Hat notation

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

In statistics, a circumflex (ˆ), called 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 ${\displaystyle {\hat {\varepsilon }}}${\displaystyle {\hat {\varepsilon }}} indicates an observable estimate (the residuals) of an unobservable quantity called ${\displaystyle \varepsilon }${\displaystyle \varepsilon } (the statistical errors).

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

In statistics, the hat matrix H projects the observed values y of response variable to the predicted values ŷ:

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

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.

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

For example, in three dimensions,

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

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

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

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}}}.

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

• Exterior algebra – Algebra associated to any vector space
• Glossary of mathematical symbols
• Top-hat filter – signal filtering techniquePages displaying wikidata descriptions as a fallback
• Circumflex – Diacritic (^) in European scripts

• Mathematical notation
• Algebra stubs

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