Hat notation explained

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}} ^{2} ^{i)} is commonly minimized via least squares by finding optimal values of \hat{\beta} _{0} and \hat{\beta} _{1} for the observed data.

Hat matrix

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

\hat{ y } = H \mathbf.

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 x b = \hat{a } \mathbf

For example, in three dimensions,

a x b = \begin{bmatrix} a _{x} \ a _{y} \ a _{z} \end{bmatrix} x \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} = \hat{a } \mathbf

Unit vector

See main article: 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 { v } (pronounced "v-hat").^[1]

Fourier transform

The Fourier transform of a function

f is traditionally denoted by \hat{f} .

Notes and References

1. Book: Barrante, James R. . Applied Mathematics for Physical Chemistry: Third Edition . 2016-02-10 . Waveland Press . 978-1-4786-3300-6 . Page 124, Footnote 1 . en.