Changeset 15443 for branches/MathNetNumerics-Exploration-2789/HeuristicLab.Algorithms.DataAnalysis.Experimental/642.f
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branches/MathNetNumerics-Exploration-2789/HeuristicLab.Algorithms.DataAnalysis.Experimental/642.f
r15442 r15443 1 MODULE MODCUBGCV 2 CONTAINS 1 3 C ALGORITHM 642 COLLECTED ALGORITHMS FROM ACM. 2 4 C ALGORITHM APPEARED IN ACM-TRANS. MATH. SOFTWARE, VOL.12, NO. 2, … … 128 130 C----------------------------------------------------------------------- 129 131 C 130 SUBROUTINE CUBGCV(X,F,DF,N,Y,C,IC,VAR,JOB,SE,WK,IER) 132 SUBROUTINE CUBGCV(X,F,DF,N,Y,C,IC,VAR,JOB,SE,WK,IER) 133 . BIND(C, NAME='cubgcv') 134 USE ISO_C_BINDING 131 135 C 132 136 C---SPECIFICATIONS FOR ARGUMENTS--- … … 247 251 150 RETURN 248 252 END 253 C ******************************************************* 249 254 SUBROUTINE SPINT1(X,AVH,Y,DY,AVDY,N,A,C,IC,R,T,IER) 250 255 C … … 529 534 RETURN 530 535 END 531 C CUBGCV TEST DRIVER 532 C ------------------ 533 C 534 C AUTHOR - M.F.HUTCHINSON 535 C CSIRO DIVISION OF WATER AND LAND RESOURCES 536 C GPO BOX 1666 537 C CANBERRA ACT 2601 538 C AUSTRALIA 539 C 540 C LATEST REVISION - 7 AUGUST 1986 541 C 542 C COMPUTER - VAX/DOUBLE 543 C 544 C USAGE - MAIN PROGRAM 545 C 546 C REQUIRED ROUTINES - CUBGCV,SPINT1,SPFIT1,SPCOF1,SPERR1,GGRAND 547 C 548 C REMARKS USES SUBROUTINE CUBGCV TO FIT A CUBIC SMOOTHING SPLINE 549 C TO 50 DATA POINTS WHICH ARE GENERATED BY ADDING A RANDOM 550 C VARIABLE WITH UNIFORM DENSITY IN THE INTERVAL [-0.3,0.3] 551 C TO 50 POINTS SAMPLED FROM THE CURVE Y=SIN(3*PI*X/2). 552 C RANDOM DEVIATES IN THE INTERVAL [0,1] ARE GENERATED BY THE 553 C DOUBLE PRECISION FUNCTION GGRAND (SIMILAR TO IMSL FUNCTION 554 C GGUBFS). THE ABSCISSAE ARE UNEQUALLY SPACED IN [0,1]. 555 C 556 C POINT STANDARD ERROR ESTIMATES ARE RETURNED IN SE BY 557 C SETTING JOB=1. THE ERROR VARIANCE ESTIMATE IS RETURNED 558 C IN VAR. IT COMPARES FAVOURABLY WITH THE TRUE VALUE OF 0.03. 559 C SUMMARY STATISTICS FROM THE ARRAY WK ARE WRITTEN TO 560 C UNIT 6. DATA VALUES AND FITTED VALUES WITH ESTIMATED 561 C STANDARD ERRORS ARE ALSO WRITTEN TO UNIT 6. 562 C 563 PARAMETER (N=50, IC=49) 564 C 565 INTEGER JOB,IER 566 DOUBLE PRECISION X(N),F(N),Y(N),DF(N),C(IC,3),WK(7*(N+2)), 567 * VAR,SE(N) 568 DOUBLE PRECISION GGRAND,DSEED 569 C 570 C---INITIALIZE--- 571 DSEED=1.2345D4 572 JOB=1 573 VAR=-1.0D0 574 C 575 C---CALCULATE DATA POINTS--- 576 DO 10 I=1,N 577 X(I)=(I - 0.5)/N + (2.0*GGRAND(DSEED) - 1.0)/(3.0*N) 578 F(I)=DSIN(4.71238*X(I)) + (2.0*GGRAND(DSEED) - 1.0)*0.3 579 DF(I)=1.0D0 580 10 CONTINUE 581 C 582 C---FIT CUBIC SPLINE--- 583 CALL CUBGCV(X,F,DF,N,Y,C,IC,VAR,JOB,SE,WK,IER) 584 C 585 C---WRITE OUT RESULTS--- 586 WRITE(6,20) 587 20 FORMAT(' CUBGCV TEST DRIVER RESULTS:') 588 WRITE(6,30) IER,VAR,WK(3),WK(4),WK(2) 