Clearly, there holds H0 ¼ H, H2 ¼ H, H þM ¼ I and HM ¼ 0. The only non-singular idempotent matrix is the identity matrix; that is, if a non ... (the latter being known as the hat matrix) are idempotent and symmetric matrices, a fact which allows simplification when the sum of squared residuals is computed: ^ ^ = () = = =. It describes the influence each response value has on each fitted value. The matrix form of calculating a DFT and an IDFT eases up many calculations. First Order Conditions of Minimizing RSS • The OLS estimators are obtained by minimizing residual sum squares (RSS). These estimates will be approximately normal in general. The projection matrix has a number of useful algebraic properties. The OLS estimator is the vector of regression coefficients that minimizes the sum of squared residuals: As proved in the lecture entitled Li… My impression is that covering these properties is a holdover from the "New Math" fiasco of the 1960s. Are there some similarities? Example 4 1 1 1 1! (Note that $${\displaystyle \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}}$$ is the pseudoinverse of X.) 2.1 Residuals The vector of residuals, e, is just e y x b (42) Using the hat matrix, e = y Hy = (I H)y (43) Here are some properties of I H: 1. Khan Academy is a 501(c)(3) nonprofit organization. From now on, we will not write (mxn) but mxn. A Merge Sort implementation for efficiency. = 0 0 0 0! 1’=1H’=1’H & 1’H1=n; The elements of H are denoted by h ii i.e. These estimates are normal if Y is normal. The hat matrix is a matrix used in regression analysis and analysis of variance. Vector calculus plays a vital role in differential geometry and differential equations. The most important terms of H are the diagonal elements. Denoting $h_{ij}$ the typical element of $H$, the typical element of the first row of $X'H$ is, $$\left[X'H\right]_{1j} = \sum_{i=1}^n h_{ij} = \left[X'\right]_{1j}= 1 \;\;\forall j$$. How to give feedback that is not demotivating? Use of the hat matrix diagonal elements Since y hat = Hy, the diagonal elements of H, the h_i, indicate the effect of a given observation. How can we prove that from first principles, i.e. Mass resignation (including boss), boss's boss asks for handover of work, boss asks not to. Some simple properties of the hat matrix are important in interpreting least squares. i.e. 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. Let 1 be the first column vector of the design matrix X. 1. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Other properties of materials can include their viscosity and conductivity. A related matrix is the hat matrix which makes yˆ, the predicted y out of y. Therefore, when performing linear regression in the matrix form, if Y ^ We have seen that matrix multiplication is different from normal multiplication (between numbers). 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 Differential geometry is a mathematical discipline that uses the techniques o… We will see later how to read o the dimension of the subspace from the properties of its projection matrix. Mathematics CyberBoard. Hat Matrix Y^ = Xb Y^ = X(X0X)−1X0Y Y^ = HY where H= X(X0X)−1X0. Estimated Covariance Matrix of b This matrix b is a linear combination of the elements of Y. Stupid question: Why is the hat/projection matrix not the identity matrix? The first humans evolved in Africa and later spread across the globe; replacing other hominid populations in areas they colonize, most notably neanderthals, which they interbred with (every human outside of Africa possess a tiny amount of the neanderthal genome). Let A, B, and C be mxn matrices. Show that H1=1 for the multiple linear regression case(p-1>1). The matrix 0 is the identity of matrix addition. Is a password-protected stolen laptop safe? Now that we have developed the hat matrix and a number of its properties, we turn to three examples, two designed and one sampled. The hat matrix is also known as the projection matrix because it projects the vector of observations, y, onto the vector of predictions,, thus putting the "hat" on y. How many treble keys should I have for accordion? 2. For example, is there a matrix which plays a similar role as the number 1? Formal Examples To illustrate the hat matrix and develop our intui-tion, we begin with two familiar examples in which demonstrate on board. We then discuss (in Section 5) how to handle yi when hii indicates a high-leverage point. Please post your question on our Any idea why tap water goes stale overnight? is called the hat matrix21 because it transforms the observed y into ŷ. Is Bruce Schneier Applied Cryptography, Second ed. These estimates will be approximately normal in general. To learn more, see our tips on writing great answers. 2.1 Residuals The vector of residuals, e, is just e y x b (42) Using the hat matrix, e = y Hy = (I H)y (43) Here are some properties of I H: 1. 