The hat operator is a mathematical notation with various uses in different branches of science and

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

Estimated value

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, a

circumflex The circumflex () is a diacritic in the Latin and Greek scripts that is also used in the written forms of many languages and in various romanization and transcription schemes. It received its English name from la, circumflexus "bent around"a ...

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

simple linear regression In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the ''x'' and ...

with observations of independent variable data

x_i

and dependent variable data

y_i

, and assuming a model of

y_i = \beta_0+\beta_1 x_i+\varepsilon_i

, can lead to an estimated model of the form

\hat_i = \hat_0+\hat_1 x_i

where

\sum_i (y_i-\hat_i)^2

is minimized via

least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...

by finding optimal values of

\hat_0

and

\hat_1

for the observed data.

Hat matrix

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

\hat = H \mathbf.

Cross product

In screw theory, one use of the hat operator is to represent the

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

operation. Since the cross product is a linear transformation, it can be represented as a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

. The hat operator takes a vector and transforms it into its equivalent matrix. :

\mathbf \times \mathbf = \mathbf \mathbf

For example, in three dimensions, :

\mathbf \times \mathbf = \begin a_x \\ a_y \\ a_z \end \times \begin b_x \\ b_y \\ b_z \end = \begin 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end \begin b_x \\ b_y \\ b_z \end = \mathbf \mathbf

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

(pronounced "v-hat").

Fourier transform

The

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

of a function

f

is traditionally denoted by

\hat

Quantum mechanics

In quantum mechanics, the capital H occurs after you partially differentiate Schrödinger's equation.

References

Mathematical notation {{Algebra-stub