MATRIX CALCULATIONS IN SAS - YOURNAME 21:17 Monday, October 17, 2005 1 NOW WE ARE IN PROC IML, SAS'S MATRIX ENVIRONMENT The matrices A (5x3) and B (3x5) are AA 5 4 1 7 9 7 0 1 1 12 -2 1 7 6 5 BB 5 4 1 4 5 4 3 1 3 4 1 3 -2 3 11 The matrices C=BA (3x3) and D=AB (5x5) are CC 136 79 63 105 62 49 139 89 78 DD 42 35 7 35 52 78 76 2 76 148 5 6 -1 6 15 53 45 8 45 63 64 61 3 61 114 EXERCISE: Check these calculations yourself! TRC TRD The traces Trace(C=BA) and Trace(D=AB) are 276 276 The matrices CINV and CINV*C and the error in inverting C are CINV CII CERROR 0.325 -0.380 -0.024 1.000 -0.000 -0.000 4.349E-14 -0.945 1.268 -0.034 0.000 1.000 0.000 0.498 -0.769 0.094 -0.000 -0.000 1.000 Note the using of formatting for proper spacing of matrix entries. The matrix C and the solution X of C X = Y are CC XX YY 136 79 63 10.84 21.71 105 62 49 -35.58 -11.31 139 89 78 21.57 21.91 MATRIX CALCULATIONS IN SAS - YOURNAME 21:17 Monday, October 17, 2005 2 NOW WE ARE IN PROC IML, SAS'S MATRIX ENVIRONMENT EXERCISE: Check that X is a solution yourself! MATRIX CALCULATIONS IN SAS - YOURNAME 21:17 Monday, October 17, 2005 3 WE ARE NOW BACK IN REGULAR SAS PROC PRINT: The dataset that we exported to SAS is: Obs CI1 CI2 CI3 XX YY 1 0.32534 -0.38014 -0.023973 10.8373 21.71 2 -0.94452 1.26781 -0.033562 -35.5798 -11.31 3 0.49795 -0.76918 0.093836 21.5657 21.91 MATRIX CALCULATIONS IN SAS - YOURNAME 21:17 Monday, October 17, 2005 4 PROC PRINT IN SAS: A DATASET WITH 5 RANDOM VECTORS Obs nn ii x1 x2 x3 x4 x5 1 30 1 8.43615 8.68449 7.93499 5.97750 0.99295 2 30 2 8.63262 4.76746 6.72594 8.42379 4.62962 3 30 3 3.14787 3.48998 5.75590 4.85829 4.09883 4 30 4 9.96177 5.41406 5.27396 1.40300 3.80778 5 30 5 4.64752 3.68508 6.66502 6.27348 6.04204 6 30 6 2.64518 3.70955 8.43493 5.55516 9.49495 7 30 7 9.49378 8.46854 4.54903 3.77442 8.17433 8 30 8 9.97245 7.32908 4.79879 1.42453 2.19269 9 30 9 1.61466 1.96173 0.55213 5.81745 3.17421 10 30 10 0.44988 4.39089 0.16766 9.18762 0.28569 11 30 11 5.15914 5.61123 9.78291 6.08304 7.79010 12 30 12 3.83466 3.26221 3.23537 2.73052 8.26581 13 30 13 3.28140 5.16328 8.40356 3.76851 6.71390 14 30 14 5.01744 2.69210 3.77251 8.21091 4.81366 15 30 15 9.49517 8.49604 5.66181 6.54274 2.64272 16 30 16 3.23091 1.73768 4.74755 7.87914 9.94022 17 30 17 0.32367 0.45983 5.75994 2.50792 9.45100 18 30 18 4.56148 2.66669 4.93272 3.12139 9.53893 19 30 19 2.59061 5.58695 4.97836 