589 30 FORMAT(/' IER =',I4/' VAR =',F7.4/ 590 * ' GENERALIZED CROSS VALIDATION =',F7.4/ 591 * ' MEAN SQUARE RESIDUAL =',F7.4/ 592 * ' RESIDUAL DEGREES OF FREEDOM =',F7.2) 593 WRITE(6,40) 594 40 FORMAT(/' INPUT DATA',17X,'OUTPUT RESULTS'// 595 * ' I X(I) F(I)',6X,' Y(I) SE(I)', 596 * ' C(I,1) C(I,2) C(I,3)') 597 DO 60 I=1,N-1 598 WRITE(6,50) I,X(I),F(I),Y(I),SE(I),(C(I,J),J=1,3) 599 50 FORMAT(I4,2F8.4,6X,2F8.4,3E12.4) 600 60 CONTINUE 601 WRITE(6,50) N,X(N),F(N),Y(N),SE(N) 602 STOP 603 END 604 DOUBLE PRECISION FUNCTION GGRAND(DSEED) 605 C 606 C DOUBLE PRECISION UNIFORM RANDOM NUMBER GENERATOR 607 C 608 C CONSTANTS: A = 7**5 609 C B = 2**31 - 1 610 C C = 2**31 611 C 612 C REFERENCE: IMSL MANUAL, CHAPTER G - GENERATION AND TESTING OF 613 C RANDOM NUMBERS 614 C 615 C---SPECIFICATIONS FOR ARGUMENTS--- 616 DOUBLE PRECISION DSEED 617 C 618 C---SPECIFICATIONS FOR LOCAL VARIABLES--- 619 DOUBLE PRECISION A,B,C,S 620 C 621 DATA A,B,C/16807.0D0, 2147483647.0D0, 2147483648.0D0/ 622 C 623 S=DSEED 624 S=DMOD(A*S, B) 625 GGRAND=S/C 626 DSEED=S 627 RETURN 628 END 629 630 CUBGCV TEST DRIVER RESULTS: 631 632 IER = 0 633 VAR = 0.0279 634 GENERALIZED CROSS VALIDATION = 0.0318 635 MEAN SQUARE RESIDUAL = 0.0246 636 RESIDUAL DEGREES OF FREEDOM = 43.97 637 638 INPUT DATA OUTPUT RESULTS 639 640 I X(I) F(I) Y(I) SE(I) C(I,1) C(I,2) C(I,3) 641 1 0.0046 0.2222 0.0342 0.1004 0.3630E+01 0.0000E+00 0.2542E+02 642 2 0.0360 -0.1098 0.1488 0.0750 0.3705E+01 0.2391E+01 -0.9537E+01 643 3 0.0435 -0.0658 0.1767 0.0707 0.3740E+01 0.2175E+01 -0.4233E+02 644 4 0.0735 0.3906 0.2900 0.0594 0.3756E+01 -0.1642E+01 -0.2872E+02 645 5 0.0955 0.6054 0.3714 0.0558 0.3642E+01 -0.3535E+01 0.2911E+01 646 6 0.1078 0.3034 0.4155 0.0549 0.3557E+01 -0.3428E+01 -0.1225E+02 647 7 0.1269 0.7386 0.4822 0.0544 0.3412E+01 -0.4131E+01 0.2242E+02 648 8 0.1565 0.4616 0.5800 0.0543 0.3227E+01 -0.2143E+01 0.6415E+01 649 9 0.1679 0.4315 0.6165 0.0543 0.3180E+01 -0.1923E+01 -0.1860E+02 650 10 0.1869 0.5716 0.6762 0.0544 0.3087E+01 -0.2985E+01 -0.3274E+02 651 11 0.2149 0.6736 0.7595 0.0542 0.2843E+01 -0.5733E+01 -0.4435E+02 652 12 0.2356 0.7388 0.8155 0.0539 0.2549E+01 -0.8486E+01 -0.5472E+02 653 13 0.2557 1.1953 0.8630 0.0537 0.2139E+01 -0.1180E+02 -0.9784E+01 654 14 0.2674 1.0299 0.8864 0.0536 0.1860E+01 -0.1214E+02 0.9619E+01 655 15 0.2902 0.7981 0.9225 0.0534 0.1322E+01 -0.1149E+02 -0.7202E+01 656 16 0.3155 0.8973 0.9485 0.0532 0.7269E+00 -0.1203E+02 -0.1412E+02 657 17 0.3364 1.2695 0.9583 0.0530 0.2040E+00 -0.1292E+02 0.2796E+02 658 18 0.3557 0.7253 0.9577 0.0527 -0.2638E+00 -0.1130E+02 -0.3453E+01 659 19 0.3756 1.2127 0.9479 0.0526 -0.7176E+00 -0.1151E+02 0.3235E+02 660 20 0.3881 0.7304 0.9373 0.0525 -0.9889E+00 -0.1030E+02 0.4381E+01 661 21 0.4126 0.9810 0.9069 0.0525 -0.1486E+01 -0.9977E+01 