Hat notation

In statistics, a circumflex (ˆ), called a "hat", is used to denote an estimator or an estimated value.{{Cite web |last=Weisstein |first=Eric W. |title=Hat |url=https://mathworld.wolfram.com/Hat.html |access-date=2024-08-29 |website=mathworld.wolfram.com |language=en}} For example, in the context of errors and residuals, the "hat" over the letter $\backslash hat\{\backslash varepsilon\}$ indicates an observable estimate (the residuals) of an unobservable quantity called $\backslash varepsilon$ (the statistical errors).

Another example of the hat denoting an estimator occurs in simple linear regression. Assuming a model of $y\_i\; =\; \backslash beta\_0+\backslash beta\_1\; x\_i+\backslash varepsilon\_i$, with observations of independent variable data $x\_i$ and dependent variable data $y\_i$, the estimated model is of the form $\backslash hat\{y\}\_i\; =\; \backslash hat\{\backslash beta\}\_0+\backslash hat\{\backslash beta\}\_1\; x\_i$ where $\backslash sum\_i\; (y\_i-\backslash hat\{y\}\_i)^2$ is commonly minimized via least squares by finding optimal values of $\backslash hat\{\backslash beta\}\_0$ and $\backslash hat\{\backslash beta\}\_1$ for the observed data.

{{Main|hat matrix}}

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

:$\backslash hat\{\backslash mathbf\{y\}\}\; =\; H\; \backslash mathbf\{y\}.$

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.

:$\backslash mathbf\{a\}\; \backslash times\; \backslash mathbf\{b\}\; =\; \backslash mathbf\{\backslash hat\{a\}\}\; \backslash mathbf\{b\}$

For example, in three dimensions,

:

{{Main|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 $\backslash hat\; \{\backslash mathbf\; \{v\}\; \}$ (pronounced "v-hat").{{Cite book |last=Barrante |first=James R. |url=https://books.google.com/books?id=_IHlCwAAQBAJ&dq=%22Hat%22+math+vectors+-wikipedia&pg=PA124 |title=Applied Mathematics for Physical Chemistry: Third Edition |date=2016-02-10 |publisher=Waveland Press |isbn=978-1-4786-3300-6 |at=Page 124, Footnote 1 |language=en}} This is especially common in physics context.

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

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

$\backslash hat\{H\}\backslash psi\; =\; E\backslash psi$

• {{Annotated link |Exterior algebra}}
• {{Annotated link |Glossary of mathematical symbols}}
• {{Annotated link |Top-hat filter}}
• {{Annotated link |Circumflex}}

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Category:Mathematical notation

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