In statistics, the projection matrix (), [1] sometimes also called the influence matrix [2] or hat matrix (), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values).

The hat matrix is also known as the projection matrix because it projects the vector of observations, y, onto the vector of predictions, y ^, thus putting the "hat" on y. The hat matrix H is defined in terms of the data matrix X: and determines the fitted or predicted values since. y …

Apr 20, 2016 · The hat matrix, H H, is the projection matrix that expresses the values of the observations in the independent variable, y y, in terms of the linear combinations of the column vectors of the model matrix, X X, which contains the observations for each of the multiple variables you are regressing on.

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").

These notes will not remind you of how matrix algebra works. However, they will review some results about calculus with matrices, and about expectations and variances with vectors and matrices. Throughout, bold-faced letters will denote matrices, as a as opposed to a scalar a. 1 Least Squares in Matrix Form

The hat matrix is a matrix used in regression analysis and analysis of variance. It is defined as the matrix that converts values from the observed variable into estimations obtained with the least squares method.

Hat Matrix Properties 1. the hat matrix is symmetric 2. the hat matrix is idempotent, i.e. HH = H Important idempotent matrix property For a symmetric and idempotent matrix A, rank(A) = trace(A), the number of non-zero eigenvalues of A.

2 H: The “hat” matrix. Suppose you calculated the predicted value of y for all of the observations. Here’s a vector: In that equation, replace ˆβ by the solution in 7. Then the predicted value is equal to.

Jun 2, 2015 · What is the usual meaning of having the symbol $\hat{}$ (i.e., a hat) over a vector name? What do vectors denoted by $\hat{\mathbf{u}}$ usually represent? For example, in this video, at min 3:00, ...

where S = Φ (Φ T Φ + λ R) − 1 Φ T is called hat matrix, as it converts the dependent variable vector X into its fitted value X ^. The number of degrees of freedom of the fit is the trace of the hat matrix