0.1440E+02 662 22 0.4266 0.7117 0.8842 0.0525 -0.1756E+01 -0.9373E+01 -0.8925E+01 663 23 0.4566 0.7203 0.8227 0.0524 -0.2344E+01 -0.1018E+02 -0.2278E+02 664 24 0.4704 0.9242 0.7884 0.0524 -0.2637E+01 -0.1112E+02 -0.4419E+01 665 25 0.4914 0.7345 0.7281 0.0523 -0.3110E+01 -0.1140E+02 -0.3562E+01 666 26 0.5084 0.7378 0.6720 0.0524 -0.3500E+01 -0.1158E+02 0.5336E+01 667 27 0.5277 0.7441 0.6002 0.0525 -0.3941E+01 -0.1127E+02 0.2479E+02 668 28 0.5450 0.5612 0.5286 0.0527 -0.4310E+01 -0.9980E+01 0.2920E+02 669 29 0.5641 0.5049 0.4429 0.0529 -0.4659E+01 -0.8309E+01 0.3758E+02 670 30 0.5857 0.4725 0.3390 0.0531 -0.4964E+01 -0.5878E+01 0.5563E+02 671 31 0.6159 0.1380 0.1850 0.0531 -0.5167E+01 -0.8307E+00 0.4928E+02 672 32 0.6317 0.1412 0.1036 0.0531 -0.5157E+01 0.1499E+01 0.5437E+02 673 33 0.6446 -0.1110 0.0371 0.0531 -0.5091E+01 0.3614E+01 0.3434E+02 674 34 0.6707 -0.2605 -0.0927 0.0532 -0.4832E+01 0.6302E+01 0.1164E+02 675 35 0.6853 -0.1284 -0.1619 0.0533 -0.4640E+01 0.6812E+01 0.1617E+02 676 36 0.7064 -0.3452 -0.2564 0.0536 -0.4332E+01 0.7834E+01 0.4164E+01 677 37 0.7310 -0.5527 -0.3582 0.0538 -0.3939E+01 0.8141E+01 -0.2214E+02 678 38 0.7531 -0.3459 -0.4415 0.0540 -0.3611E+01 0.6674E+01 -0.9205E+01 679 39 0.7686 -0.5902 -0.4961 0.0541 -0.3410E+01 0.6245E+01 -0.2193E+02 680 40 0.7952 -0.7644 -0.5828 0.0541 -0.3125E+01 0.4494E+01 -0.4649E+02 681 41 0.8087 -0.5392 -0.6242 0.0541 -0.3029E+01 0.2614E+01 -0.3499E+02 682 42 0.8352 -0.4247 -0.7031 0.0539 -0.2964E+01 -0.1603E+00 0.2646E+01 683 43 0.8501 -0.6327 -0.7476 0.0538 -0.2967E+01 -0.4132E-01 0.1817E+02 684 44 0.8726 -0.9983 -0.8139 0.0538 -0.2942E+01 0.1180E+01 -0.6774E+01 685 45 0.8874 -0.9082 -0.8574 0.0542 -0.2911E+01 0.8778E+00 -0.1364E+02 686 46 0.9139 -0.8930 -0.9340 0.0566 -0.2893E+01 -0.2044E+00 -0.8094E+01 687 47 0.9271 -1.0233 -0.9723 0.0593 -0.2903E+01 -0.5258E+00 -0.1498E+02 688 48 0.9473 -0.8839 -1.0313 0.0665 -0.2942E+01 -0.1433E+01 0.4945E+01 689 49 0.9652 -1.0172 -1.0843 0.0766 -0.2989E+01 -0.1168E+01 0.1401E+02 690 50 0.9930 -1.2715 -1.1679 0.0998 691 Documentation: 692 C COMPUTER - VAX/DOUBLE 693 C 694 C AUTHOR - M.F.HUTCHINSON 695 C CSIRO DIVISION OF MATHEMATICS AND STATISTICS 696 C P.O. BOX 1965 697 C CANBERRA, ACT 2601 698 C AUSTRALIA 699 C 700 C LATEST REVISION - 15 AUGUST 1985 701 C 702 C PURPOSE - CUBIC SPLINE DATA SMOOTHER 703 C 704 C USAGE - CALL CUBGCV (X,F,DF,N,Y,C,IC,VAR,JOB,SE,WK,IER) 705 C 706 C ARGUMENTS X - VECTOR OF LENGTH N CONTAINING THE 707 C ABSCISSAE OF THE N DATA POINTS 708 C (X(I),F(I)) I=1..N. (INPUT) X 709 C MUST BE ORDERED SO THAT 710 C X(I) .LT. X(I+1). 711 C F - VECTOR OF LENGTH N CONTAINING THE 712 C ORDINATES (OR FUNCTION VALUES) 713 C OF THE N DATA POINTS (INPUT). 