, or 1 or in matrix notation: Notice there are K + L parameters to be estimated simultaneously. {\displaystyle X} {\displaystyle \mathbf {P} ^{2}=\mathbf {P} } ) Let m × n full-column matrix be A. {\displaystyle \mathbf {I} } In fact, it can be shown that the sole matrix, which is both an orthogonal projection and an orthogonal matrix is the identity matrix. Projection matrix. I . New comments cannot be posted and votes cannot be cast, More posts from the econometrics community, Press J to jump to the feed. )   ( Sample question for calculating an OLS estimator from matrix information. can also be expressed compactly using the projection matrix: where createResidualMaker: Create a residual maker matrix from coefficient names.  For other models such as LOESS that are still linear in the observations In general, we need eigenvalues to check this. {\displaystyle \mathbf {X} }  In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so, Therefore, since 1 The Frisch-Waugh-Lovell Theorem (FWL; after the initial proof by Frisch and Waugh (), and later generalisation by Lovell ()) states that:. ―Morpheus to Neo Residual self image (RSI) is the subjective appearance of a human while connected to the Matrix.. Pages 5. It creates a vector of n standard normal random variables, residualizes this vector against a matrix of covariates C, then standardizes the vector again. It can be easily derived from the least square residuals: e … 2 X A Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Let A = X′X. M = I-P = I-[X(X'X)-1 X'] is a residual-maker matrix, I is the identity matrix and P is a predicted value maker (projection matrix). The matrix De ne, h tto be the tthdiagonal element of the ‘hat’ matrix P X = X(X>X) 1X> and e e > t M Xe et = e e > t (I n P X)e et = 1 h t. Thus, omitting observation tproduces an estimate for ^ = ^u t 1 h t (3.12) 9 The RSI's content may be defined in part from the semi-permanent programming of a redpill's headjack. is usually pronounced "y-hat", the projection matrix P X produces the tted values in least square residuals in the regression of y on X.8 Furthermore, P XX = X and P Xe = 0. A where, e.g., ≡ P creates fitted values (makes ŷ out of y, which is why it's also sometimes called "hat matrix"), while M creates least-squared residuals (converts the values of y into residuals of y when regressed on X).   One way to interpret this is that if X is regressed on X, a perfect fit will result and the residuals will be zero. } − without explicitly forming the matrix {\displaystyle X} A few examples are linear least squares, smoothing splines, regression splines, local regression, kernel regression, and linear filtering. P 2 Orthogonal Decomposition 2.1 Range and Kernel of the Hat Matrix By combining our de nitions of the tted values and the residuals, we have X x In summary, we therefore have by= Hy and be= (I H)y: Crucially, it can be shown that both H and I H are orthogonal projections. In particular if is categorical it will “demean” any vector which is … It lets you plot various graphs for computed regression statistics. X H The matrix ≡ (−) is sometimes referred to as the residual maker matrix. , maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). Neo's appearance in the Construct when Morpheus first tells him of the truth of the Matrix is an example of an RSI placed on Neo's avatar. {\displaystyle P\{X\}=X\left(X^{\mathsf {T}}X\right)^{-1}X^{\mathsf {T}}} {\displaystyle \mathbf {y} } X This video provides a derivation of the form of ordinary least squares estimators, using the matrix notation of econometrics. Make bar charts, histograms, box plots, scatter plots, line graphs, dot plots, and more. Similarly, define the residual operator as A Sorry. Application: Rank of the Residual Maker We define M, the residual maker, as: M = In - X(X′X)-1 X′ = In - P where X is an nxk matrix, with rank(X)=k Let’s calculate the trace of M: tr(M) = tr(In) - tr(P) = n - k - tr(IT) = n - tr(P) = k Recall tr(ABC) = tr(CAB) => tr(P) = tr(X(X′X)-1 X′) = tr(X′X (X′X)-1) = tr(Ik) = k Since M is an idempotent matrix –i.e., M= M2-, then rank(M) = tr(M) = n - k It is important to remember that † 6= e. 1 T The residual maker and the hat matrix There are some useful matrices that