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Hat notation

From Wikipedia, the free encyclopedia

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

Estimated value

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

Another example of the hat operator 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.

Hat matrix

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

Cross product

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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,

Unit vector

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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.

Fourier transform

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The Fourier transform of a function is traditionally denoted by .

Operator

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

See also

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References

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  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.