714 C DF - VECTOR OF LENGTH N. (INPUT/OUTPUT) 715 C DF(I) IS THE RELATIVE STANDARD DEVIATION 716 C OF THE ERROR ASSOCIATED WITH DATA POINT I. 717 C EACH DF(I) MUST BE POSITIVE. THE VALUES IN 718 C DF ARE SCALED BY THE SUBROUTINE SO THAT 719 C THEIR MEAN SQUARE VALUE IS 1, AND UNSCALED 720 C AGAIN ON NORMAL EXIT. 721 C THE MEAN SQUARE VALUE OF THE DF(I) IS RETURNED 722 C IN WK(7) ON NORMAL EXIT. 723 C IF THE ABSOLUTE STANDARD DEVIATIONS ARE KNOWN, 724 C THESE SHOULD BE PROVIDED IN DF AND THE ERROR 725 C VARIANCE PARAMETER VAR (SEE BELOW) SHOULD THEN 726 C BE SET TO 1. 727 C IF THE RELATIVE STANDARD DEVIATIONS ARE UNKNOWN, 728 C SET EACH DF(I)=1. 729 C N - NUMBER OF DATA POINTS (INPUT). 730 C N MUST BE .GE. 3. 731 C Y,C - SPLINE COEFFICIENTS. (OUTPUT) Y 732 C IS A VECTOR OF LENGTH N. C IS 733 C AN N-1 BY 3 MATRIX. THE VALUE 734 C OF THE SPLINE APPROXIMATION AT T IS 735 C S(T)=((C(I,3)*D+C(I,2))*D+C(I,1))*D+Y(I) 736 C WHERE X(I).LE.T.LT.X(I+1) AND 737 C D = T-X(I). 738 C IC - ROW DIMENSION OF MATRIX C EXACTLY 739 C AS SPECIFIED IN THE DIMENSION 740 C STATEMENT IN THE CALLING PROGRAM. (INPUT) 741 C VAR - ERROR VARIANCE. (INPUT/OUTPUT) 742 C IF VAR IS NEGATIVE (I.E. UNKNOWN) THEN 743 C THE SMOOTHING PARAMETER IS DETERMINED 744 C BY MINIMIZING THE GENERALIZED CROSS VALIDATION 745 C AND AN ESTIMATE OF THE ERROR VARIANCE IS 746 C RETURNED IN VAR. 747 C IF VAR IS NON-NEGATIVE (I.E. KNOWN) THEN THE 748 C SMOOTHING PARAMETER IS DETERMINED TO MINIMIZE 749 C AN ESTIMATE, WHICH DEPENDS ON VAR, OF THE TRUE 750 C MEAN SQUARE ERROR, AND VAR IS UNCHANGED. 751 C IN PARTICULAR, IF VAR IS ZERO, THEN AN 752 C INTERPOLATING NATURAL CUBIC SPLINE IS CALCULATED. 753 C VAR SHOULD BE SET TO 1 IF ABSOLUTE STANDARD 754 C DEVIATIONS HAVE BEEN PROVIDED IN DF (SEE ABOVE). 755 C JOB - JOB SELECTION PARAMETER. (INPUT) 756 C JOB = 0 SHOULD BE SELECTED IF POINT STANDARD ERROR 757 C ESTIMATES ARE NOT REQUIRED IN SE. 758 C JOB = 1 SHOULD BE SELECTED IF POINT STANDARD ERROR 759 C ESTIMATES ARE REQUIRED IN SE. 760 C SE - VECTOR OF LENGTH N CONTAINING BAYESIAN STANDARD 761 C ERROR ESTIMATES OF THE FITTED SPLINE VALUES IN Y. 762 C SE IS NOT REFERENCED IF JOB=0. (OUTPUT) 763 C WK - WORK VECTOR OF LENGTH 7*(N + 2). ON NORMAL EXIT THE 764 C FIRST 7 VALUES OF WK ARE ASSIGNED AS FOLLOWS:- 765 C 766 C WK(1) = SMOOTHING PARAMETER (= RHO/(RHO + 1)) 767 C WK(2) = ESTIMATE OF THE NUMBER OF DEGREES OF 768 C FREEDOM OF THE RESIDUAL SUM OF SQUARES 769 C WK(3) = GENERALIZED CROSS VALIDATION 770 C WK(4) = MEAN SQUARE RESIDUAL 771 C WK(5) = ESTIMATE OF THE TRUE MEAN SQUARE ERROR 772 C AT THE DATA POINTS 773 C WK(6) = ESTIMATE OF THE ERROR VARIANCE 774 C WK(7) = MEAN SQUARE VALUE OF THE DF(I) 775 C 776 C IF WK(1)=0 (RHO=0) AN INTERPOLATING NATURAL CUBIC 777 C SPLINE HAS BEEN CALCULATED. 778 C IF WK(1)=1 (RHO=INFINITE) A LEAST SQUARES 779 C REGRESSION LINE HAS BEEN CALCULATED. 