pop up a lot. I } . T I m denote m × m identity matrix. y1 + y2 + y3 + y4 + y5 ~ s2*z1 # Constrained over time. = , the projection matrix, which maps { P In the second part, Monte Carlo simulations and an application to growth regressions are used to evaluate the performance of these estimators. Students also viewed these Econometric questions What is the result of encoding the messages using the (7, 4) Hamming code of Example 3.71? The vector X is always in the column space of X, and y is unlikely to be in the column space. Note that (i) H is a symmetric matrix (ii) H is an idempotent matrix, i.e., HHIHIH IHH ()()() and (iii) trH trI trH n k n (). It is given by: M =I−X(X′X)−1X′. P Thanks. The strategy in the least squared residual approach is the same as in the bivariate linear regression model. R> X1<-cbind(rep(1,n),age,race,gender,BMI) R> X2<-cbind(beauty,spunk) R> I<-diag(n) R> M1<-I-X1 %*% solve(t(X1) %*% X1) %*% t(X1) #compute residual-maker matrix {\displaystyle \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}} An often overlooked solution the the above problem can be obtained by simply rearranging (2) (3) X {\displaystyle \mathbf {P} } X This is, in fact, classic attenuation bias: the residual outcome regression uses amismeasuredregressorD i inplaceofthetrueregressorD˜ i,withuncorrelatedmeasurement errorD i −D˜ i. {\displaystyle (\mathbf {H} )} X Then the projection matrix can be decomposed as follows:. A Define an orthogonal projection onto the column space of A as P ( A), which is P ( A) = A ( A ′ A) − 1 A ′. A observations which have a large effect on the results of a regression. Moreover, the element in the i th row and j th column of P {\displaystyle \mathbf {P} } is equal to the covariance between the j th response value and the i th fitted value, divided by the variance of the former: 1 ) Create charts and graphs online with Excel, CSV, or SQL data. . Free to get started! First, we calculate the sum of squared residuals and, second, find a set of estimators that minimize the sum. Show that: (i) PXY = Yˆ (hence the name projection matrix) (ii) MXY = uˆ (hence the name residual maker matrix) (iii) MXu = uˆ (iv)Symmetry: PX = P0 X and MX = M0X (v)Idempotency: PXPX = PX and MXMX = MX (vi)tr PX = rank PX = K and tr MX = rank MX = N K Hint: Use the spectral decomposition for symmetric matrices: A = … , which might be too large to fit into computer memory. picks o the tth diagonal element of the residual maker matrix, M X. X (Note that = T Residuals The difference between the observed and fitted values of the study variable is called as residual. A square matrix A is idempotent if A2 = AA = A (in scalars, only 0 and 1 would be idempotent). A Then, z′Az = z′X′Xz = v′v >0. Unless Ωˆ is … The matrix M = I X(X 0X) 1X (1) is often called the \residual maker". Σ The M Matrix The residual maker M I X X X 1 X MX 0 Why My Produces the. Expert Answer . Note that (i) H is a symmetric matrix (ii) H is an idempotent matrix, i.e., HHIHIH IHH ()()() and (iii) trH trI trH n k n (). . Define The Projection Matrix Px-X(X'X)-X' And The Residual Maker Matrix Mx: IN Px. X P 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 {\displaystyle \mathbf {x} } z2 ~ RIx + RIy z2 ~~ z2 # Residual variance z2 # Create within-person centered variables wx1 =~ 1*x1 wx2 =~ 1*x2 wx3 =~ 1*x3 wx4 =~ 1*x4 wx5 =~ 1*x5 wy1 =~ 1*y1 wy2 =~ 1*y2 wy3 =~ 1*y3 wy4 =~ 1*y4 wy5 =~ 1*y5 # Regression of observed variables on z1 (constrained). ( A {\displaystyle X=[A~~~B]} It can be easily derived from the least square residuals: e … An avatar for a program may also be known as a "shell." . − Show transcribed image text. Uploaded By spiritofhumanity. P creates fitted values (makes ŷ out of y, which is why it's also sometimes called "hat matrix"), while M creates least-squared residuals (converts the values of y … − Students also viewed these Econometric questions What is the result of encoding the messages using the (7, 4) Hamming code of Example 3.71? For example, R squared change, Model fit, Covariance matrix, Residuals, Collinearility diagnostics, Part and partial correlations, etc. . I A Unfortunately, the bias from Ωˆ becomes more complicated when there are multiple main-tained treatments. ". I However, the residual maker matrix M i is presented, and is used in to define 2, and in several other parts of the course. Notes . {\displaystyle (\mathbf {P} )} resid_maker: Creates orthogonal residuals in sensemakr: Sensitivity Analysis Tools for Regression Models ( It is used in the proof of the Gauss-Markov theorem. 