6.84635 0.13655 20 30 20 1.68841 3.79812 4.05175 0.19832 8.26155 21 30 21 3.43814 2.75790 4.45673 8.87140 3.22912 22 30 22 6.90788 8.02853 3.39588 1.05854 6.65216 23 30 23 4.57780 3.43193 7.58744 8.88934 6.15556 24 30 24 3.76581 2.56203 0.00531 4.91884 9.41169 25 30 25 2.37023 2.95570 0.90488 5.75529 4.25573 26 30 26 9.60748 9.25390 3.41809 8.87221 4.92864 27 30 27 3.89920 4.14955 5.27280 4.64732 7.05542 28 30 28 8.20558 8.51063 4.47161 9.42877 9.02749 29 30 29 5.02536 6.55638 9.57347 1.58255 9.26951 30 30 30 3.47111 2.92640 3.98111 2.02715 7.09597 MATRIX CALCULATIONS IN SAS - YOURNAME 21:17 Monday, October 17, 2005 5 WE ARE BACK IN PROC IML AGAIN Find the covariance and correlation matrix of a set of random vectors THE DATASET XX WITH FIVE COLUMN VECTORS IS XX 8.4362 8.6845 7.9350 5.9775 0.9930 8.6326 4.7675 6.7259 8.4238 4.6296 3.1479 3.4900 5.7559 4.8583 4.0988 9.9618 5.4141 5.2740 1.4030 3.8078 4.6475 3.6851 6.6650 6.2735 6.0420 2.6452 3.7096 8.4349 5.5552 9.4949 9.4938 8.4685 4.5490 3.7744 8.1743 9.9725 7.3291 4.7988 1.4245 2.1927 1.6147 1.9617 0.5521 5.8175 3.1742 0.4499 4.3909 0.1677 9.1876 0.2857 5.1591 5.6112 9.7829 6.0830 7.7901 3.8347 3.2622 3.2354 2.7305 8.2658 3.2814 5.1633 8.4036 3.7685 6.7139 5.0174 2.6921 3.7725 8.2109 4.8137 9.4952 8.4960 5.6618 6.5427 2.6427 3.2309 1.7377 4.7476 7.8791 9.9402 0.3237 0.4598 5.7599 2.5079 9.4510 4.5615 2.6667 4.9327 3.1214 9.5389 2.5906 5.5869 4.9784 6.8464 0.1365 1.6884 3.7981 4.0518 0.1983 8.2616 3.4381 2.7579 4.4567 8.8714 3.2291 6.9079 8.0285 3.3959 1.0585 6.6522 4.5778 3.4319 7.5874 8.8893 6.1556 3.7658 2.5620 0.0053 4.9188 9.4117 2.3702 2.9557 0.9049 5.7553 4.2557 9.6075 9.2539 3.4181 8.8722 4.9286 3.8992 4.1496 5.2728 4.6473 7.0554 8.2056 8.5106 4.4716 9.4288 9.0275 5.0254 6.5564 9.5735 1.5825 9.2695 3.4711 2.9264 3.9811 2.0271 7.0960 THE COVARIANCE MATRIX OF THE 5 COLUMNS OF X IS COVAR 8.6915 5.5756 1.6448 0.0696 -1.3877 5.5756 5.8195 1.4005 0.1322 -1.8653 1.6448 1.4005 6.4595 -0.5585 1.5057 0.0696 0.1322 -0.5585 7.7617 -2.5060 -1.3877 -1.8653 1.5057 -2.5060 8.7707 THE CORRELATION MATRIX OF THE 5 COLUMNS OF X IS CORR 1.0000 0.7840 0.2195 0.0085 -0.1589 0.7840 1.0000 0.2284 0.0197 -0.2611 0.2195 0.2284 1.0000 -0.0789 0.2000 0.0085 0.0197 -0.0789 1.0000 -0.3037 -0.1589 -0.2611 0.2000 -0.3037 1.0000 MATRIX CALCULATIONS IN SAS - YOURNAME 21:17 Monday, October 17, 2005 6 WE ARE BACK IN PROC IML AGAIN