780 C WK(2) IS AN ESTIMATE OF THE NUMBER OF DEGREES OF 781 C FREEDOM OF THE RESIDUAL WHICH REDUCES TO THE 782 C USUAL VALUE OF N-2 WHEN A LEAST SQUARES REGRESSION 783 C LINE IS CALCULATED. 784 C WK(3),WK(4),WK(5) ARE CALCULATED WITH THE DF(I) 785 C SCALED TO HAVE MEAN SQUARE VALUE 1. THE 786 C UNSCALED VALUES OF WK(3),WK(4),WK(5) MAY BE 787 C CALCULATED BY DIVIDING BY WK(7). 788 C WK(6) COINCIDES WITH THE OUTPUT VALUE OF VAR IF 789 C VAR IS NEGATIVE ON INPUT. IT IS CALCULATED WITH 790 C THE UNSCALED VALUES OF THE DF(I) TO FACILITATE 791 C COMPARISONS WITH A PRIORI VARIANCE ESTIMATES. 792 C 793 C IER - ERROR PARAMETER. (OUTPUT) 794 C TERMINAL ERROR 795 C IER = 129, IC IS LESS THAN N-1. 796 C IER = 130, N IS LESS THAN 3. 797 C IER = 131, INPUT ABSCISSAE ARE NOT 798 C ORDERED SO THAT X(I).LT.X(I+1). 799 C IER = 132, DF(I) IS NOT POSITIVE FOR SOME I. 800 C IER = 133, JOB IS NOT 0 OR 1. 801 C 802 C PRECISION/HARDWARE - DOUBLE 803 C 804 C REQUIRED ROUTINES - SPINT1,SPFIT1,SPCOF1,SPERR1 805 C 806 C REMARKS THE NUMBER OF ARITHMETIC OPERATIONS REQUIRED BY THE 807 C SUBROUTINE IS PROPORTIONAL TO N. THE SUBROUTINE 808 C USES AN ALGORITHM DEVELOPED BY M.F. HUTCHINSON AND 809 C F.R. DE HOOG, 'SMOOTHING NOISY DATA WITH SPLINE 810 C FUNCTIONS', NUMER. MATH. (IN PRESS) 811 C 812 C----------------------------------------------------------------------- 813 C 814 815 816 ALGORITHM 817 818 CUBGCV calculates a natural cubic spline curve which smoothes a given set 819 of data points, using statistical considerations to determine the amount 820 of smoothing required, as described in reference 2. If the error variance 821 is known, it should be supplied to the routine in VAR. The degree of 822 smoothing is then determined by minimizing an unbiased estimate of the true 823 mean square error. On the other hand, if the error variance is not known, VAR 824 should be set to -1.0. The routine then determines the degree of smoothing 825 by minimizing the generalized cross validation. This is asymptotically the 826 same as minimizing the true mean square error (see reference 1). In this 827 case, an estimate of the error variance is returned in VAR which may be 828 compared with any a priori approximate estimates. In either case, an 829 estimate of the true mean square error is returned in WK(5). This estimate, 830 however, depends on the error variance estimate, and should only be accepted 831 if the error variance estimate is reckoned to be correct. 832 833 If job is set to 1, bayesian estimates of the standard error of each 834 smoothed data value are returned in the array SE. These also depend on 835 the error variance estimate and should only be accepted if the error 836 variance estimate is reckoned to be correct. See reference 4. 