Hat operator

The hat operator is a mathematical notation with various uses in different branches of science and mathematics.
Hat matrix.
In statistics, the hat matrix "H" projects the observed values y of response variable to the predicted values ŷ: 
Cross product.
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,
Estimated value.
In statistics, the 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 with observations of independent variable data formula_4 and dependent variable data formula_5, and assuming a model of formula_6, can lead to an estimated model of the form formula_7 where formula_8 is minimized via least squares by finding optimal values of formula_9 and formula_10 for the observed data.
Fourier transform.
The Fourier transform of a function formula_11 is traditionally denoted by formula_12.
Linear Algebra.
In Linear Algebra, when examining predicted data on a line, then you must use the hat operator, too. The following is an example of a linear function:
y = mx + b
If you are examining predicted data, then the function must be turned into the following:
ŷ = mx + b
This rule is just to represent that this is not a real and written rule. It is just predicted. One example using this is when you are finding out how clean people's teeth are based on how much chocolate they eat per day. The function which you come up with may not be written. It can be like the following (this function is an example):
y = -1/2c + 8
This function is assuming that c represents the number of chocolates eaten and the teeth cleanliness is measured between 1 and 10. Since this rule may not be real and actually true, we must denote that with the following by adding a hat operator to the y:
ŷ = -1/2c + 8