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11/23/2009 10:40:54 AM
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Title
Techniques of Weather-Resources Investigations of the USGS Book 4, Chapter A1
Date
1/1/1968
Floodplain - Doc Type
Educational/Technical/Reference Information
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<br />14 <br /> <br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS <br /> <br />. <br /> <br />then substituted in the normal equations. The <br />computation of regression coefficients is shown <br />below with the explanation following. <br /> <br />N=40 <br />Normal equations (see Ezekiel, 1950, p. 198): <br />I ~(x,2)b2 +~(x,x3)b, +~(x,x.)b.=~(X,X2) <br />II ~(x,x3)b2 + ~(x,2)b, +~(x,x.)b.=~(X,X3) <br />III ~(x2x.)b2 +~(x3x.)b3 + ~(x.')b.=~(x,x.) <br />I 10.20183b, + 6.38133b, + 0.62554b.=11.74691 <br />I' -b2 -0.625508b, -0.061316b.= -1.15145 <br />II 6.38133b2 + 6.90632b, - 0.06952b.=6.57458 <br />(-0.625508) I -6.38133b2 - 3.99157b, - 0.39128b.=-7.34779 <br />~2 2.91475b, - 0.46080b.--0.77321 <br />II' -b +0.158092b.=0.26527 <br />III 0.62554b2 - 0.06952b, + 0.33512b.= 1.16244 <br />(-0.061316) I -0.62554b2 - 0.39128b, - 0.03836b.=-0.72027 <br />(0.158092) ~2 0.46080b. - 0.07285b.=-0.12224 <br />~. 0.22391b.=0.31993 <br />b,= 1.42883 <br />II' -b.+(0.158092)(1.42883) =0.26527 <br />b,= -0.03938 <br />I' -b2- (0.625508) (-0.03938)- (0.061316) (1.42883) = -1.15145 <br />-b2+O.02463-0.08761= -1.15145 <br />b2= 1.08847 <br />III (0.62454) (1.08847)- (0.06952) ( -0.03938) + (0.33512) (1.42883) = 1.16244 <br />1.16245~1.16244 Check <br /> <br />The above computation utilizes the Doolittle <br />method, asimplified method of solving simulta- <br />neous equations having a certain symmetry. <br />The normal equations are on the first three <br />lines. Next is the first normal equation with <br />converted sums from table 3 substituted in it. <br />Line 5 is obtained by dividing the equation next <br />above by its coefficient of b2 with the sign <br />changed. Line 6 is the second normal equation, <br />with converted sums from table 3 substituted <br />in it. Line 7 is obtained by multiplying the <br />equation of line 4 by the coefficient of b. in <br />line 5. Line 8 is obtained by subtracting line 7 <br />from line 6. Line 9 is line 8 divided by the <br />coefficient of b. with the sign changed. Line 10 <br />is the third normal equation. Line 11 is line 4 <br />multiplied by the coefficient of b. on line 5 with <br />the sign changed. Line 12 is line 8 multiplied <br />by the coefficient of b. on line 9 with the sign <br />changed. Line 13 is the sum of lines 10, 11, and <br />12. Lines 14-18 complete the computations of <br />the regression coefficients. Lines 20 and 21 are <br />used to check the results. Only the third normal <br />equation provides a complete check. <br /> <br />. <br /> <br />The regression constant is obtained from <br /> <br />a=X';-b2X,-b,X,-b.x. <br />a= 1.53888- (1.08847) (1.80814) <br />- (- 0.03938) (2.54489) - (1.42883) (1.89078) <br />a= -3.03061 <br /> <br />tillbstituting the computed constants in the <br />regression model gives <br /> <br />log Q20=-3.03+1.09Iog Q, <br />- 0.04 log A + 1.4:J log 8. <br /> <br />By taking antilogs this becomes <br /> <br />Q20=0.0009:lQ2'.09A -0,"8' "'. <br /> <br />The standard error of e:.;timat.e, S, is eOnl)lut.ea <br />as follows <br /> <br />8' ~(xD-b,~(XIX,)-b,~(XIX,)-b.~(x,x.), <br />N-M <br /> <br />where JvT is the number of items in die :-;ample <br />and M is the number of lost degrees of freedom <br />(one degree of freedom is lost for each constant <br /> <br />. <br />
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