P I Properties of leverages h ii: 1 0 h ii 1 (can you show this? ) {\displaystyle \mathbf {r} } Recall that H = [h ij]n i;j=1 and h ii = X i(X T X) 1XT i. I The diagonal elements h iiare calledleverages. where p is the number of coefficients in the regression model, and n is the number of observations. Then, we can take the first derivative of this object function in matrix form. Now, we can use the SVD of X for unveiling the properties of the hat matrix obtained, when performing E( ^) = E((X0X) 1X0Y) = (X0X) 1X0E(Y) = (X0X) 1X0X ~ = I n = ~ 2. The matrix X is called the design matrix. A call this matrix , the "hat matrix", because it "puts the hat on" . The projection matrix corresponding to a linear model is symmetric and idempotent, that is, The present article derives and discusses the hat matrix and gives an example to illustrate its usefulness. Then any vector of the form x = A+b+(I ¡A+A)y where y 2 IRn is arbitrary (4) is a solution of Ax = b: (5) 2 Notice here that u′uis a scalar or number (such as 10,000) because u′is a 1 x n matrix and u is a n x 1 matrix and the product of these two matrices is a 1 x 1 matrix (thus a scalar). B A. T = A. T Proof: 1. 1 Hat Matrix 1.1 From Observed to Fitted Values The OLS estimator was found to be given by the (p 1) vector, b= (XT X) 1XT y: The predicted values ybcan then be written as, by= X b= X(XT X) 1XT y =: Hy; where H := X(XT X) 1XT is an n nmatrix, which \puts the hat … In the classical application A private seller is any person who is not a dealer who sells or offers to sell a used motor vehicle to a consumer. Some facts of the projection matrix in this setting are summarized as follows:[4]. Then the eigenvalues of Hare all either 0 or 1. The matrix criterion is from the previous theorem. I Many types of models and techniques are subject to this formulation. , is (Similarly, the effective degrees of freedom of a spline model is estimated by the trace of the projection matrix, S: Y_hat = SY.) As you can see, the two x values furthest away from the mean have the largest leverages (0.176 and 0.163), while the x value closest to the mean has a smaller leverage (0.048). Moreover, the element in the ith row and jth column of , which is the number of independent parameters of the linear model. y Matrix operations on block matrices can be carried out by treating the blocks as matrix entries. First, we simplify the matrices: {\displaystyle \mathbf {Ax} } {\displaystyle A} A ] =   Hat Matrix Properties 1. the hat matrix is symmetric 2. the hat matrix is idempotent, i.e. . �GIE/T_�G�,�T����:�V��*S� !�a�(�dN$I[��.���$t���M�QXV�����(��@�KsS��˓eZFrl�Q ~�� =Ԗ�� 0G����ΐ*��ߏ�n��]��7ೌ��`G��_���&D. y P − ( His called the hat matrix and is central in regression analysis. Recall that M = I − P where P is the projection onto linear space spanned by columns of matrix X. Let H= [r1 r2 .. rn]', where rn is a row vector of H. Then r1*1=1 (scalr). {\displaystyle \mathbf {Ax} } x {\displaystyle \mathbf {r} } r We call this the \hat matrix" because is turns Y’s into Y^’s. { can be decomposed by columns as x H A Hat Matrix Y^ = Xb Y^ = X(X0X)−1X0Y Y^ = HY where H= X(X0X)−1X0. 3. is the pseudoinverse of X.) . Similarly, define the residual operator as is the identity matrix. {\displaystyle \mathbf {A} } x 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. These estimates are normal if Y is normal. The n×1 vector of ordinary predicted values of the response variable is yˆ = Hy, where the n×n prediction or Hat matrix, H, is given by (1.4) H = X(X′X)−1X′. ,[1] sometimes also called the influence matrix[2] or hat matrix T − Proof: The subspace inclusion criterion follows essentially from the deﬂnition of the range of a matrix. {\displaystyle X} {\displaystyle X} ( H plays an important role in regression diagnostics, which you may see some time. ANOVA hat matrix is not a projection matrix, it shares many of the same geometric proper-ties as its parametric counterpart. ( is on the column space of A related matrix is the hat matrix which makes yˆ, the predicted y out of y. 1 Then the projection matrix can be decomposed as follows:[9]. −− − == = == y yXβ XX'X Xy XX'X X y PXX'X X yPy H y Properties of the P matrix P depends only on X, not on y. . There are a number of applications of such a decomposition. . onto