Find the covariance and correlation matrix of a set of random vectors EIGENVALUES OF THE CORRELATION MATRIX: THE EIGENVALUES AND EIGENVECTORS OF CORR ARE EGVALS EGVECS 1.954362 0.6479 0.0971 -0.1437 0.3060 -0.6756 1.398098 0.6653 0.0410 -0.1432 0.1085 0.7235 0.861331 0.2458 0.5082 0.6711 -0.4780 -0.0504 0.579597 0.0835 -0.5774 0.7007 0.4093 0.0333 0.206611 -0.2649 0.6302 0.1324 0.7062 0.1282 (The columns of Egvecs are the eigenvectors of Corr) SUMEVALS TRCORR Sum of eigenvalues and tr(corr): 5 5 CHECKING THAT EGVECS IS AN ORTHOGONAL MATRIX: DD EGVECS`*EGVECS is 1.00 -0.00 0.00 0.00 0.00 -0.00 1.00 0.00 -0.00 -0.00 0.00 0.00 1.00 0.00 0.00 0.00 -0.00 0.00 1.00 0.00 0.00 -0.00 0.00 0.00 1.00 DDERR THE INVERSION ERROR IS 2.585E-15 C = Egvecs`*Corr*Egvecs and the diagonalization error are CC CCERROR 1.954 -0.000 0.000 -0.000 0.000 3.376E-15 -0.000 1.398 0.000 -0.000 -0.000 0.000 0.000 0.861 -0.000 0.000 -0.000 -0.000 -0.000 0.580 0.000 0.000 -0.000 0.000 0.000 0.207 A MATRIX OF T STATISTICS FOR CORR (H_0:rho=0) IS TSTAT . 6.6824 1.1906 0.0449 -0.8518 6.6824 . 1.2416 0.1041 -1.4312 1.1906 1.2416 . -0.4187 1.0804 0.0449 0.1041 -0.4187 . -1.6869 -0.8518 -1.4312 1.0804 -1.6869 . MATRIX CALCULATIONS IN SAS - YOURNAME 21:17 Monday, October 17, 2005 7 WE ARE BACK IN PROC IML AGAIN Find the covariance and correlation matrix of a set of random vectors A MATRIX OF P-VALUES FOR CORR (H_0:rho=0) IS PVALS . 0.0000 0.2438 0.9645 0.4015 0.0000 . 0.2247 0.9178 0.1635 0.2438 0.2247 . 0.6786 0.2892 0.9645 0.9178 0.6786 . 0.1027 0.4015 0.1635 0.2892 0.1027 . THE CORRELATION MATRIX AGAIN CORR . 0.7840 0.2195 0.0085 -0.1589 0.7840 . 0.2284 0.0197 -0.2611 0.2195 0.2284 . -0.0789 0.2000 0.0085 0.0197 -0.0789 . -0.3037 -0.1589 -0.2611 0.2000 -0.3037 . MATRIX CALCULATIONS IN SAS - YOURNAME 21:17 Monday, October 17, 2005 8 USING PROC CORR IN SAS: NOTE THAT CORRs and P-VALUEs ARE EXACTLY THE SAME ! The CORR Procedure 5 Variables: x1 x2 x3 x4 x5 Pearson Correlation Coefficients, N = 30 Prob > |r| under H0: Rho=0 x1 x2 x3 x4 x5 x1 1.00000 0.78398 0.21952 0.00848 -0.15894 <.0001 0.2438 0.9645 0.4015 x2 0.78398 1.00000 0.22843 0.01967 -0.26108 <.0001 0.2247 0.9178 0.1635 x3 0.21952 0.22843 1.00000 -0.07888 0.20005 0.2438 0.2247 0.6786 0.2892 x4 0.00848 0.01967 -0.07888 1.00000 -0.30373 0.9645 0.9178 0.6786 0.1027 x5 -0.15894 -0.26108 0.20005 -0.30373 1.00000 0.4015 0.1635 0.2892 0.1027