Hat notation

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 indicates an observable estimate (the residuals) of an unobservable quantity called (the statistical errors).

Another example of the hat denoting an estimator occurs in simple linear regression. Assuming a model of , with observations of independent variable data and dependent variable data , the estimated model is of the form where is commonly minimized via least squares by finding optimal values of and for the observed data.

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

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.

For example, in three dimensions,

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 (pronounced "v-hat").^[2][1] This is especially common in physics context.

The Fourier transform of a function is traditionally denoted by .

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

{{mathimg|d5b9995521be4f5e023d9cec5087f07f99fb4516|width=9.964ex|height=3.176ex|va=-0.671ex|alt={\displaystyle {\hat {H\psi =E\psi }}}

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

1. ^ a b Weisstein, Eric W. "Hat". mathworld.wolfram.com. Retrieved 2024-08-29.
2. ^ Barrante, James R. (2016-02-10). Applied Mathematics for Physical Chemistry: Third Edition. Waveland Press. Page 124, Footnote 1. ISBN 978-1-4786-3300-6.