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<br />14
<br />
<br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS
<br />
<br />then substituted in the normal equations. The .
<br />computation of regression coefficieuts is shown
<br />below with the explanation following.
<br />
<br />N=40
<br />Normal equations (see Ezekiel, 1950, p, 198):
<br />I :E(x,,')b, +:E(x,x,)b, +:E(x"x.)b.=:E(x,x,,)
<br />II :E(x"x,)b, + :E(x,,')b, + :E(x"x.)b.= :E(x,x,)
<br />III :E(x"x.)b, + :E(x,x.)b, + :E (x.') b.= :E(x,x.)
<br />I 10.20183b, + 6.38133b, + 0.62554b.=I1.74691
<br />I' -b, -0.625508b, -0.061316b.=-1.l5145
<br />II 6.38133b, + 6.90632b, - 0.06952b.=6.57458
<br />(-0.625508) I -6.38133b, - 3.99157b, - 0.39128b.=-7.34779
<br />:E, 2.91475b, - 0.46080b.= -0.77321
<br />II' -b +0.158092b.=0.26527
<br />III 0.62554b, - 0.06952b, + 0.33512b.= 1.16244
<br />(-0.061316) I -0.62554b, - 0.39128b, - 0.03836b.=-0.72027
<br />(0.158092) :E, 0.4608Ob. - O,07285b.= -0.12224
<br />:E. 0.22391b.=0.31993
<br />b.= 1.42883
<br />II' -b.+(0.158092)(1.42883)=0.26527
<br />b,= -0.03938
<br />I' -b,-(0.625508) (-0.03938)- (0.061316) (1.42883) = -1.l5145
<br />-b,+0.02463-0.08761 = -1.15145
<br />b,= 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, a simplified 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 b, 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 />The regression constant is obtained from
<br />
<br />a=X.-b.X,-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 />i:lubstituting the computed cons taut. in the
<br />regression model give.
<br />
<br />log Q20=-3.03+1.0910g IJ,
<br />-0.04 log A+1.4310!,: 8.
<br />
<br />By taking antilo!,:. this becomes
<br />
<br />Q20=0.00093Q".09A '0 ,"'8',4'.
<br />
<br />The standard error of e~timate, S, is eomputeu
<br />as follow.
<br />
<br />S' :E(xl)-b,:E(x,x,)-b,:E(x,x,)-b.:E(x,x.),
<br />N-M
<br />
<br />where N'i. the number of item. in the sample
<br />and M is the number of lo.t degrees of freedom
<br />(one degree of freedom is lost for each constant
<br />
<br />.
<br />
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