|
<br />26
<br />
<br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS
<br />
<br />.
<br />
<br />Table 4.-Data for graphical regression using highly correlated independent variables
<br />[From Riggs (UI58)J
<br />
<br />River and location
<br />
<br />IOQ-yr
<br />flood
<br />(cfB)
<br />
<br />Mean
<br />annual
<br />flood
<br />(cfB)
<br />
<br />Average
<br />discharge
<br />(cfB)
<br />
<br />1. Neosho-----Iola, Kans__________________________
<br />2. Big Blue- Randolph, Kansn __ _ _ _ _ _ _ _ __ __ __ ___
<br />3. Miami-Dayton, Ohion_n______________uu_
<br />4. Savannah-Augusta, Ga____n___________nn_
<br />5. West Branch Susquehanna-Williamsport, Pa_ __
<br />6. Susquehanna-Towanda, Pa___ __ _ _ _ _ _ __ _ __ _ - __
<br />7. Susquehanna-Harrisburg, Pa_________________
<br />8. Kanawha-Kanawha Falls, W. Va___n________
<br />9. Allegheny-Red House, N.Ynn__nu__n_n__
<br />10. Iowa, Iowa City, low&________________________
<br />11. Tennessee-Knoxville, Tenn___________________
<br />12. French Broad-Asheville, N.C_______u________
<br />13. Des Moines-Keosaugua, Iowa________________
<br />14. Connecticut-White River Junction, Vt_________
<br />15. Cumberland-Nashville, Tenn_________________
<br />16. Hudson-Mechanicville, N.Y ___n_n_nn_____
<br />17. Ohio-Cincinnati,_ Ohio_ _ __________nn__n___
<br />
<br />105, 000
<br />96,700
<br />108, 000
<br />350,000
<br />260,000
<br />236, 000
<br />594, 000
<br />276,000
<br />56,400
<br />40, 700
<br />228, 000
<br />45, 500
<br />103, 000
<br />122, 000
<br />208, 000
<br />89,300
<br />800,000
<br />
<br />28,300
<br />27,600
<br />35, 900
<br />112, 900
<br />104, 100
<br />107, 200
<br />282,600
<br />125,600
<br />24, 500
<br />13,100
<br />94, 700
<br />17, 500
<br />41,100
<br />53. 100
<br />122,200
<br />42,500
<br />443, 700
<br />
<br />1. 680
<br />1,680
<br />2,260
<br />10,540
<br />8,910
<br />10,370
<br />34,700
<br />12.670
<br />2, 795
<br />1,560
<br />12,820
<br />2,112
<br />5,351
<br />7, 190
<br />20,400
<br />7.430
<br />97. 700
<br />
<br />inoreases with QU3 it is impossible to tell by
<br />inspeotion whether use of Q., will improve
<br />the relation.
<br />The following prooedure may be used to
<br />define the effect, :if any, of Q., on the scatter
<br />of points about curve 1:
<br />1. Plot Q., against Q,.33 (as absciBsa) and draw
<br />the mean liue (curve 2).
<br />2. Divide each Q., by its value from curve 2
<br />at the same value of QU3' These divisions
<br />are shown on the graph sheet for each
<br />plotted point (curve 1). Tbey could have
<br />been obtained directly by measuring the
<br />deviations from curve 2 in percentage with
<br />dividers (only on log paper); in practice
<br />they would be obtained this way.
<br />3. Use the dividends obtained in step 2 as the
<br />third variable.
<br />4. Proceed with the graphical multiple regres-
<br />sion ItS described previously.
<br />The triangular symbols near curve 1 are the
<br />points adjusted for the effect of Q.,. The fact
<br />that they show less scatter than the original
<br />points indioates that estimates of QllJQ are im-
<br />proved by using Q., as an additional variable.
<br />It can be shown by computing the equation of
<br />the graphical relation that it is of the form
<br />
<br />log QlOo=log a+b, log Q,.33-b, log Q.,.
<br />
<br />The introduction of the dividend is merely an
<br />expedient; it cancels out of the final relation.
<br />
<br />Choice of graphical or analytical method
<br />fOf multiple regression
<br />
<br />A standard graphical method is particularly
<br />useful for exploratory work and for making
<br />preliminary estimates. The graphical method
<br />has the following advantages:
<br />
<br />1. It is rapid.
<br />2. It helps define the appropriate model.
<br />3. It points out the need for transformations,
<br />:if any.
<br />4. It brings attention to extremely wild points
<br />if they exist in the data. (~ee the wild
<br />point in fig. 18.)
<br />Disadvantages of a graphical method are:
<br />
<br />1. Small effects of independent variables cannot
<br />be identified.
<br />2. The number of independent variables is
<br />limited to about three because of the cu-
<br />mulative effect of inaccuracies in plotting
<br />and in locating the liues.
<br />3. Tests of significance of the effects of individ-
<br />ual variables are not available.
<br />4. The resulting rela tion in vol ving three 01'
<br />more variables is confusing to the user
<br />unless expressed mathematically or re-
<br />plotted in another form.
<br />An analytical method has the following
<br />advantages:
<br />
<br />1. For the model used, it gives the best estimate
<br />
<br />.
<br />
<br />.
<br />
|