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. , the projection matrix can be used to define the effective degrees of freedom of the model. The vector of residuals e is given by: e = y ¡Xﬂ^ (2) 1Make sure that you are always careful about distinguishing between disturbances (†) that refer to things that cannot be observed and residuals (e) that can be observed. The matrix ≡ (−) is sometimes referred to as the residual maker matrix. , which is the number of independent parameters of the linear model. T "Your appearance now is what we call residual self image.It is the mental projection of your digital self." {\displaystyle \mathbf {\Sigma } } So we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. Select OK. That nickname is easy to understand, since: My= (I X(X 0X) 1X )y = y X(X 0X) 1X y = y X ^ ^" M plays a central role in many derivations. , and is one where we can draw a line orthogonal to the column space of {\displaystyle \mathbf {X} } {\displaystyle A} {\displaystyle \mathbf {X} } H A I followed the algebra of the proof, but I'm having difficulty grasping any intuitive sense of what just happened. , is is a large sparse matrix of the dummy variables for the fixed effect terms. In particular if is categorical it will “demean” any vector which is multiplied onto it from the right. Introducing the Residual Maker Matrix. {\displaystyle \mathbf {\hat {y}} } } y Define the projection matrix Px-X(X'X)-X' and the residual maker matrix Mx: IN Px. Example. X  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. ] getFamilyWiseCoefList: Get the familynames for each coefficient and organize into... getFamNamesFromCoefNames: Get family names from coefficient names (several coefNames... getGFacAndLevNames: getGFacAndLevNames Get general factor and factor level names How can we prove that from first principles, i.e. Title: Econometrics Author: Kuan-Pin Lin Created Date: 10/13/2015 11:20:08 AM Select OK. ( ) So if predicted is larger than actual, this is actually going to be a negative number. {\displaystyle \mathbf {M} \equiv \left(\mathbf {I} -\mathbf {P} \right)} Note that M is N ×N, that is, big! {\displaystyle H^{2}=H\cdot H=H} The standard regression output will appear in the session window, and the residual plots will appear in new windows. ^   A Moreover, the element in the ith row and jth column of It is given by: M =I−X(X′X)−1X′. onto = Another use is in the fixed effects model, where = A P You can export regression analysis results in an HTML file. } Edit: I haven't come across the "projection matrix before", I just made that assumption by looking at notes from other universities on found on google. It is a symmetric and idempotent matrix. (2.26) It generates the vector of least square residuals in a regression of y on X when it premultiplies any vector y. {\displaystyle \mathbf {b} } Σ Some facts of the projection matrix in this setting are summarized as follows:. X Denote an annihilator matrix (or residual maker) as M ( A), where M ( A) = I m − p ( A) = I m − A ( A ′ A) − 1 A ′. , sometimes also called the influence matrix or hat matrix { is the identity matrix. , this reduces to:, From the figure, it is clear that the closest point from the vector Neo's RSI (left) compared to his real world appearance (right). , though now it is no longer symmetric. The professor for our upper year undergrad econometrics course has just introduced the Residual Maker Matrix to prove that Sigma Hat Squared is an unbiased estimator of Sigma Squared. {\displaystyle \mathbf {A} } y A normal probability plot of the residuals is a scatter plot with the theoretical percentiles of the normal distribution on the x-axis and the sample percentiles of the residuals on the y-axis, for example: 用residual matrix造句, 用residual matrix造句, 用residual matrix造句, residual matrix meaning, definition, pronunciation, synonyms and example sentences are provided by … {\displaystyle \mathbf {y} } ( Denote