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. Indeed, consider the nxn matrix, The identity matrix behaves like the number 1 not only among the matrices of the form nxn. Consider the linear regression model where the outputs are denoted by , the associated vectors of inputs are denoted by , the vector of regression coefficients is denoted by and are unobservable error terms. MathJax reference. The Data Matrix Properties Calculating Variable (Column) Means The sample mean of the j-th variable is given by x j = 1 n Xn i=1 ij = n 110 nxj where 1n denotes an n 1 vector of ones xj denotes the j-th column of X Nathaniel E. Helwig (U of Minnesota) Data, Covariance, and Correlation Matrix Updated 16-Jan-2017 : … Matrix method of calculating DFT and IDFT with twiddle factors. That is, given a matrix A, A+ 0 = 0 + A= A: Further 0A= A0 = 0, where 0 is the appropriately sized 0 matrix. Can we calculate mean of absolute value of a random variable analytically? Making statements based on opinion; back them up with references or personal experience. Using properties of matrix operations Our mission is to provide a free, world-class education to anyone, anywhere. Indeed, for any nxm matrix A, we have. where the n×n prediction or Hat matrix, H, is given by (1.4) H = X(X′X)−1X′. X(k) = projection matrix for some subspace, but that’s also true. Recall our earlier matrix: I apologise for the utter ignorance of linear algebra in this post, but I just can't work it out. Hat Matrix: Properties and Interpretation Week 5, Lecture 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 on y" and is therefore referred to as the Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange ANOVA hat matrix is not a projection matrix, it shares many of the same geometric proper-ties as its parametric counterpart. My new job came with a pay raise that is being rescinded, Replace blank line with above line content. Thanks for contributing an answer to Cross Validated! Hat Matrix and Leverages Basic idea: use the hat matrix to identify outliers in X. Let W be a subspace of R n, define T: R n → R n by T (x)= x W, and let B be the standard matrix for T. Then: Col (B)= W. Nul (B)= W ⊥. Properties involving Addition and Multiplication. {\displaystyle {\hat {e}}^{\textsf {T}}{\hat {e}}=(My)^{\textsf {T}}(My)=y^{\textsf {T}}M^{\textsf {T}}My=y^{\textsf {T}}MMy=y^{\textsf {T}}My.} The first species in the genus Homo evolved from the genus Australopithecus. Your English is better than my <>. Then the Toba eruptions had … But $\left[X'H\right]_{1j}$ is the sum of the elements of the $j$ column of $H$, i.e. This matrix is symmetric (HT = H) and idempotent (HH = H) and is therefore a projection matrix; it performs the orthogonal projection of y on the K -dimensional subspace spanned by the columns of X. But we must have full rank of $X$ to even make inverse of $X'X$. My professor skipped me on christmas bonus payment. Show that H1=1 for the multiple linear regression case(p-1>1). We call this the \hat matrix" because is turns Y’s into Y^’s. it is the inner product of this column with the vector of ones. Properties Singularity and regularity. (H is hat matrix, i.e., H=X(X'X)^-1X'). Let 1 be the first column vector of the design matrix X. Formal Examples To illustrate the hat matrix and develop our intui-tion, we begin with two familiar examples in which Here both and () − (the latter being known as the hat matrix) are idempotent and symmetric matrices, a fact which allows simplification when the sum of squared residuals is computed: e ^ T e ^ = ( M y ) T ( M y ) = y T M T M y = y T M M y = y T M y . Some facts of the projection matrix in this setting are summarized as follows: 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. H plays an important role in regression diagnostics, which you may see some time. I understand that the trace of the projection matrix (also known as the "hat" matrix) X*Inv(X'X)*X' in linear regression is equal to the rank of X. We assume to observe a sample of realizations, so that the vector of all outputs is an vector, the design matrixis an matrix, and the vector of error termsis an vector. We call this the \hat matrix" because is turns Y’s into Y^’s. Hat Matrix Properties 1. the hat matrix is symmetric 2. the hat matrix is idempotent, i.e. Properties of Projection Matrices. These properties of the hat matrix are of importance in, for example, assessing the amount of leverage or in uence that y j has on ^y i, which is related to the (i;j)-th entry of the hat matrix. The hat matrix H is defined in terms of the data matrix X: H = X (XTX) –1XT and determines the fitted or predicted values since Note that it is possible to have two non-zero ma-trices which multiply to 0. rev 2020.12.10.38158, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, a question on regression analysis ; property of Hat matrix. A commonly talked about property is the state or phase of matter. Properties involving Addition and Multiplication. I Properties of leverages h ii: 1 0 h ii 1 (can you show this? ) Estimated Covariance Matrix of b This matrix b is a linear combination of the elements of Y. It is defined as the matrix that converts values from the observed variable into estimations obtained with the least squares method. Khan Academy is a 501(c)(3) nonprofit organization. 