaystyle \sum _{i}(y_{i}-{\hat {y}}_{i})^{2}}$ is commonly minimized via least squares by finding optimal values of ${\displaystyle {\hat {\beta }}_{0}}$ and ${\displaystyle {\hat {\beta }}_{1}}$ 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:

${\displaystyle {\hat {\mathbf {y} }}=H\mathbf {y} .}$

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.

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

For example, in three dimensions,

${\displaystyle \mathbf {a} \times \mathbf {b} ={\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}}\times {\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}}={\begin{bmatrix}0&-a_{z}&a_{y}\\a_{z}&0&-a_{x}\\-a_{y}&a_{x}&0\end{bmatrix}}{\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}}=\mathbf {\hat {a}} \mathbf {b} .}$

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 ${\displaystyle {\hat {\mathbf {v} }}}$ (pronounced "v-hat").^[2][1] This is especially common in physics context.

Fourier transform

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

Operator

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

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

See also

• Exterior algebra – Algebra associated to any vector space
• Glossary of mathematical symbols
• Top-hat filter – Signal filtering technique
• Circumflex – Diacritic (◌̂) in European scripts

References

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