Hat operator

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

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

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,

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

In statistics, the 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 ε indicates an observable estimate (the residuals) of an unobservable quantity called ε (the statistical errors). In simple linear regression with observations of independent variable data ${\displaystyle x_{i}}$ and dependent variable data ${\displaystyle y_{i}}$, and assuming a model of ${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i}}$, can lead to an estimated model 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 minimized via least squares by finding optimal values of ${\displaystyle {\hat {\beta }}_{0}}$ and ${\displaystyle {\hat {\beta }}_{1}}$ for the observed data.

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

In Linear Algebra, when examining predicted data on a line, then you must use the hat operator, too. The following is an example of a linear function:

y = mx + b

If you are examining predicted data, then the function must be turned into the following:

ŷ = mx + b

This rule is just to represent that this is not a real and written rule. It is just predicted. One example using this is when you are finding out how clean people's teeth are based on how much chocolate they eat per day. The function which you come up with may not be written. It can be like the following (this function is an example):

y = -1/2c + 8

This function is assuming that c represents the number of chocolates eaten and the teeth cleanliness is measured between 1 and 10. Since this rule may not be real and actually true, we must denote that with the following by adding a hat operator to the y:

ŷ = -1/2c + 8

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