the column space of {\displaystyle \mathbf {A} } {\displaystyle \mathbf {x} } Another use is in the fixed effects model, where Just note that yˆ = y −e = [I −M]y = Hy (31) where H = X(X0X)−1X0 (32) Greene calls this matrix P, but he is alone. T {\displaystyle \mathbf {A} (\mathbf {A} ^{T}\mathbf {A} )^{-1}\mathbf {A} ^{T}\mathbf {b} }, Suppose that we wish to estimate a linear model using linear least squares. { [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. The hat matrix is calculated as: H = X (X T X) − 1 X T. And the estimated β ^ i coefficients will naturally be calculated as (X T X) − 1 X T. Each point of the data set tries to pull the ordinary least squares (OLS) line towards itself. Suppose the design matrix = b {\displaystyle \mathbf {\Sigma } =\sigma ^{2}\mathbf {I} } y So λ 2 = λ and hence λ ∈ { 0, 1 }. Three of the data points — the smallest x value, an x value near the mean, and the largest x value — are labeled with their corresponding leverages. 3 h iiis a measure of the distance between Xvalues of the ith observation and P P X X 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=992931373, Creative Commons Attribution-ShareAlike License, This page was last edited on 7 December 2020, at 21:50. {\displaystyle \mathbf {y} } (A+B)T=AT+BT, the transpose of a sum is the sum of transposes. Prove that if A is idempotent, then det(A) is equal to either 0 or 1. The projection matrix has a number of useful algebraic properties. P A For example, if there are large blocks of zeros in a matrix, or blocks that look like an identity matrix, it can be useful to partition the matrix accordingly. {\displaystyle X=[A~~~B]} In particular, U is a set of eigenvectors for XXT, and V is a set of eigenvectors for XTX.The non-zero singular values of X are the square roots of the eigenvalues of both XXT and XTX. (* inner product) without explicitly forming the matrix Or by our definition of variances, that's the variance of q transpose beta hat + the variance of k transpose y- 2 times the covariance of q transpose beta hat in k transpose y. . M The matrix These estimates will be approximately normal in general. 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. A Hat Matrix and Leverages Basic idea: use the hat matrix to identify outliers in X. is sometimes referred to as the residual maker matrix. can also be expressed compactly using the projection matrix: where A few examples are linear least squares, smoothing splines, regression splines, local regression, kernel regression, and linear filtering. ⋅ For the case of linear models with independent and identically distributed errors in which denoted X, with X as above. The matrix M is symmetric (M0 ¼ M) and idempotent (M2 ¼ M). ( 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. Residuals The residuals, … ) { 3 (c) From the lecture notes, recall the de nition of A= Q. T. W. T , where Ais an (n n) orthogonal matrix (i.e. Section 3 formally examines two ) {\displaystyle \mathbf {X} } {\displaystyle \mathbf {P} } observations which have a large effect on the results of a regression. The leverage of observation i is the value of the i th diagonal term, hii , of the hat matrix, H, where. X {\displaystyle H^{2}=H\cdot H=H} Practical applications of the projection matrix in regression analysis include leverage and Cook's distance, which are concerned with identifying influential observations, i.e. In this case, the matrix … {\displaystyle \mathbf {\hat {y}} } I is a large sparse matrix of the dummy variables for the fixed effect terms. A 1. It describes the influence each response value has on each fitted value. = demonstrate on board. is a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. The variable Y is generally referred to as the response variable. The formula for the vector of residuals A is also named hat matrix as it "puts a hat on P H X PRACTICE PROBLEMS (solutions provided below) (1) Let A be an n × n matrix. Show that H1=1 for the multiple linear regression case (p-1>1). P If you bought your used car from a private seller, and you discover that it has a defect that impairs the safety or substantially impairs the use, you may rescind the sale within 30 days of purchase, if you can prove that the seller knew about the defect but didn’t disclose it. Therefore, when