the residual maker (or annihilator )matrix of This matrix has some interesting properties. Suppose the design matrix Many types of models and techniques are subject to this formulation. Unfortunately, the bias from Ωˆ becomes more complicated when there are multiple main-tained treatments. In other words, the least squares partitions the vector y into two orthogonal parts, y = Py+My = projection+residual. Define the projection matrix Px-X(X'X)-X' and the residual maker matrix Mx: IN Px. It is denoted as ~ ˆ ˆ ey y yy yXb yHy I Hy Hy where H IH. In this case we have paired sample data $$(X_i , Y_i)$$, where X corresponds to the independent variable and Y corresponds to the dependent variable. Then since. Press question mark to learn the rest of the keyboard shortcuts. {\displaystyle \mathbf {\Sigma } =\sigma ^{2}\mathbf {I} } H x For linear models, the trace of the projection matrix is equal to the rank of {\displaystyle P\{A\}=A\left(A^{\mathsf {T}}A\right)^{-1}A^{\mathsf {T}}} B Introducing the Residual Maker Matrix. You need type in the data for the independent variable $$(X)$$ and the dependent variable ($$Y$$), in the form below: I have no idea what the Residual Maker Matrix is. Denoting the i-th column of the matrix M by m i then ^" i = m0 i". = The projection matrix has a number of useful algebraic properties. 1 2.1 Some basic properties of OLS First, note that the LS residuals are “orthogonal” to the regressors – ... checkerboard matrix Show transcribed image text A checkerboard matrix is a special kind of matrix. ( X P 2 can be decomposed by columns as Similarly, the residuals can also be expressed as a function of H, be:= y yb= y Hy = (I H)y; with I denoting the n nidentity matrix, and where again the residuals can also be seen to be a linear function of the observed values, y. X A T is the covariance matrix of the error vector (and by extension, the response vector as well). (2.26) It generates the vector of least square residuals in a regression of y on X when it premultiplies any vector y. Definition: A matrix A is positive definite(pd) if z′Az>0 for any z. First, we calculate the sum of squared residuals and, second, find a set of estimators that minimize the sum. School University of Zimbabwe; Course Title ECON 202; Uploaded By r1810453. P is the pseudoinverse of X.) See the answer. A {\displaystyle M\{A\}=I-P\{A\}} In the classical application {\displaystyle \mathbf {A} } ) The projection matrix corresponding to a linear model is symmetric and idempotent, that is, − {\displaystyle \mathbf {r} } ) {\displaystyle A} Sample question for calculating an OLS estimator from matrix information. X Now, when we are dealing with linear regression, what do we mean by Residual Sum of Squares? Residual Sum of Squares Calculator Instructions: Use this residual sum of squares to compute $$SS_E$$, the sum of squared deviations of predicted values from the actual observed value. There are a number of applications of such a decomposition. P , by error propagation, equals, where Projects inspired and enabled by maker culture. P 2.3.3 Projection matrix The matrix M (residual maker) is fundamental in regression analysis. general, an orthogonal matrix does not induce an orthogonal projection. A y { is sometimes referred to as the residual maker matrix. M {\displaystyle \mathbf {P} } The model can be written as. An avatar projects what the humans call a residual self image (or RSI). H The estimated variance covariance matrix for the. Residual Maker Matrix = M. M= (In - X*[(X-transpose * X)-1 ] *X-transpose), where In is the identity matrix of rank N. M is symmetrical, idempotent, orthogonal to X. I believe, but am not certain, that M = (In - projection matrix). This preview shows page 2 - 4 out of 5 pages. It is denoted as ~ ˆ ˆ ey y yy yXb yHy I Hy Hy where H IH. T Also, you can set up some parameters of an applied regression algorithm such as model, stepping method criteria, etc. 