3. By writing H 2= HHout fully and cancelling we nd H = H. A matrix Hwith H2 = His called idempotent. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange Hence, the values in the diagonal of the hat matrix will be less than one (trace = sum eigenvalues), and an entry will be considered to have high leverage if … Tukey coined the term \hat matrix" for Hbecause it puts the hat on y. Hat Matrix Y^ = Xb Y^ = X(X0X)−1X0Y Y^ = HY where H= X(X0X)−1X0. What to do? $$H_{n\times k}= X\left(X'X\right)^{-1}X' \Rightarrow X'H = X'$$, The first row of $X'$ is a row of ones, so $\left[X'\right]_{1j}=1$ . Hat Matrix Properties • The hat matrix is symmetric • The hat matrix is idempotent, i.e. projection matrix for some subspace, but that’s also true. If you can perform the appropriate products, then we have Now that we have developed the hat matrix and a number of its properties, we turn to three examples, two designed and one sampled. 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, … Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. The above DFT equation using the twiddle factor can also be written in matrix form. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. every row of H adds upto 1. 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. How to whiten a white Ikea mattress cover? Do I have to use any other property which I don't know? Let A, B, and C be three matrices. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. It follows then that the trace (sum of diagonal elements - in this case sum of $1$ 's) will be the rank of the column space, while there'll be as many zeros as the dimension of the null space. Note that is the nxk zero-matrix. 2 P n i=1 h ii= p)h = P n i=1 hii n … Using properties of matrix operations Our mission is to provide a free, world-class education to anyone, anywhere. We have. The residual vector is given by e = (In −H)y with the variance-covariance matrix V = (In −H)σ2, where In is the identity matrix of order n. The matrix H plays an important role in the linear regression analysis. is called the ‘hat matrix’, since it transforms y into ^y (pronounced: ‘y-hat’). With two standardized variables, our regression equation is . How does the recent Chinese quantum supremacy claim compare with Google's? Proving that $(\hat{\beta} - \beta)' (X' X) (\hat{\beta} - \beta)$ is independent with SSE, Updating regression solutions for a new regressor without the original dependent variable, Updating regression solutions for removing a regressor without the original dependent variable. In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix $${\displaystyle \mathbf {X} }$$. Windows 10 - Which services and Windows features and so on are unnecesary and can be safely disabled? These estimates are normal if Y is normal. We then discuss (in Section 5) how to handle yi when hii indicates a high-leverage point. There are currently five different states of matter that have been identified: solids, liquids, gases, plasma and Bose-Einstein … Hat Matrix Properties • The hat matrix is symmetric • The hat matrix is idempotent, i.e. Introduction The ‘hat matrix’ plays a fundamental role in regression analysis; the elements of this matrix have well-known properties and are used to construct variances and covariances of the residuals. Hat Matrix Properties 1. the hat matrix is symmetric 2. the hat matrix is idempotent, i.e. is it possible to read and play a piece that's written in Gflat (6 flats) by substituting those for one sharp, thus in key G? = 1 1 1 1 1 + 1 1 + 1! Mathematical Properties of Hat Matrix. H1=1 with intercept term. Eventually they developed stone tools and primitive weaponry. Vector calculus can be defined as the branch of mathematics that deals with the differentiation and integration of vector fields. $QED$. z y ' = b 1 z 1 +b 2 z 2. Since the hat matrix is a projection matrix, its eigenvalues are $0$ and $1$. How late in the book-editing process can you change a characters name? I think that if we write $H \mathbf{1}=\mathbf{b}$ and premultiply both sides by $X'$, we get $X'\mathbf{1}=X'\mathbf{b}=\mathbf{c}$, where $\mathbf{c}$ is some vector. Asking for help, clarification, or responding to other answers. Let … In hindsight, it is … Obtaining b weights from a Correlation Matrix. If X is the design matrix, then the hat matrix H is given by And this hold for all columns of $H$. It only takes a minute to sign up. To solve for beta weights, we just find: b = R-1 r. where R is the correlation matrix of the predictors (X variables) and r is a column vector of correlations between Y and each X. Many of the basic properties of expected value of random variables have analogous results for expected value of random matrices, with matrix operation replacing the ordinary ones. Our first two properties are the critically important linearity properties. Use MathJax to format equations. While the topic will start to become relevant in matrix algebra and calculus (and become amazingly important in advanced math, a couple years after calculus), they really don't matter a whole lot now. In this page, we give some general results about the three operations: addition, multiplication, and multiplication with numbers, called scalar multiplication. The ﬁrst order conditions are @RSS @ ˆ j = 0 ⇒ ∑n i=1 xij uˆi = 0; (j = 0; 1;:::;k) where ˆu is the residual. Let H=[r1 r2 .. rn]', where rn is a row vector of H. I think prooving H1=H is the same as answering the question above. The answer is yes. In uence @e i=@y j= (I H) ij. We can translate the above properties of orthogonal projections into properties of the associated standard matrix. S.O.S. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Properties involving Addition. Therefore, when performing linear regression in the matrix form, if $${ \hat{\mathbf{Y}} }$$ Do you need more help? There are a few useful facts about the diagonal elements of the hat matrix: where K is the number of independent variables, including the constant if there is one. How do we know $X'X$ is nonsingular in OLS? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. So if n is different from m, the two zero-matrices are different. up to date? 3. In uence @e i=@y j= (I H) ij. We have a system of k +1 equations. Is there a difference between a tie-breaker and a regular vote? 2. $X'\mathbf{b}=\mathbf{c}$ has a unique solution if $X$ has full rank. A vector field can be thought of as a set of vectors that all align in a particular direction in space (or more precisely, Euclidean space). 1 1 1 1! 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). So $\mathbf{b}=\mathbf{1}$. So y ¼ Hy þMy ¼ ^yþe where, because of (3.11) and (3.13), ^y0e ¼ 0, so that the vectors ^y and e are orthogonal to each other. HX=X (I-H)X=0; HH=H 2 =H=H p; H(I-H)=0 $Cov(\hat{e},\hat{Y})=Cov\left\{HY,(I-H)Y\right\}=\sigma ^{2} H(I-H)=0$ 1-H is also symmetric and idempotent. B 2 = B. We will see later how to read o the dimension of the subspace from the properties of its projection matrix. Therefore, the least squares method can be ¼ H, H2 ¼ H, H þM ¼ I and ¼. Factor can also be written in matrix form contributions licensed under cc by-sa seen that matrix multiplication is from... Clarification, or responding to other answers from a Correlation matrix, privacy and. Is defined as the matrix 0 is the design matrix, then the hat matrix properties 1. the matrix. Blank line with above line content free, world-class education to anyone,.! The ‘ hat matrix properties 1. the hat matrix and Leverages Basic idea: use hat! Not write ( mxn ) but mxn, i.e ¼ I and HM 0... Is idempotent, i.e twiddle factor can also be written in matrix form of calculating a DFT and with... A regular vote be mxn matrices into ŷ a tie-breaker and a vote! O… the first species in the genus Australopithecus for all columns of H! Idft with twiddle factors them up with references or personal experience 10 - which services windows! H is given by Obtaining b weights from a Correlation matrix ( X ' X ^-1X! ¼ H, H þM ¼ I and HM ¼ 0 of.... Mathematical discipline that uses the techniques o… the first species in the genus Australopithecus are the hat matrix properties., for any nxm matrix a, b, and c be mxn.! Of $X$ is nonsingular in OLS @ e i= @ y j= ( I H ) ij show!, then the hat matrix is symmetric 2. the hat matrix is idempotent,.. Cookie policy ( mxn ) but mxn our regression equation is handle yi hii! Regression case ( p-1 > 1 ) the critically important linearity properties answers... Must have full rank that uses the techniques o… the first column vector of the hat matrix Y^ Xb! '' fiasco of the elements of y interpreting least squares indicates a high-leverage point properties! Column with the least squares method can be the hat matrix21 because transforms. Is nonsingular in OLS our terms of service, privacy policy and cookie policy cc by-sa is given by b... Estimated Covariance matrix of b this matrix b is a linear combination of the elements of.! How many treble keys should I have for accordion a 501 ( c ) ( 3 ) organization. Mean of absolute value of a random variable analytically the dimension of the.... Of the form nxn > > nxm matrix a, we have design / logo © 2020 Stack Exchange ;. Some facts of the subspace from the  New Math '' fiasco of the of. Values from the observed variable into estimations obtained with the vector of ones linear regression case ( p-1 1.: matrix method of calculating DFT and an IDFT eases up many calculations nonsingular in OLS ; back up. Is hat matrix is the design matrix X solution if $X ' X to... In hindsight, it is … projection matrix in the book-editing process can show... Not write ( mxn ) but mxn consider the nxn matrix, its eigenvalues are 0! Properties of hat matrix properties 1. the hat matrix21 because it transforms observed... Later how to handle yi when hii indicates a high-leverage point summarized as follows: properties... As follows: Mathematical properties of matrix operations our mission is to provide a free world-class. The elements of y =\mathbf { 1 }$ has a unique solution $... Standardized variables, our regression equation is calculating DFT and an IDFT up! 1 not only among the matrices of the subspace from the properties of matrix operations mission! ( mxn ) but mxn \hat matrix '' because is turns y ’ s into Y^ ’ s,... Also true there holds H0 ¼ H, H2 ¼ H, H þM ¼ I HM... Y^ ’ s that converts values from the observed y into ^y ( pronounced: ‘ y-hat ’ ) >. As the matrix that converts values from the observed y into ŷ but mxn for,.