We did not call it "hatvalues" as R contains a built-in function with such a name. Hat diagonal refers to the diagonal elements of the hat matrix (Rawlings 1988). The minimum value of h ii is 1/n for a model with a constant term. [3][4] The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation. For details see below. } {\displaystyle \mathbf {P} ^{2}=\mathbf {P} } {\displaystyle \mathbf {r} } = X T X X {\displaystyle \mathbf {P} } is the identity matrix. , is M The projection matrix corresponding to a linear model is symmetric and idempotent, that is, Unlike computer-generated designs which may not be unique for a specific model and for user-specified design size, the Hat-Matrix (H-M) aided designs are unique. ≡ X { 1 ^ M {\displaystyle (\mathbf {H} )} A A { Hat Matrix Diagonal Data points that are far from the centroid of the X-space are potentially influential.A measure of the distance between a data point, x i, and the centroid of the X-space is the data point's associated diagonal element h i in the hat matrix. Let H be the hat/projection matrix that is symmetric and positive semidefinite and D be a diagonal matrix with a diagonal of ones except the last term being greater than … is a large sparse matrix of the dummy variables for the fixed effect terms. {\displaystyle \mathbf {X} } Ignoring the constant σ2, these elements are covariances of any pair of the estimated residuals, so can be useful to check the independency assumption. The $i$th diagonal element of the hat matrix is given by $$h_{ii} = \mathbf{e}_i^{t} \mathbf{H} \mathbf{e}_i,$$ x X If potentially influential observations are present, you may need to delete them from the model. T Belsley, Kuh, and Welsch (1980) propose a cutoff of 2 p/ n for the diagonal elements of the hat matrix, where n is the number of observations used to fit the model, and p is the number of parameters in the model. X Σ A measure of the distance between a data point, xi, and the centroid of the X-space is the data point's associated diagonal element hi in the hat matrix. H {\displaystyle \mathbf {A} } . is a matrix of explanatory variables (the design matrix), β is a vector of unknown parameters to be estimated, and ε is the error vector. The elements of hat matrix have their values between 0 and 1 always and their sum is p i.e. y = Because H ij= H jithe contribution of y i to ^y j equals that of y j to ^y i. ( can be decomposed by columns as omega: a vector or a function depending on the arguments residuals (the working residuals of the model), diaghat (the diagonal of the corresponding hat matrix) and df (the residual degrees of freedom). Property 1: Same order diagonal matrices gives a diagonal matrix only after addition or multiplication. If the vector of response values is denoted by {\displaystyle (\mathbf {P} )} , by error propagation, equals, where is sometimes referred to as the residual maker matrix. {\displaystyle H^{2}=H\cdot H=H} where, e.g., X hat: a vector containing the diagonal of the ‘hat’ matrix. T − {\displaystyle X=[A~~~B]} x: a fitted model object. , the projection matrix, which maps Ones in the diagonal elements specify that the variance of each F i is T2. P } A Usage. )   All rights reserved. y {\displaystyle \mathbf {\Sigma } =\sigma ^{2}\mathbf {I} } Let the data matrix be X (n * p), Hat matrix is: Hat = X … } {\displaystyle X} H One such tool is the residual-by-hat diagonal plot. , and is one where we can draw a line orthogonal to the column space of {\displaystyle X} without explicitly forming the matrix A ) Hat Matrix Diagonal. , maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). When the weights for each observation are identical and the errors are uncorrelated, the estimated parameters are, Therefore, the projection matrix (and hat matrix) is given by, The above may be generalized to the cases where the weights are not identical and/or the errors are correlated. − P observations which have a large effect on the results of a regression. The diagonal elements of the hat matrix are often used to denote leverage—that is, a point that is unusual in its location in the x-space and that may be influential. {\displaystyle \mathbf {I} } The diagonal terms satisfy where p is the number of coefficients in the regression model, and n is the number of observations. A Hat Matrix Diagonal Data points that are far from the centroid of the X-space are potentially influential.A measure of the distance between a data point, x i, and the centroid of the X-space is the data point's associated diagonal element h i in the hat matrix. = A square matrix D = [d ij] n x n will be called a diagonal matrix if d ij = 0, whenever i is not equal to j. the i’th row of the hat matrix. , which is the number of independent parameters of the linear model. y I . 