Crucially, this is where Stata and the packages and modules in R and Python disagree. \[ \hat{y} = H y \] The diagonal elements of this matrix are called the leverages Note. For hatvalues, dfbeta, and dfbetas, the method for linear models also works for generalized linear models.. In statistics, the projection matrix (), sometimes also called the influence matrix or hat matrix (), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). Thus a look at the hat matrix can reveal sensitive points in the Although OP has not been active for more than a year, I still decide to post an answer. (2.5) For the data analyst, the element hij of H has a direct interpretation as the amount of leverage or influence exerted on 9i by yj (regardless of the actual value of y3, since H depends only on X). The hat matrix, is a matrix that takes the original \(y\) values, and adds a hat! A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in ^ (pronounced "v-hat").. Theorem 2.2. The matrix Z0Zis symmetric, and so therefore is (Z0Z) 1. Both rely on the hat matrix. The hat matrix is used to project onto the subspace spanned by the columns of \(X\). Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 11, Slide 21 Hat Matrix Properties • The hat matrix is symmetric • The hat matrix is idempotent, i.e. The Rhat function produces R-hat convergence diagnostic, which compares the between- and within-chain estimates for model parameters and other univariate quantities of interest. It is also simply known as a projection matrix. • The hat matrix plans an important role in diagnostics for regression analysis. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. Let Hbe a symmetric idempotent real valued matrix. This answer assumes the context of ordinary least squares. write H on board. Value. If chains have not mixed well (ie, the between- and within-chain estimates don't agree), R-hat is larger than 1. The function hat() exists mainly for S (version 2) compatibility; we recommend using hatvalues() instead.. Then the eigenvalues of Hare all either 0 or 1. References. HAT MATRIX 5 The matrix Hin (2) is a projection matrix and thus by de nition, His idempotent, H2 = H. By construction, H is symmetric, HT = H (in this case H is an orthogonal projection matrix). It follows that the hat matrix His symmetric too. It describes the influence each response value has on each fitted value. I would use X instead of S, as in statistics, we often want projection matrix in linear regression context, where X is the model matrix, y is the response vector, while H = X(X'X)^{-1}X' is hat / projection matrix so that Hy gives predictive values.. The hat matrix H is defined in terms of the data matrix X: H = X(X T X) –1 X T. and determines the fitted or predicted values since . When weights are specified, Stata estimates the hat matrix as \[ \mathbf{H}_{Stata} = \mathbf{X} (\mathbf{X}^{\top}\mathbf{W}\mathbf{X})^{-1} … The hat matrix is also helpful in directly identifying outlying X observation. , Z n τ, where H = H(h) is the n × n hat matrix, depending only on the X-covariate and the δ-censoring indicator, and the superscript r denotes the transpose. The hat matrix plays an important role in determining the magnitude of a studentized deleted residual and therefore in identifying outlying Y observations. Author(s) Several R core team members and John Fox, originally in his ‘ car ’ package. Since premultiplying yby this matrix Hputs a ‘hat’ on yit is often called the hat A symmetric idempotent matrix such as H is called a perpendicular projection matrix. Belsley, D. A., Kuh, E. and Welsch, R. E. (1980). 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. r = y - y: var() = c2H, (2.4) var(r) = -2 - H).

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