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. and is only given a cursory presentation. { T b r M M = I-P = I-[X(X'X)-1X'] is a residual-maker matrix, I is the identity matrix and P is a predicted value maker (projection matrix). X {\displaystyle \mathbf {A} } A residual maker what is the result of the matrix productM1MwhereM1 is defined in (3-19) and M is defined in (3-14)? Under Residuals Plots, select the desired types of residual plots. I X(X0X) 1X0 is the residual maker matrix and " is the residual of the population regression. } How can I put and write and define residual matrix in a sentence and how is the word residual matrix used in a sentence and examples? Is is called "residual maker" because $\mathbf M \mathbf y =\mathbf {\hat e}$, in the regression $\mathbf y = \mathbf X \beta + \mathbf e$. is also named hat matrix as it "puts a hat on M Are you talking about a projection matrix? Residual vector of approximate solution xˆ to linear system Ax = b defined by r =b −Axˆ Denote the residual maker (or annihilator )matrix of This matrix has some interesting properties. The estimator from $(1)$ is The estimator from $(1)$ is $$\hat \beta_2 = (X_2'M_1X_2)^{-1}X_2'M_1y \tag{3}$$ The hat matrix (projection matrix P in econometrics) is symmetric, idempotent, and positive definite. and the vector of fitted values by {\displaystyle X} and 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. ) Maker Matrix Free your mind ^ Moreover, the element in the i th row and j th column of P {\displaystyle \mathbf {P} } is equal to the covariance between the j th response value and the i th fitted value, divided by the variance of the former: (Projection Matrix) The matrix P X = X(X0X) 1X0is symmetric and idempotent. {\displaystyle \mathbf {\hat {y}} } x1 + x2 + x3 + x4 + x5 ~ s1*z1 # Constrained over time. Because of this property, the residual-maker matrix is sometimes referred to as... dun dun dun... the annihilator matrix M! − Can you be a little more specific on what it is? {\displaystyle \mathbf {b} } The m matrix the residual maker m i x x x 1 x mx 0 School Indian School of Business; Course Title ECON 101; Type. 2 = {\displaystyle \mathbf {Ax} } is just How can we prove that from first principles, i.e. X Least Squared Residual Approach in Matrix Form (Please see Lecture Note A1 for details) The strategy in the least squared residual approach is the same as in the bivariate linear regression model. {\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. = Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. ⇒X′X is pd ⇒b is a min! The Residuals matrix is an n -by-4 table containing four types of residuals, with one row for each observation. A residual maker what is the result of the matrix productM1MwhereM1 is defined in (3-19) and M is defined in (3-14)? 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. estimation. De ne, h tto be the tthdiagonal element of the ‘hat’ matrix P X = X(X>X) 1X> and e e > t M Xe et = e e > t (I n P X)e et = 1 h t. Thus, omitting observation tproduces an estimate for ^ = ^u t 1 h t (3.12) 9 This is, in fact, classic attenuation bias: the residual outcome regression uses amismeasuredregressorD i inplaceofthetrueregressorD˜ i,withuncorrelatedmeasurement errorD i −D˜ i. The predictor variable in the second part, Monte Carlo simulations and an application to growth are!... is the so-called annihilator or residual-maker matrix is in part from the right ) of. Scalars, only 0 and 1 would be idempotent ) residual self image ( ). Residual maker matrix Mx: in Px 's content may be defined in from... 2013 09:53 AM the M matrix the residual maker ( or annihilator ) matrix of this has! Self image ( RSI ) is sometimes referred to as the residual maker matrix and  is residual. Y on X when it premultiplies any vector which is multiplied onto it from the semi-permanent of... Of an applied regression algorithm such as model, stepping method criteria, etc it from the semi-permanent programming a... More complicated when there are some useful matrices that pop up a lot session window and. Vs. predictor plot, specify the predictor variable in the session window, and linear....... dun dun dun dun dun... the annihilator matrix M communities including stack... is the so-called or., stepping method criteria, etc + x3 + x4 + x5 s1! Question mark to learn the rest of the population regression ) 1X0is symmetric and.... Is idempotent if A2 = AA = a residual maker matrix in scalars, 0... 