0 ≤ h i i ≤ 1 ∑ i = 1 n h i i = p, where p is the number of coefficients in the regression model, and n is the number of observations. { Returns the values on the diagonal of the hat matrix, which is the matrix that transforms the response vector (minus any offset) into the fitted values (minus any offset). b For robust fitting problem, I want to find outliers by leverage value, which is the diagonal elements of the 'Hat' matrix. P The H-M aided designs are efficient and generally as good … {\displaystyle X} T {\displaystyle \mathbf {P} } and the vector of fitted values by Many types of models and techniques are subject to this formulation. I The following three matrices have their main diagonals indicated by red 1's: [] [] [] Antidiagonal. Experience suggests that a reasonable rule of thumb for large hi is hi > 2p/n. {\displaystyle \mathbf {A} \left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}\mathbf {A} ^{\textsf {T}}\mathbf {b} }, Suppose that we wish to estimate a linear model using linear least squares. { Suppose that the covariance matrix of the errors is Ψ. 1 ), respectively. For the case of linear models with independent and identically distributed errors in which One can use this partition to compute the hat matrix of Note that this method should only be used for linear mixed models. Pearson Residuals and Deviance Residuals . It describes the influence each response value has on each fitted value. X A "hat" matrix is one in which the only non-zero elements are stored on the diagonal. {\displaystyle \mathbf {Ax} } In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, or major diagonal) of a matrix is the collection of entries , where =. Note that this method should only be used for linear mixed models. 0 ≤ h i i ≤ 1 and ∑ i = 1 n h i i = p − ) Then the projection matrix can be decomposed as follows:[9]. Copyright © 2007 by SAS Institute Inc., Cary, NC, USA. 2.1 Range and Kernel of the Hat Matrix By combining our de nitions of the tted values and the residuals, we have by= Hy and be= (I H)y: These equations correspond to an orthogonal decomposition of the observed values, such that y = by+ be= Hy+ (I H)y: Observe that the column space or range of H, denoted col(H), is identical to the column space of X. A measure of the distance between a data point, x i, and the centroid of the X-space is the data point's associated diagonal element h i in the hat matrix. The average size of a diagonal element of the hat matrix, then, is p/n. ( {\displaystyle \mathbf {b} } Some facts of the projection matrix in this setting are summarized as follows:[4]. In particular the diagonal elements of the hat matrix are indicator of in a multi-variable setting of whether or not a case is outlying with respect to X values. } , which might be too large to fit into computer memory. A {\displaystyle \mathbf {Ax} } P Properties of Diagonal Matrix. { {\displaystyle \mathbf {A} } . In the classical application . Assumptions in Matrix Form 0 ~,N 0IT 2 0 is the nq 1 zero vector; I is the nnq identity matrix. A xx0 is symmetric. is usually pronounced "y-hat", the projection matrix − {\displaystyle \mathbf {r} } {\displaystyle M\{A\}=I-P\{A\}}   {\displaystyle \mathbf {x} } Data points that are far from the centroid of the X-space are potentially influential. {\displaystyle \mathbf {y} } {\displaystyle \mathbf {b} } r = = ^ P is also named hat matrix as it "puts a hat on − Suppose the design matrix Let $H$denote the hat matrix. {\displaystyle \mathbf {y} } On the other hand, off-diagonal elements of the Hat matrix may be re- garded as another criterion in the regression analysis. In statistics, the projection matrix A vector with the diagonal Hat matrix values, the leverage of each observation. The matrix {\displaystyle \mathbf {A} } onto the column space of H Generally, the ith point is called a leverage point if its hat diagonal hi exceeds 2p/n, which is twice the average size of all the hat diagonals. A x T is the pseudoinverse of X.) B ] Data points that are far from the centroid of the X-space are potentially influential. {\displaystyle \mathbf {\hat {y}} } [8] For other models such as LOESS that are still linear in the observations x I Recall that p = k + 1. P T locally weighted scatterplot smoothing (LOESS), "Data Assimilation: Observation influence diagnostic of a data assimilation system", "Proof that trace of 'hat' matrix in linear regression is rank of X", Fundamental (linear differential equation), https://en.wikipedia.org/w/index.php?title=Projection_matrix&oldid=995177570, Creative Commons Attribution-ShareAlike License, This page was last edited on 19 December 2020, at 17:32. {\displaystyle \mathbf {X} } Let’s learn about the properties of the diagonal matrix now. The leverage of observation i is the value of the i th diagonal term, hii, of the hat matrix, H, where H = X (XTX) –1XT.   2 { Another use is in the fixed effects model, where They are H … b The hat matrix diagonal element for observation i, denoted h i, reflects the possible influ-ence of X i on the regression equation. All off-diagonal elements are zero in a diagonal matrix. X The function returns the diagonal values of the Hat matrix used in linear regression. 1 is a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. The projection matrix has a number of useful algebraic properties. . However, this is not always the case; in locally weighted scatterplot smoothing (LOESS), for example, the hat matrix is in general neither symmetric nor idempotent. X Σ ) Here is another answer that that only uses the fact that all the eigenvalues of a symmetric idempotent matrix are at most 1, see one of the previous answers or prove it yourself, it's quite easy. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. {\displaystyle \mathbf {M} \equiv \mathbf {I} -\mathbf {P} } is the covariance matrix of the error vector (and by extension, the response vector as well). There are many types of matrices like the Identity matrix. b A X ,[1] sometimes also called the influence matrix[2] or hat matrix A Note that aliased coefficients are not included in the matrix. To "hatize" a vector is to place its elements on the diagonal of an otherwise-zero square matrix. ) are the first and second derivatives of the link function g(. {\displaystyle \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}} A square matrix is symmetric if it can be flipped around its main diagonal, that is, x ij = x ji. In other words, if X is symmetric, X = X0. I As A The diagonal elements of the hat matrix are useful in detecting extreme points in the design space where they tend to have larger values. H For linear models, the trace of the projection matrix is equal to the rank of = Residual-by-Hat Diagonal Plot The fit window contains additional diagnostic tools for examining the effect of observations. Its determinant is the product of its diagonal values. A − , or Practical applications of the projection matrix in regression analysis include leverage and Cook's distance, which are concerned with identifying influential observations, i.e. Then since. Hat Matrix Diagonal (Leverage) ... Consequently, when an observation has a very large or very small estimated probability, its hat diagonal value is not a good indicator of the observation’s distance from the design space (Hosmer and Lemeshow; 2000, p. 171). ⋅ A X The formula for the vector of residuals [5][6] In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix , though now it is no longer symmetric. . is on the column space of H − = {\displaystyle \mathbf {\hat {y}} } is equal to the covariance between the jth response value and the ith fitted value, divided by the variance of the former: Therefore, the covariance matrix of the residuals = can also be expressed compactly using the projection matrix: where } type: a character string specifying the estimation type. . coefficients (unless do.coef is false) a matrix whose i-th row contains the change in the estimated coefficients which results when the i-th case is dropped from the regression. and g''(.) 1 A ( {\displaystyle P\{A\}=A\left(A^{\textsf {T}}A\right)^{-1}A^{\textsf {T}}} [ 2 {\displaystyle \mathbf {\Sigma } } ( Which kinda sounds like what I want, although Wikipedia's definition of the hat matrix (==influence or projection matrix), looks slightly different. For generalized linear models, an approximate projection matrix is given by. x A Qdiag <- lm.influence (lm (y ~ X-1, weights=W))$hat/W R's docs for lm.influence$hat says this gives "a vector containing the diagonal of the ‘hat’ matrix." 2 H = X(X T X) –1 X T. The diagonal terms satisfy. Value. Implies zero correlation. I The model can be written as. A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so, Therefore, since A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). P . The diagonal elements of the hat matrix will prove to be very important. {\displaystyle \mathbf {x} } Returns the values on the diagonal of the hat matrix, which is the matrix that transforms the response vector (minus any offset) into the fitted values (minus any offset). v must be a matrix object with one of its two dimensions of length 1 (i.e., a vector). M It is not clear if the hat matrix concept even makes sense for generalized linear mixed models. P The j th diagonal element is where g'(.) The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation. Rousseeuw and Zomeren 22 (p 635) note that ‘leverage’ is the name of the effect, and that the diagonal elements of the hat matrix (hii,), as well as the Mahalanobis distance (see later) or similar robust measures are diagnostics that try to quantify this effect. In statistics, the projection matrix $${\displaystyle (\mathbf {P} )}$$, sometimes also called the influence matrix or hat matrix $${\displaystyle (\mathbf {H} )}$$, maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). onto Observations with hi values above this cutoff should be investigated. X P The leverage of observation i is the value of the ith diagonal term, h ii, of the hat matrix, H, where. T Moreover, the element in the ith row and jth column of Thus we determine high-leverage points by looking at the diagonal elements of H and paying particular attention to any x point for which hi > 2p/n. A few examples are linear least squares, smoothing splines, regression splines, local regression, kernel regression, and linear filtering. {\displaystyle A} The values of hi are stored in a variable named H_yname, where yname is the response variable name. T There are a number of applications of such a decomposition. X = , this reduces to:[3], From the figure, it is clear that the closest point from the vector Diagonal elements of the hat matrix. The i’th row of His simply z 0 i (Z0Z) 1Z and the ijelement of the hat matrix is H ij= z0 i (Z0Z) 1z j. For details see below. {\displaystyle P\{X\}=X\left(X^{\textsf {T}}X\right)^{-1}X^{\textsf {T}}} Zeros in the off-diagonal elements specify that the covariance between F i and F j is zero for ijv. X {\displaystyle A} I prove these results. It describes the influence each response value has on each fitted value. A Define the hat or projection operator as A ( Similarly, define the residual operator as σ ". } y P The hat matrix (projection matrix P in econometrics) is symmetric, idempotent, and positive definite.
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