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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
A "hat" (circumflex (ˆ)), placed over a symbol is a mathematical notation with various uses.
Estimated value
[ edit ]
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
ε
^
{\displaystyle {\hat {\varepsilon }}}
indicates an observable estimate (the residuals) of an unobservable quantity called
ε
{\displaystyle \varepsilon }
(the statistical errors).
Another example of the hat operator denoting an estimator occurs in simple linear regression . Assuming a model of
y
i
=
β
0
+
β
1
x
i
+
ε
i
{\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i}}
, with observations of independent variable data
x
i
{\displaystyle x_{i}}
and dependent variable data
y
i
{\displaystyle y_{i}}
, the estimated model is of the form
y
^
i
=
β
^
0
+
β
^
1
x
i
{\displaystyle {\hat {y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}x_{i}}
where
∑
i
(
y
i
−
y
^
i
)
2
{\displaystyle \sum _{i}(y_{i}-{\hat {y}}_{i})^{2}}
is commonly minimized via least squares by finding optimal values of
β
^
0
{\displaystyle {\hat {\beta }}_{0}}
and
β
^
1
{\displaystyle {\hat {\beta }}_{1}}
for the observed data.
Hat matrix
[ edit ]
In statistics, the hat matrix H projects the observed values y of response variable to the predicted values ŷ :
y
^
=
H
y
.
{\displaystyle {\hat {\mathbf {y} }}=H\mathbf {y} .}
Cross product
[ edit ]
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
×
b
=
a
^
b
{\displaystyle \mathbf {a} \times \mathbf {b} =\mathbf {\hat {a}} \mathbf {b} }
For example, in three dimensions,
a
×
b
=
[
a
x
a
y
a
z
]
×
[
b
x
b
y
b
z
]
=
[
0
−
a
z
a
y
a
z
0
−
a
x
−
a
y
a
x
0
]
[
b
x
b
y
b
z
]
=
a
^
b
{\displaystyle \mathbf {a} \times \mathbf {b} ={\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}}\times {\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}}=\mathbf {\hat {a}} \mathbf {b} }
Unit vector
[ edit ]
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
v
^
{\displaystyle {\hat {\mathbf {v} }}}
(pronounced "v-hat").[1]
[ edit ]
The Fourier transform of a function
f
{\displaystyle f}
is traditionally denoted by
f
^
{\displaystyle {\hat {f}}}
.
See also
[ edit ]
References
[ edit ]
t
e
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Hat_notation&oldid=1223584042 "
C a t e g o r i e s :
● M a t h e m a t i c a l n o t a t i o n
● A l g e b r a s t u b s
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a
● A r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s f r o m M a y 2 0 2 4
● A l l a r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s
● P a g e s d i s p l a y i n g w i k i d a t a d e s c r i p t i o n s a s a f a l l b a c k v i a M o d u l e : A n n o t a t e d l i n k
● A l l s t u b a r t i c l e s
● T h i s p a g e w a s l a s t e d i t e d o n 1 3 M a y 2 0 2 4 , a t 0 2 : 4 9 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
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