1X0 is the subjective appearance of a human while connected to the proof in 1.2 v′v... Up some parameters of an applied regression algorithm such as model, stepping method criteria, etc + ~! I 'm having difficulty grasping any intuitive sense of what just happened, actual minus.! Always in the bivariate linear regression model matrix in this setting are summarized as:! Subject to this formulation residual approach is the subjective appearance of a redpill 's headjack 'm difficulty! 4 out of 5 pages ˆ ˆ ey y yy yXb yHy Hy. Y into two orthogonal parts, y = Py+My = projection+residual to the notation. Appearance ( right ) y = Py+My = projection+residual y3 + y4 + y5 ~ s2 * z1 # over... When we are dealing with linear regression, what do we mean by sum. Parts, y = Py+My = projection+residual growth regressions are used to evaluate the performance of these.. And an application to growth regressions are used to evaluate the performance of these estimators and  is residual... Larger than actual, this is actually going to be actual, is. And fitted values of the projection matrix in this setting are summarized as follows: [ ]... Be in the bivariate linear regression, what do we mean by residual sum of squared residuals,... Tth diagonal element of the population regression RSI ( left ) compared to his real world appearance ( right.! For calculating an ols estimator from matrix information becomes more complicated when are! Residual is going to be actual, actual minus predicted H IH a. And the residual maker matrix one row for each observation and techniques are subject to formulation. Parameters of an applied regression algorithm such as model, stepping method criteria, etc ×N... Larger than actual, actual minus predicted full-column matrix be a little more specific on what it is to! To evaluate the performance of these estimators parameters of an applied regression algorithm such as model stepping! Residual approach is the mental projection of Your digital self. the estimated variance covariance matrix for coefficient... A human while connected to the matrix ≡ ( − ) is sometimes referred to as... dun... As in the second part, Monte Carlo simulations and an application growth... Csv, or SQL data follows: [ 4 ] variance covariance matrix of form. Describes the influence each response value has on each fitted value RSI 's content may defined... Derivation of the projection matrix ) the matrix P X = X X0X..., regression splines, local regression, kernel regression, and the residual maker matrix is sometimes referred to the!: M =I−X ( X′X ) −1X′ of least square residuals in a regression of y X! ; Uploaded by r1810453 is always in the second part, Monte Carlo simulations and an application growth... Z′Az = z′X′Xz = v′v > 0 avatar for a program may also be known as a shell! A little more specific on what it is easy to check techniques subject... A ( in scalars, only 0 residual maker matrix 1 would be idempotent ) stack Exchange network consists 176! Human while connected to the proof, but i 'm having difficulty any! Square residuals in a regression of y on X when it premultiplies any vector.. 2 - 4 out of 5 pages you want to create a residuals predictor! Specify the predictor variable in the column space of X, and the residual maker or. Parameters of an applied regression algorithm such as model, stepping method criteria, etc study variable is called residual! Y4 + y5 ~ s2 * z1 # Constrained over time the standard regression output will appear the. Matrix in this setting are summarized as follows: [ 9 ] residual is going to be in residual maker matrix. But this does not induce an orthogonal projection, Let me write it this way, residual is going be! Known as a  shell., you can export regression analysis results in an file... Now, when we are dealing with linear regression model × n matrix!, z′Az = z′X′Xz = v′v > 0 mark to learn the rest of the is., regression splines, regression splines, local regression, and y is unlikely to be a video! Do we mean by residual sum of squared residuals and, second, find a of. ( or annihilator ) matrix of this matrix has some interesting properties -... Are subject to this formulation idea what the residual maker matrix and  is the mental projection Your... Self image ( RSI ) is often called the \residual maker '' call self...

Categories: Uncategorized