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MatrixComputationsViaIML
Course: STAT 701, Fall 2009School: Los Angeles Southwest...
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701 Stat Handout Matrix Computations Using PROC IML in SAS The following is an implementation of the computations in the lecture example using PROC IML in SAS. The SAS Program: proc iml; /* Defining the design matrix, X */ X={1 85 22, 1 83 23, 1 88 24, 1 86 23, 1 100 24, 1 94 26, 1 91 23} ; /* Defining the Y vector */ Y={12, 6, 25, 10, 29, 21, 19}; /* Obtaining X'X */ XtX = t(X)*X; /* Obtaining X'Y */ XtY =...
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701 Stat Handout Matrix Computations Using PROC IML in SAS The following is an implementation of the computations in the lecture example using PROC IML in SAS. The SAS Program:
proc iml; /* Defining the design matrix, X */ X={1 85 22, 1 83 23, 1 88 24, 1 86 23, 1 100 24, 1 94 26, 1 91 23} ; /* Defining the Y vector */ Y={12, 6, 25, 10, 29, 21, 19}; /* Obtaining X'X */ XtX = t(X)*X; /* Obtaining X'Y */ XtY = t(X)*Y; /* Inverting X'X */ XtXinv = inv(XtX); /* Obtaining the estimates of the regression coefficients */ b = XtXinv*XtY; /* Obtaining the H-matrix */ H=X*XtXinv*t(X); /* Obtaining the fitted values */ Yhat = H*Y; /* Obtaining the residuals */ e = Y - Yhat; /* Computing the sum of squared of residuals, which is the SSE */ norme = t(e)*e; /* Defining a vector of 1's */ v1 = {1, 1, 1, 1, 1, 1, 1}; /* Obtaining the J matrix, which is a square matrix of 1's */
J = v1*t(v1); /* Symmetric matrix needed for the quadratic form to obtain SYY */ /* i(n) creates an n by n identity matrix */ P = i(7)-J/7; /* Symmetric matrix needed for the quadratic form to obtain SSR */ P1 = H - J/7; /* Symmetric matrix needed for the quadratic form to obtain SSE (another way) */ P2 = i(7)-H; /* Taking product of P1 and P2 */ P1P2 = P1*P2; /* computing SYY */ SYY = t(Y)*P*Y; /* computing SSR */ SSR = t(Y)*P1*Y; /* Obtaining SSE by taking difference between SYY and SSR */ SSE = SYY - SSR; /* Computing the MSE */ /* Divisor is (n-1) - p, where p is the number of predictor variables MSE = SSE/(6-2); /* Printing the matrices obtained */ print X, Y, XtX, XtY, XtXinv, b, H, Yhat, e, norme P, P1, P2, P1P2, SYY, SSR, SSE, MSE; run; The Output:
X= Design Matrix 1 1 1 1 1 1 1 85 83 88 86 100 94 91 22 23 24 23 24 26 23
Y = Vector of Responses 12 6 25 10 29
21 19
XTX = X'X 7 627 165 627 165 56371 14806 14806 3899
XTY = X'Y 122 11181 2911
XTXINV = (X'X) -1 61.834107 -0.181651 -1.926929 -0.181651 0.0073394 -0.020183 -1.926929 -0.020183 0.1584458
B = (X'X) -1(X'Y) -96.5742 1.146789 0.4786832
H = X (X'X) -1 X' 0.3975175 0.2444684 0.0785753 0.2389638 0.0565569 -0.245872 0.2297895 0.2444684 0.3599568 0.2185645 0.2498651 -0.221802 0.0825688 0.0663788 0.0785753 0.2185645 0.2172693 0.158014 -0.024933 0.2954128 0.0570966 0.2389638 0.2498651 0.158014 0.2058284 -0.018133 0.0330275 0.1324339 0.0565569 -0.221802 -0.024933 -0.018133 0.7897464 0.0972477 0.3213168 -0.245872 0.0825688 0.2954128 0.0330275 0.0972477 0.787156 -0.049541 0.2297895 0.0663788 0.0570966 0.1324339 0.3213168 -0.049541 0.2425256
YHAT = HY = vector of fitted values 11.433891 9.6189962 15.831624 13.059363 29.593092 23.669725 18.793308
E = Y - YHat = (I - H)Y = vector of residuals 0.566109 -3.618996 9.1683756 -3.059363 -0.593092
-2.669725 = 0.2066919
NORME sum of squares of the residuals 114.35834 P = I - J/n 0.8571429 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 0.8571429 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 0.8571429 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 0.8571429 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 0.8571429 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 0.8571429 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 -0.142857 0.8571429
P1 = H - J/n 0.2546604 0.1016113 -0.064282 0.0961067 -0.0863 -0.388729 0.0869324 0.1016113 0.2170997 0.0757073 0.1070079 -0.36466 -0.060288 -0.076478 -0.064282 0.0757073 0.0744122 0.0151569 -0.16779 0.1525557 -0.085761 0.0961067 0.1070079 0.0151569 0.0629712 -0.16099 -0.10983 -0.010423 -0.0863 -0.36466 -0.16779 -0.16099 0.6468892 -0.045609 0.1784596 -0.388729 -0.060288 0.1525557 -0.10983 -0.045609 0.6442988 -0.192398 0.0869324 -0.076478 -0.085761 -0.010423 0.1784596 -0.192398 0.0996685
P2 = I - H 0.6024825 -0.244468 -0.078575 -0.238964 -0.056557 0.2458716 -0.22979 -0.244468 0.6400432 -0.218564 -0.249865 0.2218025 -0.082569 -0.066379 -0.078575 -0.218564 0.7827307 -0.158014 0.0249325 -0.295413 -0.057097 -0.238964 -0.249865 -0.158014 0.7941716 0.0181328 -0.033028 -0.132434 -0.056557 0.2218025 0.0249325 0.0181328 0.2102536 -0.097248 -0.321317 0.2458716 -0.082569 -0.295413 -0.033028 -0.097248 0.212844 0.0495413 -0.22979 -0.066379 -0.057097 -0.132434 -0.321317 0.0495413 0.7574744
P1P2 = matrix product of P1 and P2 -6.54E-15 -3.97E-15 6.76E...
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