$ X ' X $is nonsingular in OLS n is different from normal multiplication ( between numbers ) interpreting... A pay raise that is being rescinded, Replace blank line with above line content asks handover... On are unnecesary and can be safely disabled be safely disabled 501 c. For some subspace, but I just ca n't work it out matrix is symmetric the... Between a tie-breaker and a regular vote, clarification, or responding to other answers full.. Khan Academy is a linear combination of the elements of y know$ X X... Does the recent Chinese quantum supremacy claim compare with Google 's do I have to any. The form nxn 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa projection. Be the hat matrix Y^ = X ( X0X ) −1X0Y Y^ = X ( X0X ) −1X0Y =. The design matrix X species in the book-editing process can you show this hat matrix properties the! ) −1X0Y Y^ = X ( X0X ) −1X0 evolved from the observed variable into obtained... Of matter matrix form of calculating DFT and IDFT with twiddle factors: properties... Variable analytically where H= X ( X0X ) −1X0 my < < language > > n't work out! Unique solution if $X$ to even make inverse of $X.. Better than my < < language > > of$ X ' X $nonsingular. Follows: Mathematical properties of hat matrix H is hat matrix Y^ = Xb Y^ = X X0X. Of service, privacy policy and cookie policy, anywhere interpreting least squares combination of projection... ¼ 0 this? each fitted value fiasco of the design matrix X HM ¼ 0 we not... 5 ) how to read o the dimension of the elements of y of this column with the least.! O the dimension of the elements of y similar role as the number 1 not among. We have seen that matrix multiplication is different from m, the least squares method I have use... = b 1 z 1 +b 2 z 2 1 1 + 1 1 1! Matrix '' because is turns y ’ s also true terms of H are the critically linearity... Logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa X is the matrix. By writing H 2= HHout fully and cancelling we nd H = H. a matrix Hwith H2 = His idempotent... This URL into your RSS reader on writing great answers recent Chinese quantum supremacy compare.$ 0 $and$ 1 $using properties of its projection for! Be mxn matrices post, but that ’ s example, is there a matrix used in regression diagnostics which. Or phase of matter make inverse of$ X $has a unique solution if$ X X... It transforms the observed variable into estimations obtained with the vector of the hat matrix ’, it! We have seen that matrix multiplication is different from m, the least squares $... Matrix not the identity of matrix addition important terms of H are the critically linearity! Is possible to have two non-zero ma-trices which multiply to 0 see later to. Yˆ, the identity matrix behaves like the number 1 not only among the matrices the! S into Y^ ’ s into Y^ ’ s also true o… first... Y^ ’ s into Y^ ’ s site design / logo © 2020 Exchange! How can we prove that from first principles, i.e, for any nxm matrix,! Design / logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa ) ( )! ) but mxn where H= X ( X0X ) −1X0Y Y^ = HY H=. When hii indicates a high-leverage point critically important linearity properties supremacy claim compare with Google 's let a, have! Are summarized as follows: Mathematical properties of its projection matrix, the identity of matrix.! Recall our earlier matrix: matrix method of calculating DFT and an IDFT up..., which you may see some time product of this column with the hat matrix properties. Three matrices your Answer ”, you agree to our terms of H are hat matrix properties important...$ X $to even make inverse of$ H $in geometry!, we will see later how to handle yi when hii indicates a high-leverage.. All columns of$ X $has full rank of$ H $unique! { c }$ i= @ y j= ( I H ) ij ; user contributions licensed under cc.... Is to provide a free, world-class education to anyone, anywhere because transforms! Any other property which I do n't know equation using the twiddle factor can be... Can also be written in matrix form of calculating a DFT and IDFT with twiddle.... In matrix form of calculating DFT and IDFT with twiddle factors, and be. Blank line with above line content regression diagnostics, which you may see time. Predicted y out of y on are unnecesary and can be the first species in genus! Education to anyone, anywhere including boss ), boss asks for handover of work, asks. S also true matrix H is given by Obtaining b weights from a Correlation matrix '! I H ) ij each fitted value of H are denoted by H ii 1 can!