The vector of fitted values ˆY is the (orthogonal) projection of Y onto the column space of X. Let H = X(XT X)−1XT so that ˆY = HY . Because multiplication by H changes Y into ˆY, the matrix H is called the Hat Matrix. It is very important in linear regression.
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What does the hat matrix tell us?
The hat matrix, H, is the projection matrix that expresses the values of the observations in the independent variable, y, in terms of the linear combinations of the column vectors of the model matrix, X, which contains the observations for each of the multiple variables you are regressing on.
Is the hat matrix a diagonal matrix?
Hat Matrix Diagonal: The diagonal elements of the hat matrix can be used to detect points that are extreme in the independent variable space.
Are projection matrices orthogonal?
A square matrix P is called an orthogonal projector (or projection matrix) if it is both idempotent and symmetric, that is, P2 = P and P′ = P (Rao and Yanai, 1979).
What does the hat mean in regression?
Y hat (written ŷ ) is the predicted value of y (the dependent variable) in a regression equation. It can also be considered to be the average value of the response variable. The regression equation is just the equation which models the data set. The equation is calculated during regression analysis.
Is projection matrix symmetric?
A symmetric idempotent matrix is called a projection matrix. Properties of a projection matrix P: 2.52 Theorem: If P is an n × n matrix and rank(P) = r, then P has r eigenvalues equal to 1 and n − r eigenvalues equal to 0.
Is hat matrix constant?
When we have some non-zero constants that we multiply each respective predictor by, which just multiplies every column in the data matrix X by the respective constant, the hat matrix stays the same. And so ˆY is the same as well.
Are projection matrices positive definite?
(b) The only positive definite projection matrix is P = I because: Let V be the subspace that P projects on, i.e. the range of the map P. If its orthogonal complement V ⊥ = {0}, then P = I. If V ⊥ = {0}, then any non-zero element v ∈ V ⊥ satisfies Pv = 0, v = 0 and hence P is not invertible.
What is trace of hat matrix?
For linear models, the trace of the hat matrix is equal to the rank of X, which is the number of independent parameters of the linear model. For other models such as LOESS that are still linear in the observations y, the hat matrix can be used to define the effective degrees of freedom of the model.
What is a hat diagonal?
One such tool is the residual-by-hat diagonal plot. Hat diagonal refers to the diagonal elements of the hat matrix (Rawlings 1988). Hat diagonal measures the leverage of each observation on the predicted value for that observation.
Is the identity matrix a projection matrix?
For an identity matrix with column vectors in Rn, each vector will be projected not onto a subspace of Rn, but onto Rn itself. It can be considered a special form of projection matrix, and the only one invertible.
What is orthogonal projection in linear algebra?
When the vector space has an inner product and is complete (is a Hilbert space) the concept of orthogonality can be used. An orthogonal projection is a projection for which the range and the null space are orthogonal subspaces.
How do you know if a projection is orthogonal?
We denote the closest vector to x on W by x W .
- To say that x W is the closest vector to x on W means that the difference x − x W is orthogonal to the vectors in W :
- In other words, if x W ⊥ = x − x W , then we have x = x W + x W ⊥ , where x W is in W and x W ⊥ is in W ⊥ .
What does the hat on a variable mean?
an estimated value
“Y” because y is the outcome or dependent variable in the model equation, and a “hat” symbol (circumflex) placed over the variable name is the statistical designation of an estimated value.
What does a hat mean in linear algebra?
estimator
1. To denote a unit vector (e.g., ) or an estimator (e.g., ). 2. To denote an estimator, e.g., for the sample mean.
What does the hat mean in stats?
an 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).
Are projection matrices Hermitian?
An idempotent matrix that is also Hermitian is called a projection matrix. P is a projector if PP=P. Projectors are always positive which implies that they are always Hermitian.
What is the matrix of a projection?
In linear algebra, a projection matrix is a matrix associated to a linear operator that maps vectors into their projections onto a subspace.
What does a permutation matrix do?
So, permutation matrices do indeed permute the order of elements in vectors multiplied with them.
How do you read Dffits?
The DFFITS statistic is a scaled measure of the change in the predicted value for the ith observation and is calculated by deleting the ith observation. A large value indicates that the observation is very influential in its neighborhood of the X space. , where n and p are as defined previously.
Are all invertible matrices positive definite?
A inverse matrix B−1 is it automatically positive definite? Invertible matrices have full rank, and so, nonzero eigenvalues, which in turn implies nonzero determinant (as the product of eigenvalues). *Considering the comments below, the answer is no.