performing linear regression in the matrix form, if $${ \hat{\mathbf{Y}} }$$ {\displaystyle A} − } 2. A A Useful Multivariate Theorem P = (H is hat matrix, i.e., H=X (X'X)^-1X') The followings are my reasoning so far. {\displaystyle \mathbf {I} } The matrix Z0Zis symmetric, and so therefore is (Z0Z) 1. Define the hat or projection operator as ) } where, e.g., In statistics, the projection matrix ( P ) {\displaystyle (\mathbf {P} )} , sometimes also called the influence matrix or hat matrix ( H ) {\displaystyle (\mathbf {H} )} , maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). (2) Let A be an n×n matrix. A However, the points farther away at the extreme of … Suppose that the covariance matrix of the errors is Ψ. and {\displaystyle \mathbf {P} }   Let A be a symmetric and idempotent n × n matrix. T − 2 is usually pronounced "y-hat", the projection matrix 1 ) 2 {\displaystyle \mathbf {y} } X {\displaystyle (\mathbf {H} )} and the vector of fitted values by ^ is an unbiased estimator of ~ . Let Hbe a symmetric idempotent real valued matrix. {\displaystyle \mathbf {X} } {\displaystyle M\{X\}=I-P\{X\}} ≡ A The aim of regression analysis is to explain Y in terms of X througha functional relationship like Yi = f(Xi,∗). M ^ has a multivariate normal distribution. , maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). Trace of a matrix is equal to the sum of its characteristic values, thus tr(P) = … { In statistics, the projection matrix X I ) Estimated Covariance Matrix of b This matrix b is a linear combination of the elements of Y. P , though now it is no longer symmetric. 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. ( = H Properties of ^ Theorem 4.2. H Exercise problem/solution in Linear Algebra. {\displaystyle \mathbf {A} } ) The following properties hold: (AT)T=A, that is the transpose of the transpose of A is A (the operation of taking the transpose is an involution). [5][6] In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix Theorem 2.2. ^ Since it also has the property MX ¼ 0, it follows from (3.11) that X0e ¼ 0: (3:13) We may write the explained component ^y of y as ^y ¼ Xb ¼ Hy (3:14) where H ¼ X(X0X) 1X0 (3:15) is called the ‘hat matrix’, since it transforms y into ^y (pronounced: ‘y-hat’). σ . This column should be treated exactly the same as any other column in the X matrix. } x = It follows that the hat matrix His symmetric too. P Σ Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 11, Slide 22 Residuals • The residuals, like the fitted values of \hat{Y_i} can be expressed as linear A Hat Matrix Properties • The hat matrix is symmetric • The hat matrix is idempotent, i.e. , or X 1 GDF is thus defined to be the sum of the sensitivity of each fitted value, Y_hat i, to perturbations in its corresponding output, Y i. A A For linear models, the trace of the projection matrix is equal to the rank of A symmetric idempotent matrix such as H is called a perpendicular projection matrix. y X   The model can be written as. where OLS in Matrix Form 1 The True Model † Let X be an n £ k matrix where we have observations on k independent variables for n observations. {\displaystyle \mathbf {P} ^{2}=\mathbf {P} } . , by error propagation, equals, where } For every n×n matrix A, the determinant of A equals the product of its eigenvalues. {\displaystyle (\mathbf {P} )} An idempotent matrix M is a matrix such that M^2=M. , this reduces to:[3], From the figure, it is clear that the closest point from the vector A , and is one where we can draw a line orthogonal to the column space of ( T tion of the observed values yj. Kutner et al. ^ H = X ( XTX) –1XT. { } y { b = X The least-squares estimators are the fitted values, y ^ = X β ^ = X ( X T X) − 1 X T y = X C − 1 X T y = P y. P is a projection matrix. T {\displaystyle \mathbf {x} } ) onto It describe A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so, Therefore, since 1 T [8] For other models such as LOESS that are still linear in the observations 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. {\displaystyle \mathbf {\Sigma } } The minimum value of hii is 1/ n for a model with a constant term. 2 I is the covariance matrix of the error vector (and by extension, the response vector as well).
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