837 838 The number of arithmetic operations and the amount of storage required by 839 the routine are both proportional to N, so that very large data sets may be 840 analysed. The data points do not have to be equally spaced or uniformly 841 weighted. The residual and the spline coefficients are calculated in the 842 manner described in reference 3, while the trace and various statistics, 843 including the generalized cross validation, are calculated in the manner 844 described in reference 2. 845 846 When VAR is known, any value of N greater than 2 is acceptable. It is 847 advisable, however, for N to be greater than about 20 when VAR is unknown. 848 If the degree of smoothing done by CUBGCV when VAR is unknown is not 849 satisfactory, the user should try specifying the degree of smoothing by 850 setting VAR to a reasonable value. 851 852 References: 853 854 1. Craven, Peter and Wahba, Grace, "Smoothing noisy data with spline 855 functions", Numer. Math. 31, 377-403 (1979). 856 2. Hutchinson, M.F. and de Hoog, F.R., "Smoothing noisy data with spline 857 functions", Numer. Math. (in press). 858 3. Reinsch, C.H., "Smoothing by spline functions", Numer. Math. 10, 859 177-183 (1967). 860 4. Wahba, Grace, "Bayesian 'confidence intervals' for the cross-validated 861 smoothing spline", J.R.Statist. Soc. B 45, 133-150 (1983). 862 863 864 Example 865 866 A sequence of 50 data points are generated by adding a random variable with 867 uniform density in the interval [-0.3,0.3] to the curve y=sin(3*pi*x/2). 868 The abscissae are unequally spaced in [0,1]. Point standard error estimates 869 are returned in SE by setting JOB to 1. The error variance estimate is 870 returned in VAR. It compares favourably with the true value of 0.03. 871 The IMSL function GGUBFS is used to generate sample values of a uniform 872 variable on [0,1]. 873 874 875 INPUT: 876 877 INTEGER N,IC,JOB,IER 878 DOUBLE PRECISION X(50),F(50),Y(50),DF(50),C(49,3),WK(400), 879 * VAR,SE(50) 880 DOUBLE PRECISION GGUBFS,DSEED,DN 881 DATA DSEED/1.2345D4/ 882 C 883 N=50 884 IC=49 885 JOB=1 886 VAR=-1.0D0 887 DN=N 888 C 889 DO 10 I=1,N 890 X(I)=(I - 0.5)/DN + (2.0*GGUBFS(DSEED) - 1.0)/(3.0*DN) 891 F(I)=DSIN(4.71238*X(I)) + (2.0*GGUBFS(DSEED) - 1.0)*0.3 892 DF(I)=1.0D0 893 10 CONTINUE 894 CALL CUBGCV(X,F,DF,N,Y,C,IC,VAR,JOB,SE,WK,IER) 895 . 896 . 897 . 898 END 899 900 OUTPUT: 901 902 IER = 0 903 VAR = 0.0279 904 GENERALIZED CROSS VALIDATION = 0.0318 905 MEAN SQUARE RESIDUAL = 0.0246 906 RESIDUAL DEGREES OF FREEDOM = 43.97 907 FOR CHECKING PURPOSES THE FOLLOWING OUTPUT IS GIVEN: 908 909 X(1) = 0.0046 F(1) = 0.2222 Y(1) = 0.0342 SE(1) = 0.1004 910 X(21) = 0.4126 F(21) = 0.9810 Y(21) = 0.9069 SE(21) = 0.0525 911 X(41) = 0.8087 F(41) = -0.5392 Y(41) = -0.6242 SE(41) = 0.0541 912 536 END MODULE
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